Algebra Readiness (Pre-Algebra) B Course Summary This is the second of two courses that comprise Algebra Readiness. In this course, the student will explore basic algebraic principles. The student will also examine and evaluate two-step and multistep equations and inequalities and then explore and use graphs to solve linear relations and functions. Next, the student will be introduced to basic concepts of geometry including angle relationships, parallel lines, polygons, circles, and transformations. The student will continue to apply his knowledge of geometry and algebra to solve area and volume problems. Then the student will explore nonlinear functions and polynomials. Finally, the student will examine properties of right triangles, data analysis, and probability. Course Directions and Tips You may see a reference to “your Journal Page” while completing activities from the digits™ website. You do not need to access this resource in order to complete these activities. You may use a digital document or a separate sheet of paper to complete your work. Grading Suggested Grading: Graded Credits: 0.5 Units 1. Measurement 2. Functions 3. Using Graphs to Analyze Data 4. STAAR Review 5. Polynomials and Properties of Exponents 6. Probability 7. Personal Financial Literacy Key Discussion: This lesson has a Discussion. Quiz: This lesson has a Quiz. Portfolio Item: This lesson has a Portfolio Item. Practice: This lesson has a Practice. Reflection: This lesson has a Reflection. Quick Check: This lesson has a Quick Check. Resource Packet: This unit has a Resource Packet, which includes materials to support and supplement assessments. Test: This lesson has a Test. © 2015 Connections Education LLC. This content is protected by copyright and owned by Connections Education LLC, and/or owned by, and used with permission from third party content owners. It cannot be used, reproduced, in whole or in part, without express consent of the owner. Users cannot modify, publish, participate in the transfer or sale of, distribute, sub-license, rent, reproduce or transmit in any form or means, create derivative works from, copy, reproduce, display or in any way exploit any of the content, in whole or in part without permission in writing from the owner. For permissions and owner information, send an email to pobllegal2@pearson.com. The Connections Education name and logo are registered trademarks of Connections Education LLC. Any use of these marks without the express written consent of the owner of the mark is strictly prohibited. All rights reserved. Lesson 1: Polyhedrons, 3-D Figures, and Solids CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Identify solids, parts of solids, and skew line segments Desmos™ Graphing Calculator Before you begin your lessons, access the Desmos graphing calculator through the Web Links resource by selecting the backpack. You may access the calculator from the backpack at any time. To get started, type your desired equation or expression into the first field in the list. If you need help, select the question mark icon below the calculator. If you have already purchased a graphing calculator, you do not need to access the Desmos calculator. Extension: Gizmo activities are used throughout this and other courses. You may bookmark this reference guide to all Gizmo activities if you choose. Tip: This list is lengthy and may take some time to load. Gizmo Reference Guide Adding a Third Dimension In your last unit, you worked with two-dimensional shapes. You learned about many different polygons, circles, and irregular shapes. These objects had length and width and could cover a surface, but they didn’t take up any space. See the example below of a two-dimensional shape, a square. In this unit, you will begin working with three-dimensional shapes. These objects also have length and width, but they have an added third dimension of depth or height. These objects take up space. See the example below of a three-dimensional shape, a cube. Write the first letter of your name. Now write it again, but see if you can make it look 3-D by adding depth to the letter. While studying polygons, you learned that they have different names based on their characteristics. The same is true of 3-D figures. In today’s lesson, you will learn about the different types of 3-D figures, how to identify them, and the specific characteristics that separate them from each other. Objective Identify solids, parts of solids, and skew line segments Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words cone cylinder faces lateral edge polyhedron prism pyramid skew line solids sphere Classifying Solids All three-dimensional figures are classified as solids. These objects do not lie in a plane, meaning they are not flat. Three-dimensional figures have a length, width, and height. Can you name the following prisms and pyramids? Click on the Show Answer button below to check your answers. Answer: A is a trapezoidal prism. B is a rectangular pyramid. C is a triangular prism. There are two other types of solids with bases that are not polygons. There is one remaining solid with no bases. Because solids are three-dimensional objects, they have lines that do not all lie in the same plane. Lines that are not in the same plane, are not parallel, and do not intersect are called skew lines. The red line segments in the given diagram are skew lines. Complete the following activities. 1. Click on the link below to watch the "Polyhedrons" movie from the Brain Pop website. While watching the movie, make a list of the four different types of polyhedrons that are shown. Next to each one, write the number of faces and the shape of its base. Polyhedrons 2. Read pp. 354–355 in Mathematics: Course 3. 3. Complete problems 6–20 on p. 356–357. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. 1. Copy the following diagram into your notebook. Identify one pair of parallel lines with a green marker, one pair of intersecting lines with a purple marker, and one pair of skew lines with a red marker. Click on the Show Answer button below to check your work. Answer: 2. Click on the link below to access the Solids graphic organizer. Fill in the name and definition for the five types of solids that you learned about today. Solids Polyhedrons, 3-D Figures, and Solids Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. What is the shape of the bases for the following polyhedron? triangle square rectangle circle (1 point) 2. What is the best name for the given solid figure? (1 point) rectangular pyramid rectangular cone rectangular prism rectangle 3. How many lateral faces are there in the given polyhedron? (1 point) 7 5 2 4 4. Identify one pair of skew lines in the following figure. (1 point) segment AC and AB segment DH and BF segment CG and DB segment EF and GH 5. How many lateral edges are there in a triangular prism? (1 point) 6 3 9 5 DreamCalcTM Graphing Calculator Before you begin your lessons, download the DreamCalcTM graphing calculator. If you have already purchased a graphing calculator, please continue with today's lesson. Click on the link below to view the "DreamCalc Graphing Calculator." Follow the instructions to download and save the DreamCalc graphing calculator to your computer. DreamCalc Graphing Calculator Remember: the DreamCalc graphing calculator is only for students enrolled in this course, and the license will expire after a period of one year.* At the end of one year, you will receive notification from DreamCalc about purchasing the software. DO NOT download the DreamCalc graphing calculator from the public Web site because that download will only be valid for 30 days. * The DreamCalc graphing calculator will be available to download each year for all high school math courses. Mac Users Note: DreamCalc is not compatible with a Mac computer. Please use Mac Grapher or Graphing Calculator to solve problems that require a graphing calculator. Mac Grapher or Graphing Calculator is an application that is already installed on Mac computers. Click on the Applications folder, and then click on the Utilities folder to find this application. © 2015 Connections Education LLC. Lesson 2: 3-D Views CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Draw 3-D figures, base plans, and isometric views Changing Perspective If you have ever flown in an airplane, you know that the world looks very different from above than it does at ground level. Architects and designers often draw plans from many different perspectives to create a better understanding of their vision. They may include views from the top—which are also know as aerial or bird's-eye views—and views from the front or side. Each of the images below is an aerial view. Try to picture what they would look like at ground level. Click on the Show Answer box below to check your answers. Answer: field of tulips oak tree in field urban center building In today’s lesson, you will learn how to draw a base plan, isometric view, and front and right views for a given three-dimensional figure. You will also learn how to construct a threedimensional figure from a given plan. Objective Draw 3-D figures, base plans, and isometric views Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words base plan isometric view Drawing Views of Three-Dimensional Figures There are several different ways to draw this three-dimensional figure to show what it looks like from different perspectives. Complete the following activities. 1. Click on the link below to access Isometric Dot Paper. You will use this during the movie to complete the first part of today's activity. Isometric Dot Paper 2. Read the information about drawing three-dimensional figures. As you read, follow the instructions to practice drawing the cubes from different perspectives on your own paper. Drawing Three-Dimensional Figures 3. Read through Section 8-2 on pp. 358–359 of Mathematics: Course 3. Be sure you understand all of the key words from the lesson. 4. Complete problems 5–19 on p. 360 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activity. Click on the link below to complete the "3D and Orthographic Views - Activity A" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide to investigate building a model when given the base plan, front view, and right view. Take the quiz at the end to check your understanding of the key ideas. 3-D and Orthographic Views - Activity A 3-D Views Charles Washington is not permitted to take this assessment again. These answers will not be saved. Use the following model. 1. How many cubes are in the base of the model? 3 4 7 2 Use the following model. (1 point) 2. How many cubes would be shown in the right view of the model? (1 point) 3 6 7 4 3. Which of the following is the top view for the model? (1 point) 4. Which view allows you to see the top, front, and right sides of a model? (1 point) front top base plan isometric 5. How many total blocks would be needed to build the model for the given base (1 point) plan? 5 11 6 12 Drawing Three-Dimensional Figures You can use isometric dot paper to draw three-dimensional figures. Look at the top of this connecting cube. How can you draw this figure on the dot paper? Plot the 4 vertices of the top face, then connect the segments. Then draw segments 1 unit downward. Then connect to complete the cube! Now, try looking at the cube from the bottom, instead of the top. Plot the 4 vertices of the bottom face first. Then connect them to form edges. Then connect the edges at the top to complete the cube. Try one more. Think of 3 cubes snapped together. Plot the vertices of the base and connect. Then draw lines upward, but this time, they should be 3 units tall. Then connect to complete the prism. You can draw lines to show the three separate cubes, but you don’t have to. You can use these skills to help you represent three-dimensional figures on flat paper! © 2015 Connections Education LLC. Lesson 3: 3-D Figures and Nets CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Identify nets of solids Moving Between Dimensions Most cardboard containers such as boxes, tubes, and cylinders start out as one flat piece of cardboard that gets folded into the appropriate three-dimensional shape. Manufacturers spend a great deal of time designing packaging that is the right size and shape for whatever product they need to ship. Although they are usually creating a box for something that is three-dimensional, the plan will eventually get converted into a two-dimensional design to make the box. flat pack cardboard boxes many kind of boxes Think about a cereal box. How many faces does it have? Are any of them congruent? Which faces are parallel? What would it look like if you gently disconnected the seams and unfolded it to make a flat piece of cardboard? This unfolded pattern is called a net. In today’s lesson, you will learn to identify the net of a solid figure. Objective Identify nets of solids Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Word net Nets of Solids The two-dimensional pattern that is formed when a solid shape is unfolded is called a net. A net shows all the surfaces of a three-dimensional shape. Following is one box shown from many different perspectives. The pattern in the middle of the boxes is the net that is used to make the box. Although every rectangular prism has six faces, the net for this box has eight sections because of the flaps on the top. If the top was one section, what would the net look like? Click on the Show Answer button below to check your answer. Answer: You learned in an earlier lesson that solids can have bases that are different shapes, and a different number of faces depending on the shape of the base. See if you can predict the solid that would be formed from the following nets. Click on the Show Answer button below to check your answer. Answer: A is a cylinder, B is a square pyramid, C is a cone, and D is a triangular prism. Complete the following activity. Complete Activity Lab 8-3a on p. 363 of Mathematics: Course 3. You will need graph paper, a compass, scissors, and tape for this activity. Complete the opening activity and problems 1–8. Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activities. 1. Read pp. 364–365 in Mathematics: Course 3. Be sure you understand what a net is and can recognize the nets of different solids. 2. Complete problems 4–14 on pp. 365–366 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 3-D Figures and Nets Quiz Charles Washington is not permitted to take this assessment again. These answers will not be saved. 1. Name the solid according to its description: The figure has one base that is a hexagon. pentagonal pyramid hexagonal pyramid hexagonal prism pentagonal pyramid (1 point) 2. Draw the base plan for the set of stacked cubes. Assume that no cubes are hidden from view. 3. Identify the solid formed by the given net. (1 point) triangular prism triangular pyramid cone triangle 4. Identify the solid formed by the given net. (1 point) (1 point) cylinder cone rectangular pyramid rectangular prism 5. Identify the solid formed by the given net. (1 point) cylinder triangular prism square pyramid rectangular pyramid © 2015 Connections Education LLC. Lesson 4: Surface Area of Prisms and Cylinders CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Find surface area of prisms and cylinders using nets and formulas Note: This lesson should take 2 days. Covering a Surface In order to be sure you have enough wrapping paper to cover these gifts, you need to know their surface area. Surface area is the total area of all the surfaces of a solid, including both bases and all of the faces. In today’s lesson, you will learn how to calculate the surface area of prisms and cylinders by using nets and formulas. Objective Find surface area of prisms and cylinders using nets and formulas Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words congruent lateral area surface area Tip: You will have 2 days to complete this lesson. Calculating Surface Area of Prisms Using Nets You can find the surface area of a solid by finding the area of its net. Prisms are made up of two parallel and congruent bases and three or more lateral faces. This rectangular prism has two bases and four lateral faces. Use your experience from Lesson 3 to sketch a net for the prism. Add the dimensions for length, width, and height. Click on the Show Answer box to check your answer. To find the surface area of the entire net, first find the area of each rectangle by using the formula of A = l • w. To find the total area of the net, find the sum of the areas of all six rectangles. What is the area of this net? Click on the Show Answer button below to check your answer. Answer: 10 cm2 + 10 cm2 + 45 cm2 + 45 cm2 + 18 cm2 + 18 cm2 = 146 cm2 or 2(10 cm2) + 2 (45 cm2) + 2(18 cm2) = 146 cm2 Calculating Surface Area of Prisms Using a Formula You can also use a formula to calculate the surface area of a prism. You will need to understand the definition of lateral area in order to apply the formula successfully. The lateral area of a solid figure is the sum of the area of each of its lateral faces. Consider a pentagonal prism. This pentagonal prism has five lateral faces. To find the lateral area, find the sum of all five faces. A simple way to calculate the lateral area is to multiply the the prism. of the base by the of If each side of the regular pentagon pictured above is 6 cm, the perimeter of the base is 6 cm • 5 sides or 30 cm. If the height of the prism is 2 cm, the lateral area is 30 cm • 2 cm or 60 cm2. This is the lateral area. What would the lateral area be if the pentagon had side lengths of 7 cm and a height of 3 cm? Click on the Show Answer button below to check your answer. Answer: perimeter is 7 cm • 5 = 35 cm/LA = perimeter • height/35 cm • 3 cm = 105 cm2 Once you have calculated the lateral area, the only remaining step to find the surface area of the prism is to calculate the area of the bases. The triangular prism below has two bases that are triangles and a height of 2 cm. Lateral Area = perimeter • height L.A. = (3 + 4 + 5) cm • 2 cm L.A. = 12 cm • 2 cm L.A. = 24 cm2 Since the right triangles are the bases, A = Area of base = bh • 4 cm • 3 cm A=6 cm2 Surface Area = Lateral Area + 2(area of the base) 24 cm2 + 2 ( base • height) 24 cm2 + 2 ( • 4 cm • 3 cm) 24 cm2 + 12 cm2 SA = 36 cm2 Suppose you are covering a box in the shape of a triangular prism with paper. Each end of the box is an equilateral triangle with side lengths of 4 inches and a height of 3.5 inches. The length of the box is 30 inches. Because the bases of the triangular prism are equilateral triangles, the prism has three congruent faces. Use the formula S.A. = L.A. + 2(area of the base) to find the surface area of the triangular prism. Click on the Show Answer button to review your answer. Answer: S.A. = 360 + 14 Calculating Surface Area of Cylinders Using Nets You will follow the same steps you used to find the area of the net for a prism to find the area of the net for a cylinder. Let’s review those steps: 1. Draw a net. 2. Add measurements to your net. 3. Find the area for each part of the net using the appropriate formulas. 4. Find the sum of all the individual areas. r = 4 ft h = 3 ft Now that you know the areas for each portion of the net, you can find the surface area by adding all areas together. What is the total area of this net? Click on the Show Answer button below to check your answer. Answer: SA = area of base + area of base + area of rectangle/SA = 50.2 ft2 + 50.2 ft2 + 75.4 ft2/SA = 175.8 ft2 Calculating Surface Area of Cylinders Using a Formula You can also calculate the surface area of a cylinder by using a formula. The formula is very similar to the one used for prisms. Lateral Area = circumference • height C ≈ 2 • 3.14 • 3 cm = 18.8 cm LA = 18.8 cm • 7 cm LA = 131.6 cm2 Surface Area = Lateral Area + 2(area of the base) S.A. = 131.6 cm2 + 2(π • r2) S.A. ≈ 131.6 cm2 + 2(3.14 • 32) S.A. = 131.6 cm2 + 56.5 cm2 S.A. = 188.1 cm2 Complete the following activities. 1. Click on the link below to complete the "Surface and Lateral Area of Prisms and Cylinders" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide to investigate how the surface area of a model changes when its dimensions change. Take the quiz at the end to check your understanding of the key ideas. Surface and Lateral Area of Prisms and Cylinders 2. Read pp. 368–370 of Mathematics: Course 3. 3. Complete problems 6–14 and 18–20 on pp. 371–372. Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activities. 1. You can demonstrate your understanding of finding the area of cylinders. Click on the links below to complete the Surface Area of Cylinders activity from the digits™ website. Example 1 Example 2 Example 3 Key Concept 2. Click on the link below to watch the “Surface Area” BrainPOP® movie. Surface Area Tip: When working with cylinders, you should use 3.14 for π and round your answers to the nearest tenth. Surface Area of Prisms and Cylinders Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Find the surface area for the given prism. (1 point) 564 in2 664 in2 1,120 in2 1,080 in2 2. Find the lateral area for the given prism. (1 point) 1,092 cm2 998 cm2 1,272 cm2 870 cm2 3. Find the surface area for the given cylinder. Use 3.14 for nearest whole number. and round to the (1 point) 180 ft2 720 ft2 433 ft2 135 ft2 4. Find the lateral area for the given cylinder. Use 3.14 for nearest whole number. 144 yd2 288 yd2 2,712 yd2 904 yd2 5. Find the surface area for the following net. 184 cm2 200 cm2 120 cm2 174 cm2 © 2015 Connections Education LLC. (1 point) and round to the (1 point) Lesson 5: Surface Area of Pyramids and Cones CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Find surface area of pyramids and cones using nets and formulas Which Solid is Larger? If you wanted to know which of these objects would be less expensive to paint, you would need to find a way to accurately compare them to each other. By calculating the surface area for the cone and the pyramid, you would have exactly the information you need to make a decision. In today’s lesson, you will continue your investigation into surface area by learning about pyramids and cones. Objective Find surface area of pyramids and cones using nets and formulas Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Word slant height Calculating Surface Area of Pyramids Using Nets In Lesson 4, you used the net for a prism and cylinder to calculate surface area. You can also find the surface area of a pyramid by finding the area of its net. This square pyramid has a square base and four congruent triangular faces. The of a pyramid is the distance of the perpendicular line from the center of the base to the vertex of the pyramid. In order to find the surface area, you will need to know the of a pyramid’s lateral faces. , which is the height Draw a net for the square pyramid. Add the measurements of the base (4 inches) and the slant height (7 inches) to your net diagram. Click on the Show Answer button below to check your answers. Answer: To find the total area of the net, multiply the area of one triangle by four and add the area of the base. What is the area of this net? Click on the Show Answer button below to check your answers. Answer: Area of triangle = b•h Area of one triangle = × 4 in • 7 in Area of one triangle = 14 in2 Area of four triangles=14 in2 • 4 triangles Area of four triangles = 56 in2 Area of square = b • b Area of square = 4 in • 4 in Area of square = 16 in2 Surface Area = Area of 4 triangles + Area of 1 square Surface Area= 56 in2 + 16 in2 Surface Area= 72 in2 Calculating Surface Area of a Pyramid Using a Formula It might be simpler to apply a formula to find the surface area of a square pyramid. Once again, you will to calculate the area of the lateral faces and add that area to the area of the base. Lateral Area = 4 ( base • slant height) LA = 4 ( • 6 cm • 7 cm) LA = 84 cm2 ℓ = 7 cm base = 6 cm • 6 cm Surface Area = Lateral Area + Area of the Base S.A. = 84 cm2 + 6 cm • 6 cm S.A. = 84 cm2 + 36 cm2 S.A. = 120 cm2 Calculating Surface Area of Cones Using Nets Just like pyramids, cones have a and a . In order to calculate the surface area of a cone, you will need to know the radius of the base and the slant height. You will use the same steps that you used for all the other solids. 1. Draw a net. 2. Add measurements to your net. 3. Find the area for each part of the net using the appropriate formulas. 4. Find the sum of all the individual areas. Draw a net for a cone with a radius of 5 cm and a slant height of 9 cm. Include the appropriate measurements on your net. Click on the Show Answer button below to check your answers. Answer: Use πrl to calculate the lateral area. Use πr2 to calculate the area of the base. What is the surface area of this cone? Use 3.14 for π and round to the nearest tenth. Click on the Show Answer button below to check your answers. Answer: Base ≈ 3.14 • 52 Base = 78.5 cm2 Lateral Area ≈ 3.14 • 5 • 9 Lateral area = 141.3 cm2 Surface area = 78.5 + 141.3 Surface Area = 219.8 cm2 Calculating Surface Area of a Cone Using a Formula As with all of the other solids, you can use a formula to calculate the surface area of a cone. Although the formula might seem simpler, it is important that you understand why you are doing each step. Just like with pyramids, you will calculate the lateral area and add it to the area of the base. Lateral Area = π • r • ℓ LA ≈ 3.14 • 5 cm • 12 cm LA = 188.4 cm2 Surface Area = Lateral Area + Area of the Base SA = 188.4 cm2 + πr2 SA ≈ 188.4 cm2 + 3.14 • 52 SA = 188.4 cm2 + 78.5 cm2 SA = 266.9 cm2 Find the surface area of the cone using the formula S.A. = L.A. + area of the base. The radius of the base is 11 centimeters and the slant height is 20 centimeters. Click on the Show Answer button to review your answer. Answer: area of the base (A) = S.A. = 690.8 + 379.94 Complete the following activities. 1. Click on the link below to complete the "Surface and Lateral Area of Pyramids and Cones" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide to investigate how the surface area of a model changes when its dimensions change. Take the quiz at the end to check your understanding of the key ideas. Surface and Lateral Area of Pyramids and Cones 2. Read pp. 374–376 of Mathematics: Course 3. 3. Complete problems 6–15 on p. 377 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activities. 1. Click on the links below to complete the Surface Areas of Cones activity from the digits™ website. Example 1 Example 2 Example 3 Key Concept 2. To demonstrate your understanding of finding the surface area of pyramids, complete the following problems. Click on the Show Answer button to review your answers. Answer: a. SA = 799.4 m2 b. SA = 11899 cm2 Tip: When working with cylinders, you should use 3.14 for π and round your answers to the nearest tenth. Surface Area of Pyramids and Cones Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Use a formula to find the surface area of the square pyramid. (1 point) 45 ft2 81 ft2 36 ft2 72 ft2 2. Find the lateral area of the pyramid to the nearest whole unit. (1 point) 176 m2 352 m2 704 m2 416 m2 3. Find the surface area of the cone to the nearest whole unit. (1 point) 226 in2 88 in2 377 in2 138 in2 4. Find the lateral area of the cone. Use 3.14 for pi and round the result to the nearest whole unit. (1 point) 1,319 cm2 2,639 cm2 707 cm2 2,026 cm2 © 2015 Connections Education LLC. Lesson 6: Using Pythagorean Theorem with 3-D Figures CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Apply the Pythagorean Theorem in three-dimensional figures Will it Fit? In previous mathematics classes, you have learned to use the Pythagorean Theorem to find the measurement of a missing side in a right triangle. The Pythagorean Theorem can also be used to solve many real world problems involving solids. For example, if you wanted to choose a box that is the right size to ship a baseball bat, you would need to decide if the bat would fit into a box on the diagonal. In today’s lesson, you will learn how to apply the Pythagorean Theorem to find diagonals in prisms and the slant height in pyramids and cones. Objective Apply the Pythagorean Theorem in three-dimensional figures Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words Pythagorean Theorem surface area Pythagorean Theorem in Solid Figures You have already used the Pythagorean Theorem to find the missing measurement in a right triangle. In the given triangle, sides a and b are the legs, and side c is the hypotenuse (or longest side). a2 + b2 = c2 In three-dimensional figures, you can use the Pythagorean Theorem to find two different types of missing measurements. To find the slant height in a pyramid or a cone, use the following steps: Once you know the slant height, you can use it to find the surface area of a pyramid or cone. To find the diagonal of a rectangular prism, use the following steps: Modification: If you would like to review how to use the Pythagorean Theorem in right triangles, click on the link below to watch the "Pythagorean Theorem" BrainPOP® movie. Pythagorean Theorem Complete the following review activity. Answer each question below. Then click on the Show Answer button to check your answer. Round all answers to the nearest tenth. 1. What is the length of the diagonal in a prism with a length of 8 cm, a width of 4 cm, and a height of 5 cm? Answer: 10.2 cm 2. What is the slant height for a cone with a base of 6 cm and a height of 4 cm? Answer: 5 cm 3. What is the slant height for a pyramid with a base of 10 cm and a height of 2 cm? Answer: 5.4 cm 4. What is the length of the diagonal in a prism with a length of 3 cm, a width of 4 cm, and a height of 5 cm? Answer: 7.1 cm 5. What is the slant height for a square pyramid with a base of 8 cm and a height of 4 cm? Answer: 5.7 cm 6. What is the slant height for a cone with a base of 6 cm and a height of 5 cm? Answer: 5.8 cm Using Pythagorean Theorem with 3-D Figures Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Calculate the slant height for the given cone. Round to the nearest tenth. (1 point) 10.2 cm 11.4 cm 9.8 cm 12.0 cm 2. Calculate the slant height for the given cone. Round to the nearest tenth. (1 point) 11.2 cm 12.5 cm 14.8 cm 11.4 cm 3. Calculate the slant height for the given square pyramid. Round to the nearest (1 point) tenth. 6.2 cm 5.8 cm 7.8 cm 7.2 cm 4. Calculate the length of the diagonal for the given rectangular prism. Round to (1 point) the nearest tenth. 14.7 cm 10.8 cm 12.2 cm 15.6 cm 5. Calculate the length of the diagonal for the given rectangular prism. Round to the nearest tenth. 16.7 cm 14.3 cm 14.8 cm 15.6 cm (1 point) Reflection Charles Washington is not permitted to take this assessment again. These answers will not be saved. 1. How would you describe your understanding of the relationship between two- (1 point) and three-dimensional figures? I can describe the relationship between a base shape and its solid. I can also draw the base plan, isometric view, and side views for 3-D figures. I can usually match base shapes with their solids. I can draw at least two views of a 3D figure. I can usually match base shapes with their prisms. I have some trouble drawing 3-D figures. I need help better understanding 2-D and 3-D figures. 2. Which best describes your ability to calculate the surface area of prisms, (1 point) cylinders, pyramids, and cones? Select all that apply. I can use the net of a figure to determine its surface area. I know the correct formulas to find the surface area of all these figures. I know the correct formulas to find the surface area of at least half of these figures. I can find the surface area of these figures when the formulas are given to me. 3. Which of these skills do you think you could teach someone else? Select all that apply. identifying solids, parts of solids, and skewed line segments drawing 3-D figures, base plans, and isometric views identifying nets of solids finding the surface area of 3-D figures applying the Pythagorean Theorem in 3-D figures 4. Which of these skills do you need more help with? Select all that apply. identifying solids, parts of solids, and skewed line segments drawing 3-D figures, base plans, and isometric views identifying nets of solids finding the surface area of 3-D figures applying the Pythagorean Theorem in 3-D figures Complete the following activities. (1 point) (1 point) 1. Review the steps for using the Pythagorean Theorem with three-dimensional figures on p. CC16 of Mathematics: Course 3. 2. Complete problems 1–9 (odd) on p. CC17 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 © 2015 Connections Education LLC. Lesson 7: Volumes of Prisms and Cylinders CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Calculate the volume of prisms and cylinders The Space Inside For the last several lessons, you have been working with surface area, which is the area of all the surfaces of a three-dimensional object. In today’s lesson, you will begin to investigate volume, which is the amount of space filled by the object. If you filled a box with sand, the box would represent the surface area, and the amount of sand that fills the box is the volume. Which of these two swimming pools do you think holds more water? What dimensions would you need to know before calculating the amount of water in each pool? Objective Calculate the volume of prisms and cylinders Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words cylinder prism volume Understanding Volume Volume is the number of cubic units it takes to fill in a space. Volume is the amount of soda in a bottle, the amount of space inside a suitcase, or the amount of air in a balloon. When working with volume, the units are always cubed because it takes three dimensions to take up space. Inches3, feet3, and cm3 are all examples of cubic measurements. You can find the volumes of prisms and cylinders by using the formula V = Bh, where B is the area of the base. The juice carton shown in the image is a rectangular prism. The length of the carton is 6 inches, the width of the carton is 3 inches, and the height of the carton is 8 inches. To find the volume of the carton, substitute the given values into the formula. The same formula can be used to find the volume of the triangular prism. However, because the bases are triangles, the formula is used to find the area of the base for this problem. Each triangular base has a base length of 6 feet and a height of 8 feet. The height of the prism is 7 feet. What is the volume of the prism? Click on the Show Answer button to review your answer. Answer: Complete the following activities. 1. Click on the link below to complete the "Prisms and Cylinders - Activity A" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide to investigate how the volume of a model changes when its dimensions change. Be sure to explore rectangular, triangular, and circular bases. Take the quiz at the end to check your understanding of the key ideas. Prisms and Cylinders - Activity A 2. Read pp. 380–382 of Mathematics: Course 3. 3. Complete problems 6–18 on p. 383 of Mathematics: Course 3. 4. Click on the link below to watch the "Volume of Rectangular Solids" Teachlet® tutorial. Volume of Rectangular Solids Click on the link below to access the Volume of Rectangular Solids Transcript. Volume of Rectangular Solids Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activities. 1. Click on the links below to complete the Volume of a Cylinder activity from the digits™ website. Example 1 Example 2 Example 3 Key Concept 2. Click on the links below to watch the "Volume of Cylinders" and "Volume of Prisms" BrainPOP® movies. While watching the movies, write down the steps used to find the volume of the drum and the volume of the triangular prism. Volume of Cylinders Volume of Prisms Volumes of Prisms and Cylinders Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Find the volume of the rectangular prism. (1 point) 23 in.3 297 in.3 318 in.3 159 in.3 2. Find the volume of the triangular prism. 864 ft3 432 ft3 216 ft3 492 ft3 3. Find the volume of the cylinder. (1 point) (1 point) 16,990 mm3 8,491 mm3 4,247 mm3 2,369 mm3 4. Find the volume of a cylinder with a diameter of 16 mm and a height of 5.7 mm. (1 point) 2,292 mm3 4,584 mm3 689 mm3 1,146 mm3 © 2015 Connections Education LLC. Lesson 8: Volumes of Pyramids and Cones CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Calculate the volume of pyramids and cones Which Holds More? If you could fill either the ice cream cone or pyramid with ice cream, which do you think would hold more? What information do you need to know before making your decision? How do you think you can use what you learned in the last lesson to help you find the volume of the cone or pyramid? In today’s lesson you will continue your exploration of volume by looking at cones and pyramids. Objective Calculate the volume of pyramids and cones Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words cone pyramid volume of cone formula volume of pyramid formula Volume of Pyramids In the last lesson, you learned the formula for finding the volume of a prism or cylinder is V = Bh where V is the volume, B is the area of the base and h is the height of the pyramid. The square pyramids shown below have the same base as the square prism. Their height is also the same. However, it would take three pyramids to hold the same volume as that of the prism. Volume of + + = volume of This means that the formula for finding the volume of a pyramid will be very similar to that of a prism. What is the difference between this formula and the one you learned for prisms? Click on the Show Answer button below to check your answer. Answer: In this formula, B • h is multiplied by Use the formula to find the volume of a square pyramid. The sides of the square base are 9 feet, and the height of the pyramid is 6 feet. Click on the Show Answer button to review your answer. Answer: Volume of Cones The same idea also applies to cones. The pictured cones have the same base and height as the cylinder, but it would take the volume of three cones to equal the volume of the cylinder. Volume of + + = volume of What do you think the formula is for finding the volume of a cone? Click on the Show Answer button below to check your answer. Answer: Volume = • area of base • height Use the formula to find the volume of a cone. The radius of the cone’s base is 3 feet, and the height of the cone is 6 feet. Click on the Show Answer button below to check your answer. Answer: Complete the following activities. 1. Click on the link below to complete the "Pyramids and Cones - Activity A" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide to investigate how the volume of a model changes when its dimensions change. Be sure to explore square, triangular, and circular bases. Take the quiz at the end to check your understanding of the key ideas. Pyramids and Cones - Activity A 2. Read pp. 388–389 of Mathematics: Course 3. 3. Complete problems 5–20 on pp. 390–391 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3Complete the following review activities. 1. You can demonstrate your understanding of finding the volume of cones. Click on the links below to complete the Volume of a Cone activity from the digits™ website. Example 1 Example 2 Example 3 Key Concept 2. Find the volume for the following pyramids. Click on the Show Answer button below to check your answer. Answer: The volume for pyramid A is 6,760 in3; the volume for pyramid B is 252 cm3 Volumes of Pyramids and Cones Quiz Charles Washington is not permitted to take this assessment again. These answers will not be saved. 1. Find the lateral area of a cone with a radius of 7 ft. and a slant height of 13 ft. (1 point) Use 3.14 for π and round to the nearest tenth. 439.6 ft2 324.5 ft2 571.5 ft2 285.7 ft2 2. Find the volume of a square pyramid with a base length of 14.2 cm and a height of 3.9 cm. (1 point) Cylinder Diagram Use the diagram of the cylinder to answer question. Use 3.14 for π and round to the nearest tenth. 3. Find the surface area of the cylinder. (1 point) 2009.6 in.2 401.9 in.2 803.8 in.2 602.9 in.2 4. Find the volume of the cylinder. (1 point) 1607.7 in.3 2,845.7 in.3 6,430.7 in.3 401.9 in.3 5. Find the volume of a rectangular prism with the following dimensions: Length = 5 mm Width = 7 mm Height = 3 mm (1 point) 142 mm3 105 mm3 126 mm3 130 mm3 6. Find the volume of the given pyramid. (1 point) 147 yd3 175 yd3 221 yd3 441 yd3 7. Find the volume of a square pyramid with a base length of 9 cm and a height of 4 cm. 324 cm3 108 cm3 36 cm3 152 cm3 8. Find the volume of the given cone. 320 in3 1,244 in3 415 in3 622 in3 (1 point) (1 point) 9. Find the volume of a cone with a radius of 10 mm and a height of 6 mm. (1 point) 628 mm3 600 mm3 1,884 mm3 1,254 mm3 © 2015 Connections Education LLC. Lesson 9: Spheres CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Find the surface area and volume of a sphere What is a Sphere? Spheres are easy to find in the world around you. A sphere is different from the other solids you have studied in this unit, because it does not have a base. If you lined up a pyramid, prism, cone, cylinder, and sphere at the top of a hill, the sphere would be the solid that would roll away down the hill. However, a sphere is still a three-dimensional object that takes up space. In today’s lesson, you will learn to find the surface area and volume of a sphere. Can you think of five spheres that are different than the ones pictured below? Objective Find the surface area and volume of a sphere Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Word sphere Surface Area of Spheres Recall that a circle is the set of all points in a given plane that are the same distance from the center point. A sphere is the set of all points in space that are the same distance from the center point. A sphere is a three-dimensional solid. The formula for finding the surface area of a sphere is 4πr2. blue sphere 1. Click on the links below to complete the Finding Surface Area of Spheres activity from the digits™ website. Watch the Key Concept portion and then record the steps leading to the formula for finding the surface area of a sphere in your notebook. Once you understand the formula, continue with Examples 1–3 of the lesson. Example 1 Example 2 Example 3 Key Concept 2. What is the relationship between the surface area of a sphere and the lateral area of a cylinder if they have the same radius and the height of the cylinder is 2r? Click on the Show Answer button below to check your answer. Answer: The surface area of a sphere is equal to the lateral area of a cylinder if they share the dimensions above. Volume of Spheres 1. Click on the links below to complete the Volume of a Sphere activity from the digits™ website. Watch the Key Concept portion and then record the steps leading to the formula for finding the volume of a sphere in your notebook. Once you understand the formula, continue with Examples 1–3 of the lesson. Example 1 Example 2 Example 3 Key Concept 2. What is the relationship between the volume of a sphere and the volume of a cylinder? Click on the Show Answer button below to check your answer. Answer: The volume of a sphere is the volume of a cylinder with a height and diameter equal to that of the diameter of the sphere. Complete the following activities. 1. Read pp. 393–394 of Mathematics: Course 3. 2. Complete problems 7–15 on p. 395 of Mathematics: Course 3. 3. Click on the link below to watch the "Volumes of Cylinders, Cones, and Spheres Teachlet® tutorial Volume of Cylinders, Cones, and Spheres 4. To review the volume formulars for cylinders, cones, and spheres, click on the link below to access the Volume of Cylinders, Cones, and Spheres Transcript. Volumes of Cylinders, Cones, and Spheres Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activity. 1. Find the surface area and volume for the following sphere. 2. Find the surface area and volume for a sphere with a diameter of 9 in. 3. A sphere has a surface area of 1,256 ft2. Work backwards to find the radius. Click on the link below to check your answers. Answers Spheres Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Find the surface area for a sphere with a radius of 10 feet. Round to the nearest (1 point) whole number. 1,256 ft2 4,189 ft2 1,089 ft2 1,568 ft2 2. Find the volume of a sphere with a radius of 10 feet. Round to the nearest whole (1 point) number. 1,257 ft3 4,187 ft3 1,089 ft3 1,568 ft3 3. Find the surface area for a sphere with a radius of 7 cm. Round to the nearest (1 point) whole number. 307 cm2 1,436 cm2 1,020 cm2 615 cm2 4. Find the volume for a sphere with a radius of 7 cm. Round to the nearest whole number. 307 cm3 1,436 cm3 1,020 cm3 615 cm3 5. Find the radius of a sphere with a surface area of 804 cm2. (1 point) (1 point) 9 cm 8 cm 64 cm 204 cm Answers 1. surface area = 113.04 m2, volume = 113.04 m3 2. surface area = 254.3 in2, volume = 381.5 in3 3. 10 feet © 2015 Connections Education LLC. Lesson 10: Similar Solids CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Apply proportional reasoning to find the missing measurement in similar solids Note: The content you are trying to access is not formatted properly. Think Big Each of the objects in the images shown is very big. They are recognizable as a chair, a faucet, and fork because they were closely modeled after an original object. In order to keep each of these works of art proportional, the artists had to make a multitude of careful measurements and then use the same scale factor. How do you think increasing the dimensions of length, width, and height would affect the surface area of the large fork compared to the original? How would the volume of the gigantic faucet compare with the volume of the original? In today’s lesson, you will learn how to find missing measures in solids that are similar and how increasing linear measurements affects the surface area and volume in solids that are similar. Objective Apply proportional reasoning to find the missing measurement in similar solids Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words proportional similar solids surface area volume Calculations for Similar Solids In the last unit, you learned that similar polygons had proportional side lengths and congruent corresponding angles. Two solids are similar if all of their corresponding side lengths are proportional. Cube with side lengths of 15 cm Cube with side lengths of 30 cm Why are the model of the space craft Voyager and the life size replica of the Voyager similar solids? Click on the Show Answer button below to check your answer. Answer: Each of the measurements on the life size model is 16 times larger than the corresponding measurements on the model. If you know that two solids are similar, you can set up a proportion to solve for any missing measurement. = 4x = 36 r = 4 cm r = 6 cm h = 6 cm h = x cm x=9 The height of the larger cylinder is 9 cm. The radius and height in the cylinders above each increased 1.5 times. Do you think that the surface area and volume would also increase 1.5 times? To make things simpler, let’s look at several different cubes. Copy the following table into your notebook. Fill in the information for surface area and volume of the 2 x 2 cube and the 3 x 3 cube. Click on the Show Answer button below to check your answer. Answer: 2 x 2 cube surface area = 24 units2, volume =8 units3 3 x 3 cubed surface area = 54 units2, volume = 27 units3 Now let’s compare the ratios of the side lengths, surface areas, and volumes for the 1 x 1 cube and the 2 x 2 cube. The side length is two times larger, the surface area is four times larger (22), and the volume is eight times larger (23). Therefore, you know that an increase of x in side length results in an increase in surface area of x2 and an increase in volume of x3. Use the ratios of , , , to solve the following: Assume the two prisms are similar. We can use the ratio from above to find the volume of the larger prism even though we don’t have all of the dimensions. The ratio of the volumes is , the volume of the smaller prism is 8 mm2. You can find the missing volume by setting up a proportion. Ratio of side lengths 4:12 or 1:3 Ratio of volumes 1:33 or 1:27 = 4 mm 12 mm Volume = 8 mm3 Volume = x mm3 = v = 216 mm3 Complete the following activities. 1. Read pp. 398–399 of Mathematics: Course 3. 2. Complete problems 1-17 (odd) on pp. 400–401 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Ice Design The relationship between an object’s surface area and its volume is one of the most powerful and important ratios in the world. It is a major factor in the shape and size of everything from plants and animals to tin cans and skyscrapers. It’s why all living cells are small and why your intestines are enormous (if you could stretch out their surfaces, that is). Designers use the volume to surface area ratio to choose the best shapes for all kinds of objects— even ice cubes! In fact, designer ice cubes are a really hot trend right now. There are all kinds of crazy ice cube shapes available, but what shape is best? You will answer that question in this unit's portfolio item. Begin thinking about and working on the portfolio project now by reviewing the portfolio worksheet and rubric. You will submit the portfolio at the end of the next lesson. Click on the link below to access the Ice Design worksheet. Ice Design Click on the link below to access the Ice Design rubric. Ice Design RubricComplete the following review activities. 1. Find the missing measurements: base = 10 in. base = 8 in. height = 2 in. base = 10 in. height = 6 in. height = x in. base = x in. height = 3 in. 2. Fill in the missing information. Remember to use proportions to solve for the surface area and volume. Original Shape New Dimensions Surface Area Volume Dimensions Surface Area Volume radius = 2 cm 88 cm2 63 cm3 radius = 4 cm Rectangular Prism length = 6 in 104 in2 60 in3 20 cm2 5 cm3 height = 12 cm Cylinder Pyramid height =4 cm length = 12 in Click on the link below to check your answers. Answers Similar Solids Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice For numbers 1–3, find the indicated measurement of the figure described. Use 3.14 for and round to the nearest tenth. 1. Find the surface area of a sphere with a radius of 8 cm. (1 point) 267.9 cm2 803.8 cm2 2143.6 cm2 201.0 cm2 2. Find the surface area of a sphere with a diameter of 12 m. (1 point) 452.2 m2 150.7 m2 113.0 m2 904.3 m2 3. Find the volume of a sphere with a radius of 4 ft. (1 point) 33.5 ft3 67.0 ft3 267.9 ft3 803.8 ft3 4. For the pair of similar solids, find the value of the variable. (1 point) 3 cm 18 cm 16 cm 2 cm 5 F th i f i il lid fi d th l f th i bl (1 i t) 5. For the pair of similar solids, find the value of the variable. (1 point) 12 mm 48 mm 20 mm 3 mm 6. A pyramid has a height of 5 in. and a surface area of 90 in2. Find the surface (1 point) area of a similar pyramid with a height of 10 in. Round to the nearest tenth, if necessary. 360 in2 180 in2 22.5 in2 3.6 in2 7. A rectangular prism has a width of 92 ft and a volume of 240 ft3. Find the volume of a similar prism with a width of 46 ft. Round to the nearest tenth, if necessary. 30 ft3 40 ft3 60 ft3 120 ft3 Answers 1. rectangular prism: x = 15 in triangular prism: x = 7.5 in 2. new cylinder surface area = 352 cm2 and volume = 504 cm3 new rectangular prism surface area = 416 in2 and volume = 480 in3 new pyramid surface area = 180 cm2 and volume = 135 cm3 © 2015 Connections Education LLC. (1 point) Lesson 11: Surface Area and Volume in the Real World CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Calculate the surface area and volume (in appropriate units) of three-dimensional objects found in the home, such as a can of food, box of tissues, shoebox, etc. Note: This lesson should take 2 days. The content you are trying to access is not formatted properly. Measuring the Outside and the Inside Most of the containers in your home include some sort of measurement indicating how much food or material they contain. Although the measurement is often given in weight (grams, ounces, pounds), each box or can also has a distinct surface area and volume. Careful planning goes into deciding the dimensions of a can or box. In this two-day lesson, you will have the opportunity to use the formulas you have learned to calculate the measurements (including surface area and volume) of two objects from the real world. The assessment will be a portfolio format. Objective Calculate the surface area and volume (in appropriate units) of three-dimensional objects found in the home, such as a can of food, box of tissues, shoebox, etc. Tip: You will have two days to complete this lesson. Measuring Solids If you created a new type of breakfast cereal, you would also need to design the packaging for the product. You would want to create a box that was eye-catching, but you would also need to know how much the box would cost to make and how much cereal it would hold. You could design boxes of different sizes. dimensions: dimensions: length: 10 in. length: 10 in. width: 3 in. width: 4 in. height: 12 in. height: 9 in. Calculate the volume for each of the cereal boxes. Which one holds more? Which one do you think makes a better cereal box? Click on the Show Answer button below to check your answers. Answer: Both of the boxes have a volume of 360 in3. Now calculate the surface area for both of the boxes. If cardboard costs $.02 per square inch, how much would each box cost to make? Click on the Show Answer button below to check your answers. Answer: The surface area for the first box is 372 in2 and would cost $7.44 to make. The surface area for the second box is 332 in2 and would cost $6.64 to make. Complete the following activity. Complete problems 15–20 on p. 405 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activities. 1. 2. Find the surface area and volume for the composite shape. Click on the link below to check your answers. Answers Ice Design You will now submit the portfolio that you started working on in the Similar Solids lesson on Slide 4. The relationship between an object’s surface area and its volume is one of the most powerful and important ratios in the world. It is a major factor in the shape and size of everything from plants and animals to tin cans and skyscrapers. It’s why all living cells are small and why your intestines are enormous (if you could stretch out their surfaces, that is). Designers use the volume to surface area ratio to choose the best shapes for all kinds of objects— even ice cubes! In fact, designer ice cubes are a really hot trend right now. There are all kinds of crazy ice cube shapes available, but what shape is best? Click on the link below to access the Ice Design worksheet. Ice Design Click on the link below to access the Ice Design rubric. Ice Design Rubric This is a portfolio item. When you are finished, please submit your answers to your teacher using the Drop Box below. Complete and submit the Ice Design Portfolio assessment. Answers 1. 1. Polyhedron 2. Prism 3. Surface Area 4. Volume 5. Bh 6. 7. Pyramid 8. Cone surface area 2. volume base = πr2 = 3.14 × 32 cylinder volume = Bh = (3.14 × 32)5 = 28.26 in2 = 141.3 in3 cylinder lateral area = 2πr × h = 2 x 3.14 × 3 × 5 = 94.2 in2 cone lateral area = πrℓ ℓ = 5 in cone volume = = (Bh) (3.14 × 32)4 = 37.68 in3 = 3.14 × 3 × 5 = 47.1 in2 28.28 + 94.2 + 47.1 = 169.56 in2 © 2015 Connections Education LLC. 141.3 + 37.68 = 178.98 in3 Lesson 12: Measurement Unit Review CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Objective: Review previously studied material Note: This lesson should take 2 days. Measurement Unit Review The test at the end of a unit is an opportunity for you to demonstrate everything you have learned while studying these concepts. In this lesson, you will review test-taking strategies that will help you to be successful taking the unit test and showing your teacher all you have learned in this unit. You will also have the chance to practice what you learned during previous lessons in this unit by using various review activities. Key Words base plan cone congruent cylinder faces front view isometric view lateral area lateral edge net polyhedron prism proportional pyramid Pythagorean Theorem right view similar solids skew line slant height solids sphere surface area top view volume volume of cone formula volume of pyramid formula Objective Review previously studied material Test-Taking Strategies In the next lesson, you will take the test on the skills that you have learned in this unit. In preparation for this test, review the following test-taking strategies. Multiple-Choice Questions 1. Read through the question and all of the answer choices before selecting your response. 2. Find any key words in the question. 3. Find out what the question is asking. There may be choices that look like the correct answer, but do not answer the question. 4. Eliminate any choices that are incorrect. 5. After you make your choice, re-read the question again to check that the answer you chose is the best answer. 6. In questions that involve calculations, double check your work. Short Answer Questions 1. Read through the question. 2. Find any key words and determine what the question is asking. 3. Show all of the steps you used to find your answer. 4. Check over your work to be sure that your computation is correct. 5. Re-read the question and make sure that your response properly answers the question. 6. Be sure that you have included the units (ft, in2) in your answer. Complete the following activities. 1. Read through the “Vocabulary Review” section on p. 404 of Mathematics: Course 3. Be sure you know the meaning of each of the words under Vocabulary Review and are able to answer problems 1–5. 2. Work through the “Skills and Concepts” section on pp. 404–405 of Mathematics: Course 3. 3. Fill in the following table. You can double check the formulas by using your textbook. IMPORTANT FORMULAS FROM UNIT 2: Measurement SOLID Prism Pyramid Cylinder SURFACE AREA VOLUME Cone Click on the link below to access the online textbook. Mathematics: Course 3Complete the following review activities. 1. Click on the link below to complete the "Plane and Solid Figures" activity. Plane and Solid Figures 2. Click on the link below to complete the "Surface Area" activity. Surface Area 3. Click on the link below to complete the "Volume of Prisms and Cylinders" activity. Volume of Prisms and Cylinders 4. Click on the link below to complete the "Volumes of Solid Figures" activity. Volumes of Solid Figures Click on the link below to access the Measurement Unit Review Practice. Measurement Unit Review Practice Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Plane and Solid Figures A plane figure has two dimensions: length and width. A polygon is a closed figure made of three or more line segments. Some common polygons are shown below: There are numerous types of each polygon, some of which have specific names: 1. Select the figure that is a trapezoid. a. b. c. d. 2. Click on the true statement. a. All polygons are rhombuses. b. All rectangles are squares. c. Triangles have three equal sides. d. Trapezoids have one set of parallel sides. 3. Which shape is a rhombus? a. b. c. d. 4. Which name best describes this shape? a. rhombus b. trapezoid c. parallelogram d. quadrilateral 5. Select the polygon that can have sides with different lengths. a. rhombus b. square c. trapezoid d. equilateral triangle Solid figures have three dimensions: length, width, and height. Some of these have special names: Prisms’ shapes are determined by their bases. Triangular prisms have triangular bases. Rectangular prisms have rectangular bases. The bases are connected by rectangular faces. Some examples of prisms are shown below: Cones and pyramids are similar to prisms. They are also three-dimensional and have a base that is a plane figure, such as a circle, triangle, or square. The difference is that while prisms have two plane figure bases and parallel sides, cones and pyramids have only one plane figure base and the sides slant in and meet at a point. Pyramids are also determined by their bases. Triangular pyramids have triangle bases and square pyramids have square bases. 6. Which term best describes this shape? a. prism b. sphere c. cylinder d. cone 7. Which figure is a prism? a. b. c. d. 8. Which solid figure does this tent most closely resemble? a. cone b. prism c. triangle d. cylinder 9. Which figure has all triangular faces? a. cone b. triangular prism c. cylinder d. triangular pyramid 10. What combination of figures is seen here? a. spheres and cylinder b. spheres and cone c. cylinders and pyramid d. cylinders and prism Select the Show Answers button to check your answers. Answer: 1. a. 2. d. Trapezoids have one set of parallel sides. 3. d. 4. d. quadrilateral 5. c. trapezoid 6. c. cylinder 7. c. 8. b. prism 9. d. triangular pyramid 10. b. spheres and cone Surface Area The area of a shape is the amount of space it takes up on a two-dimensional plane. Volume is the amount of space a three-dimensional object takes up. Surface area is the sum of the areas of all of the sides of a three-dimensional object. For triangular prisms, calculate the area of the 5 sides and add them together. For rectangular prisms, calculate the area of all 6 sides and add them together. For cylinders, calculate the area of the circular ends, and then multiply the circumference of the circle by the height of the cylinder. Add those two products together. Complete the following review. 1. What is the surface area of a cube with sides that each measure 5 centimeters? a. 30 square centimeters b. 250 square centimeters c. 150 square centimeters d. 300 square centimeters 2. You are painting the walls (not the floor or ceiling) of your bedroom. If your room has the dimensions shown above, what total area will you paint? a. 23.2 square meters b. 33.5 square meters c. 56.7 square meters d. 77.0 square meters 3. Jared is making a play tent as a present for his little sister. If he uses the dimensions above, how much fabric will he need? a. 8.8 square meters b. 10.6 square meters c. 23.4 square meters d. 77.0 square meters 4. The Great Pyramid of Giza has a base of 230 meters and a height of 146.5 meters. What is the surface area of the four triangular faces? a. 16,847.5 square meters b. 67,390 square meters c. 120,290 square meters d. 187,680 square meters 5. What is surface area of a soup can with a diameter of 60 mm, a circumference of 188.5 mm, and a height of 98 mm, rounded to the nearest whole number? a. 5,654 square millimeters b. 11,310 square millimeters c. 18,473 square millimeters d. 24,128 square millimeters Use proper formatting and standard language when providing answers. Select the Show Answer button to check your answers. Answer: 1. c. 150 square centimeters 2. c. 56.7 square meters 3. b. 10.6 square meters 4. b. 67,390 square meters 5. d. 24,128 square millimeters Volume of Prisms and Cylinders Remember, to find the volume of a prism or cylinder, use the formula: In this formula, V means volume, A means area of the base, and h means height. Also remember the formulas for the areas of different shapes: rectangle: triangle: circle: Once you find the area of the base, multiply it by the height of the prism or cylinder to get the volume. 1. What is the volume of a cube if its sides each measure 5 centimeters? a. 15 cubic centimeters b. 20 cubic centimeters c. 25 cubic centimeters d. 125 cubic centimeters 2. You are helping your mom build a sandbox for your little brother. It is a 2.5 meter by 2.5 meter square with a height of half a meter. How much area of the backyard will the sandbox occupy? a. 1.25 square meters b. 5.5 square meters c. 3.125 square meters d. 6.25 square meters 3. How many cubic meters of sand will you need to fill the box to the top? a. 1.25 cubic meters b. 5.5 cubic meters c. 3.125 cubic meters d. 6.25 cubic meters 4. A scout is setting up her tent. The front and back are triangles and the sides and floor are rectangles. The tent is 1.5 meters wide, 1.5 meters tall, and 2.5 meters long. What is the volume of the tent? a. 2.81 cubic meters b. 5.5 cubic meters c. 5.63 cubic meters d. 9.32 cubic meters 5. Which is the correct formula for finding the volume of this prism? a. b. c. d. 6. Brad was working on finding the volume of a triangular prism. The triangular face had a base of 3 cm and a height of 4 cm. The length of the prism is 5 cm. He concluded that the area of the triangular base is 12 square centimeters, so the volume is 60 cubic centimeters. What was his error? a. He should have divided 4 by 3. b. His multiplication was incorrect. c. He multiplied 3 by 4 and forgot to divide it in half. d. He mixed up square and cubic centimeters. 7. A drinking glass has a diameter of 10 centimeters and a height of 15 centimeters. What volume of water can it hold, rounded to the nearest whole number? a. 1,500 cubic centimeters b. 1,178 cubic centimeters c. 750 cubic centimeters d. 150 cubic centimeters 8. Mario is helping to fill in an old well on his grandfather’s farm. The well is 1 meter by 15 meters. What volume of dirt will he need, rounded to the nearest whole number? a. 5 cubic meters b. 25 cubic meters c. 50 cubic meters d. 100 cubic meters 9. Which of these shapes would have the largest volume? a. cylinder b. rectangular prism c. triangular prism d. All are equal. Select the Show Answer button to check your answers. Answer: 1. d. 125 cubic centimeters 2. d. 6.25 square meters 3. c. 3.125 cubic meters 4. a. 2.81 cubic meters 5. a. 6. c. He multiplied 3 by 4 and forgot to divide it in half. 7. b. 1,178 cubic centimeter 8. a. 5 cubic meters 9. b. rectangular prism Volumes of Solid Figures Remember that the volume of prisms and cylinders can be calculated using the formula This formula works for solid figures that have two parallel and congruent bases. For pyramids, no matter the shape of the base, the formula is For cones, the formula is For spheres, the formula is . . 1. What is the volume of this equilateral triangular pyramid? . . a. 3 cubic meters b. 5 cubic meters c. 7 cubic meters d. 9 cubic meters 2. What information is missing to allow you to find the volume of this pyramid? a. the area b. the base c. the hypotenuse d. the height 3. If the missing value is 7, what is the volume of the pyramid? a. 30 cubic mm b. 210 cubic mm c. 170 cubic mm d. 70 cubic mm 4. If this ice cream cone were packed full of ice cream and flat across the top, how much ice cream would it hold? a. 419 cubic cm b. 79 cubic cm c. 1,257 cubic cm d. 240 cubic cm 5. What is the volume of a cone that has a diameter of 20 meters and a height of 35 meters? a. 4,660 cubic meters b. 3,665 cubic meters c. 25,656 cubic meters d. 19,242 cubic meters 6. A basketball has radius of approximately 13 centimeters. What is its volume? a. 575 cubic cm b. 9,203 cubic cm c. 13,322 cubic cm d. 54,575 cubic cm 7. The globe in the public library has a diameter of 25 centimeters. What is its volume? a. 65,450 cubic cm b. 78,125 cubic cm c. 524,000 cubic cm d. 897,375 cubic cm 8. Which object has the greatest volume? a. the pyramid b. the cone c. the sphere d. They’re all equal. Select the Show Answer button to check your answers. 1. c. 7 cubic meters 2. d. the height 3. d. 70 cubic mm 4. a. 419 cubic cm 5. b. 3,665 cubic meters 6. b. 9,203 cubic cm 7. a. 65,450 cubic cm 8. c. the sphere © 2015 Connections Education LLC. Lesson 13: Measurement Unit Test CE 2015 Algebra Readiness (Pre-Algebra) B Unit 1: Measurement Measurement Unit Test Part 1 Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Which solid has one base that is a triangle and three lateral surfaces that are (1 point) triangles? triangular pyramid triangular prism rectangular prism rectangular pyramid 2. A solid with two parallel and congruent bases cannot be which of the following? cylinder cube prism (1 point) pyramid 3. Which of the following are considered skew lines? (1 point) AC and CD DH and BF EG and FH AE and GH 4. What is the base plan for the set of stacked cubes? (1 point) 5. Which of the following is the front view for the model? 6. Which solid does the net form? (1 point) hexagonal prism hexagonal pyramid rectangular prism rectangular pyramid 7. Which solid does the net form? (1 point) (1 point) drawing not to scale square pyramid triangular prism triangular pyramid cube 8. What is the surface area of the given figure? (1 point) 2,564 cm2 2,276 cm2 2,184 cm2 1,160 cm2 9. Use the net to find the approximate surface area of the cylinder to the nearest square meter. 440 m2 314 m2 283 m2 214 m2 10. What is the volume of the prism to the nearest whole unit? (1 point) (1 point) 0. at s t e o u e o t e p s to t e ea est o e u t? ( point) 23 in.3 297 in.3 318 in.3 159 in.3 11. What is the volume of the triangular prism to the nearest whole unit? (1 point) 864 ft3 432 ft3 216 ft3 492 ft3 12. What is the volume of the cone to the nearest whole unit? (1 point) 452 in.3 339 in.3 226 in.3 151 in.3 13. What is the volume of the pyramid to the nearest whole unit? (1 point) 147 yd3 175 yd3 221 yd3 441 yd3 14. What is the slant height for the given pyramid to the nearest whole unit? (1 point) 7 cm 5 cm 9 cm 8 cm 15. What is the length of the diagonal for the given rectangular prism to the nearest whole unit? 10 cm 11 cm 6 cm 13 cm 16. The cones below are similar, although not drawn to scale. (1 point) (1 point) What is the length of the radius of the larger cone? 5 ft 6 ft 7 ft 8 ft 17. A cone has a radius of 15 cm and a volume of 540 cm3. What is the volume of a (1 point) similar cone with a radius of 10 cm? 54 cm3 240 cm3 160 cm3 360 cm3 18. What is the surface area of a sphere with a radius of 6 meters rounded to the (1 point) nearest square meter? 226 m2 905 m2 113 m2 452 m2 19. What is the volume of a sphere with a radius of 6 meters rounded to the nearest square meter? 905 m3 679 m3 452 m3 226 m3 Take the assessment. Measurement Unit Test Part 2 © 2015 Connections Education LLC. (1 point) Unit 2: Functions Algebra Readiness (Pre-Algebra) B Unit Summary This unit focuses on the concept of functions. By the end, you will be able to identify a function as an equation having one output for every input, and you will be able to graph linear and nonlinear functions on a coordinate plane. Objectives Describe a sequence Identify and graph functions and determine slope and y-intercept Determine the solution of two functions by graphing Lessons 1. Sequences 2. Relating Graphs and Events 3. Functions 4. Understanding Slope 5. Slope and Similar Triangles 6. Graphing Linear Functions 7. Graphing Proportional Relationships 8. Writing Rules for Linear Functions 9. Solving Systems of Equations 10. Nonlinear Functions 11. Comparing Functions 12. Functions Unit Review 13. Functions Unit Test Lesson 1: Sequences CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objective: Write rules for sequences and use the rules to find terms Mathematical Patterns A sequence is an arrangement of things in a certain order. There are many different types of sequences in the world. Books are arranged in a particular order at the library, CDs and movies are arranged alphabetically at a store, the colors in a rainbow always follow the pattern red, orange, yellow, green, blue, indigo, violet. There are also many different mathematical sequences, and they can be very simple or very complex. The set of numbers 1, 2, 3, 4, 5… is an example of a simple sequence. The numbers 1, 1, 2, 3, 5, 8, 13… form a very famous mathematical sequence that you will learn about later in this lesson. This sequence is famous partly because of the way it shows up in nature. The number of petals shown in each flower is an example of the sequence. In this lesson, you will learn how to write rules for several different types of mathematical sequences. You will also learn how to use those rules to find a term of the sequence. Objective Write rules for sequences and use the rules to find terms Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words arithmetic sequence common difference common ratio geometric sequence sequence term Determining the Link A mathematical sequence is a list of numbers in a particular order. Each number in a sequence is called a term. An arithmetic sequence is a list where consecutive numbers have the same difference. The amount that each term increases (or decreases) by is called the common difference. The sequence of 0, 2, 4, 6, 8… is an arithmetic sequence because each term increases by 2. When each term is formed by multiplying the previous term by the same number it is called a geometric sequence. The number that each term is multiplied by is called the common ratio. The sequence of 1, 2, 4, 8, 16… is a geometric sequence because each term is found by multiplying the previous term by two. Is the following sequence arithmetic or geometric? What are the next two terms in the sequence? Position (n) 1 2 3 Term 4 8 12 4 16 Click on the Show Answer button to review your answer. Answer: The sequence is arithmetic because there is a common difference between the terms of 4. The next two terms will be 20 and 24. You can use an algebraic expression to represent a sequence by using n to describe a term’s position in the sequence. An expression can be used to find the value of any term in the sequence without knowing the value of the previous term. The expression 4n represents the arithmetic sequence shown in the table. Verify that the expression matches the sequence by substituting the term number for n and evaluating the expression. Position (n) 1 2 3 4 8 12 16 4n Term 4 To determine the value of the 10th term in the sequence, evaluate the expression for n = 10. So, the 10th term is 40. The pattern shown below starts with a shape made of three segments. Two segments are added each time to get the next design. You can find the number of segments for the fourth design by taking the number of segments in the third design, which is 7, and adding 2. Since 7 + 2 = 9, there will be a total of 9 segments in the fourth design. The pattern can also be represented with an algebraic expression. Notice that the number of twos added to each term totals one less than the position number, n. The expression is Position (n) 1 2 Term 3 4 3 5 7 9 3 3+2 3+2+2 3+2+2+2 . Use the expression to find the value of the 10th term in the sequence. What does the value of the term mean in this sequence? Click on the Show Answer button to review your answer. Answer: 18 + 3 = 21 There will be 21 segments in the 10th design. Extension: A grid may also be used to generate interesting patterns and sequences of numbers. Click on the link below Patterns Click on the link below to access the Patterns Transcript. Patterns Transcript Complete the following activities. 1. Read pp. 512–514 in Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 8–21 and 28–33 (all) on p. 515. 3. Read the “Exploring Sequences” Activity Lab 11-1b on p. 517. Complete problems 1–6. You do not need a graphing calculator to complete this activity. With a regular calculator, you can repeatedly press = instead of ENTER to achieve the same results. Click on the link below to access the online textbook. Mathematics: Course 3 Extension: If you would like to explore a more in-depth explanation of geometric sequences, including how to find the nth term of a sequence and write the algebraic formula for a geometric sequence, you can watch the following video. Click on the link below to access the Geometric Sequences (Introduction) video on the Khan Academy website. Geometric Sequences (Introduction) Complete the following review activities. 1. Click on the link below to access the Math Patterns Example 1 video on the Khan Academy website. Math Patterns Example 1 After watching the video, determine how many people would be able to sit if there are 6 tables. Number of Tables 12 3 How Many People Can Sit 61014 Click on the Show Answer button below to check your answer. Answer: 26 people will be able to sit if there are 6 tables. 2. Click on the link below to access the Math Patterns Example 2 video on the Khan Academy website. Math Patterns Example 2 After watching the video, determine how many toothpicks would be in the 35th figure. Click on the Show Answer button below to check your answer. Answer: 176 toothpicks, 3. Click on the link below to watch the "Fibonacci Sequence" BrainPOP® movie. Take the quiz at the end of the movie to see how much you learned about Fibonacci and his famous sequence of numbers. Fibonacci Sequence Select the image and complete the Sequences interactive review. Lesson Answers Click on the link below to check your answers to the Exploring Sequences activity lab. Answers Sequences Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice Determine whether each sequence is arithmetic or geometric. Find the next three terms. 1. 14, 19, 24, 29, . . . (1 point) geometric, 34, 39, 44 arithmetic, 32, 36, 41 arithmetic, 34, 39, 44 The sequence is neither geometric nor arithmetic. 2. –4, 8, –16, 32, . . . (1 point) arithmetic, 64, 128, 256 geometric, –64, 128, –256 geometric, –48, 64, –80 The sequence is neither geometric nor arithmetic. 3. 81, 27, 9, 3, . . . (1 point) arithmetic, 0, –3, –6 geometic, 0, –3, –6 geometric, 1, , The sequence is neither geometric nor arithmetic. 4. What are the first four terms of an arithmetic sequence if the common (1 point) difference is 1.5 and the first term is 15? 15, 30, 45, 60 15, 16.5, 18, 19.5 15, 22.5, 33.75, 50.625 none of the above 5. What are the first four terms of a geometric sequence if the common ratio is 10 and the first term is 4.5? 4.5, .45, .045, .0045 4.5, 9.0, 13.5, 18.0 4.5, 14.5, 24.5, 34.5 none of the above Answers 1. –3.5, –2.8, –2.1, –1.4, –0.7 2. 900, 817, 734, 651, 568 3. 5, 6, 7, 8, 9; start with 5 and add 1 repeatedly. 4. 15, 45, 135, 405, and 1,215; start with 15 and multiply by 3 repeatedly. 5. 26, 22, 18, 14, 10; start with 26 and add –4 repeatedly. 6. 6, 8, 10, 12, 14; start with 6 and add 2 repeatedly. © 2015 Connections Education LLC. (1 point) Lesson 2: Relating Graphs and Events CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objective: Interpret and sketch graphs that represent real-world situations The Shape of Graphs Graphs appear everywhere in our everyday lives: newspapers, books, magazines, and the Internet. For example, it is common to see the results of a survey on various social networking sites. Graphs can help us examine relationships between variables in a visual way. This often makes the data easier to understand. Even without knowing what the variables in the graph shown represent, you can see that overall the graph is going up. Some sections of the graph are steeper than others, and there is also one small section where the graph goes down slightly. This graph could be showing the value of an investment over time, the accumulation of inventory at a store, or the total points of a player during a game. In this lesson, you will learn to interpret and sketch the general shape of graphs to represent real-world situations. Objective Interpret and sketch graphs that represent real-world situations Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words function function notation function rule input output Using Graphs to Tell a Story You may have learned about different types of graphs, how to create them, and how to interpret them. This lesson will focus on line graphs, which show change over time. Instead of using actual data to create a graph, you will work on sketching the general shape of a line graph to represent certain situations. For example, if you took your dog for a walk and then sketched a graph to represent the relationship between time spent walking and the distance traveled. Your graph could look something like this: Read the "Line Graphs" Activity Lab 11-2b, and complete problems 1–3 on p. 522 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following activities. 1. Read pp. 518–519 of Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 6–11 and 13–16 on p. 520 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3Complete the following review activities. 1. Click on the link below to complete the “Distance-Time Graphs” Gizmo to practice the concepts from this lesson. Distance-Time Graphs The student exploration sheet found within this simulation can help you review important terms and concepts. Click on the Lesson Materials link to access the student exploration sheet. 2. Take the Checkpoint Quiz (all problems) on p. 522 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Lesson Answers Click on the link below to check your answers to the even-numbered problems on p. 520 of Mathmatics: Course 3. Even-Numbered Answers Click on the link below to check your answers to problems 1–3 of Activity Lab 11-2b on p. 522. Activity Lab Answers Relating Graphs and Events Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice Use the following graphs to answer questions 1–4. 1. Which graph might show the temperature during a 24-hour period? (1 point) A B C D 2. Which graph might show the amount of money in a bank account if money is never taken out? A B C (1 point) D 3. Which graph might show the amount of money in a bank account if money is (1 point) saved for several months and then taken out for a vacation? A B C D 4. Which graph might show the depth of water in a bathtub after the drain is (1 point) pulled? A B C D Even-Numbered Answers 6. 11 weeks 8. fourth and fifth weeks 10. Graphs may vary, but should show a graph over 24 hours that rises during daytime hours and declines during night hours. 14. Al 16. Al ran the same distance in a shorter period of time; Al won. Activity Lab Answers 1. a. about $15 b. Sept. and Oct. 2. –3. Discuss your work on these problems with your Learning Coach. © 2015 Connections Education LLC. Lesson 3: Functions CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objectives: Identify functions; Represent functions with equations, tables, and function notation What Comes Out? Have you ever put a coin into a gumball machine? Usually, for every coin you put in, one gumball comes out. You could create an inputoutput table to show this relationship. If the machine were modified so that two gumballs came out for each coin you put in, what would the table look like? You could also consider the amount of money you put into the gumball machine instead of the number of coins. What would the input-output table look like if you put $0.25 into the machine to get one gumball? Each of these relationships is considered a function. In this lesson, you will learn to identify functions and to represent them with equations, tables, and function notation. Objectives Identify functions Represent functions with equations, tables, and function notation Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words function function notation function rule input output Machines that Make Numbers A function is a special kind of relationship between variables. In a function, each input value has exactly one output value. A function rule describes how the input and output values are related. The graph of a function is the set of ordered pairs each consisting of an input and the corresponding output. Consider the following example: Situation For every hour that you work, you earn $10. Input value hours worked (independent variable) Output value total income (dependent variable) Function rule Equation Income = 10 × hours worked y = 10x The table shows possible input and output values. Notice that for each input value (the number of hours worked) there is exactly one output value (total income). input value output value (x) (y) 10 100 15 150 25 250 Tables of input and output values can be used to identify functions. The following table of x- and y-values does not represent a function because each input value does not correspond to exactly one output value. The input value of 2 results in output values of 12 and 16. input value output value (x) (y) 1 8 2 12 input value output value (x) (y) 2 16 3 20 Determine if the table of values represents a function. How do you know? input value output value (x) (y) 0 0 1 7 2 14 3 21 Click on the Show Answer button to review your answer. Answer: The input and output values do represent a function because for each input value, there is exactly one output value. The function rule for the table is each output value is 7 times the input value. The description can be written a shorter way by using the equation y = 7x. It can also be written using function notation: . The function notation is read “f of x equals 7x.” Notice the similarities between the equation and the function notation. To evaluate , substitute 5 for x in the expression7x. is the output value of the function when the input value is 5. The function represents the number of markers each student will get if a package of 24 markers is divided evenly between x students. What is ? What does represent in terms of this context? Click on the Show Answer button to review your answer. Answer: If there are 8 students, each student will get 3 markers. 1. Click on the links below to complete the Recognizing a Function activity from the digits™ website. Topic Opener Launch Example 1 Key Concept Example 2 Example 3 Example 4 Close and Check 2. Click on the links below to complete the Representing a Function activity from the digits™ website. Launch Key Concept Example 1 Example 2 Close and Check 3. Click on the link below to watch the "Function Tables" Teachlet® tutorial. Function Tables 4. Click on the link below to access the Function Tables Transcript. Function Tables Complete the following activities. 1. Read pp. 523–524 of Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 6–15 on p. 525 of Mathematics: Course 3. 3. Click on the link below to complete the Recognizing a Function questions from the MathXL® for School website. Recognizing a Function 4. Click on the link below to complete the Representing a Function questions from the MathXL® for School website. Representing a Function Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activities. 1. Click on the link below to complete the "Function Machines 1 (Functions and Tables)" Gizmo to practice the concepts from today's lesson. While interacting with the Gizmo, try to figure out the rule for each of the function machines. What happens if you stack two function machines on top of each other? You should also create several functions of your own with the blank machines. Observe the relationship between the input, the function rule, and the output. Take the quiz, and check your answers at the end. Function Machines 1 (Functions and Tables) 2. Read the "Rate of Change" Activity Lab 11-4a, and complete problems 1–4 on p. 527 of Mathematics: Course 3. Click on the link below to review your answers. Answers 3. Review Lessons 1–3 in preparation for the quiz at the end of this lesson. Click on the link below to access the online textbook. Mathematics: Course 3 Functions Quiz Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Find the next three terms in the sequence. (1 point) 30, 22, 14, 6, . . . –3, –12, –21 –1, –8, –15 –1, –2, 3 –2, –10, –18 2. Write a rule for the sequence. (1 point) 4, 8, 16, 32, . . . Start with 4, and multiply by 2 repeatedly. Start with 4, and add 8 repeatedly. Start with 4, and add 2 repeatedly. Start with 2, and multiply by 4 repeatedly. 3. Find the first four terms of the sequence represented by the expression. 3n + 5 3, 8, 14, 20 8, 11, 14, 17 3, 6, 9, 12 0, 8, 11, 14 (1 point) 4. The graph below shows your speed at different times while riding a bicycle. (1 point) For how many minutes did your speed remain constant? 3 4 7 10 5. The graph shows the amount of gas in the tank of Sharon’s car during a trip to her mom’s house. At what time did she stop to buy gas? about 9:00 P.M. about 7:25 P.M. about 7:15 P.M. about 8:00 P.M. 6. Which is a table of values for y = x – 6? (1 point) (1 point) 7. A gas station charges $2.19 per gallon of gas. Use function notation to describe (1 point) the relationship between the total cost C(g) and the number of gallons purchased g. C(g) = –2.19g g = 2.19C(g) C(g) = g + 2.19 C(g) = 2.19g 8. Use the function rule . Find the output . (1 point) 2.5 –2.5 6.5 –6.5 Short Answer Note: Your teacher will grade your response to ensure you receive proper credit for your answer. 9. A relation contains the points function? Explain. , , , and . Is this a (2 points) Lab Answers 1. Rate of Change = 2 in/yr. Every year, the subject grows two inches. 2. Rate of Change = 3 mm/hr. Three inches of rain falls every hour. 3. Rate of Change = 6.25 m/s. The runner runs 6.5 meters every second. 4. –0.006 C°/m. For every meter above sea level the subject climbs, the temperature drops 0.006° C.Click on the links below to complete questions 1–16. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 Question 16 Click on the links below to complete questions 1–12. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 © 2015 Connections Education LLC. Lesson 4: Understanding Slope CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objective: Calculate slope of a line from different representations Slope Look at the difference in the slopes on these two ski runs. If you were just learning to ski, you would probably prefer the slope that is less steep. If you were an advanced skier, you would probably be bored with the gentle slope and would prefer to spend your time on the run that is steeper. In mathematics, slope describes the steepness of a line. Specifically, it is the amount of vertical change compared to the amount of horizontal change. In this lesson, you will learn how to find the slope of a line from a graph, a table, or a set of given points. Objective Calculate slope of a line from different representations Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words coordinates rise run slope Describing Steepness Slope in mathematics refers to the steepness of the graph of a function. Slope can be positive (meaning the line slants uphill from left to right) or negative (meaning the line slants downhill from left to right). It can also be a horizontal or vertical line. You can think of slope in the following ways: Sometimes, you will calculate the slope based on the points in a graph. You can also calculate the slope from a given table of values, or a set of ordered pairs. If you were given the points (3, 5) and (7, 10), you could find the slope between them as follows. You can use points from a table of x- and y-values to find the slope of a function in two ways. x y 0 1 1 5 2 9 3 13 You could use the values to plot points and make the graph of the function. Then you could use two points from the graph to calculate the slope. You could also use two pairs of points from the table to calculate the slope directly. You can choose any two pairs of points. The slope of the function shown in the table is 4. An increase of 1 in the x-coordinate results in a change of 4 in the y-coordinate. In the Getting Started section of the lesson, the slope examples literally represented the amount of vertical change compared to the amount of horizontal change. Although the slope of a line can always be described as rise over run, a linear function often represents a relationship between values that is not related to the steepness of a hill or mountain. For example, the following graph represents the relationship between the number of hours of boat rental, x, and the total cost of renting the boat, y. Use the two points shown on the graph to calculate the slope. The slope is 15, but what does that mean in the context of this function? Each time the x-value increases by 1, the y-value increases by 15. Since x is the number of hours, for each additional hour, the total cost of renting the boat increases by $15. The slope represents the hourly cost of renting the boat. A table of values can also be used to interpret the slope of a line. Click on the link below to watch the "Slope of Linear Functions" Teachlet® tutorial. Slope of Linear Functions Click on the link below to access the Slope of Linear Functions Transcript. Slop of Linear Functions Complete the following activities. 1. Read pp. 528–530 of Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 5–12 on pp. 530–531 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activities. 1. Click on the link below to complete the "Slope Activity - B" Gizmo to practice the concepts from today's lesson. Slope - Activity B The student exploration sheet found within this simulation can help you review important terms and concepts. Click on the Lesson Materials link to access the student exploration sheet. Take the quiz at the end to check your understanding of these conecepts. 2. Click on the links below and complete the Slope from a Graph activity and the Slope from Two Solutions activity on the Khan Academy website. You should work through the problems for each activity until you get 10 problems correct. Slope from a Graph Slope from Two Solutions Understanding Slope Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Find the slope of the following graphs. 2 (1 point) –2 – 2. (1 point) 1 –1 – 3. (1 point) 3 – 4. Find the slope of the line that goes through the following points. (1 point) –1 1 –4 –7 5. (1 point) –5 –3 3 © 2015 Connections Education LLC. Lesson 5: Slope and Similar Triangles CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objectives: Demonstrate that the slope between any two points on a line is the same; Derive the equation for a line in slopeintercept form: y = mx + b Note: This lesson should take 2 days. Slope in Triangles You can use the hypotenuse in a right triangle to demonstrate the slope of a line between two points. In this case, the two points are the vertices of the acute angles of the triangle. If you marked off a smaller triangle inside of the first, it would be similar to the original triangle. Remember that similar triangles have congruent angles and proportional side lengths. You can see that the slope of the hypotenuse of the smaller triangle would be the same as the slope of the original triangle. Look at the way the mainsail on the boat is divided into similar triangles by the horizontal lines. What could you say about the slope along the edge of the sail for each of these triangles? In this lesson, you will use what you already know about similar triangles to help you understand the concept of slope. Objectives Demonstrate that the slope between any two points on a line is the same Derive the equation for a line in slope-intercept form: y = mx + b Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words congruent parallel ratio similar similar triangles Tip: You will have 2 days to complete this lesson. Representing Slope Another Way Recall that similar triangles have corresponding angles that are congruent and corresponding side lengths that are proportional. Similar right triangles can be used to understand slope. Start by choosing a point on the given line. A line with a slope of will have a rise of 3 units and a run of 5 units between points. You can find a new point by counting up 3 units (the rise) and right 5 units (the run). If you used your pencil to trace the movements you made, you would end up with a right triangle. You could then find another point by counting up 3 units and right 5 units again, creating another right triangle. Notice that the slope on the two triangles is the same. Notice that each of the vertices from the first triangle has been translated the same distance and in the same direction to form the vertices of the second triangle. Both triangles have the same slope. Now go back to your original starting point: What would happen if you counted up 6 (3 + 3) and right 10 (5 + 5)? Would you land on your third point? If you traced your movements, how would this triangle compare to your original triangle? This larger triangle and the original triangle you drew are similar triangles. Triangles are similar when they have proportional side lengths. The ratio of vertical height to horizontal length in the small triangle is Since the ratio . The ratio of vertical height to horizontal length in the larger triangle is in simplified form is . , both ratios are the same. Any two points along the same transversal line that can be used to create right triangles will create triangles that are similar. The triangles in the next diagram are on opposite sides of the transversal line. To determine if they are similar, compare the ratios of the vertical height to the horizontal length for each triangle. The ratio of vertical height to horizontal length for the small triangle is height to horizontal length for the large triangle is . The ratio of vertical , which can be simplified to . The triangles have proportional side lengths and are therefore similar. They also have the same slope. One right triangle in a graph has a vertical height of 2 and a horizontal length of 3. Does a triangle with a vertical height of 8 and a horizontal length of 12 have the same slope? Click on the Show Answer button to review your answer. Answer: Since the ratio of the vertical height to the horizontal length in the large triangle can be simplified to , the two triangles are similar. Similar triangles have the same slope. Click on the link below to complete the Relating Similar Triangles and Slope activity from the digits™ website. Work through the Launch, Key Concept, and Parts 1 and 2 of the Example section of the lesson. Launch Example 1 Example 2 Key Concept Close and Check Click on the link below to access the Graphical Slope of a Line video on the Khan Academy website. Graphical Slope of a Line Complete the following activity. Read the “Parallel and Perpendicular Lines” Extension on p. 532. Complete problems 1–9 on p. 532. Click on the link below to access the online textbook. Mathematics: Course 3Complete the following review activity. 1. Click on the link below to complete the Relating Similar Triangles and Slope questions from the MathXL® for School website. Relating Similar Triangles and Slope Extension: To see how problem solving strategies can be used to solve a problem involving a census taker, click on the link below to watch the "Problem Solving" Teachlet® Tutorial. Problem Solving Click on the link below to access the Problem Solving Transcript. Problem Solving Transcript Lesson Answers Click on the link below to check your answers to problems 1–9 on p. 532. p. 532 Answers Slope and Similar Triangles Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Write a ratio in simplified form of the vertical length to the horizontal length (1 point) for the red triangle. 2:3 3:2 1:1 none of the above 2. How does the slope of the red triangle compare to the slope of the blue triangle? (1 point) The slope of the blue triangle is twice the slope of the red triangle. The slope of the red triangle is steeper than the slope of the blue triangle. The slopes of the two triangles are the same. The slope of the red triangle is half the slope of the blue triangle. 3. Would a triangle with a vertical length of 6 and a horizontal length of 10 have (1 point) the same slope as the blue and red triangles shown in the graph? Yes, the slopes would be the same. No, the slope of the new triangle would be 2 : 4. No, the slope of the new triangle would be 2 : 5. No, the slope of the new triangle would be 3 : 5. 4. Which equation would you use to find out if the two lines in the graph are parallel? 5. How can you determine if the given lines are perpendicular? (1 point) (1 point) determine if they have slopes with opposite values determine if they have the same slope determine if the product of their slopes is 1 determine if the product of their slopes is –1 p. CC24 Answers 1. Yes 2. because they are corresponding angles 3. Yes, they are corresponding angles, which are congruent. 4. Yes, the triangles are similar because corresponding angles are congruent. 5. ; ; the ratios are equivalent 6. The ratio of side lengths is equal to the slope. 7. The slopes of the two line segments are equal because the ratios of the vertical to horizontal lengths are equal. 8. The rise-run ratio between any two points of the line can be represented by a triangle. All such rise-run ratios will form similar triangles with a ratio of the side lengths equal to m. p. CC25 Answers 1. Yes, all corresponding angles are congruent. 2. = 3. The ratios and slope are all equal to 2. 4. 2 5. y = mx 6. The second triangle is translated three units upward. 7. no 8. 2; 3 9. y = mx + b; the slope of a line is the rate of change of the y-coordinate divided by the corresponding change in the x-coordinate. The number b represents the value of y when x = 0. p. 532 Answers 1. perpendicular 2. neither 3. neither 4. perpendicular 5. parallel 6. parallel, 7. 1; –1 8. , –4 9. – , Click on the links below to complete questions 1–12. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 © 2015 Connections Education LLC. Lesson 6: Graphing Linear Functions CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objective: Use tables and equations to graph linear functions Picture the Graph Imagine that you and your friends go out for ice cream. The amount of money you pay for your cone will depend on the number of scoops you decide to order. If the price is $1.50 per scoop, and each of you decides to order a different number of scoops, what would the graph of this situation look like? Would the points make a line? If so, would the line go up or down? Does this example represent a function? In this lesson, you will build on your previous knowledge about graphs and functions and learn how to create a graph using tables and equations. Objective Use tables and equations to graph linear functions Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words continuous data discrete data linear function slope-intercept form y-intercept Representing Functions Graphically You previously learned that a function is a relationship between two things, and each x-value has exactly one corresponding y-value. Sometimes it is easier to see this relationship by creating a graph. The graph of a linear function will have data points that create a straight line. The line may either be dashed or solid, depending on the type of data. A dashed line is sometimes used to indicate that only whole number values have meaning, such as choosing a number of scoops of ice cream. Data Type Discrete Date Continuous Data Description Data involves a count of items, where the numbers in between values do not make sense Uses data in which the values in between any two points have meaning Example Number of scoops of ice cream and cost Hours worked and total income Line Every linear function can be written in the form y = mx + b. In this equation, m is the slope of the line, and b is the y-intercept. The y-intercept is where the line crosses the y-axis. Both the slope and the y-intercept can be positive or negative values. The equation for the ice cream graph is y = 1.5x + 0. The slope of the line is 1.5 because the ice cream costs $1.50 per scoop. The line crosses the y-axis at 0, which means that the y-intercept is 0. This is because if you buy no scoops of ice cream, you pay nothing. If the cost per scoop of ice cream was $2.00 rather than $1.50, the slope of the line would increase, making the line steeper. Can you figure out what the equation would be for the graph above of the hours worked and income? Click on the Show Answer button below to check your answer. Answer: y = 7.5x + 0 In both of the previous graphs, the y-intercept was 0. If you work zero hours, you have zero income. If you buy zero scoops of ice cream, you owe zero dollars. But what if you wanted to create an equation and graph to represent the amount of money in a savings account after a given number of months if there is $150 in the account to start with and $25 is added to the account each month? The slope for this scenario will be the amount that the account balance increases each month, which is $25. The y-intercept will be the value of the account after zero months, which is $150. The equation will be y = 25x + 150. The following graph represents the equation. Functions That Are Not Linear A linear function can always be represented by the equation y = mx + b, where m is the slope and b is the y-intercept. Linear functions always make a straight line. However, not all functions are linear. Each graph below gives an example of a non-linear function and the type of equation that represents it. Notice that none of these functions make a straight line. Which of the following equations represent a linear function? A. B. y + 2 = 4x C. D. Click on the Show Answer button to review your answer. Answer: Equations B and D are both linear. Both equations can be changed into the y = mx + b format. B: y + 2 = 4x is equivalent to D: . is equivalent to y = 3x + 1. Since equations A and C both have exponents that are greater than 1, they do not represent linear functions. 1. Click on the links below to complete the Linear Functions activity from the digits™ website. Launch Key Concept Example 1 Example 2 Example 3 Close and Check 2. Click on the links below to complete the Linear Equations activity from the digits™ website. Launch Example 1 Example 2 Example 3 Key Concept Close and Check 3. Click on the link below to watch the "Rate of Change" Teachlet® tutorial. Rate of Change 4. Click on the link below to access the Rate of Change Transcript. Rate of Change Complete the following activities. 1. Read pp. 534–536 of Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 6–15 and 17 on p. 537. 3. Read Activity Lab 11-5a on p. 533 of Mathematics: Course 3. Then complete Exercises 1–6. 4. Click on the link below to watch the "Graphing Lines" Teachlet® tutorial. Graphing Lines Click on the link below to acces sthe Graphing Lines Transcript. Graphing Lines Click on the link below to access the online textbook. Mathematics: Course 3Complete the following review activities. 1. Click on the link below to complete the Graphing Linear Functions questions from the MathXL® for School website. Graphing Linear Functions 2. Review Lessons 4 through 6 in preparation for the quiz at the end of this lesson. Reflection Charles Washington is not permitted to take this assessment again. These answers will not be saved. 1. How comfortable are you working with sequences? (1 point) I can find a rule for a sequence and then extend it. I can tell if a sequence is arithmetic, geometric, both, or neither. This comes easily to me. I can find a rule for a sequence and then extend it. I need to use trial and error. I can find and extend simple sequences, but I have trouble with those that are more complex. I am not able to find or extend sequences. 2. Which best describes your understanding of slope? (1 point) I understand how a line’s rise is related to its run, and how to use this information to find slope. I can use slope to explain relationships between quantities. I understand how a line’s rise is related to its run, and how to use this information to find slope. I can find slope, but I am not quite certain what slope represents. I do not understand the concept of slope. 3. How comfortable are you using a graphing calculator? (1 point) I am very comfortable using a graphing calculator. I understand how to use the calculator to graph and find solutions. I am comfortable using a graphing calculator. I generally understand how to use the calculator to graph and find solutions, but sometimes I need to be reminded of the steps to take. I am sometimes comfortable using a graphing calculator, but not always. I understand some of the features, but I often need to be reminded of how to use them to graph or find solutions. I am not comfortable working with a graphing calculator at all. 4. Which of these skills do you think you could teach someone else? Select all that apply. writing rules for sequences and using the rules to find terms interpreting and sketching graphs that represent real-world situations identifying functions representing functions with equations, tables, and function notation calculating slope from different representations deriving the equation for a line in slope-intercept form using tables and equations to graph linear functions 5. With which of these skills do you need more help? Select all that apply. (1 point) writing rules for sequences and using the rules to find terms interpreting and sketching graphs that represent real-world situations identifying functions representing functions with equations, tables, and function notation (1 point) calculating slope from different representations deriving the equation for a line in slope-intercept form using tables and equations to graph linear functions Graphing Linear Functions Quiz Charles Washington is not permitted to take this assessment again. These answers will not be saved. Mulitple Choice 1. Find the slope. (1 point) 2 – –2 Use the graph below to answer the following question. 2. Find the slope of the line. Describe how one variable changes in relation to the other. 2; distance increases by 2 miles per hour 2; distance decreases by 2 miles per hour (1 point) ; distance increases 1 mile every 2 hours ; distance decreases 1 mile every 2 hours Use the graph below to answer the following question. 3. Find the slope of the line. Describe how one variable changes in relation to the other. ; the amount of water decreases by 2 gallons every 3 minutes. ; the amount of water decreases by 2 gallons every 3 minutes. ; the amount of water decreases by 3 gallons every 2 minutes. –1 ; the amount of water decreases by 1 gallon per minute. 4. The data in the table are linear. Use the table to find the slope. x 2 4 6 8 y 1 -2 -5 -8 5. Graph the linear function in questions 5 and 6. (1 point) (1 point) (1 point) 6. y = –2x + 3 (1 point) 7. Find the slope of a line that is parallel to the line containing the points (3, 4) and (2, 6). m=1 m=2 m = –2 (1 point) m= 8. Find the slope of a line that is perpendicular to the line containing the points (– (1 point) 2, –1) and (2, –3). m = –2 m=2 m= m = –1 Click on the links below to complete questions 1–15. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 © 2015 Connections Education LLC. Lesson 7: Graphing Proportional Relationships CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objectives: Draw the graph of proportional relationships; Distinguish between proportional and non-proportional relationships and identify examples of each; Identify the relationship between unit rate and the slope of the graphed line; Solved proportional relationships and direct variation problems Note: This lesson should take 2 days. Proportions and Graphs If you came across the following sign at your favorite gocart track, how would you decide which amount of time is the best deal? Is it cheaper to race for one hour at a time, or four hours all together? Are the rates proportional? What would a graph of the rates look like? Previously, you learned about proportional relationships and unit rates. You learned how to solve for missing values in a proportion, determine missing side lengths in proportional figures, and apply proportion to scale. In this lesson, you will discover what a proportional relationship looks like when graphed and about the connection between unit rate and slope. As with a previous unit on proportions, you will see that these problems have many real-world applications. Objectives Draw the graph of proportional relationships Distinguish between proportional and non-proportional relationships and identify examples of each Identify the relationship between unit rate and the slope of the graphed line Solved proportional relationships and direct variation problems Key Words direct variation proportional relationship unit rate Tip: You will have 2 days to complete this lesson. The Unit Rate as the Slope The rates for the go cart rentals given in the Getting Started section are proportional. By finding the unit rate, which is the rental rate for one hour, you can compare all three. Set up a proportion where x represents the charge to rent a go cart for one hour: = . Using cross products, you find that x = $12.50. Is this unit rate the same for all three prices shown on the sign? Test them to see. The unit rate for the go cart rental is $12.50 per hour in all three cases. This means the rates are proportional. For every additional hour of rental time, the price increases by $12.50. What would a graph of the prices look like? If you graphed points showing the rates for 1 hour, 2 hours, and 3 hours, the points would form a straight line. The rise would be 12.5, and the run would be 1. The slope (rise over run) is 12.5 and is the same as the unit rate. The line would go through the origin, because if you rent the go cart for zero hours, it would cost you zero dollars. The equation would be y = 12.5x (where y = cost and x = hours). Comparing Graphs, Equations, and Tables The cost of going to see a movie at three different theaters is shown below. The information for each theater is presented in a different format. For all three theaters, x represents the number of tickets and y represents the total cost. Theater 1: Theater 2: y = 10x Theater 3: x y 2 19 4 38 6 57 Think about what you have learned about unit rate and slope to put the theaters in order from the one with the least expensive tickets to the one with the most expensive tickets. Click on the Show Answer button to review your answer. Answer: The tickets at Theater #1 are $8 each. This is the slope of the graph. The tickets at Theater #2 are $10 each. This is m in the equation y = mx + b. The tickets at Theater #3 are $9.50. This is the ratio between each set of points. In order, the theater prices from least to greatest are #1, #3, #2. 1. Click on the links below to complete the Graphing Proportional Relationships activity from the digits™ website. Topic Opener Launch Example 1 Key Concept Example 2 Example 3 Close and Check 2. Click on the links below to complete the Unit Rates and Slope activity from the digits™ website. Launch Key Concept Example 1 Example 2 Example 3 Close and Check Complete the following review activities. 1. Click on the link below to complete the Graphing Proportional Relationships questions from the MathXL® for School website. Graphing Proportional Relationships 2. Click on the link below to complete the Unit Rates and Slope questions from the MathXL® for School website. Unit Rates and Slope Graphing Proportional Relationships Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice Use the table, graph, and equation to answer the following questions about three different sub shops. 1. What is the unit rate for Fred’s Sub Shop? (1 point) $10 for 2 subs $5 for 1 sub $1 for of a sub $30 for 6 subs 2. What is the slope-intercept equation for the cost of a sub at Fred’s Sub Shop? (1 point) y = 2x y = 10x y = 5x y = 5x + 2 3. What is the slope of the line for the cost of subs at Sam’s Sub Shop? (1 point) Sam's Sub Shop Cost $11 $22 $44 Subs 2 subs 4 subs 8 subs 5.5 5 4. The equation for the cost for subs at Anne’s Restaurant is y = 4.75x. If the cost (1 point) for subs at all three sandwich places were graphed, which would have the steepest line? Fred’s Sam’s Anne’s The lines would have the same slope. 5. Which of the following equations represents a proportional relationship? (1 point) y = 6x – 4 y = 4x + 0 y = 1.7x + 9 y = –3x + 1 p. CC27 Answers 1 a. The January blizzard; the slope for December is inches per hour, and the slope for January is inches per hour. b. inches per hour; inches per hour 2. The giant sea kelp plant had a faster growth rate. Compare the slopes (unit rates) in inches per hour. bamboo: ; bull kelp: ; kudzu: ; giant sea kelp: . 3. Basketball burns more calories per hour (750) than cross-country skiing (660) because 11 calories per minute equals a rate of 660 calories per hour. Unit Rates and Slope Answers 1 a. The January blizzard; the slope for December is inches per hour, and the slope for January is inches per hour. b. inches per hour; inches per hour 2. The giant sea kelp plant had a faster growth rate. Compare the slopes (unit rates) in inches per hour. bamboo: ; bull kelp: ; kudzu: ; giant sea kelp: . 3. Basketball burns more calories per hour (750) than cross-country skiing (660) because 11 calories per minute equals a rate of 660 calories per hour. Problems 1–7 on p. CC26 1. yes, because the ratios of the weight to the cost are the same for all three pricing structures. 2. 3. (0,0) 4. 5. $1.50 per pound; the same as the cost of 1 pound of tomatoes: $1.50 per pound 6. Slope is equivalent to unit rate. 7. The ration used to find rate is equivalent to the slope, , of the line containing the points. Problems 1–6 on p. CC27 1. 30 miles per hour 2. The speed of train A 3. 45 miles per hour 4. Train B is moving faster. The unit rate for train B (45 miles per hour) is faster than the unit rate for train A (30 miles per hour). 5. 35 miles per hour 6. The speed of train C is 35 miles per hour, so it is faster than train A and slower than train B. Proportional vs. Non-Proportional Situations Not all equations are proportional situations. Remember, for a situation to be proportional (and for an equation to be a direct variation) each x and y pair must be proportional to every other x and y pair. Consider the following scenario: You and a group of friends are going to an amusement park. The cost to get into the park is $4.00 per person, and each ride costs $0.50 to get on. How much money will you spend if you ride 4 rides? What if you ride 10 rides? In this situation, you will spend more money if you ride more rides, but there is an added cost: The admission price. Even if you plan to ride zero rides, you’ll still spend $4.00 just to get into the park. This extra cost makes the x and y pairs not proportional. The general equation for a situation such as this one is y = mx + b, where m is the unit rate (this is the same as k in the equation you learned earlier) and b is the constant value that does not depend on the input value x. If you were to apply this general equation to the situation with the amusement park described above, m would stand for the cost per ride ($0.50), b would stand for the admission price ($4.00), x would be the number of rides you ride, and y would be the total amount spent. The equation would be y = 0.5x + 4. The table below shows some of the x and y pairs for this equation. x (number of rides) y (money spent) 0 $4.00 1 $4.50 2 $5.00 3 $5.50 4 $6.00 You can probably see from the table already that the x and y pairs are not proportional to each other, but it can be tested just to be sure. Set up a proportion with the pairs (2, 5) and (4, 6). The graph of this situation should also tell you that this equation is not a direct variation. Recall that the graph of a direct variation will always be a straight line that goes through the origin. While this graph produces a straight line, it does not go through the origin. This makes sense in terms of the problem. Even if you ride zero rides, you still need to pay $4.00 to get into the park. Look at the following graphs and determine which line, if any, represents a proportional relationship and which represents a non-proportional relationship. Click on the Show Answer button to check your answer. Answer: Line a is a non-proportional relationship because it does not go through the origin. Line b is a proportional relationship because it is a straight line that goes through the origin. Look at the table below. Does the data in the table represent a proportional relationship? Explain why or why not. x y 0 0 1 3 2 6 3 9 4 12 Click on the Show Answer button to check your answer. Answer: Yes, the data in the table represents a proportional relationship because the x and y pairs are proportional and the value for y is zero when x = 0. Look at the equations below. Identify which ones represent a proportional relationship and explain your answer. a. y = 2.3x + 0 b. y = 3x + 8 c. y = x –­ 6 d. y = 100x e. y = x Click on the Show Answer button to check your answers. Answer: Equations a, d, and e represent proportional relationships because they do not have a constant term that is independent of the x term. They follow the form y = kx. Equations b and c are not proportional relationships because they follow the form y = mx + b. The b term is a constant that is independent of the x term. Complete the following activities. 1. Complete problems 1–7 and problems 1–6 of each Activity (Graphing Proportional Relationships) on pp. CC26–CC27 of Mathematics: Course 3. 2. Complete exercises 1–3 on p. CC27. Click on the link below to access the online textbook. Mathematics: Course 3 Lesson Answers Click on the link below to check your answers to problems 1–7 on p. CC26 and problems 1–6 on p. CC27 of Mathematics: Course 3. pp. CC26–CC27 Answers Click on the link below to check your answers to problems 1–3 on p. CC27. p. CC27 Answers Click on the links below to complete questions 1–15. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 Click on the links below to complete questions 1–15. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 © 2015 Connections Education LLC. Lesson 8: Writing Rules for Linear Functions CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objective: Write function rules from words, tables, and graphs Solving Problems Gus wants to get a job delivering newspapers. One company will pay him $10 per day plus $0.15 per newspaper. Another company will pay him $20 per day plus $0.05 per newspaper. He wants to make a smart decision about which company to work for, but he isn’t sure how to use mathematics to solve the problem. In this lesson, you will learn to write linear equations when given facts in words, tables, or graphs. This skill will help you choose the better deal in many different real-world situations. Objective Write function rules from words, tables, and graphs Key Words intercept slope slope-intercept form Describing Situations as Linear Functions Often, it is useful to have a rule for a linear function written as an equation. You can use the example from the Getting Started section of the lesson. First, make a chart that shows how much money Gus would make from each company for delivering different numbers of newspapers. 10 Papers Company #1 Company #2 10 + (10 × 0.15) = 11.5 20 + (10 × 0.05) = 20.5 20 Papers 10 + (20 × 0.15) = 13 20 + (20 × 0.05) = 21 30 Papers 50 Papers 10 + (30 × 0.15) 10 + (50 × 0.15) = 14.5 = 17.5 20 + (30 × 0.05) 20 + (50 × 0.05) = 21.5 = 22.5 You can determine the amount you will be paid for a certain number of papers at each company by multiplying the number of papers by the amount you will be paid for each paper. Then you must add the result to the fixed daily amount ($10 or $20). You could also write a function rule for each company in the form y = mx + b. For a function rule in this form, x represents the input value. It is the part of the scenario that can change and you can usually attach it to the word “per.” In this scenario, x = the number of papers. The value of m represents the rate of change, or slope. In this scenario, m = 0.15 for Company #1 and m = 0.05 for Company #2. The y-variable represents the output value. It is the part of the scenario that will be affected as the x-value changes. In this scenario, y = your total income. The value of b represents the starting value in the scenario. It is fixed, so it does not change. In this scenario, b = $10 for Company #1 and b = $20 for Company #2. You would then be able to use your equations to decide which company is better to work for. Your total pay for Company #1 can be determined using the equation y = 0.15x + 10. Your total pay for Company #2 can be determined using the equation y = 0.05x + 20. In this lesson, it will be important to understand the relationship between words, tables, graphs, and linear equations. A linear equation in the form y = mx + b can be derived from words, tables, or graphs. Writing a Function Rule Given Words At a certain pizza restaurant, the manager plans to purchase 3 new work shirts for each employee, plus 50 additional shirts to keep on hand. The function rule is y = 3x + 50. Writing a Function Rule Given a Table The table below shows pairs of x- and y-values. Notice that the values for the input, x, increase by 3 while the values for the output, y, increase by 4. x 0 3 6 9 y 4 8 12 16 The difference between the x-values and the difference between the y-values is consistent throughout the table. The ratio of will be m in the function rule. The initial value, or y–intercept, is the y-value when x = 0. In this problem, the y-intercept is 4. The function rule is . Writing a Function Rule Given a Graph To find the rate of change, count the rise over the run between two points. The y-intercept is the point where the graph crosses the y-axis. The function rule is y = 2x + 5. 1. Click on the links below to complete the Defining a Linear Function Rule activity from the digits™ website. Topic Opener Launch Example 1 Example 2 Key Concept Example 3 Close and Check 2. Click on the link below to watch the "Deriving the Equation of a Line" Teachlet® tutorial. Deriving the Equation of a Line 3. Click on the link below to access the Deriving the Equation of a Line Transcript. Deriving the Equation of a Line Complete the following activities. 1. Read pp. 540–541 of Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 3–11 on pp. 542–543. 3. Read the "Linear Functions" Guided Problem Solving on p. 544. 4. Complete problems 1–6 on p. 545. Click on the link below to access the online textbook. Mathematics: Course 3Complete the following review activity. 1. Click on the link below to complete the Defining a Linear Function Rule questions from the MathXL® for School website. Defining a Linear Function Rule Lesson Answers Click on the link below to check your answers to the questions from the Guided Problem Solving section on p. 545. Guided Problem Solving Answers Writing Rules for Linear Functions Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Write a function rule for finding the amount of daily pay, p, in the following (1 point) situation: A bus driver gets paid $100 each day plus $0.20 per kilometer, k. 100 = 0.20 × k p = 0.20 × 100 × k p = 0.20k + 100 0.20k = 100 + p 2. Do the values in the table represent a linear function? If so, what is the function rule? The values do not show a linear function. Yes, they show a linear function; y = x + 4. Yes, they show a linear function; y = 2x + 2. Yes, they show a linear function; y = 2x. 3. Write an equation for the line shown in the graph. 10 = 4x y = 2x + 10 y = 10x – 2 y = –­ x + (1 point) (1 point) 10 4. Write a function rule for the total cost. (1 point) One frozen yogurt store sells frozen yogurt for $3.00 per cup and $1.25 per topping. Write a linear equation to show the total cost of a cup of frozen yogurt. Then calculate the total price for one cup of frozen yogurt with 4 toppings. y = 1.25x + 3; $5.00 y = 3x + 1.25; $13.25 y = 1.25x + 3; $8.00 y = 5x + 3; $8.00 Click on the links below to complete questions 1–15. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15Guided Problem Solving Answers 1. Yes, the function rule works for any positive number. 2. The conclusion was made by using the table and graph. You can see from the graph that for some distances Plan 1 is better, and for some distances Plan B is better. It depends on whether you drive more or less than 35 miles. If you drive exactly 35 miles, both plans will cost the same. 3. about 53 million senior citizens 3a. Graph is a line with the points (2004, 36.3) and (2050, 86.7). 3b. y = 1.1x + 31.9 4. 8.8 miles 4a. 3,000 ft/mi 4b. A = 3,000m 5. About 287 hats will sell if hats are priced at $5 each. 6. T = 20,000 + 500m; 60 months © 2015 Connections Education LLC. Lesson 9: Solving Systems of Equations CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objective: Solve systems of equations by graphing and by substitution Note: The content you are trying to access is not formatted properly. Will You Meet? Imagine each of these roads as long and straight, and extending indefinitely in both directions. If you were travelling along one of the roads and a friend was travelling along the other, you would meet each other at the point where the roads cross. In previous lessons, you learned that the graphs of linear equations form straight lines. In this lesson, you will begin to investigate the relationships of a set of linear equations, known as systems of equations. You will also learn how to solve a system of equations through graphing and substitution. Objective Solve systems of equations by graphing and by substitution Key Word system of equations Solving Systems of Equations with Graphing You can think of a graph in the same way you thought about the crossroads in the Getting Started section of this lesson. All of the points along the red line are solutions to the equation y = 2x + 1. All of the points along the blue line are solutions to the equation y = –x – 2. The point where the two lines meet (–1, –1) is a solution to both equations. A system of equations is a collection of two or more equations that have the same variables. You can find the point where two linear equations intersect through graphing. The point where the graphs intersect is called the solution of the system of equations. There are three possible types of solutions for systems of linear equations: no solution, one solution, or an infinite number of solutions. The graph above shows one solution, (–1, –1). The equations y = 3x + 4 and are represented by the two lines in the graph below. Use the graph to find the solution to this system of equations. Click on the Show Answer button to review your answer. Answer: Since the two lines on the graph never intersect, this system has no solution. Click on the links below to complete the Solving Systems of Linear Equations by Graphing activity from the digits™ website. Launch Example 1 Example 2 Example 3 Close and Check Solving Systems of Equations Using Substitution Another method of solving systems of equations is substitution. The first step for this method is to solve one of the equations for one of the variables, x or y. y+x=3 y–x=1 y=1+x Solve for y. (Now the y is isolated on one side of the = sign.) Since you know that y = 1 + x, you can substitute 1 + x into the other equation for y. The rest of the equation remains the same. y+x=3 1+x+x=3 1 + 2x = 3 2x = 2 x=1 So, the x-coordinate of the solution is 1. Now substitute x = 1 into either equation to solve for y. y+x=3 y+1=3 y=2 The solution is (1, 2). When you graph the two equations, (1, 2) is the point where the two lines intersect. Use substitution to solve the following system of equations: and Click on the Show Answer button to review your answer. Answer: The solution to the system of equations is x=2 . y=0 1. Click on the link below to watch the "Systems of Equations" Teachlet® tutorial Systems of Equations 2. Click on the link below to access the Systems of Equations Transcript Systems of Equations Transcript 3. Click on the links below to complete the Solving Systems of Linear Equations Using Substitution activity from the digits™ website. Launch Key Concept Example 1 Example 2 Close and Check Fair Race You probably have heard about the classic fable "The Tortoise and the Hare." In the fable, the two animals race—which is totally unfair because a hare is much faster than a tortoise. Still, the tortoise wins the race thanks to its slow-and-steady strategy. "The Tortoise and the Hare" fable is the inspiration for a new video game called Animal Tracks. In the game, two players can choose any two animals to race. The computer automatically adjusts the race to make it fair. As a result, the outcome of the race depends not on speed, but on the player's skill and strategy in the race. In this unit's portfolio item, you will use equations and graphs to show how the computer will make the race fair. Begin thinking about and working on the portfolio project now by reviewing the portfolio worksheet and rubric. You will submit the portfolio at the end of the next lesson. Click on the link below to access the Fair Race worksheet. Fair Race Click on the link below to access the Fair Race rubric. Fair Race RubricComplete the following review activities. 1. Click on the link below to complete the Solving Systems of Equations by Graphing questions from the MathXL® for School website. Solving Systems of Equations by Graphing 2. Click on the link below to complete the Solving Systems of Equations Using Substitution questions from the MathXL® for School website. Solving Systems of Equations Using Substitution 3. Review Lessons 7 through 9 in preparation for the quiz at the end of this lesson. Solving Systems of Equations Quiz Charles Washington is not permitted to take this assessment again. These answers will not be saved. 1. You buy 3 pounds of organic apples for $7.50. The graph shows the price for (1 point) regular apples. What is the unit rate for each type of apple? organic $2.50/pound; regular $3.00/pound organic $0.40/pound; regular $0.50/pound organic $2.50/pound; regular $2.00/pound none of the above 2. The price for pears is y = 2.75x. Which line would have the steepest slope if organic apples and pears were added to the graph? organic apples pears regular apples We need more information to answer this question. 3. What is the rule for the function shown in the table? (1 point) (1 point) y = 3x + 1 4. What is the function rule for the following situation? Rex paid $20 for a (1 point) membership to the pool and pays $3.00 each time he goes to the pool. y = 20x + 3 20 = 3x + y y = x + 20 none of these 5. Find the solution to the system of equations by using either graphing or (1 point) substitution. y = 6 – x and y = x – 2 (2, 4) (–4, 2) (4, 2) no solutions 6. y = 2x – 1 and y = x + 3 (1 point) (4, 7) (7, 4) (–7, –4) infinite solutions 7. y = 4x and y + x = 5 (1 point) (–4, 1) (1, 4) (–3, 2) (2, 3) 8. What will the graph look like for a system of equations that has no solution? The lines will be perpendicular. (1 point) The lines will cross at one point. Both equations will form the same line. The lines will be parallel. p. CC31 Answers 1. no solution 2. infinitely many solutions 3. (–1, 0) 4. 1.3, 0.4) 5. (3.5, 0.5) 6. (0.7, –3.8) 7. (1, 3) 8. (5, 4) 9. (7, –2) 10. (5, 8) 11. infinitely many solutions 12. no solution 13. 5; $24 14. The lines intersect at (3, 3.5) y = 0.5x + 2 and y = –1.5x + 8. 15. Yes; the lines intersect at (2.5, 2). 16. Multiply 5x – 6y = 8 by a factor of 3 to make it 15x – 18y = 24. Then multiply –3x + 11y = 10 by a factor of 5 to make it –15x + 55y = 50. Add the two new equations together to eliminate the x-terms, resulting in 37y = 74. Divide both sides by 37 to get y = 2. Finally, solve for x by substituting the value of 2 in for y into either of the equations. x = 4. Lesson Answers Click on the link below to check your answers to problems 1–16 on p. CC31. p. CC31 Answers Complete the following activities. 1. Read "Solving Systems of Equations" on pp. CC28–CC30 of Mathematics: Course 3. Be sure you understand how to solve systems of equations with the graphing and substitution methods. 2. Complete problems 1–16 on p. CC31. Click on the link below to access online textbook. Mathematics: Course 3 Click on the links below to complete questions 1–15. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 Click on the links below to complete questions 1–12. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 © 2015 Connections Education LLC. Lesson 10: Nonlinear Functions CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objective: Graph and write quadratic functions and other nonlinear functions Note: The content you are trying to access is not formatted properly. Are All Functions Linear? Earlier in this unit, you learned how to sketch a graph to match a particular situation. Instead of sketching a graph to get a general idea of what it would look like, you could record actual values that would make the graph more exact. For example, if you wanted to make a graph to represent the jump of this dog, you could plot a set of points that showed the dog’s distance for x and the dog’s height for y. Would this graph be linear? If not, what would the overall shape of the graph look like? Would it still represent a function? In this lesson, you will learn to recognize and graph nonlinear functions. Although many of the steps will be familiar to you, the look of the graphs will be different. Objective Graph and write quadratic functions and other nonlinear functions Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words nonlinear function parabola quadratic function Quadratic Functions Not all functions have graphs that are straight lines; therefore, not all functions are linear functions. Functions that are not linear are called nonlinear functions. One important nonlinear function is the quadratic function. A quadratic function is one where the greatest exponent of any variable is 2. For example, y = 2x2 – 3 is a quadratic function. The graph of a quadratic function is a special U-shaped curve called a parabola. The curve of a parabola can open either up or down. Unlike a linear function, which has a constant rate of change, a quadratic function does not have a constant rate of change. Its slope does not remain constant, but changes continually. To graph the quadratic function , make a table of x- and y-values. Choose values to substitute into the equation for x, and then solve for y. When graphing a quadratic function, it is important to choose negative and positive x-values. x 0 1 2 y 3 0 0 3 Since the parabola opens up, it has a minimum value of . The minimum value is the point with the least y-value. The graph has intercepts at and . To graph the quadratic function , make a table of x- and y-values. Choose values to substitute into the equation for x and then solve for y. x y 0 3 0 1 2 4 3 0 Since the parabola opens down, it has a maximum value of y = 4. The maximum value is the point with the greatest y-value. The graph has intercepts at and . Click on the links below to complete the Nonlinear Functions activity from the digits™ website. Work through the Key Concept and Part 1 of the Example section of the lesson. Launch Key Concept Example 1 Example 2 Example 3 Close and Check Click on the link below to access the Graphing a Quadratic Function video on the Khan Academy website. As you watch the video, please note that the quadratic function is written in function notation, which uses f(x) (read as “F of X”) instead of the variable y. However, these two values are interchangeable, and the graph does not change at all as a result. Graphing a Quadratic Function Click on the link below to watch the Graphing Nonlinear Functions Teachlet® tutorial. Notice how a table of values is used to graph the nonlinear functions. Graphing Nonlinear Functions Complete the following activities. 1. Read pp. 546–547 of Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 4–7, 10, 11, 14, 15, 21, and 22 on pp. 548–549. 3. Read the "Changing Representations" Activity Lab on p. 550. Complete problems 1–4 on p. 550. Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activity. 1. Click on the link below to complete the Nonlinear Functions questions from the MathXL® for School website. Nonlinear Functions Lesson Answers Click on the link below to check your answers to the Activity Lab. Activity Lab Answers Fair Race You will now submit your portfolio that you started working on in the Solving Systems of Equations lesson on Slide 4. You probably have heard about the classic fable "The Tortoise and the Hare." In the fable, the two animals race—which is totally unfair because a hare is much faster than a tortoise. Still, the tortoise wins the race thanks to its slow-and-steady strategy. "The Tortoise and the Hare" fable is the inspiration for a new video game called Animal Tracks. In the game, two players can choose any two animals to race. The computer automatically adjusts the race to make it fair. As a result, the outcome of the race depends not on speed, but on the player's skill and strategy in the race. In this unit's portfolio item, you will use equations and graphs to show how the computer will make the race fair. Click on the link below to access the Fair Race worksheet. Fair Race Click on the link below to access the Fair Race rubric. Fair Race Rubric This is a portfolio item. When you are finished, please submit your answers to your teacher using the Drop Box below. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Activity Lab Answers 1. Function Rule: y = 3 + 2x2 x y –2 11 –1 5 0 3 1 5 2 11 2. n p 1 15 2 45 3 135 4 405 5 1,215 3. Function Rule: y = 10 + 8x 4. It is easier to use a graph when looking for an estimate or prediction, and it is easier to use a function rule when you have a known value.Click on the links below to complete questions 1–15. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 © 2015 Connections Education LLC. Lesson 11: Comparing Functions CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objective: Compare functions represented in various ways Looking at Things in Different Ways In mathematics, there is often more than one way to represent the same thing. When you first learned to add, you probably started by using pictures. + = Soon, you learned to substitute numbers for the pictures: 2+3=5 Later you understood that there are words that could go with the pictures of the equation: Lucy picked two oranges on the first day and three more on the second day. How many did she pick altogether? You also learned that you could show this information with a graph, such as a pictograph. In this unit, you learned how to represent the same function in different ways. In this lesson, you will compare functions. Objective Compare functions represented in various ways Key Words algebraic representation graphic representation numerical representation verbal description Functions: How Do They Compare? Just as the example of the oranges in the Getting Started section can be expressed in different forms, functions can also be expressed in a variety of ways. In this unit, you learned to use function notation, algebraic equations, tables, graphs, and verbal descriptions to show a function. The ability to compare functions that are expressed in any of these ways is a key step in understanding how functions work and what they do. The same relationship between 2 variables is shown 4 different ways in the following table: Form Example Benefits of This Form A verbal description gives verbal Andre received a $60 gift card to purchase online music. meaning to the relationship description Each download costs $1.20. when it is presented in other ways. Form Example Benefits of This Form When a relationship is shown algebraically, it is easy to identify the rate of change and starting value. algebraically It is also easy to identify whether the graph will have a positive or negative slope. An equation can be used to make a graph by graphing the y–intercept and slope. A table of values gives you points that can be used to make x y 0 60 10 48 20 36 30 24 40 12 a graph. Two pairs of values from the table can be used to find the table of values rate of change, or slope. The points from the table can be used to understand the scenario better. For example, after 10 downloads, Andre will have $48 left on his gift card. 50 0 If the table of values includes a 0 for x, it shows the starting value. Two points from the graph can be used to find the rate of change. graphically The graph shows the starting value. The graph can be used to identify the value of both variables at various points. Although the two functions that follow are presented using two different methods, they can still be compared to determine which expresses a greater rate of change. Function 1: Function 2: You can identify the rate of change in the graph by determining the slope between two points. Since the line shows a rise of 2 and a run of 1, the rate of change in the graph is 2. The rate of change in the equation is 2.5, so Function 2 has a greater rate of change. Click on the links below to complete the Comparing Two Linear Functions activity from the digits™ website. Launch Example 1 Example 2 Key Concept Close and CheckComplete the following review activities 1. Click on the link below to complete the Comparing Two Linear Functions questions from the MathXL® for School website. Comparing Two Linear Functions 2. Review Lessons 10 and 11 in preparation for the quiz at the end of this lesson. Comparing Functions Quiz Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice Determine which function has the greater rate of change in problems 1−3. 1. (1 point) The rates of change are equal. The graph has a greater rate of change. The table has a greater rate of change. none of the above 2. (1 point) As x increases by 1, y increases by 3. The slopes are equal. The graph has a greater slope. The function rule has a greater slope. none of the above 3. What would the graph of y = x2 + 1 look like? (1 point) a straight line a parabola a dotted line none of the above 4. Which of the following equations represent nonlinear functions? (2 points) y=3 5. What would the graph of y = x– look like? (1 point) a straight line a parabola a curve none of the above 6. Complete the table for the given function. (1 point) –3, –2, 1, 6 3, 4, 7, 12 0, 1, 4, 9 none of the above Complete the following activities. 1. Read p. CC32 of Mathematics: Course 3. Be sure you understand how to compare linear and nonlinear functions. 2. Complete problems 1–10 on p. CC33. Click on the link below to access the online textbook. Mathematics: Course 3 Lesson Answers Click on the link below to check your answers to problems 1–10 on p. CC33. p. CC33 Answers Click on the links below to complete questions 1–12. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 p. CC33 Answers 1. the function given as an equation (3 > 2) 2. the function given as an equation ( > ) 3. the function given as a table (5 > 4) 4. the function given as a table (2.75 > 2.5) 5. Both functions are continuous. The function in the graph decreases until x = 30 and y = 4, then it increases. The graph does not have a constant slope. The other function always increases at a constant rate. 6. Both functions are continuous and neither increases or decreases at a constant rate. The function in the graph decreases until x = 30 and y = 4, then it increases. The other function always decreases, and gets closer and closer to zero. 7. The slopes are A: , B: , C: 1, and D: . By order of increasing slope: B, D, C, A 8. T, W, E, G 9. Gamma, Inc.; Alpha, Inc.; Beta Co.; Delta Corp 10. The base fee for subcompact cars is $30 plus $25 per day. The base fee for compact cars is $40 plus $28 per day. The base fee for luxury cars is $40 plus $30 per day. The subcompact plan is always cheaper than the other two plans. © 2015 Connections Education LLC. Lesson 12: Functions Unit Review CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Objective: Review for unit test Note: This lesson should take 2 days. Getting Ready The test at the end of a unit is an opportunity for you to demonstrate everything you have learned while studying the concepts from this unit. In this lesson, you will review testtaking strategies that will help you successfully take the unit test and demonstrate to your teacher all you have learned in this unit. You will also have the chance to practice what you learned during previous lessons in this unit by using various review activities. Objective Review for unit test Key Words algebraic representation arithmetic sequence common difference common ratio congruent continuous data coordinates discrete data function function notation function rule geometric sequence graphic representation inductive reasoning input intercept linear function nonlinear function numerical representation output parabola parallel proportional relationship quadratic function ratio rise run sequence similar similar triangles slope slope-intercept form system of equations term transversal unit rate verbal description y-intercept Test-Taking Strategies In the next lesson, you will take the test on the skills that you have learned in this unit. Before taking any big test, it is a good idea to review test-taking strategies. Multiple Choice Questions 1. Read through the question and all of the answer choices before selecting your response. 2. Find any Key Words in the question. 3. Find out what the question is asking. There may be choices that look like the correct answer, but do not answer the whole question. 4. Eliminate any choices that are incorrect. 5. After you make your choice, re-read the question again to check that the answer you chose is the best answer. 6. In questions that involve calculations, double check your work. Short Answer Questions 1. Read through the question. 2. Find any Key Words, and determine what the question is asking. 3. Show all of the steps you used to find your answer. 4. Ask yourself if your answer makes sense. 5. Check over your work to be sure that your computation is correct. 6. Re-read the question, and make sure that your response properly answers the question. What to Study Go back through the unit and review any concepts that you are still struggling with. You may review any Teachlet® tutorial, video, Gizmo, or online lesson that you need to in order to prepare yourself for the upcoming test. Complete the following activities. 1. Read through the "Vocabulary Review" section on p. 552 of Mathematics: Course 3. Be sure you can answer questions 1–5. 2. Work through the "Skills and Concepts" portion on pp. 552–553. Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activities. 1. For each of the scenarios below, complete the following: a. Wendy goes bowling at Al’s Bowling. It costs $5.00 to rent the lane and $3.00 per game. b. Jamil goes bowling at Family Bowling Center, where it costs $2.00 to rent the lane and $4.00 per game. Create a graph of both functions on the same coordinate grid. Make an input-output table for each function. Find the slope for each scenario. Find the equation for each scenario. Click on the link below to check your answers. Answers 2. Answer the following questions based on the scenarios in Part 1: a. Do Wendy’s y-values show an arithmetic or geometric sequence? b. Is Jamil’s graph a linear or nonlinear graph? c. How much would it cost Wendy to bowl 25 games? d. Write an equation for a line that would be parallel to Wendy’s. e. Which bowling alley’s cost function has a greater rate of change—Al’s Bowling or Family Bowling Center? f. Is there a point where the lines for both bowling alleys would cross? What does this point represent? g. Use substitution to find the solution for the system of equations formed using both functions. Click on the link below to check your answers. Answers 3. Take the Chapter 11 Test on p. 554 of Mathematics Course 3. Check your answers in the back of the book, and go back and review any concepts that you didn’t get correct on the test. Click on the link below to access the Functions Unit Review Practice. Functions Unit Review Practice Click on the link below to access the online textbook. Mathematics: Course 3 Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Answers a. arithmetic, the common difference is 3 b. linear c. $80 d. Sample: y = 3x + 8 (They should have the same slope.) e. Family Bowling Center; it has a slope of 4 f. yes; the lines cross at (3, 14); it is the place where the same number of games cost the same amount g. y = 3x + 5, y = 4x + 2 4x + 2 = 3x + 5 x=3 Answers 1a. and 1b. 1c. The graph of Wendy’s line has a slope of 3; the graph of Jamil’s line has a slope of 4. © 2015 Connections Education LLC. Lesson 13: Functions Unit Test CE 2015 Algebra Readiness (Pre-Algebra) B Unit 2: Functions Functions Unit Test Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. What are the next three terms in the sequence? (1 point) –3, 6, 15, 24, … 35, 46, 57 34, 44, 54 33, 44, 56 33, 42, 51 2. Geoff planted dahlias in his garden. Dahlias have bulbs that divide and reproduce underground. In the first year, Geoff’s garden produced 8 bulbs. In the second year, it produced 16 bulbs, and in the third year it produced 32 bulbs. If this pattern continues, how many bulbs should Geoff expect in the sixth year? 64 bulbs 512 bulbs 128 bulbs 256 bulbs 3. Which graph represents y as a function of x? (1 point) (1 point) 4. A car maintains a speed of 23 mi/h for 5 seconds. It then accelerates to a speed of 46 mi/h in 5 seconds. It maintains that speed for the next 5 seconds. Which graph shows the car’s speed over time? (1 point) 5. Use the graph below to answer the question that follows. (1 point) Which statement describes the speed of the remote-control car over time? The speed of the car decreases from 4 mi/h to 2 mi/h in the first 3 seconds, increases to 5 mi/h in the next 2 seconds, and then remains at 5 mi/h for the last 5 seconds. The speed of the car increases from 4 mi/h to 2 mi/h in the first 3 seconds, decreases to 5 mi/h in the next 2 seconds, and then remains at 5 mi/h for the last 5 seconds. The speed of the car decreases from 4 mi/h to 2 mi/h in the first 3 seconds, increases to 6 mi/h in the next second, and then remains at 6 mi/h for the last 6 seconds. The speed of the car decreases from 4 mi/h to 2 mi/h in the first 3 seconds, increases to 5 mi/h in the next 5 seconds, and then remains at 5 mi/h for the last 10 seconds. 6. Given the function rule f(x) = x² – 4x + 3, what is the output of f(–3)? (1 point) 24 21 0 –3 7. Suppose you earn $10 each hour you babysit. Which function describes the (1 point) relationship between your total earnings E and the number of hours you babysit, h? E(h) = 10h E(h) = h + 10 E(h) = h – 10 h = 10E 8. The data in the table illustrate a linear function. x –3 0 3 6 y –5 –3 –1 1 What is the slope of the linear function? Which graph represents the data? (1 point) 9. Which hill described in the table is the steepest? Explain. Street Horizontal Vertical Rise Distance (ft) of Street (ft) Dixie Hill 80 40 Bell Hill 80 20 (1 point) Liberty Hill 80 60 Bell Hill; it rises 1 foot for every 4 feet of horizontal travel. Dixie Hill; it rises 2 feet for every 1 foot of horizontal travel. Liberty Hill; it rises 4 feet for every 3 feet of horizontal travel. Liberty Hill; it rises foot for every 1 foot of horizontal travel. 10. Which graph represents the linear function y = x – 4? (1 point) 11. Which graph represents the linear function y = –5x + 2? (1 point) 12. Which function rule represents the data in the table below? Input (x) 1 2 3 4 5 Output (y) 9 15 21 27 33 y = 4 + 5x y = 3 + 6x y = 5 + 4x y = 6 + 3x 13. Which representation shows y as a function of x? (1 point) (1 point) 14. The sale price of ground beef at a local grocery store is $1.49 for the first pound and $1.09 for each additional pound. Which function rule shows how the cost of ground beef, y, depends on the number of pounds, x? y = 1.49x + 1.09 y = 1.09(x – 1) + 1.49 y = (1.09 + 1.49)x y = 1.09x + 1.49 15. Which function rule represents the data in the table? x –3 –2 –1 y 1 0 (1 point) 1 –2 –5 –8 –11 y = –3x – 8 y= x–8 y= x+8 y = 3x + 8 16. Which quadratic rule represents the data in the table? x –1 0 1 2 y 4 3 5 4 1 –4 y = –2x² + 5 y = –x² + 5 y = x² – 5 y = x² + 5 Short Answer (1 point) (1 point) Note: For questions 17–20, your teacher will grade your responses to ensure you receive proper credit for your answers. 17. A 154-lb person burns 420 calories per hour riding an exercise bicycle at a rate of 15 mi/hr. Write a function rule to represent the total calories burned over time by that person. Explain how the information in the problem relates to the function. (3 points) 18. Explain how to write a function rule from the table below. Then write a function rule. x 0 2 4 6 y 2 1 0 –1 (3 points) 19. The graph below shows the average daily temperature over the period of a year. Explain how each labeled section of the graph relates to the four seasons. (3 points) 20. The following table represents the total cost, in dollars (y) to join a gym for x number of months. The cost includes a one-time joining fee of $10. Does the data in the table represent a proportional relationship or a non-proportional relationship? How do you know? x 1 2 3 4 5 y 25 40 55 70 85 (2 points) © 2015 Connections Education LLC. Unit 3: Using Graphs to Analyze Data Algebra Readiness (Pre-Algebra) B Unit Summary In this unit of the course, you will examine, analyze, and construct scatter plots and tables. With these skills, you will be able to roughly predict the strength and direction of a pattern of association between two things. You will also find measures of central tendency and determine which graph and measure of central tendency best represents a data set. Objectives Calculate the mean, median, and mode of a data set and explain the best use of each Determine the best type of graph to display a data set Identify patterns of association— indicating strength and direction—of two factors and make predictions based upon a scatter plot Lessons 1. Measures of Center 2. Frequency 3. Venn Diagrams 4. Stem-and-Leaf Plots 5. Box-and-Whisker Plots 6. Scatter Plots 7. Bivariate Data 8. Modeling Data with Lines 9. Circle Graphs 10. Choosing the Right Graph 11. Relative Frequency 12. Using Graphs to Analyze Data Unit Review 13. Using Graphs to Analyze Data Unit Test Lesson 1: Measures of Center CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objective: Compute mean, median, mode, range and select appropriate measure of tendency Finding the Middle In previous mathematics classes, you have learned how to calculate the mean, median, mode, and range of a set of data. If you knew the height of each of the kids shown in the photograph, you would be able to determine the height that is the mean, median, mode, as well as the height range for the group. But which value represents the group most accurately? And what would happen to our mean, median, and mode if we added a 6'4'' basketball player to the group? In today’s lesson, you will review how to find the mean, median, and mode for a set of data; and learn how to choose the best measure to represent that data. Objectives Compute mean, median, mode, range and select appropriate measure of tendency Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words mean measure of central tendency median mode outlier range The Distribution of a Set of Data When given a set of data, whether it is favorite flavor of ice cream among your friends or amount of time it takes to run a mile for each member of your family, it is interesting to find a way to summarize the data. When given a set of data, whether it is favorite flavor of ice cream among your friends or amount of time it takes to run a mile for each member of your family, it is interesting to find a way to summarize the data. One way to summarize a set of data is to describe the features of its distribution. Range The range of a set of data is the difference between the least and greatest values in the data set. Range is one measure of the spread of the numbers in a data set. Consider the following sets of data: Heights of 13-year-old boys (in inches) 55, 56, 57, 57, 59, 60, 60, 61 range = 6 inches Heights of kids playing at a park (in inches) 30, 32, 37, 38, 40, 55, 64, 70 range = 40 inches The range of the heights of 13-year-old boys is relatively small since the heights are all similar. The range in heights for total kids playing at a park is much greater because the data set includes a greater variety of heights. Sometimes a data set can include an outlier, which is a value that is much different from the other values in the set. If a data set has an outlier, then the range will not be a clear representation of the data. For example, if a height of 75 inches was added to the data set for 13year-old boys, the range would be 20, even though most of the numbers are clustered more closely together. Measures of Central Tendency Another way to summarize a set of data is by finding a value that is somewhere in the middle that represents all of the given values. This value is known as a measure of central tendency, and can be the mean, median, or mode. You can use the following data set to find each measure of central tendency: Number of sit-ups in one minute: 24, 28, 34, 35, 36, 38, 38, 39, 40, 45 Mean The mean is the sum of the data values divided by the number of values. The mean is generally not one of the numbers from the data set. Adding an outlier to the data set will cause an increase or decrease in the mean. For example, if someone did 8 sit-ups, the mean would decrease to 33.2. If someone did 70 sit-ups, the mean would increase to 38.8. Median The median is the value in the middle of the data set when the numbers are arranged from least to greatest. The median may or may not be a number from the data set. Since this set of data has an even number of values, the median value falls between the fifth and sixth Numbers. The median is the mean of 36 and 38. The median is 37. If the data set had an odd number of values, the median would be one of the numbers in the Set. For this data set, the median is 36. Adding an outlier to the data set will cause the median to shift one number to the left or right of the current median. So, an outlier does not have a strong impact on the median. Mode The mode is the value that occurs most often. It is possible to have no mode or more than one mode in a set of data. If there is a mode, it will always be a number from the data set. no mode: 24, 28, 34, 35, 36, 37, 38, 39, 40, 45 Each number occurs only once. one mode: The only value that occurs more than once is 38. more than one mode: Both 38 and 40 are modes because both occur twice. Adding an outlier to the data set will not impact the mode because it will not change which value occurs most often. Copy the following table into your notebook. While watching the movie, fill in the definition for each measure of central tendency, step-by-step instructions for how to calculate it, and the type of data it works best with. Mean Median Mode Definition How to Calculate With an odd number of values With an even number of values When is it Most Useful Click on the link below to watch the “Mean, Median, & Mode Example” video from the Khan Academy website. Mean, Median, & Mode Example Click on the link below to access the Mean, Median, & Mode Example transcript. Mean, Median, & Mode Example Transcript Certain measures of central tendency can be affected by an outlier in the data. An outlier is a value in a set of data that is much larger or smaller than the other values. The following movie will help you understand how an outlier impacts the mean, median, and mode. Mean Absolute Deviation Sometimes, it is not enough to know the center of a sample of data. It can also be necessary to see how far spread out the sample is. For example, a smaller spread of data will mean that the population is more uniform. One way of measuring a sample of data’s spread is to use the mean absolute deviation. In order to find the mean absolute deviation, first find the mean of the data sample. Then, calculate the difference between the mean and each data point. Take the absolute value of each difference. Finally, take the mean of the absolute value of the differences. This can be confusing, so work along with the following example. Rachel’s owns a business where she knits mittens. On some days, she is able to knit more mittens than other days. For the past five days, Rachel has been able to the following amounts of mittens each day: 10, 15, 23, 12, and 15. We will now follow the steps to find the mean absolute deviation. 1. Find the mean of the data sample. Since there are five data points, find the sum of the data points and divide it by five. 2. Calculate the difference between the mean and each data point. 3. Take the absolute value of each difference. 4. Calculate the mean of the absolute value of the differences. Following these steps, the mean absolute deviation for Rachel’s business is 3.2. The mean absolute deviation only has meaning when compared to another set of data. Let’s look at another business. Barry has a competing business wherein he knits mittens. For the past five days, Barry has been able to knit the following amounts of mittens each day: 3, 20, 11, 30, and 11. What is the mean absolute deviation for Barry’s business? What does the difference between the mean absolute deviations for Rachel’s and Barry’s businesses mean in the context of this problem? Click on the Show Answer button to check your answer. Answer: The mean absolute deviation for Barry’s business is 8. Since Rachel’s mean absolute deviation is 3.2, it means that her set of data is more concentrated around the mean of 15. This would make it easier to predict the number of mittens that Rachel knits on any given day, since there is a smaller spread than there is for Barry’s business. Complete the following activities. 1. Click on the link below to complete the "Mean, Median and Mode" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide to investigate how the measures of central tendency change depending on your data set. Take the quiz at the end to check your understanding of the key ideas. Mean, Median and Mode 2. Read pp. 412–414 of Mathematics: Course 3. 3. Complete problems 7–18 on p. 415 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. 1. Click on the link below to watch the "Mean, Median, Mode, and Range" BrainPOP® movie. After watching the movie, take the quiz to check your understanding of these measures of central tendency. Mean, Median, Mode, and Range 2. To check your understanding of mean, see if you can work backwards. Shakey Jake drinks a lot of coffee. His mean number of cups this week (7 days) is 4. If he drank 3, 2, 6, 7, 2, and 2 cups Sunday through Friday, how many cups did he drink on Saturday? If Jake had 10 cups of coffee on Saturday, how would that affect the mean? Which measure of central tendency do you think best represents Jake’s coffee consumption? Click on the Show Answer button below to check your answers. Answers: 6 cups on Saturday, 10 cups of coffee on Saturday raises the mean to approximately 4.6 cups of coffee. The mean represents the data better than the other measures of central tendency. The mode is 2, and 4 of the values are higher than that (some much higher). The median is 3, which doesn’t reflect the high values of 6, 6, and 7. Measures of Center Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Find the mean, median, and mode of the data set. (1 point) 15, 16, 21, 23, 25, 25, 25, 39 mean = 23.6, median = 25, mode = 24 mean = 24, median = 23.6, mode = 25 mean = 24.6, median = 24, mode = 25 mean = 23.6, median = 24, mode = 25 2. Find the outlier in the data set and tell how it affects the mean. 4, 4, –6, –2, 14, 1, 1 –6; it raises the mean by about 1. –6; it lowers the mean by about 1. (1 point) 14; it raises the mean by about 1.9. 14; it lowers the mean by about 1.9. 3. Find the mean, median, mode, and range of the data set. (1 point) Daily temperature in degrees Celsius: 24, 24, 25, 27, 31, 32, 38, 39 mean = 24, median = 30, mode = 29, range = 16 mean = 29, median = 30, mode = 24, range = 15 mean = 30, median = 29, mode = 24, range = 15 mean = 30, median = 24, mode = 29, range = 16 4. Pat recorded the weights of the first ten fish she caught and released at Mirror (1 point) Lake this season. The weights were 8 lb, 6 lb, 9 lb, 6 lb, 7 lb, 5 lb, 7 lb, 6 lb, 23 lb, and 6 lb. What is the median of the data set? 6 6.5 -7 -7.5 5. 22.6 is an outlier for which of the following sets of data? (1 point) 22.6, 21.5, 23.7, 22.6, 28.9, 22.6, 20.9 2.4, 5.3, 3.5, 22.6, 1.8, 2.1, 4.6, 1.9 20.5, 20.8, 21.6, 22.6, 23.7, 24.5, 25.1 13.6, 31.7, 25.8, 22.6, 18.9, 21.6, 30.5 6. During a week in Santa Fe, the following temperatures are recorded in degrees Fahrenheit: 75, 83, 77, 61, 82, 67, and 45. What is the mean absolute deviation of this set of temperatures? 13.2 7.5 10.6 6.7 © 2015 Connections Education LLC. (1 point) Lesson 2: Frequency CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objective: Utilize line plots, frequency tables, and histograms to display data Displaying Specific Values While a set of data usually holds a great amount of information, a long list of values is not very useful. In the last lesson, you learned that measures of central tendency can be used to find the center of the data, but most of the information about the specific data values is lost. Graphing is a method of representing a large group of data in a way that retains more of the original information and summarizes the data in a visual way. It is difficult to determine the most common responses by reading through the list below. By graphing the data, you can easily determine that the majority of people would feel either disappointed or angry about their favorite team losing. Question posed to people. Responses to the question “How would you feel if your favorite team lost the big game?” How would you feel...? disappointed, angry, confused, disgusted, angry, upset, disgusted, disappointed, angry, upset, disappointed, angry, disappointed, upset, angry Objectives Utilize line plots, frequency tables, and histograms to display data Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words distribution frequency frequency table histogram line plot Displaying Recurring Instances Organizing a set of data by the frequency of the values can help you easily assess how the data is grouped. You can determine if there are any gaps or outliers, and which value (or values) occur most frequently. How often each value occurs is known as frequency. One way you can show frequency is with a line plot. You can create a frequency table to organize your data and turn it into a line plot. A frequency table shows how many times each value occurs in the data set. The following frequency table shows the heights of 10 8th grade students. Height Frequency 5'2" 5'3" 5'4" 5'5" 5'6" To create a line plot, draw a number line that shows each of the values in the frequency table. Then place one X above each value for each time the value occurs. For example, there are two students who are 5’2”, so there are 2 Xs above 5’2” on the line plot. You can also use a line plot to find the mean of the data by multiplying each value by the number of times it occurs and then dividing by the total number of values in the data set. The first step in calculating the mean is to convert each measurement into inches. Since there are 12 inches in each foot, 5’2” is equal to 62” (5 × 12 + 2 = 62). You can also use a frequency table to represent groups of values rather than individual values. When using a frequency table, the data is clumped together into equal intervals. Scores earned by students on a math test: 80, 69, 65, 95, 78, 74, 72, 66, 62, 90, 94, 75, 68, 71, 94, 68, 60, 65, 64, 77, 87, 93, 92, 88 Once you have added your data to a frequency table, you can easily create a histogram. A histogram is similar to a bar graph, except each bar represents a range of values. To create a histogram, place each interval from the frequency table along the horizontal axis of your graph. For each interval, the height of the bar will correspond to the frequency on that interval from the frequency table. For example, 9 students scored 60 –– 69, so the bar for the 60 – 69 interval goes up to 9. You cannot calculate the median or the mode from a frequency table or histogram because you can no longer see the individual values of the data. Both line plots and histograms can help you understand the distribution and shape of the data. You can identify the range of the data as well as any outliers. If the left and right sides of the graph are the same, the data is symmetrical. The following graph shows symmetrical data. The range is 4 inches and there are no outliers. Complete the following activities. 1. Click on the link below to complete the "Exploring Data Using Histograms" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide to investigate how the histogram changes depending on the size of your interval. Take the quiz at the end to check your understanding of the key ideas. Exploring Data Using Histograms 2. Read pp. 418–419 in Mathematics: Course 3. 3. Complete problems 5–21 (odd) in Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. Which of the histograms would best represent the following scenario? 1. Students were asked to do as many sit-ups as they could in 2 minutes. The range was 0–59 sit-ups. Most of the students did between 10–19 or 50–59 sit-ups. What would each interval be in the histogram? Which value occurs least frequently? How many students did between 0–9 sit-ups? 2. Students were asked how many minutes they spend eating dinner. The histogram covers a range of 0–60 minutes, with 15 minute intervals. Most responded between 16–30 minutes. How many students spent between 31–45 minutes? How many total students are represented by the graph? How would the graph changed if you made the intervals smaller? What would the graph look like if you added a student who spent 65 minutes eating? 3. Can you think of a scenario to go with the remaining histogram? A B C Click on the link below to check your answers. Answers Frequency Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Which line plot matches the set of data? 61, 58, 57, 64, 59, 57, 64, 58, 56, 57 (1 point) 2. Which frequency table shows the test times (in minutes) for a reading test? (1 point) 81, 63, 61, 58, 72, 70, 79, 68, 82, 64, 54, 82, 72, 63, 64, 76, 57, 65, 73, 58 3. Which histogram for drivers’ ages matches the data from the frequency table below? Drivers' Ages Age Frequency 17–19 2 20–22 3 23–25 5 26 28 6 (1 point) 26–28 6 4. There was a berry-picking contest at the Earth Day celebration this year. The (1 point) line plot below shows the number of pints of berries collected by the people participating in the contest. a. What is the median of the data displaced on the line plot? b. How many people participated in the contest? 9 pints; 12 people 8 pints; 11 people 7 pints; 13 people 8 pints; 13 people Answers Question 1 matches histogram B. The intervals are 0–9, 10–19, 20–29, 30–39, 40–49, 50–59. Interval 40–49 occurs least frequently. Four students did between 0–9 sit-ups. Question 2 matches histogram C. Four students spend 31–45 minutes eating dinner. If the intervals were smaller, the histogram would be more spread out and there would be fewer students in each interval. If a student spent 65 minutes eating, you would need to add another column to the histogram, representing 61–75 minutes. Question 3: A sample answer might be the number of baskets made in each game throughout the basketball season. The graph shows gradual improvement during the season. © 2015 Connections Education LLC. Lesson 3: Venn Diagrams CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objective: Examine relationships between data with Venn diagrams How Does Data Relate? Sometimes, it is more important to show the relationships between data rather than the measures of central tendency. In today’s lesson, you will explore how a Venn diagram shows the relationships between data. Click on the link below to access the Survey of Student Pets Venn diagram. The Venn diagram shows the results of a survey in which students were asked about their pets. Survey of Student Pets What does each circle represent? What does the “4” in the middle of the diagram represent? Click on the Show Answer button below to check your answer. Answer: Each circle shows the number of students who have a dog, cat, or goldfish. The “4” is the number of students who have all three kinds of pets. Objective Examine relationships between data with Venn diagrams Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Word Venn diagram Venn Diagrams A Venn diagram is an illustration broken into different regions to show the relationships between sets of information. It is usually made up of overlapping circles. Take another look at a simplified version of the Venn diagram from the “Getting Started” section of the lesson. Click on the link below to access the Dogs vs. Cats simplified Venn diagram. Dogs vs. Cats Complete the following activities. 1. Read p. 424 in Mathematics: Course 3. 2. Complete problems 4–11 on p. 425 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activity. Copy the Venn diagram into your notebook. Add the numbers 0–25 to the diagram. How many values would there be in the area where all three circles intersect? Create your own Venn diagram with three circles that shows the relationship between people in your life. Example categories might be people who like movies, people you see every day, people who like sports, people who are older than you, or people who live in another state. Be sure you know how many total people are shown in your diagram and what each of the intersecting areas represents. Click on the Show Answer button below to check your answer. Answer: Only the number 5 would be placed in the intersection of all three circles. Venn Diagrams Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Eighteen students in a class play baseball. Seventeen students in the class play (1 point) basketball. Thirty students in the class play either or both sports. Select the Venn diagram that shows the number of students who play basketball and baseball. 2. In a school of 464 students, 89 students are in the band, 215 students are on (1 point) sports teams, and 31 students participate in both activities. How many students are involved in neither band nor sports? 160 students 191 students 249 students 433 students 3. In a marketing survey involving 1,000 randomly chosen people, it is found that (1 point) 3. In a marketing survey involving 1,000 randomly chosen people, it is found that (1 point) 630 use brand P, 420 use brand Q, and 210 use both brands. How many people in the survey use brand P and not brand Q? 210 people 420 people 630 people none of these 4. Forty people were surveyed about their favorite flavor of ice cream. How many like just chocolate? 24 16 31 none of these © 2015 Connections Education LLC. Lesson 4: Stem-and-Leaf Plots CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objective: Represent and interpret data using stem-and-leaf plots Note: This lesson should take 2 days. Why Use a Stem-and-Leaf Plot? The given histogram shows the finish times for a 5K race. From the histogram, you can tell that most people finished the race in 30–39 minutes and that 100 people participated in the race. However, you cannot tell the winning time or the time of the slowest runner because you don’t have those exact values. Ten people finished the race in a time of 10–19 minutes, but you can’t tell if their times were closer to 10 or 19 minutes. (1 point) The histogram provides a good visual summary of the data, but some specific information is lost. In today’s lesson, you will learn the advantages of presenting data in a stem-and-leaf plot and how to interpret the data from this type of graph. Objectives Represent and interpret data using stem-and-leaf plots Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words stem-and-leaf plots Tip: You will have 2 days to complete this lesson. Creating Stem-and-Leaf Plots There are several advantages to using a stem-and-leaf plot: The original data stays intact. You can identify gaps and clusters in the data. You can calculate the mean, median, mode, and range. This data set represents the numbers 3, 4, 10, 17, 23, 23, 26, 38, 40, 52, and 59. The tens digit is under “stem” and the ones digit is under “leaf.” If your data set is made up of three digit numbers, the stem will include the first two digits. The leaves will still be a single digit. Sometimes, you may want to use a back-to-back stem-and-leaf plot to compare sets of data. The following plot shows men’s 5K times on the left, and women’s 5K times on the right. Complete the following activities. 1. Click on the link below to complete the "Stem-and-Leaf Plots" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide to compare how a set of values looks in a line plot and stem-and-leaf plot. Take the quiz at the end to check your understanding of the key ideas. Stem-and-Leaf Plots 2. Read pp. 433–435 in Mathematics: Course 3. 3. Complete problems 6–13 on pp. 435–436 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. State the stems that you would use to plot each set of data: 1. 17, 32, 41, 34, 42, 35 2. 9, 12, 24, 51, 33, 14, 18, 26 3. 294, 495, 272, 153, 240, 427 4. 7.5, 5.4, 8.6, 6.3, 7.1, 5.9, 8.2 Find the median and mode of the data in each stem-and-leaf plot. 5. median = ____ ; mode = ____ 6. median = ____ ; mode = ____ 7. median = ____ ; mode = ____ The data at the right are the ages of people who attended a play. Use this data to solve each problem. 8. Construct a stem-and-leaf plot 9. What was the age of the youngest person attending the play? 10. What was the age of the oldest person attending the play? 11. With what age group was the play most popular? Click on the link below to check your answers. Answers Stem-and-Leaf-Plots Quiz Part 1 Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. What data are represented by the stem-and-leaf plot below? (1 point) 37, 38, 39, 41, 43, 47, 52, 54 73, 83, 93, 14, 34, 74, 25, 45 7, 8, 9, 1, 3, 7, 2, 4 37, 38, 39, 14, 34, 74, 25, 45 2. Find the mode and the median of the data in the stem-and-leaf plot below. (1 point) no mode; 73 63; 73.5 54; 73 no mode; 73.5 3. The stem-and-leaf plot below could NOT represent which of the following? the average number inches of rain in June (1 point) the height of plant seedlings, in inches the weight of phonebooks, in pounds the height of NBA basketball players, in meters 4. Make a histogram for drivers’ ages using the data from the table below. Drivers' Ages Age Frequency 17–19 8 20–22 7 23–25 9 26–28 4 (1 point) 5. In a recent survey of middle school students about pizza toppings, it was found (1 point) that 25 students like pepperoni pizza, 31 like banana peppers pizza, and 5 liked both pepperoni and banana peppers on their pizza. If 66 students were surveyed, how many students do not like banana peppers on their pizza? 20 34 31 35 6. Which is the best measure of central tendency for the type of data below–the (1 point) mean, the median, or the mode? Explain. Hours of sleep each night Median; there will be outliers Range; there are no outliers Mode; the data are non-numeric Mean; the outliers are limited Short Answer 7. The back-to-back stem-and-leaf plot below shows the ages of patients seen by two doctors in a family clinic in one day. Compare the ages of the patients of Doctor 1 and Doctor 2 using the mean and the median of each data set. (4 points) Take the assessment. Stem-and-Leaf-Plots Quiz Part 2Answers 1. 1, 2, 3, 4 2. 0, 1, 2, 3, 4, 5 3. 15 – 49 4. 5, 6, 7, 8 5. median = 60, mode = no mode 6. median = 97.5, mode = 82 7. median = 32, mode = 27 8. 9. 6 10. 42 11. 10–19 © 2015 Connections Education LLC. Lesson 5: Box-and-Whisker Plots CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objective: Represent and interpret data using box-and-whisker plots Looking for Clusters If you were planning to spend the month of January in either Phoenix, Arizona or Juneau, Alaska, you might want to know some information about the temperatures to expect. It would be easy to find out the high and low temperatures for the month, but it would be more valuable to know whether the rest of the days are grouped closer to the high or the low temperature. In this case, the individual data values are less important than how the data clusters together. In today’s lesson, you will learn how to create and interpret box-and-whisker plots. Objective Represent and interpret data using box-andwhisker plots Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words box-and-whisker plots interquartile range lower quartile quartiles upper quartile Box-and-Whisker Plots A box-and-whisker plot is a graph that splits a large amount of data into groups and organizes it along a number line. Following are some of the important elements of a box-and-whisker plot: Each group is called a quartile because it contains one-fourth of the data. The middle half of the data is included inside the box. The difference of the highest and lowest of these values is called the interquartile range. The median is also easily identified inside the box. One-fourth of the remaining data is located between the lowest value and the lower quartile. The remaining one-fourth is located between the upper quartile and the highest value. The range is also easily identified in the graph by finding the difference between the lowest value and the highest value. To create a box-and-whisker plot, first arrange the values in the data set in order from least to greatest. Then follow these steps to find the values that are important for creating the plot: 1. Find the median, which is the number in the middle. 2. Find the lower quartile, which is the median of the lower half of the values. 3. Find the upper quartile, which is the median of the upper half of the values. Each of these values has been circled below: The lower and upper quartiles form the left and right sides of the box. The median is the line inside the box. The whiskers extend to the numbers that are the lower and upper limits. The range is the difference between the greatest value and the least value in a data set. The range for this box plot is 16 because . The interquartile range ( IQR) is the difference between the upper quartile and the lower quartile in a data set. The interquartile range for this box plot is 9 because . Complete the following activities. 1. Click on the link below to complete the "Box-and-Whisker Plots" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide and complete the Displaying Data and Making a Box-and-Whisker Plot sections. Take the quiz at the end to check your understanding of the key ideas. Box-and-Whisker Plots 2. Read pp. 438–439 of Mathematics: Course 3. 3. Complete problems 6–16 on pp. 440–441 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. 1. Use the box-and-whisker plot below to solve each problem. 1. What are the ranges for both sets of data? 2. What percent of ice cream has more than 270 calories? 3. What percent of chocolate has more than 230 calories? 2. Answer the following questions about the box-and-whisker plot. 1. What is the greatest data point? 2. Between what two data points is the middle half of the data? What is this range called? 3. What is the range of the data? 4. What part of the data is less than 25? Click on the link below to check your answers. Answers Box-and-Whisker Plots Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Make a box-and-whisker plot of the data. (1 point) 21, 21, 22, 20, 13, 13, 27, 24 2. Make a box-and-whisker plot of the data. (1 point) 60, 63, 53, 66, 65, 58, 51, 55, 58, 51, 58, 62, 53, 66, 61, 51, 65, 52, 54, 50 3. The box-and-whisker plots show data for the test scores of four groups of students in the same class. Which plot represents a group with a median grade below 65? (1 point) 4. Use the two box-and-whisker plots shown below to determine which of the (1 point) following statements is true. The lower quartiles are equal. The upper quartiles are equal. They both have the same median. The range is the same for both sets of data. Reflection Charles Washington is not permitted to take this assessment again. These answers will not be saved. 1. How comfortable are you working with a data set? (1 point) I can find the mean, median, mode, and range of a data set, and I can explain what these values mean. I can find the mean, median, mode, and range of a data set, but I sometimes make mistakes. I occasionally mix up mean, median, mode, and range. I can find all or some of these values with help. I do not understand how to interpret data sets. 2. How would you describe your ability to model data? (1 point) I can choose whether a frequency table, line plot, histogram, stem-and-leaf plot, boxand-whisker plot, or Venn diagram is best suited for a data set. I can display data in these formats without making any mistakes. I usually choose whether a frequency table, line plot, histogram, stem-and-leaf plot, box-and-whisker plot, or Venn diagram is best suited for a data set. I can display data in these formats, but sometimes I make mistakes. I can display data in most formats, but I am not always confident that I picked the best model for the data. I need help better understanding how to model different types of data. 3. Which of these skills do you think you could teach someone else? Select all that apply. (1 point) finding the mean, median, mode, and range of a data set choosing the best measure to represent a data set making and interpreting line plots, frequency tables, and histograms using Venn diagrams to examine relationships making and interpreting stem-and-leaf plots making and interpreting box-and-whisker plots 4. With which of these skills do you need more help with? Select all that apply. finding the mean, median, mode, and range of a data set choosing the best measure to represent a data set making and interpreting line plots, frequency tables, and histograms using Venn diagrams to examine relationships making and interpreting stem-and-leaf plots making and interpreting box-and-whisker plots Answers Question 2 1. chocolate 280-220 = 60, ice cream 330-120 = 210 2. 50% 3. 75% Question 3 1. 50 2. 10 and 45; interquartile range 3. 50-0 = 50 4. 50% © 2015 Connections Education LLC. Lesson 6: Scatter Plots CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objective: Create scatter plots and analyze trends to make predictions Two Sets of Data Sometimes, you might want to analyze two sets of data at once to determine how closely they are related. For (1 point) example, you might want to investigate the relationship between the number of rings in a tree and the height of the tree. You could easily graph either the height of the trees or number of tree rings with a stem-and-leaf plot or boxand-whisker plot. But if you wanted to see whether the two sets of data are related, you would need a scatter plot. In today’s lesson, you will review scatter plots and learn how to analyze them to see if there is a relationship between the two quantities. Objective Create scatter plots and analyze trends to make predictions Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words negative trend pattern of association positive trend scatter plot Using Scatter Plots to Analyze Data Scatter plots are used to look for a pattern of association, or trend, between two sets of values. One set of values is shown on the horizontal (x) axis and the other is shown on the vertical (y) axis of a coordinate grid. In the following example, the points are plotted using a G for girls and a B for boys. Boys' Results Hours of TV 10 20 0 30 35 25 10 28 Grade Point Average 2.8 2.25 3.6 1.5 1.5 1.9 1.8 1.9 Girls' Results Hours of TV 25 0 8 18 15 3 7 15 18 Grade Point Average 2.25 3.1 3.3 3.0 2.7 3.4 2.9 2.25 3.3 In general, what would you say happens to students’ GPAs as their television hours increase? Click on the Show Answer button below to check your answer. Answer: Their GPAs go down as their TV hours go up. There are three types of relationships that can be shown in a scatter plot. Click on the links below to complete the Interpreting a Scatter Plot activity from the digits™ website. Topic Opener Launch Key Concept Example 1 Example 2 Close and Check Complete the following activities. 1. Click on the link below to watch the "Scatter Plots" Teachlet® tutorial Scatter Plots Click on the link below to access the Scatter Plots Transcript Scatter Plots 2. Click on the link below to complete the "Scatter Plots - Activity A" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide and be sure to investigate positive and negative trends. Take the quiz at the end to check your understanding of the key ideas. Scatter Plots - Activity A 3. Read pp. 444–445 of Mathematics: Course 3. 4. Complete problems 6–14 on pp. 446–447 of Mathematics: Course 3. Click on the link below to access the online textbook Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. 1. Complete the Activity Lab 9-7a on p. 443 of Mathematics: Course 3 which investigates the relationship between arm span and height. For this activity, you will need a centimeter tape measure and graph paper. In step 4 of the activity, you will be asked to exchange your data (height and arm span) with your classmates. Instead, you will need to add your own data to the following table: height (cm) 102 143 128 117 107 136 115 151 arm span (cm) 100 143 122 121 104 136 109 156 2. Continue with steps 5–10 of the activity. Click on the link below to access the online textbook. Mathematics: Course 3 Scatter Plots Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. Which of the scatter plots above shows a negative trend? (1 point) II III I none of these 2. The scatter plot below shows the population of a village (P) over time (t). (1 point) Describe the relationship between the population of the village and time. The population remains roughly the same over time. The population is increasing over time. The population is decreasing over time. none of these 3. Which of the following examples would show a negative trend? (1 point) height and weight of students test scores and height of students outside temperature and heating bill none of these 4. The scatter plot below shows the relationship between the time spent learning a piece of music for the guitar and the score at the annual solo competition. Predict the score for 15 weeks of practice. (1 point) about 61 about 41 about 29 about 56 © 2015 Connections Education LLC. Lesson 7: Bivariate Data CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objectives: Analyze bivariate data; Describe patterns of association in bivariate data What Would the Data Look Like? In today’s lesson, you will analyze data that has two sets of values and look for patterns and associations in the data. Picture the scatter plots for the following sets of data. Would each show a positive or negative trend? Would the values be spread out or clustered together? What would an outlier look like on either graph? temperature and the amount of water a person drinks hours spent watching TV and hours of sleep Objectives Analyze bivariate data Describe patterns of association in bivariate data Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words bivariate data cluster linear association nonlinear association Scatter Plot Patterns Bivariate data is a data set with two quantities that can vary—or change. Every scatter plot is created from bivariate data. In the last lesson, most of the scatter plots you looked at had some sort of general associations in the data, such as positive or negative trends. Often in the real world, graphs have more complicated trends that may be more difficult to recognize and make sense of. Today you will look at some of these more complex relationships such as linear and nonlinear relationships, clustering, and outliers. 1. Click on the links below to complete the “Constructing a Scatter Plot" activity from the digits™ website. Pay careful attention to Example 3, which will teach you how to use the data and graphs tool. Launch Key Concept Example 1 Example 2 Example 3 Close and Check 2. Click on the links below to complete the “Investigating Patterns – Association" activity from the digits™ website. Launch Key Concept Example 1 Example 2 Key Concept Example 3 Close and Check Complete the following activity. Click on the link below to complete the "Scatter Plots - Activity B" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide and take the quiz at the end to check your understanding of the key ideas. Scatter Plots - Activity B Complete the following review activity. Go back and invent a table of values to go with the two graphs in the Getting Started section of this lesson. Make a graph from your table of values and describe any associations in the data. Bivariate Data Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. What is the term for data that are grouped closely together? (1 point) outlier linear positive clustering 2. What association would you expect if graphing height and weight? (1 point) positive nonlinear negative none of these 3. What association is shown in the given scatter plot? (1 point) clustering linear negative none of these 4. What association would you expect if graphing number of hours worked and money earned? negative linear nonlinear none of these © 2015 Connections Education LLC. (1 point) Lesson 8: Modeling Data with Lines CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objective: Examine data to determine accuracy of models One Line from Many Data Points If you plotted the elevation every half mile on this winding mountain road you could use that data to make a scatter plot. Although the road goes downhill overall, there are also places where it flattens out or goes slightly uphill. You could create a scatter plot that shows the elevation on the y axis and the distance driven down from the top on the x axis. It might look something like the graph below. The overall trend of the data would be a linear association with a negative trend. Could you approximate the overall shape of the data with one straight line? In today’s lesson, you will investigate trend lines and learn to evaluate how well they fit the data. Objective Examine data to determine accuracy of models Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Word trend line Approximating a Linear Trend This scatter plot shows data that is linear with a negative trend. You can draw a line through the points, making an effort to get it as close to each point as possible. This line is called the trend line, which is sometimes referred to as the line of best fit. The trend line can be used to make predictions about the data. But how can you decide which line fits the data best? Which of the above lines do you think represents all of the data points better? What makes you think it is a better fit? Click on the links below to complete the “Linear Models – Fitting a Straight Line” activity from the digits™ website. Launch Key Concept Example 1 Example 2 Example 3 Close and Check Complete the following activity. Click on the link below to complete the "Solving Using Trend Lines" Gizmo to practice the concepts from today's lesson. Follow the steps in the Exploration Guide and take the quiz at the end to check your understanding of the key ideas. Solving Using Trend Lines Complete the following review activities. Go back and look at the graph in the Getting Started section of the lesson. What would a trend line for the data look like? Click on the Show Answer button to check your answer. Answer: A line with a downward trend with the data points distributed evenly on each side of the line. Modeling Data With Lines Quiz Charles Washington is not permitted to take this assessment again. These answers will not be saved. Use the graph. 1. Which line models the data points better and why? (1 point) blue, because it is longer blue, because the data points are all close to the line red, because it goes through one of the points red, because there are three points above the line and three points below the line 2. According to the blue line, what would you estimate the score was after 3 weeks of practice? (1 point) about 40 about 18 about 8 about 22 3. According to the blue line, about how many weeks of practice are required to (1 point) achieve a score of 50? 15 weeks 17 weeks 19 weeks 21 weeks 4. Which of the following is true about a trend line for data? (1 point) The minimum data point always lies on the trend line. Every data point must lie on the trend line. The trend line describes the pattern in the data if one exists. The trend line includes the effect of all outliers in the data. 5. The scatterplot shows the number of visitors to the zoo on eight different days and the high temperatures on those days. Based on the scatterplot, what is the best prediction of the number of visitors the zoo will receive on a day with a high temperature of 106? 200 425 445 620 (1 point) 6. The scatter plot below shows the population of a village (P) over time (t). (1 point) Describe the relationship between the population of the village and time. The population is decreasing over time. The population is increasing over time. The population remains roughly the same. none of these Short Answer 7. Describe the trend in the scatter plot. Explain your answer. (2 points) © 2015 Connections Education LLC. Lesson 9: Circle Graphs CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objectives: Interpret data using circle graphs; Present data by creating circle graphs Breaking a Whole Into Parts A circle graph is a very visual way of presenting data. What information is shown by this circle graph? What would you title the graph? What does each sector of the circle show? What do the different sized sectors mean? Can you think of any food that would not fit in the circle? In today’s lesson, you will interpret data using circle graphs. You will also practice creating a circle graph of your own based on given data. Objectives Interpret data using circle graphs Present data by creating circle graphs Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words central angle circle graph Comparing a Part to the Whole A circle graph is a graph in which the whole is shown by a circle and each part is shown by a sector of the circle. Each sector is a percent of the whole circle and all the sectors together always add up to 100%. For example, if you surveyed 1,000 people about their favorite animal, the circle represents all of the people surveyed. Each sector of the circle (elephant, giraffe, lion, or armadillo) represents the number of people who chose that animal. What if you were given a set of data and asked to create a circle graph? What steps would you use to find the size of each sector? Click on the link below to access the How to Create a Circle Graph directions. How to Create a Circle Graph Complete the following activities. 1. Review section 9-8 “Circle graphs” on pp. 450–451 of Mathematics: Course 3. 2. Complete problems 4–14 on pp. 452–453 of Mathematics: Course 3. You will need a compass and protractor for this activity. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activity. Practice making some circle graphs of your own with the data below. Types of Vehicles Owned Type Percentage Sedan 45% Wagon 12% SUV 27% Minivan 16% Chemical Composition of the Human Body Element Percentage Oxygen 65% Carbon 18% Hydrogen 10% Nitrogen 3% Other 4% Click on the link below to review your answers. Answers Circle Graphs Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. The following circle graph was published in the Cane County annual report. If there are 1,000 registered voters in Cane County, how many are 35–45 years old? (1 point) 150 voters 250 voters 300 voters 350 voters 2. Grade 7 students were surveyed to determine how many hours per day they (1 point) spent on various activities. The results are shown in the circle graph below. About how many hours per day altogether were spent on watching TV and homework? about 6 hours about 7 hours about 4 hours about 5 hours 3. The circle graph shows data on the suitability of land for farming. Which three categories together account for exactly half of the land? (1 point) suitable and too dry suitable, too dry, and chemical problems too wet and too shallow too wet, too shallow, and chemical problems 4. All 500 students at Robinson Junior High were surveyed to find their favorite (1 point) sport. How many more students played baseball than soccer? 50 students 175 students 75 students 125 students 5. Theo made the table below to show the number of middle school students who attended the last football game. If this data were displayed in a circle graph, how many degrees would be in the sector representing the 8th graders? Grade Number of Students in Attendance 6 375 7 275 8 350 145° 126° 35° 65° (1 point) Answers Types of Vehicles Owned circle graph Chemical Composition of the Human Body circle graph © 2015 Connections Education LLC. Lesson 10: Choosing the Right Graph CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objective: Choose the best graph to represent various data Which Graph is Best? Each of the following graphs shows the number of points scored in each game of the season by the star player on the basketball team. Which graph do you think represents the data most clearly and accurately? Are any of the graphs misleading? Which graphs could you use to find the mean, median, and mode? In which graph is it easiest for you to tell that the star usually earned between 10 and 19 points? Which graph shows the outlier most clearly? Objective Choose the best graph to represent various data Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words bar graph box-and-whisker plot circle graph frequency table histogram line plot scatter plot stem-and-leaf plot Choosing the Right Graph to Use Create a table like the one below to review the graphs you have studied in this unit and a few that you have studied in previous mathematics classes. Fill in the table with the information you have studied. Type of Graph bar graph box-and-whisker plot circle graph frequency table histogram line plot Purpose Type of Graph Purpose scatter plot stem-and-leaf plot Venn diagram Click on the link below to watch the “Graphs” BrainPOP® movie. While watching the movie, continue to fill in the table for bar graphs, line graphs, and circle graphs. Graphs After watching the movie, fill in the purpose column for the remaining graphs in the table. You can use your textbook and work from previous lessons to help you. Click on the Show Answer button below to check your answers. Answer: Type of Purpose Graph bar graph to compare quantities box and to display data where clustering and medians are important, but whisker plot exact numbers are not circle graph to compare parts of a whole frequency table to record the number of times a data items occurs histogram to display data divided into intervals and describe frequency line plot to display each frequency of a number scatter plot to show relationships between sets of data and use trends to make predictions stem-and-leaf to display data where values are fairly close together, and exact plot values are important Venn diagram to show the relationships between data Complete the following activities. 1. Read pp. 456–457 of Mathematics: Course 3. 2. Complete problems 6–17 on pp. 458–459 of Mathematics: Course 3. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. 1. In the following table, mark which graph or graphs would be appropriate for the given question. line plot histogram stem- box-and- and-leaf whisker circle scatter graph plot Your friends’ favorite type of music Average scores on a history test Times for running one mile The daily high temperature in your town Hours spent practicing the guitar and score on the guitar test 2. Why wouldn’t a scatter plot be appropriate for graphing shoe size? 3. Which graphs can you use to calculate mean, median, and mode? Choosing the Right Graph Charles Washington is not permitted to take this assessment again. These answers will not be saved. Multiple Choice 1. What type of graph does not show the number of times a response was given? (1 point) box-and-whisker plot line plot stem-and-leaf plot bar graph 2. Which of the following types of information is suited for display on a scatter (1 point) plot? the types of car models in your neighborhood the numbers of pets in neighborhood households your average daily minutes of exercise the relationship between hours in the car and distance traveled 3. You want to make a graph to show how you spend your time each day. What is (1 point) an advantage of choosing a circle graph for this data? A circle graph shows how each category of time relates to the total amount of time. A circle graph is easier to make. It is easy to calculate the mean, median, and mode with a circle graph. A circle graph will show the times when you are the busiest. Use the following two ways to display the test scores received on Mr. Alexander's math test. Use these displays to solve each problem. 4. Which graph shows the lowest score on the test? (1 point) both graphs only the stem-and-leaf plot only the circle graph Neither of the graphs shows this information. Use the following two ways to display the test scores received on Mr. Alexander's math test. Use these displays to solve each problem. 5. Which graph shows that most of the students earned between 80–89 on the test? both graphs only the stem-and-leaf plot only the circle graph Neither of the graphs shows this information. © 2015 Connections Education LLC. (1 point) Lesson 11: Relative Frequency CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objective: Create and analyze data using a two-way table Note: This lesson should take 2 days. Organizing Data Suppose you conducted a survey of your friends in which you asked about their favorite activity on weekdays and on weekends. You could show the results from your survey in two frequency charts as follows. Is there any information that is duplicated on both graphs? Is there a way that you could present this data more clearly? Click on the Show Answer button below to check your answers. Answer: There might be a way to combine the two tables into one. In today’s lesson, you will learn how to create and analyze data using a two-way table. By creating a two-way table, you will be able to quickly analyze the data more completely. Objective Create and analyze data using a two-way table Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words relative frequency two-way table Constructing Two-Way Frequency Tables A two-way frequency table shows the data for two different variables in one table. One variable is shown in the columns, and the other in the rows. 1. Click on the links below to complete the "Constructing Two-Way Frequency Tables" activity from the digits™ website. Launch Key Concept Example 1 Example 2 Example 3 Close and Check 2. Construct a two-way frequency table from the information about favorite activities in the "Getting Started" section of this lesson. What conclusions can you draw from the table? Click on the Show Answer button below to check your answer. Sample Answer: Activity Weekdays Weekends Total TV/video games 7 6 13 Sports 8 6 14 Friends 5 8 13 Total 20 20 40 Constructing Relative Two-Way Frequency Tables This type of table shows each value as a ratio of the number in each group compared to the total population. The value is frequently listed as a percent, but can also be listed as a fraction or decimal. You can create ratios that compare a value to the total number of participants, or to the total in a row, or to the total in a column. 1. Click on the links below to complete the Constructing Two-Way Relative Frequency Tables activity from the digits™ website. Launch Example 1 Example 2 Example 3 Key Concept Close and Check 2. Construct a relative two-way frequency table from the information about favorite activities in the "Getting Started" section of this lesson. The table should show percentages for each category relative to the total number of responses. For example, people chose TV/video games on weekdays, which is equal to 17.5%. Click on the Show Answer button below to check your answer. Answer: Activity Weekdays Weekends Total TV/video games 17.5% 15% 32.5% Sports 20% 15% 35% Friends 12.5% 20% 32.5% Total 50% 50% 100% 3. Using the same data, now construct a relative two-way frequency table that shows the percentages for each category relative to the total number of people who chose that activity. For example, of the 13 people who chose TV/video games, 7 of them chose weekdays. This is equal to 53.8%. Click on the Show Answer button below to check your answer. Answer: Activity Weekdays Weekends Total TV/video games 53.8% 46.2% 100% Sports 57.1% 42.9% 100% Friends 38.5% 61.5% 100% Total 50% 50% 100% 4. How are the two tables different? Click on the Show Answer button below to check your answer. Answer: One shows percentages based on total responses, and the other shows percentages based on the number of people who chose each activity. Complete the following review activities. 1. Click on the links below to complete the "Interpreting Two-Way Frequency Tables" activity from the digits™ website. Launch Example 1 Example 2 Example 3 Close and Check 2. Click on the links below to complete the "Interpreting Two-Way Relative Frequency Tables" activity from the digits™ website. Launch Example 1 Example 2 Example 3 Close and Check Relative Frequency Quiz Part 1 Charles Washington is not permitted to take this assessment again. These answers will not be saved. Activity Sixth Seventh Graders Graders Walk 3 6 Bike 5 3 Skateboard 2 1 Total Total Use the frequency table about preferred methods of transportation to answer the assessment questions. 1. How many students chose walking as their preferred method of transportation? 6 3 9 none of these Activity Sixth Seventh Graders Graders Walk 3 6 Bike 5 3 Skateboard 2 1 Total Total 2. How many total students participated in the survey? (1 point) (1 point) 10 20 15 none of these Activity Sixth Seventh Graders Graders Walk 3 6 Bike 5 3 Skateboard 2 1 Total Total 3. What percentage of the total students chose skateboarding? (1 point) 10% 20% 30% none of these Activity Sixth Seventh Graders Graders Walk 3 6 Bike 5 3 Skateboard 2 1 Total Total 4. What percentage of the sixth graders chose walking? (1 point) 30% 45% 25% none of these Activity Sixth Seventh Graders Graders Walk 3 6 Bike 5 3 Skateboard 2 1 Total Total 5. What percentage of the students who chose biking were seventh graders? (1 point) 30% 37.5% 40% none of these 6. All 500 students at Robinson Junior High were surveyed to find their favorite (1 point) sport. How many more students played football than basketball? 325 students 135 students 190 students 55 students 7. Which of the following types of information is most likely to display no trend (1 point) on a scatterplot? relationship between age and number of books read in a year relationship between height and foot length relationship between calories consumed and body weight relationship between height and hair color Take the assessment. Relative Frequency Quiz Part 2 Complete the following activity. Complete CC-9 “Relative Frequency” on pp. CC22–CC23 of Mathematics: Course 3. Work through the activity and exercise problems. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. © 2015 Connections Education LLC. Lesson 12: Using Graphs to Analyze Data Unit Review CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Objective: Review for unit test Note: This lesson should take 2 days. Getting Ready The test at the end of a unit is an opportunity for you to demonstrate everything you have learned while studying these concepts. In this lesson, you will review test taking strategies that will help you to be successful taking the unit test and showing your teacher all you have learned in this unit. You will also have the chance to practice what you learned during previous lessons in this unit by using various review activities. Objective Review for unit test Key Words bivariate data box-and-whisker plots central angle circle graph clustering frequency frequency table histogram interquartile range line plot linear relationship mean measure of central tendency median mode nonlinear relationship outlier quartiles range relative frequency scatter plot stem-and-leaf plot trend line two-way frequency table Venn Diagram Tip: You will have 2 days to complete this lesson. Test-Taking Strategies In the next lesson, you will take the test on the skills that you have learned in this unit. In preparation for this test, review the following test-taking strategies. Multiple-Choice Questions 1. Read through the question and all of the answer choices before selecting your response. 2. Find any key words in the question. 3. Find out what the question is asking. There may be choices that look like the correct answer, but do not answer the question. 4. Eliminate any choices that are incorrect. 5. After you make your choice, re-read the question again to check that the answer you chose is the best answer. 6. In questions that involve calculations, double check your work. Short Answer Questions 1. Read through the question. 2. Find any key words and determine what the question is asking. 3. Show all of the steps you used to find your answer. 4. Check over your work to be sure that your computation is correct. 5. Re-read the question and make sure that your response properly answers the question. Complete the following activities. 1. Read through the "Vocabulary Review" section on p. 462 of Mathematics: Course 3. Be sure you know the meaning of each of the words and are able to answer problems 1–4. 2. Work through the "Skills and Concepts" portion on pp. 462–463 of Mathematics: Course 3. You may skip question 9 on p. 463. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to all of the review problems in the back of your textbook in the Selected Answers section. Complete the following review activities 1. To review circle graphs, click on the link below to complete the "Circle Graphs" activity. Circle Graphs 2. To review how to calculate the mean, median, and mode for a set of data, click on the link below to complete the “Averages and Measures of Central Tendency" activity. Averages and Measures of Central Tendency Click on the link below to access the Using Graphs to Analyze Data Unit Review Practice. Using Graphs to Analyze Data Unit Review Practice Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Averages and Measures of Central Tendency Complete the following review questions. 1. Which term is a synonym for the term mean? a. average b. mode c. median d. divide 2. What term describes the middle number when a set of data is arranged in order from least to greatest? a. average b. mode c. median d. divide 3. How do you find the mode in a set of data? a. Add up all of the values and divide by the number of values. b. Arrange the data from least to greatest and find the number in the middle. If there are two, add them up and divide them in half. c. Count how many times each value occurs in a data set and identify which occurs the most. d. Subtract the least value from the greatest value. 4. Look at the following steps. Which item below fits the missing step? 1. Find the sum of all of the numbers. 2. Count how many values you added up. 3. ? 4. Check your answer. a. Multiply the sum by the number of values. b. Multiply the sum by the median number. c. Divide the sum by the number of values. d. Divide the number of values by the sum. 5. A group of Girl Scouts has the following ages: 10, 13, 11, 12, 14. What is their average age? a. 5 b. 13 c. 12 d. 14 6. What is the mean of the ticket prices listed for local movie theaters? $11.00, $5.00, $11.00, $8.00, $11.00, $10.00, $7.00 a. $6.00 b. $7.00 c. $8.00 d. $9.00 7. Terrence is 6 feet 3 inches tall. Keisha and Emily are both 5 feet tall. Manuel is 5 feet 9 inches tall. What is their average height? a. 5 feet 6 inches b. 5 feet 5 inches c. 4 feet 3 inches d. 6 feet 2 inches 8. The following list shows the temperatures for a city from April 23–April 30: 70°F, 68°F, 78°F, 61°F, 84°F, 68°F, 73°F, 78°F. What is the average temperature? a. 64°F b. 72.5°F c. 70°F d. 580°F 9. The finalists in the small dog division of the dog show have the following weights: 34 lbs., 45 lbs., 39 lbs., 18 lbs., 26 lbs., 39 lbs., 23 lbs. What is their mean weight? a. 112 lbs. b. 32 lbs. c. 224 lbs. d. 39 lbs. 10. Felicity is on the track team and was practicing for the 100-meter dash. Her times for the practice are listed below. What is her median run time? Monday Tuesday Wednesday Thursday Friday 35 seconds 34 seconds 29 seconds 31 seconds 31 seconds a. 29 seconds b. 31 seconds c. 32 seconds d. 35 seconds 11. The temperatures for May 16 are listed for the past five years. What is the median temperature? 43°F, 39°F, 81°F, 50°F, 62°F a. 55°F b. 62°F c. 43°F d. 50°F 12. Mrs. Abdullah posted the science test grades for her study group. What is the median grade? 35%, 77%, 96%, 90%, 83%, 89%, 85%, 85% a. 77% b. 85% c. 80% d. 89% 13. Identify the mode(s) of the ticket prices listed for local movie theaters. $11.00, $5.00, $11.00, $8.00, $11.00, $10.00, $7.00 a. $7.00 and $9.00 b. $8.00 and $5.00 c. $10.00 d. $11.00 14. Ming-Na was visiting her relatives and recorded the ages of the adults in the house in the following list: 45, 94, 61, 27, 58. Identify the mode(s). a. 57 b. 58 and 61 c. 94 d. There is no mode. 15. Felicity is on the track team and was practicing for the 100-meter dash. Her times for the practice are listed below. What is her median run time? Week Monday Tuesday Wednesday Thursday Friday Week 1 35 secs 34 secs 29 secs 31 secs 31 secs Week Monday Tuesday Wednesday Thursday Friday Week 2 31 secs 30 secs 29 secs 30 secs 29 secs a. 34 and 35 secs. b. 31 and 29 secs. c. 30 secs. d. 29 secs. Select the Show Answer button to check your answers. Answer: 1. a. average 2. c. median 3. c. Count how many times each value occurs in a data set and identify which occurs the most. 4. c. Divide the number of items by the sum. 5. c. 12 6. d. $9.00 7. a. 5 feet 6 inches 8. b. 72.5°F 9. b. 32 lbs. 10. b. 31 seconds 11. a. 55°F 12. b. 85% 13. d. $11.00 14. d. There is no mode. 15. b. 31 and 29 secs. Circle Graphs You are doing a survey of all of the students in your school to see how frequently each has visited a public library in the last month. You divide the responses into four categories: zero visits, 1 or 2 visits, 3 to 5 visits, and more than 5 visits. You want to show how many responses are in each category and how each relates to the total number of students, so you create a circle graph. 15% of the students went to the library 0 times. 10% went 1 or 2 times. 20% went 3 to 5 times. 55% went to the library more than 5 times. Use this information and the circle graph below to answer questions 1–5. 1. Which section represents the most common number of visits? a. 0 b. 1–2 c. 3–5 d. more than 5 2. Which two sections combined represent of the students? a. 0 and 1–2 b. 1–2 and 3–5 c. 3–5 and more than 5 d. 0 and more than 5 3. What fraction of students visited the library 3–5 times in the last month? a. b. c. d. 4. If you surveyed a total of 160 students, how many students went to the library more than 5 times? a. 55 b. 66 c. 77 d. 88 5. If the number of students who went to the library 1–2 times was 25, how many total students did you survey? a. 100 b. 200 c. 250 d. 325 A new online streaming movie service looked at its data to see how popular 6 genres were among its viewers. The circle graph below show the percentage of views each genre received. 6. Which genre was least popular? a. adventure b. comedy c. documentary d. thriller 7. Which three genres combined were viewed by of the respondents a. thriller, documentary, and adventure b. documentary, adventure, and comedy c. adventure, comedy, and drama d. comedy, drama, and children’s 8. What fraction of movies viewed were in the thriller genre? a. b. c. d. 9. If the total number of movies viewed is 300 million, how many of those movies were comedies? a. 50 million b. 60 million c. 70 million d. 80 million 10. If the number of adventure movie views is 80 million, how many total movie views were there? a. 320 million b. 360 million c. 480 million d. 600 million Select the Show Answer button to check your answers. Answer: 1. d. more than 5 2. a. 0 and 1–2 3. c. 4. d. 88 5. c. 250 6. c. documentary 7. b. documentary, adventure, and comedy 8. b. 9. b. 60 million 10. a. 320 million © 2015 Connections Education LLC. Lesson 13: Using Graphs to Analyze Data Unit Test CE 2015 Algebra Readiness (Pre-Algebra) B Unit 3: Using Graphs to Analyze Data Using Graphs to Analyze Data Unit Test Part 1 Charles Washington is taking this assessment. This assessment was assigned 0 points by Connie Aphonephanh. Once taken, the actual score will be used instead. Multiple Choice 1. What are the mean, median, mode, and range of the data set given the altitude (1 point) of lakes in feet: –11, –28, –17, –25, –28, –39, –6, and –46? mean = –25; median = –26.5; mode = –28; range = 40 mean = –25; median = –40; mode = –26.5; range = 28 mean = –26.5; median = –25; mode = –28; range = 28 mean = –26.5; median = –28; mode = –25; range = 40 2. Given the data 14, 26, 23, 19, 24, 46, 15, 21: (1 point) a. What is the outlier in the data? b. What is the mean with the outlier? c. What is the mean without the outlier? 14; 20.3; 23.5 14; 23.5; 20.3 46; 20.3; 23.5 46; 23.5; 20.3 3. Which frequency table represents the set of data below related to how each student in a class traveled to school in the morning? family car bus car pool biked biked walked family car biked bus biked walked bus biked family car car pool biked bus bus walked walked walked walked walked car pool (1 point) 4. The data below represent the ages of the first ten people in line at the movie theater. 25, 23, 25, 29, 28, 22, 29, 29, 30, 23 Which line plot represents the data? (1 point) 5. Which stem-and-leaf plot represents the data set below? (1 point) 56, 113, 89, 85, 96, 104, 65, 67, 72, 88, 97 6. A back-to-back stem-and-leaf plot showing the points scored by each player on two different basketball teams is shown below. (1 point) What is the median number of points scored for each team? Median for Team 1: 12 Median for Team 2: 10 Median for Team 1: 13.5 Median for Team 2: 12 Median for Team 1: 12 Median for Team 2: 11 Median for Team 1: 11 Median for Team 2: 10 7. Which box-and-whisker plot shows the scores of ten students on a mathematics (1 point) exam? 89, 78, 93, 90, 75, 81, 91, 80, 89, 79 8. Which box-and-whisker plot shows the high temperatures in Philadelphia, Pennsylvania, during the first two weeks of March: 39, 41, 33, 57, 34, 30, 47, 33, 49, 52, 32, 53, 37, 43 (1 point) 9. Ms. Alison drew a box-and-whisker plot to represent her students’ scores on a midterm test. Jason received 81 on the test. How does Jason’s score compare to his classmates? About 25% scored higher; about 75% scored lower. About 50% scored higher; about 50% scored lower. About 75% scored higher; about 25% scored lower. No one scored higher. 10. Which scatter plot represents the given data? x 1.2 1.3 3.4 3.9 5 6.1 7.9 8.4 8.6 y 1 2.5 4 6 2.5 4 1 2 2 (1 point) (1 point) 11. What type of trend does the scatter plot below show? What type of real-world situation might the scatter plot represent? (1 point) positive trend; weight and height negative trend; weight and height no trend; the number of pets owned and the owner’s height negative trend; the water level in a tank in the hot sun over time 12. Given the following values, which point would be considered an outlier? x 1 2 3 4 5 6 7 8 9 y 0.9 2.1 3.2 3.9 7.4 5.8 7.2 8 9.1 (1 point) (2, 2.1) (9, 9.1) (8, 8) (5, 7.4) 13. A recording artist released a compilation of songs on the Internet. The scatter (1 point) plot below shows the number of downloads for her album, in the thousands, over the course of nine days. If this trend continues, approximately how many thousands of downloads occurred on day 10? 30 40 50 60 14. Carol has a collection of 100 stamps. The graph below shows the percentage of stamps she has from each country. (1 point) How many more of Carol’s stamps are from France than from England? 22 stamps 2 stamps 24 stamps 46 stamps 15. A survey about the student government program at a school finds the following results: 110 students like the program. 120 students think the program is unnecessary. 210 students plan on running for student government next year. If a circle graph were made from the data, what is the measure of the central angle for the group that plans on running for student government next year? 39° 90° 98° 172° Type of Cookie Adults Children Total Chocolate chip 5 10 Peanut butter 8 6 Oatmeal 7 4 Use the table below. Type of Cookie Adults Children Total Chocolate chip 5 10 Peanut butter 8 6 (1 point) Oatmeal 7 4 16. Based on the two-way frequency table, how many adults were surveyed? (1 point) 13 15 20 40 Use the table to answer the question. 17. What percentage of the children chose oatmeal? (1 point) 20% 30% 50% 80% 18. Which scatterplot does NOT suggest a linear relationship between x and y? (1 point) 19. Which Venn diagram correctly represents the relationship between rational numbers and irrational numbers? Explain your answer. (3 points) Take the assessment. Using Graphs to Analyze Data Unit Test Part 2 © 2015 Connections Education LLC. Unit 4: STAAR Review Algebra Readiness (Pre-Algebra) B Unit Summary This unit will help you prepare for the STAAR test. Lessons 1. STAAR Review Lesson 1 2. STAAR Review Lesson 2 3. STAAR Review Lesson 3 4. STAAR Review Lesson 4 5. STAAR Review Lesson 5 6. STAAR Review Lesson 6 7. STAAR Review Lesson 7 8. STAAR Review Lesson 8 9. STAAR Review Lesson 9 10. STAAR Review Lesson 10 Lesson 1: STAAR Review Lesson 1 Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review Note: The content you are trying to access is not formatted properly. Refer to your math course's message board under "STAAR Review" for the instructions and assignments for this lesson. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Lesson 2: STAAR Review Lesson 2 Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review Note: The content you are trying to access is not formatted properly. Refer to your math course's message board under "STAAR Review" for the instructions and assignments for this lesson. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Lesson 3: STAAR Review Lesson 3 Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review Note: The content you are trying to access is not formatted properly. Refer to your math course's message board under "STAAR Review" for the instructions and assignments for this lesson. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Lesson 4: STAAR Review Lesson 4 Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review Note: The content you are trying to access is not formatted properly. Refer to your math course's message board under "STAAR Review" for the instructions and assignments for this lesson. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Lesson 5: STAAR Review Lesson 5 Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review Note: The content you are trying to access is not formatted properly. Refer to your math course's message board under "STAAR Review" for the instructions and assignments for this lesson. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Lesson 6: STAAR Review Lesson 6 Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review Note: The content you are trying to access is not formatted properly. Refer to your math course's message board under "STAAR Review" for the instructions and assignments for this lesson. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Lesson 7: STAAR Review Lesson 7 Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review Note: The content you are trying to access is not formatted properly. Refer to your math course's message board under "STAAR Review" for the instructions and assignments for this lesson. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Lesson 8: STAAR Review Lesson 8 Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review Note: The content you are trying to access is not formatted properly. Refer to your math course's message board under "STAAR Review" for the instructions and assignments for this lesson. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Lesson 9: STAAR Review Lesson 9 Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review Note: The content you are trying to access is not formatted properly. Refer to your math course's message board under "STAAR Review" for the instructions and assignments for this lesson. You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test Lesson 10: STAAR Review Lesson 10 Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review Refer to your math course's message board under "STAAR Review" for the instructions and assignments for this lesson. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Unit 5: Polynomials and Properties of Exponents Algebra Readiness (Pre-Algebra) B Unit Summary In this final unit, you will be working with expressions called polynomials. By the end, you will be able to add, subtract, and multiply these expressions. You will also simplify powers and use negative and zero exponents. Objectives Add, subtract, and multiply polynomials Multiply and divide powers with the same base, including numbers in scientific notation Lessons 1. Polynomials 2. Adding and Subtracting Polynomials 3. Exponents and Multiplication 4. Multiplying Polynomials 5. Exponents and Division 6. Polynomials and Properties of Exponents Review 7. Polynomials and Properties of Exponents Unit Test Lesson 1: Polynomials CE 2015 Algebra Readiness (Pre-Algebra) B Unit 5: Polynomials and Properties of Exponents Objective: Write algebraic expressions and simplify polynomials What Goes Together? Have you ever heard the phrase, “That’s like comparing apples to oranges?” It is an idiom that means you are trying to compare two things that are completely different. Just as you cannot accurately compare two things that are different, you also cannot combine them mathematically. This picture could be represented by the equation a + p = ?. You learned previously that because a and p are different variables, they cannot be combined. There is no way to make this equation any simpler. If you added another apple to the equation, you would have a + a + p = ?. In this case, you can combine the a’s, which are like terms. You would end up with 2a + p = ?. In this lesson, you will learn to write expressions for polynomials and to simplify polynomials. Objective Write algebraic expressions and simplify polynomials Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words coefficient constant monomial polynomial Working with Several Terms In a previous unit, you learned that you can add or subtract like terms in an algebraic expression. The following expression has two like terms and a constant. If terms have the same variable with the same exponent, they are considered like terms. Terms that don’t have any variable are called constants. All constants are like terms. When you combine terms with addition or subtraction, the resulting expression is called a polynomial. A polynomial is one term, or the sum or difference of two or more terms. A polynomial with only one term is called a monomial. Following are several examples of polynomials. 3x2 + y + 4z –6x2 – 4 x2 + 5x + 2x2 + 17 Polynomial expressions can be used to represent real-world situations. For example, the following diagram shows the dimensions of a backyard that has a square patio at one end. To find the area of the grass, you could use the polynomial expression In the expression, 15x represents the area of the entire rectangle and . represents the area of the patio. To find the area of the grass, subtract the area of the patio from the area of the rectangle. The expression represents the following situation. A group purchases x adult tickets and y student tickets to the school play. The adult tickets cost $11 each and the student tickets cost $8 each. There is a discount of $10 for purchasing the tickets early. What does each term in the polynomial expression represent? Click on the Show Answer button to review your answer. Answer: Each term in the polynomial expression is explained below. 11x – cost of the adult tickets 8y – cost of the student tickets 10 – amount of the discount Click on the link below to access the Terms coefficients and exponents in a polynomial video on the Khan Academy website. Terms coefficients and exponents in a polynomial You saw in the video that a polynomial such as 3x2 – x + 7 can be written as 3x2 – x + 7x0. However, it is customary to simply refer to 7 as a constant. In the polynomial 3x2 – x + 7, what are the coefficients? Click on the Show Answer button below to check your answer. Answer: 3, –1 To simplify a polynomial, you combine like terms using addition or subtraction. Remember, like terms have the same variable with the same exponent. If given the expression 2x2 + 5x + 3x2 + 4 +2x+ 7, the first step is to rearrange the problem so that like terms are together. (2x2+ 3x2) + (5x + 2x) + (4 + 7). By combining like terms, you end up with 5x2 + 7x + 11. Each term with a variable has a degree. The degree is the value of that term’s exponent. When you simplify and expression, the term with the highest degree is listed first. The degree of a polynomial expression is the greatest exponent, so the equation above has a degree of 2. Click on the link below to access the Simply a Polynomial video on the Khan Academy website. Simply a Polynomial Complete the following activities. 1. Read pp. 561–563 in Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 8–14, 16–20, and 22–25 on pp. 564–565. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. 1. Click on the link below to watch the "Polynomials" BrainPOP® movie. After you watch the movie, click on the Take the Quiz icon. Then select the review quiz to see how well you understand polynomials. Polynomials 2. Use the properties of numbers to simplify the following polynomials. a. 2x + 5 + 9 b. c. d. e. f. g. h. i. j. Click on the Show Answer button to check your answers. Answers: a. 2x +14 b. c. d. e. f. g. h. 9a +2b i. j. You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 © 2015 Connections Education LLC. Lesson 2: Adding and Subtracting Polynomials CE 2015 Algebra Readiness (Pre-Algebra) B Unit 5: Polynomials and Properties of Exponents Objective: Add and subtract polynomials More Than One Polynomial If you looked at this skate ramp from above, it would have a rectangular shape with a platform on each end. The dimensions of the skate ramp could be given in polynomial expressions rather than exact values. By learning to add polynomials, you will be able to simplify the dimensions of this skate ramp as well as other geometric figures. You have learned how to combine like terms within a polynomial expression to make it easier to understand. In this lesson, you will learn how to add and subtract two separate polynomials. Objective Add and subtract polynomials Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Word coefficient Simplifying Large Polynomial Expressions You can add and subtract polynomials by combining like terms. When you combine like terms, you add or subtract the coefficients of the terms. A coefficient is a number that is multiplied by a variable. For example, in the term 2x, the coefficient is 2. You can add the polynomials (8a2 + 6a + 3) + (5a2 – 3a + 7) by using the following steps. = (8a2 + 5a2) + (6a – 3a) + (3 + 7) Use the Commutative Property of Addition to rearrange the terms. = (8 + 5)a2 + (6 –3)a + (3 + 7) Distributive Property = 13a2 + 3a + 10 Combine like terms. Tip: You can also add polynomials by lining the terms up vertically and combining the like terms. When subtracting polynomials, remember to distribute the subtraction sign to each term in the second polynomial. This is the same as adding the opposite of the second polynomial. You can use the following steps to subtract (8a2 + 6a + 3) – (5a2 – 3a + 7). = (8a2 + 6a + 3) + (–5a2 + 3a + – Use the Distributive Property to change the problem into addition of the 7) opposite. = (8a2 + –5a2) + (6a + 3a) + (3 + – Use the Commutative Property of Addition to rearrange the terms. 7) = (8 + –5)a2 + (6 + 3)a + (3 + –7) Distributive Property = 3a2 + 9a – 4 Combine like terms. Tip: Just as with adding polynomials, you can subtract polynomials by writing them vertically and combining like terms. 1. Click on the link below to access the Addition and Subtraction of Polynomials video on the Khan Academy website. Before watching the video, make a table like the one shown below. Fill in an explanation and example for each word or term while watching the video. key word example explanation coefficient constant degree standard form variable Addition and Subtraction of Polynomials 2. Click on the link below to access the Adding and Subtracting Polynomials 1 video on the Khan Academy website. Pause the video when the problem comes onscreen, complete the problem, and then check your answer against the instructor’s work. Adding and Subtracting Polynomials 1 Complete the following activities. 1. Read pp. 566–567 in Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 7–23 on pp. 568–569. Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Click on the link below to access the online textbook. Mathematics: Course 3 Complete the following review activities. 1. Click on the link below and complete the Adding and subtracting polynomials activity on the Khan Academy website. Work to receive 20 leaves in a single stack. Adding and subtracting polynomials 2. Remember the skateboard ramp problem in the Getting Started section of this lesson? To find the perimeter of the ramp, you need to add the polynomial from each side of the rectangle. What is the sum of the sides? Click on the Show Answer button below to check your answer. Answer: (3 x + 2) + (3 x + 2) + (2 x – 1) + (2 x –1) = (3 x + 3 x + 2 x + 2 x ) + (2 + 2 – 1 – 1) = 10 x + 2 You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials © 2015 Connections Education LLC. Lesson 3: Exponents and Multiplication CE 2015 Algebra Readiness (Pre-Algebra) B Unit 5: Polynomials and Properties of Exponents Objectives: Multiply powers with the same base; Multiply numbers in scientific notation Note: This lesson should take 2 days. Very Large Numbers You can easily think about the number of programs on your computer or even the number of files. But sometimes you need to think about numbers that are much larger. For example, the number of bits of information that are stored on your hard drive is a very large number. Very large numbers are sometimes written in scientific notation. Because scientific notation uses powers of 10, if you want to multiply numbers in scientific notation, you will need to multiply powers. In this lesson, you will learn how to multiply numbers that have the same base but different exponents. You will also learn how to multiply two numbers that are written in scientific notation. Objectives Multiply powers with the same base Multiply numbers in scientific notation Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words base power scientific notation Tip: You will have 2 days to complete this lesson. Multiplying Exponents with the Same Base Recall that a power consists of a base and an exponent. The base is the factor that you are multiplying. The exponent tells you how many times to multiply the base by itself. Look at the example below. Both powers have the same base, 5, but different exponents. How can you multiply these powers together? Start by expanding each power. Then count the total number of times you are multiplying the base by itself. 53 • 55 = 5 • 5 • 5 • 5 • 5 • 5 • 5 • 5 = 58 The example illustrates a property of powers. If you multiply two powers with the same base, you keep the base and add the exponents. This can be written algebraically as am • an = am + n. Because scientific notation uses numbers multiplied by powers of 10 to represent very large or very small values, the same rules for multiplying exponents apply. Look at the steps used to multiply (3 × 109)(4 × 103). (3 × 4)(109 × 103) = Use the Commutative and Associative Properties to rearrange the problem. 12 × 1012 = Multiply 3 and 4 and add the exponents. 1.2 × 101 × 1012 = Write 12 in scientific notation. 1.2 × 1013 Add the exponents. 1. Try these problems on your own. 1. 35 • 3 2. 143 • 142 3. (2 • 107)(7 • 104) Click on the Show Answer button below to check your answers. Answer: 1. 36 2. 145 3. 1.4 • 1012 2. Click on the link below to complete the Exponents and Multiplication activity from the digits™ website. Complete the Key Concept and Part 1 of the Example section. Launch Example 1 Example 2 Example 3 Key Concept Close and Check 3. Click on the link below to access the Scientific Notation 2 video on the Khan Academy website. After watching the video, write the following product in scientific notation: (4 • 105) • (7 • 103) Scientific Notation 2 Click on the Show Answer button below to check your answer. Answer: (4 • 105) • (7 • 103) = (4 • 7) • (105 • 103) = (4 • 7) • (105 + 3) = 28 • 108 = 2.8 • 101 • 108 = 2.8 • 109 Complete the following activities. 1. Read pp. 571–572 in Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 9–23 on p. 573. 3. Complete the Exploring Exponents Activity Lab on p. 570. Complete problems 1–3 and then check your answers. 4. Complete the Scientific Notation Activity Lab on p. 575. Read through the explanation and examples and then do problems 1–8. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. 1. Click on the link below to complete the Exponents and Multiplication questions from the MathXL® for School website. Exponents and Multiplication 2. Review lessons 1, 2, and 3 in preparation for the quiz at the end of this lesson. Click on the link below to access the online textbook. Mathematics: Course 3 Lesson Answers Click on the link below to check your answers to the Exploring Exponents Activity Lab. Exploring Exponents Activity Lab Answers Click on the link below to check your answers to the Scientific Notation Activity Lab. Scientific Notation Activity Lab Answers You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exploring Exponents Activity Lab Answers 1. two exponents product as a repeated factor standard form single exponent 21 • 21 2•2 4 22 21 • 22 2•2•2 8 23 21 • 23 2•2•2•2 16 24 21 • 24 2 • 2• 2 • 2 • 2 32 25 2. a. The sum of the exponents in the first cell is equal to the exponent in the last cell. b. Yes, the relationship holds for the other rows in the table. 3. am • an = am + n Scientific Notation Activity Lab Answers 1. 8.05 • 1021 2. 1.30364• 1029 3. 7.5625 • 108 4. 6.92 • 106 5. 2.01 • 1013 6. 7.84 • 1035 (rounded) 7. 1.0 • 1020 8. 2.25 • 108 Click on the links below to complete questions 1–15. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 © 2015 Connections Education LLC. Lesson 4: Multiplying Polynomials CE 2015 Algebra Readiness (Pre-Algebra) B Unit 5: Polynomials and Properties of Exponents Objective: Multiply monomials and binomials What About Multiplication? You already know how to find the area of many different geometric shapes. For instance, you might recall that to find the area of a rectangle, you multiply the base by the height. The solution to an area problem will always have square units (units2). So far, all of your experience with area problems has involved actual measurements for each of the sides. Some problems, however, use polynomial expressions to represent the side lengths. How would you go about finding the area of a garden bed with the given polynomial dimensions? In this lesson, you will learn how to multiply polynomials. Objective Multiply monomials and binomials Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words binomial Distributive Property Simplifying Polynomial Expressions Recall that a monomial is a polynomial with only one term, such as 5a2 or 2x. A binomial is a polynomial with two terms, such as 4x2 + 4 or 2y – 5. To multiply monomials, multiply the coefficients and add the exponents of powers with the same base. 5a2 • 2a3 = (5 • a2) • (2 • a3) = (5 • 2) (a2 • a3) Use the Commutative Property of Multiplication to rearrange the factors. = (10) (a2 • a3) Multiply the coefficients. = 10a5 Add the exponents to multiply the powers. Click on the link below to access the Multiplying Monomials video on the Khan Academy website. Multiplying Monomials To multiply a monomial and a binomial, you can use the Distributive Property to distribute the monomial to each of the terms in the binomial. 2y3 (y + 3) = (2y3 • y) + (2y3 • 3) Use the Distributive Property. = 2y4 + 6y3 Simplify. To multiply two binomials, each term in the second binomial must be multiplied by each term in the first binomial. You can use the steps below to multiply (2x + 1) (x + 2). (2x • x) + (2x • 2) + (1 • x) Use the Distributive Property to multiply each term in the second binomial by + (1 • 2) each term in the first binomial. 2x2 + 4x + x + 2 Simplify. 2x2 + 5x + 2 Combine like terms. Tip: You might hear the method used to solve multiplying binomial problems as the FOIL method. FOIL is an acronym that helps you remember the order to use when you multiply the terms. (2y + 1) (3y – 4) F – first terms (2y • 3y) = 6y2 O – outside terms (2y • –4) = –8y I – inside terms (1 • 3y) = 3y L – last terms (1 • –4) = –4 = 6y2 + (–8y + 3y) + (– 4) = 6y2 – 5y – 4 Find the product of (3x + 4)(2x + 2). Click on the Show Answer button below to check your answer. Answer: (3x • 2x) + (3x • 2) + (4 • 2x) + (4 • 2) = 6x2 + 6x + 8x + 8 = 6x2 +14x + 8 Click on the link below to access the Multiplying Binomials video on the Khan Academy website. Multiplying Binomials Complete the following activities. 1. Read pp. 576–577 of Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 7–21 on p. 578. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. 1. Click on the link below to access the Multiplying expressions 0.5 activity on the Khan Academy website. Work to receive 20 leaves in a single stack. Multiplying expressions 0.5 2. Use what you have learned about multiplying binomials to find the area of the garden presented in the Getting Started section of this lesson. Click on the Show Answer button below to check your answer. Answer: (3x – 2) (2x + 5) = (3x • 2x) + (3x • 5) + (–2 • 2x) + (–2 • 5) = 6x2 + 15x + (–4x) + (–10) = 6x2 + 11x – 10 You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication © 2015 Connections Education LLC. Lesson 5: Exponents and Division CE 2015 Algebra Readiness (Pre-Algebra) B Unit 5: Polynomials and Properties of Exponents Objectives: Divide powers with the same base; Simplify expressions with negative exponents Note: This lesson should take 2 days. Very Small Numbers When working with things that are microscopic, such as cells, viruses, and bacteria, the numbers involved are often very small. The length or mass of a cell is generally expressed with scientific notation, and the exponents are usually negative. When working with these small numbers, there will be situations in which you need to divide powers with the same base. Previously, you learned how to simplify expressions by multiplying powers. In this lesson, you will learn to divide powers with the same base. Objectives Divide powers with the same base Simplify expressions with negative exponents Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words exponent expression simplify Tip: You will have 2 days to complete this lesson. Dividing Power Terms Just as with multiplying powers with the same base, expanding powers can help you understand the rules for dividing powers with the same base. Rewrite the expression 64 ÷ 62 in factor form to see what happens to the exponents. As illustrated in the example, if you are dividing two nonzero numbers or variables with the same nonzero base, keep the base and subtract the exponents. This can be written algebraically as . You can apply the same rules for dividing powers with the same base to scientific notation problems that involve division. = • 102 = 7 • 102 Click the links below to complete the Exponents and Division activity from the digits™ website. Launch Example 1 Example 2 Concept Close and Check Zero and Negative Numbers as Exponents Look for patterns in the following table. Try a few problems of your own. Write each expression with a single exponent. 1. 2. 3. Click on the Show Answer button below to check your work. Answer: 1. n4 2. 4w3 3. (–5)2 Click the links below to complete the Zero and Negative Exponents activity from the digits™ website. Launch Example 1 Example 2 Example 3 Key Concept Close and Check Complete the following activities. 1. Read pp. 581–583 of Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 6–28 on p. 584. 3. Complete the Power Rules Extension activity on p. 586. Read through the explanation and examples, and complete problems 1–8. Click on the link below to access the online textbook. Mathematics: Course 3Complete the following review activities. 1. Click on the link below to complete the Exponents and Division questions from the MathXL® for School website. Exponents and Division 2. Click on the link below to complete the Zero and Negative Exponents questions from the MathXL® for School website. Zero and Negative Exponents 3. Review Lessons 4 and 5 in preparation for the quiz at the end of this lesson. Click on the link below to access the online textbook. Mathematics: Course 3 You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Power Rules Extension Activity Answers 1. 321 2. 9–10 3. w12 4. r6 5. 9x2 6. a8b12 7. 100x10 8. 256y8 Click on the links below to complete questions 1–12. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Click on the links below to complete questions 1–15. Work through each question until you find the correct answer. Once you answer the question, you can solve similar questions by clicking on the Similar Exercise button at the bottom of the screen. Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 © 2015 Connections Education LLC. Lesson 6: Polynomials and Properties of Exponents Review CE 2015 Algebra Readiness (Pre-Algebra) B Unit 5: Polynomials and Properties of Exponents Objective: Review problems and terms Note: This lesson should take 2 days. Getting Ready The test at the end of a unit is an opportunity for you to demonstrate everything you have learned while studying these concepts. In this lesson, you will review test-taking strategies that will help you be successful on the unit test. You will also have the chance to practice what you learned in previous lessons of this unit by doing several review activities. By thoroughly preparing for the unit test, you will rely on your skill rather than luck to be successful. Objective Review problems and terms Key Words base binomial coefficients constant Distributive Property exponent expression monomial polynomial power scientific notation simplify Tip: You will have two days to complete this lesson. Test-Taking Strategies In the next lesson, you will take a test on the skills and concepts you learned in this unit. To prepare for the test, start by reviewing the following test-taking strategies. Multiple-Choice Questions 1. Read through the question and all the answer choices before selecting your response. 2. Find any key words in the question. 3. Be sure you understand what the question is asking. There may be choices that answer part of the question correctly, but not the whole question. 4. Eliminate any choices that are incorrect. 5. After you make your choice, reread the question again to check that the answer you chose is the best answer. 6. In questions that involve calculations, double-check your work. Short-Answer Questions 1. Read through the question. 2. Find any key words and determine what the question is asking. 3. Show all the steps you used to find your answer. 4. Check your work to be sure that your computation is correct. 5. Reread the question and make sure that your response properly answers the whole question. 6. Be sure that you have included the units (people, months) in your answer. Content Review You may go back and review any previous videos, problems sets, vocabulary, or concepts that you did not understand in this unit. In addition, use the following video links to practice simplifying polynomial expressions and multiplying polynomials. Pause the video when a problem appears, solve it on your own, and then check your answer with the instructor’s solution. 1. Click on the link below to access the Adding and Subtracting Polynomials 3 video on the Khan Academy website. Adding and Subtracting Polynomials 3 2. Click on the link below to access the Multiplying Monomials by Polynomials video on the Khan Academy website. Multiplying Monomials by Polynomials Complete the following activity. Click on the link below to complete the “Addition of Polynomials - Activity B” Gizmo to practice the concepts from this unit. Use the algebra tiles to model several addition problems. After completing the activity, answer the assessment questions. Addition of Polynomials - Activity B Complete the following review activities. 1. Read through the Vocabulary Review section on p. 590 of Mathematics: Course 3. Be sure you know the meaning of each of the words and are able to answer questions 1–5. 2. Work through the “Skills and Concepts” portion on pp. 590–591. Complete exercises 6–45. Click on the link below to access the Polynomials and Properties of Exponents Review Practice. Polynomials and Properties of Exponents Review Practice Click on the link below to access the online textbook. Mathematics: Course 3 Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. © 2015 Connections Education LLC. Lesson 7: Polynomials and Properties of Exponents Unit Test CE 2015 Algebra Readiness (Pre-Algebra) B Unit 5: Polynomials and Properties of Exponents You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Take the assessment. Polynomials and Properties of Exponents Unit Test Part 2 © 2015 Connections Education LLC. Unit 6: Probability Algebra Readiness (Pre-Algebra) B Unit Summary In this unit of the course, you will find probabilities and odds of events. Objectives Calculate odds and probabilities of dependent and independent events, and make predictions using those calculations Calculate permutations and combinations of sets of objects Lessons 1. Theoretical and Experimental Probability 2. Independent and Dependent Events 3. Making Predictions 4. Permutations 5. Combinations 6. Unit Review 7. Probability Unit Test Lesson 1: Theoretical and Experimental Probability CE 2015 Algebra Readiness (Pre-Algebra) B Unit 6: Probability Objective: Find theoretical probabilities, experimental probabilities, and odds Note: This lesson should take 2 days. What Will Happen? You may have previously used graphs to make predictions. Another way to make predictions in mathematics is to find the probabilities of events. Probability is the likelihood that a certain event will occur. For example, how likely is it that you will get a red gumball from the gumball machine? If you’ve ever rolled a number cube or guessed at a test question, you have experienced probability. In this lesson, you will learn the difference between theoretical and experimental probability, and learn how to calculate both. You will also learn how odds are expressed and calculated. Objective Find theoretical probabilities, experimental probabilities, and odds Key Words complements experimental probability odds against odds in favor theoretical probability Tip: You will have two days to complete this lesson. Probability Probability is the ratio of the number of ways an event can occur to the total number of possible outcomes. It is a value between zero and 1 and can be expressed as a fraction, decimal, or percent. A probability of 1 means that the event is certain to occur, whereas a probability of zero means that the event will certainly not occur. The closer the value is to 1, the more likely the event will occur. Theoretical probability is based on the following mathematical formula: theoretical probability of an event = If you are finding the probability of getting heads when flipping a coin, heads is called a favorable outcome. Since the coin has two sides (heads and tails), there are two possible outcomes. The theoretical probability of getting heads is . The following cube has two red sides, two yellow sides, and two white sides. There are two ways to get the favorable outcomes of rolling red, because there are two red sides. There are six possible outcomes since the cube has six sides. The theoretical probability of rolling a red side— expressed by the notation P(red)—is calculated as follows: P(red) = or A probability distribution table shows the possible outcomes for an event and the probability of each outcome. The probability distribution for the red, yellow, and white cube is shown below. Notice that the combined probabilities from the table have a sum of 1; there is a 100% probability that red, yellow, or white will be rolled on the cube. Outcome RedYellowWhite Probability The probability distribution can also be shown graphically. The bar graph shows the probability distribution for rolling the colored cube. If one of the white sides of the cube was changed to blue, how would the probability distribution table and the bar graph change? How would they stay the same? Click on the Show Answer button to review your answer. Answer: The probability of rolling blue and the probability of rolling white would both be in the table and the graph. The probabilities for red and for yellow would stay the same. The total of the probabilities would still be 1. Experimental probability is based on the actual results from an experiment and can be different each time you run the experiment. For example, if you rolled the colored cube six times and kept track of the results, you might get the following: According to these results, you have a (0.5 or 50%) chance of rolling red. Experimental probability can be deceiving. The results show that you have a 0% chance of rolling white. You know this cannot be correct, because it is not impossible to roll white. The more times an experiment is conducted, the more accurate the experimental probability will be. If you rolled the cube 100 times instead of just six times, your results would likely be closer to the theoretical probabilities of the events. Complements The opposite of an event is called its complement. If two events are complementary, the sum of their probabilities is 1. The complement to rolling red on the colored cube is rolling a color that is not red. You can calculate the probability of not rolling red by subtracting the probability of rolling red from 1. P(red) = P(not red) = 1 – = An event and a complement of an event are both subsets of the entire sample space. An event and its complement represent all of the outcomes that are possible for an event. Intersection and Union of Two Events The intersection of two events is the outcomes that occur for both events. The intersection of rolling an even number on a six-sided number cube and rolling a number greater than 4 on a number cube is shown below. even numbers: numbers greater than 4: outcomes that are even AND and greater than 4: Because there is only one outcome that is in both sets, the probability of rolling a number that is an even number and a number greater than 4 is . The union of two events is the outcomes that occur for either event. The intersection of rolling an even number on a six-sided number cube or rolling a number greater than 4 on a number cube is shown below. even numbers : numbers greater than 4: outcomes that are even OR or greater than 4: Because there are four outcomes that are in either set, the probability of rolling a number that is even or greater than 4 is . Odds If you have ever heard someone say, "We had a two-to-one chance of winning," you have heard someone talking about odds. Unlike probability, odds do not involve the total number of possible outcomes. Instead, odds are a ratio of favorable outcomes and unfavorable outcomes. Odds are always written as a ratio using either a colon (2:1) or the word to (2 to 1). To express odds, you have to think about whether you are giving the odds something will happen or the odds something will not happen. They can be written in the following ways: Odds in favor of an event = number of favorable outcomes : number of unfavorable outcomes Odds against an event = number of unfavorable outcomes : number of favorable outcomes Tip: You might be interested in a review of how to find the probability of rolling a certain number on a number cube. Click on the link below to watch the "Basic Probability" BrainPOP® movie. Basic Probability Complete the following activities. 1. Read pp. 470–471 of Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 5–10 and 12–19 on pp. 472–473. 3. Read through the Extension on p. 479. 4. Complete problems 1–3 on p. 479. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activity. Click on the link below to complete the "Probability Simulations" Gizmo to practice the concepts from today’s lesson. Follow the steps in the Exploration Guide to investigate how theoretical probability and experimental probability compare. After completing the activity, answer the assessment questions. Probability Simulations You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Polynomials and Properties of Exponents Unit Test Polynomials and Properties of Exponents Unit Test © 2015 Connections Education LLC. Lesson 2: Independent and Dependent Events CE 2015 Algebra Readiness (Pre-Algebra) B Unit 6: Probability Objective: Find the probabilities of independent and dependent events Does Probability Change? What if these colorful cookies were put in a bag and offered to you and a friend? You each get to choose a cookie at random, but your friend will go first. Suppose you want an orange cookie. If your friend doesn’t choose an orange cookie, how does that affect your chance of getting an orange cookie? If your friend does get an orange cookie, how does that affect your chance of getting an orange cookie? What if another cookie, exactly like the one your friend chooses, is put back into the bag before you choose? Would your chance of choosing an orange cookie be affected at all by what your friend chose first? In the last lesson, you learned how to find the probability of an event, such as choosing a cookie of a certain color. For example, the probability of choosing an orange cookie is (2 favorable outcomes and 11 possible outcomes). In this lesson, you will learn how to calculate the probability when there are two events. Objective Find the probabilities of independent and dependent events Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words compound events dependent events independent events Taking the Past into Account Rolling a number cube, flipping a coin, and choosing a card are all simple events. What if you were flipping two coins or rolling a number cube and spinning a spinner? A compound event involves at least two simple events. The events can be independent—the outcome of one event does not affect the probability of the other event—or the events can be dependent—the outcome of one event will affect the probability of the other event. Flipping a coin is an example of an independent event. What is the probability that it will land tails up? If you flip the coin again, will the probability of getting tails depend on what you flipped earlier? If the outcome of one event does not affect the probability of another event, then the events are independent. Click on the link below to watch the “Independent & Dependent Events” BrainPOP® movie. Why is the coin toss at the beginning of an ultimate game an independent event? What is the probability that both coins will land on the same side? Independent and Dependent Events Click on the Show Answer button below to check your answer. Answer: Coin tosses are independent events because the outcome of one toss does not affect the outcome of another toss. The four possible outcomes for two tosses are heads­heads, heads­tails, tails­tails, and tails­heads. In two of the four possible outcomes, the coins land with the same side up (heads­-heads and tails-­tails). So the probability of that event is , or . In the video, the result of one coin toss did not affect the result of the other coin toss, so the two events are independent. If two events are independent, the formula can be used to find the probability of both events occurring. The probability of rolling a 1, followed by a 2, on a six-sided number cube can be determined using this formula. The events are independent because the first number that is rolled does not affect the second number that is rolled. The probability of rolling a 1 and the probability of rolling a 2 are both . The probability of rolling a 1, followed by a 2, is . You can also use the formula to determine if two events are independent. Two events, A and B, are independent if the probability of A and B occurring together is the product of their individual probabilities. One shape is chosen at random from the following shapes. Let R represent choosing a red shape, and let T represent choosing a triangle. Are the events R and T independent events? If the events are independent, the . You can see that the probability of R and T is since one of the four shapes is both red, and a triangle. and . Since , the events are independent. Let R represent choosing a red shape and let T represent choosing a triangle. Are the events R and T independent events? The probability of R and T is still since only one of the four shapes is red, and a triangle. and . Since , the events are not independent. Conditional Probability Suppose you have a bag that contains 5 blue marbles, 8 white marbles, and 7 red marbles. You will choose 2 marbles at random. If you want to determine the probability that you will choose 2 blue marbles, you will need to use conditional probability. When you choose the first marble, the probability of selecting a blue one is . If you do grab a blue marble, there are now 19 marbles left in the bag, and 4 of them are blue. The probability of choosing another blue marble is . This probability is called a conditional probability because it depends on the condition that you have already selected 1 blue marble. You are finding the probability, given that the first marble was blue. The probability of choosing 2 blue marbles is then the product of both of the individual probabilities, or . What is the probability of choosing 2 white marbles? Click on the Show Answer button to review your answer. Answer: The probability of getting a white marble twice in a row is . Complete the following activities. 1. Read pp. 486–487 of Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 6–22 (odd) on pp. 488–489. 3. Read through the Activity Lab on p. 485. Complete problems 1–9. You can use marbles, colored slips of paper, or candies if you don’t have any blocks. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. 1. Click on the link below to complete the "Compound Independent and Dependent Events" Gizmo to practice the concepts from today’s lesson. Follow the steps in the Exploration Guide to investigate how independent and dependent events affect probability. Experiment with different numbers of blue and green marbles in the bag. After completing the activity, answer the assessment questions. Independent and Dependent Events 2. Click on the link below to watch the "Independent and Dependent Events" BrainPOP® movie. After you watch the movie, click on the Take the Quiz icon. Then select the review quiz to see how well you understand independent and dependent events. Independent and Dependent Events Lesson Answers Click on the link below to check your answers to the Activity Lab. Answers You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Polynomials and Properties of Exponents Unit Test Polynomials and Properties of Exponents Unit Test Theoretical and Experimental Probability Answers 1. Check students' work. 2. 3. 4. 5. greater than 6–9. Your answers will depend on the results of your trials. © 2015 Connections Education LLC. Lesson 3: Making Predictions CE 2015 Algebra Readiness (Pre-Algebra) B Unit 6: Probability Objective: Make predictions based on theoretical and experimental probabilities How Does Probability Apply to the Real World? Suzanna is running for eighth grade class president. She asks everyone in her math class how they plan to vote, and finds out 15 out of 25 students will vote for her. The probability that Suzanna will get a vote in her math class is . She can use this probability to make a prediction about the overall results of the election. If there are 125 students in the eighth grade class who will vote, you can calculate how many total votes Suzanna will receive by multiplying by 125. Both theoretical and experimental probabilities are used to make predictions every day about weather patterns, shopping habits, and voting trends. In today’s lesson, you will learn to make predictions based on theoretical and experimental probabilities. Objective Make predictions based on theoretical and experimental probabilities Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Using Probability to Predict Success or Failure Probabilities can be used to make predictions about future events. Sometimes fast-food restaurants have contests where a game piece on a carton reveals a prize. Soft-drink companies will hold similar contests with the winning message under the cap. These contests have stated probabilities of winning, so you can use those probabilities to figure out approximately how many winning tokens exist in a group of tokens. For example, suppose your favorite fast-food restaurant is having a contest where a game token is affixed to every soft-drink cup. If the rules state that the probability of winning a prize is , how many winning game tokens would you expect to be in an unopened sleeve of 100 cups? To find this, multiply the probability of a winning token by the total amount of tokens. Since each cup has 1 token and each sleeve has 100 cups, there are 100 game tokens. Multiplying by 100 means that you can expect there to be approximately 20 prize-winning tokens in each unopened sleeve of soft-drink cups. Suppose the prize that you really want is a new sports car. The rules state that the probability of winning the sports car is . How many sports car-winning tokens would you expect there to be in 2,500 sleeves of cups? Click on the Show Answer button below to check your answers. Answer: Every sleeve has 100 tokens, so in 2,500 sleeves there would be 250,000 tokens. If you multiply 250,000 tokens by the probability of , or 0.00001, you will find that in 2,500 sleeves, you might find 2 or 3 tokens that win a sports car. A water park is considering raising their rates. The park surveys 50 customers about how much more they would be willing to pay for a full-day pass at the park. The results of the survey are shown in the table. $2.00 $3.00 $4.00 $5.00 more more more more Totals Adults 10 5 5 3 23 Teens 14 9 2 2 27 Totals 24 14 7 5 50 The results of the survey can be used to make predictions about how larger numbers of the water park’s customers would feel about raising the rates. For example, 5 out of 50, or , of the people surveyed said they would be willing to pay $5.00 more for a full-day pass. The number of customers in a group of 300 who would feel the same way can be predicted by multiplying 300 by . So, it could be predicted that 30 out of 300 customers would be willing to pay an additional $5.00 for a full-day pass. Complete the following activities. 1. Read pp. 475–476 of Mathematics: Course 3. Be sure you understand the difference between using experimental and theoretical probability to make predictions. 2. Complete problems 6–18 on p. 477. 3. Read Fair Games (Activity Lab 10-2a) on p. 474. 4. Complete problems 1–7 on p. 474. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activity. The assessment for this lesson will cover Lessons 1–3. To prepare for the quiz, use the Checkpoint quizzes on p. 479 (1–6) and p. 490 (3–10). Click on the link below to access the online textbook. Mathematics: Course 3 Lesson Answers Click on the link below to check your answers to the Fair Games Activity Lab. Fair Games Activity Lab Answers Click on the link below to check your answers to the Checkpoint Quizzes. Checkpoint Quizzes Answers You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Polynomials and Properties of Exponents Unit Test Polynomials and Properties of Exponents Unit Test Theoretical and Experimental Probability Independent and Dependent Events You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Polynomials and Properties of Exponents Unit Test Polynomials and Properties of Exponents Unit Test Theoretical and Experimental Probability Independent and Dependent Events Checkpoint Quizzes Answers Checkpoint Quiz 1 1. 0.95 2. 0.94, 3. 0.77, 4. experimental; the results are based on a survey, 5. 10 heads, 6. 9 toys Checkpoint Quiz 2 3. , 4. – , 5. , 6. , 7. , 8. Fair Games Activity Lab Answers , 9. , 10. 1. You should have a table showing results for heads and tails. Since your friend’s team had already won three games, the table should start with three heads. Since your team had already won one game, the table should start with one head. The table should show either four heads or four tails. 2. Another table showing results for heads and tails. This time the table should show the results of 40 tosses. 3. Based on the results for question 2, write a ratio of heads to tails. 4. The series ends when a team wins 4 games, so Team H needs to win one more game or Team T needs to win all 3 remaining games. outcome probability H-H-H H-H-T H-T-H H-T-T T-H-H T-H-T T-T-H T-T-T 6. ; 7. No; he should be willing to do your chores for 7 weeks since his team is 7 times more likely to win. © 2015 Connections Education LLC. Lesson 4: Permutations CE 2015 Algebra Readiness (Pre-Algebra) B Unit 6: Probability Objective: Find the number of permutations of a set of objects How Many Arrangements? What if you forgot the string of numbers needed to open your bike lock? How long do you think it would take you to guess the right numbers in the correct order? How many possible arrangements are there for four numbers? If the lock only had three numbers, would the number of arrangements change? In this lesson, you will learn how to find the number of ways a set of objects can be arranged by using the fundamental counting principle. The rules you will learn can be applied to the arrangement of things like letters, numbers, seats, or places in a contest. Objective Find the number of permutations of a set of objects Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Words counting principle factorial permutation The Order Matters! A permutation is an arrangement of a set of objects in which the order of the objects matters. In the bike lock problem, the order of the numbers matters. The set of numbers 4789 is different from 7498, even though they both use the same digits. One way you can determine the total number of possible permutations is with the fundamental counting principle. The fundamental counting principle says that all possible outcomes for a series of events can be found by multiplying the number of outcomes for each event. If the first event can happen x ways and the second event can happen y ways, then you can find the total number of outcomes by multiplying x · y. With some problems, you can use a tree diagram or organized list to find the total number of possible permutations. If you wanted to find all the possible numbers with non-repeating digits that can be made from the digits 2, 4, and 6, you could use one of the following strategies. Tree diagrams and organized lists work best when the problem is fairly simple and there is not a large number of possible permutations. The counting principle states that if one event can occur in m ways, and a second event can occur in n ways, then the two events can occur together in ways. Here is another example of how to apply the principle. A pizza restaurant offers pizzas in 3 different sizes with a choice of 12 different toppings. How many different choices of pizzas with 1 topping does the restaurant offer? There are 3 options for the size of the pizza and 12 options for the toppings. The total number of possible outcomes is , or 36. If the restaurant also offers 2 different types of crust, the number of possible outcomes is , or 72. You can apply the fundamental counting principle to the bike problem from the Getting Started section of the lesson. There are four numbers on the lock and each of the numbers can be 10 different digits (0–9). Find the total possible outcomes as follows: × × × = 10,000 possibilities The fundamental counting principle can also be applied to any sort of competition in which the order of the finishers matters. If you and five friends are having a foot race, how many possible outcomes are there for the finish 1st place 2nd place 3rd place 4th place 5th place 6th place 6 people could 5 people could 4 people could 3 people could 2 people could 1 person could place first place third place second place fourth place fifth finish sixth (since one already placed first) To find the total possible outcomes, multiply 6 × 5 × 4 × 3 × 2 × 1 = 720. A simpler way of representing this multiplication problem is 6!, which is read, “six factorial.” A factorial is the product of all positive integers less than or equal to the number itself. Alice is arranging pictures of her best friends on a bookshelf in her room. How many possible arrangements are there for 8 photographs? g p g p Click on the Show Answer button to review your answer. Answer: , or There are 40,320 possible arrangements of 8 photographs on a bookshelf. You can use permutation notation to express permutation problems in a simplified way. The number of permutations of n objects chosen r at a time can be represented by the expression . For example, means that from a group of 10 objects, 3 are being chosen and arranged in order. Which has more possible arrangements, 10P3 or 7P4? Click on the Show Answer button below to check your answer. Answer: 10P3 = 10 × 9 × 8 = 720, 7P4 = 7 × 6 × 5 × 4 = 840; 7P4 has more possible arrangements. Complete the following activities. 1. Read pp. 491–493 of Mathematics: Course 3. Be sure you understand each of the key words and concepts from the lesson. 2. Complete problems 6–12 (all) and 18–26 (odd) on p. 494. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activities. 1. Click on the link below to complete the Counting, Probability, and Predicting activity. Counting, Probability, and Predicting 2. Click on the link below to complete the "Permutations" Gizmo to practice the concepts from today’s lesson. Follow the steps in the Exploration Guide to investigate permutations, tree diagrams, and the counting principle with different numbers of objects. After completing the activity, answer the assessment questions. Permutations You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Polynomials and Properties of Exponents Unit Test Polynomials and Properties of Exponents Unit Test Theoretical and Experimental Probability Independent and Dependent Events Making Predictions Making Predictions © 2015 Connections Education LLC. Lesson 5: Combinations CE 2015 Algebra Readiness (Pre-Algebra) B Unit 6: Probability Objective: Find the number of combinations of a set of objects using lists and combination notation Note: This lesson should take 2 days. Does Order Always Matter? In the previous lesson you learned how to find the total number of possible outcomes in situations where the order matters. The number 1234 is different from the number 4321 on a bike lock. But does the order of events always matter? If you had four pairs of shoes but could only take two pairs on vacation, how many ways could you pick two pairs to bring? If you made an organized list of the possible outcomes, would they all be different from each other? Complete the tree diagram below. After completing the tree diagram, see if there are any outcomes that appear more than once. Click on the link below to access the Tree Diagram. Tree Diagram Objective Find the number of combinations of a set of objects using lists and combination notation Objectives derived from Pearson Education, Inc. © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Key Word combination Tip: You will have 2 days to complete this lesson. Order Doesn't Matter! A combination is an arrangement of items in which the order does not matter. For example, you are going to a concert and have three extra tickets to give to your friends. If you have five friends who want to go, how many combinations of friends can you take? In this scenario, the order does not matter. Deciding to give the extra tickets to Joe, Lauren, and Mike is the same as giving the tickets to Mike, Joe, and Lauren—the same three friends will go with you to the concert, no matter who was chosen first. One way to find the total number of possible combinations is to make an organized list. This is the method you used to find shoe combinations in the Getting Started section. This method works best when you have a small number of choices and events. Another method is to start by finding the total number of permutations, and then divide that number by the number of duplicates. You can apply this method to the concert ticket problem. To find the total number of permutations, multiply 5 × 4 × 3 = 60 (you have five choices for the first ticket, four choices for the second ticket, and three choices for the third ticket). Now you need to eliminate the arrangements that use the same three people (remember, Joe, Lauren, and Mike is the same as Mike, Joe, and Lauren). To eliminate the duplicates, divide the number of permutations by the number of ways 3 people can be arranged, which is 3!. Just like with permutations, there is a special combination notation that can be used to write problems in a simplified way. the number of nCr general form combinations of n objects taken r at a time the number of 6C2 You can choose 2 toppings for combinations of your pizza. There are 6 toppings 6 objects taken to choose from. 2 at a time Ellis is making plans for Saturday. His choices are to go on a hike, take his dog to the dog park, go to a movie, catch up on chores, play basketball, and read a good book. Ellis thinks he will have time for 3 of his choices. In this problem, there are 6 activity choices, taken 3 at a time. The simplified notation for the problem is . To find the number of possible combinations, divide the permutations by the number of different orders for 3 activities. There are 20 different combinations of activities. Complete the following activities. 1. Read pp. 496–497 of Mathematics: Course 3. Be sure you understand the key words and concepts from the lesson. 2. Complete problems 7–14, 19, 21, and 22 on p. 498. 3. Read Guided Problem Solving: Permutations, Combinations, and Probability on p. 501. 4. Complete problems 1–6 on pp. 501–502. Click on the link below to access the online textbook. Mathematics: Course 3 Tip: Be sure to check your answers. You can find answers to the odd-numbered problems in the back of your textbook in the Selected Answers section. Complete the following review activity. Click on the link below to complete the "Permutations and Combinations" Gizmo to practice the concepts from today’s lesson. Follow the steps in the Exploration Guide to investigate the difference in the number of possible outcomes when you find permutations and when you find combinations. After completing the activity, answer the assessment questions. Permutations and Combinations Lesson Answers Click on the link below to check your answers to problems 1–6 on pp. 501–502. Answers You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Polynomials and Properties of Exponents Unit Test Polynomials and Properties of Exponents Unit Test Theoretical and Experimental Probability Independent and Dependent Events Making Predictions Making Predictions Permutations Answers 1. The number of ways you can pick 3 from 6 gymnasts where order does not matter. 2. Dependent; the selection of the second gymnast depends on the selection of the previous gymnast. 3. Because order does not matter, the total number of permutations is divided by the number of ways of arranging three gymnasts in order to remove the duplicate groups. 4. 5. 6. © 2015 Connections Education LLC. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. Lesson 7: Probability Unit Test CE 2015 Algebra Readiness (Pre-Algebra) B Unit 6: Probability You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Polynomials and Properties of Exponents Unit Test Polynomials and Properties of Exponents Unit Test Theoretical and Experimental Probability Independent and Dependent Events Making Predictions Making Predictions Permutations Combinations Take the assessment. Probability Unit Test Part 2 © 2015 Connections Education LLC. Unit 7: Personal Financial Literacy Algebra Readiness (Pre-Algebra) B Unit Summary In this unit, students will apply mathematical processes to develop an economic way of thinking and problem solving. Students will calculate interest and costs, compare different payment methods, analyze different financial situations, and solve real-world finance problems. Lessons Materials 1. Personal Financial Literacy Lesson 1 Crayons or colored pencils* Drawing compass* 2. Personal Financial Literacy Lesson 2 Markers, assorted colors (1 pack)* 3. Personal Financial Literacy Lesson 3 Measuring tape* Paper, white (1 pack)* 4. Personal Financial Literacy Lesson 4 Pencils (1 box)* 5. Personal Financial Literacy Lesson 5 Protractor* 6. Personal Financial Literacy Lesson 6 Ruler* Scissors* Tape (1 roll)* * You need to supply Lesson 1: Personal Financial Literacy Lesson 1 CE 2015 Algebra Readiness (Pre-Algebra) B Unit 7: Personal Financial Literacy Materials: Crayons or colored pencils, Drawing compass, Markers, assorted colors (1 pack), Measuring tape, Paper, white (1 pack), Pencils (1 box), Protractor, Ruler, Scissors, Tape (1 roll) Note: This is not a mandatory unit. If you would like to complete this unit for additional attendance hours or extra credit, please follow the instructions in the attachment tab In this unit, you will learn the following terms: ATM borrow checks compound interest credit debit financial aid grants invest loans need room and board savings scholarships simple interest store-value card tuition work study For today's lesson, refer to the Section Message Boards. Contact your teacher for additional information regarding lesson pacing and assessments. For this lesson, you will submit designated financial literacy activities using a Drop Box. Contact your teacher for additional information regarding the lesson assessment. Complete and submit the Repaying Loans portfolio assessment. You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Polynomials and Properties of Exponents Unit Test Polynomials and Properties of Exponents Unit Test Theoretical and Experimental Probability Independent and Dependent Events Making Predictions Making Predictions Permutations Combinations Probability Unit Test Probability Unit Test © 2015 Connections Education LLC. Lesson 2: Personal Financial Literacy Lesson 2 CE 2015 Algebra Readiness (Pre-Algebra) B Unit 7: Personal Financial Literacy Note: This is not a mandatory unit. If you would like to complete this unit for additional attendance hours or extra credit, please follow the instructions in the attachment tab For today's lesson, refer to the Section Message Boards. Contact your teacher for additional information regarding lesson pacing and assessments. For this lesson, you will submit designated financial literacy activities using a Drop Box. Contact your teacher for additional information regarding the lesson assessment. Complete and submit the Saving and Investing portfolio assessment. You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Polynomials and Properties of Exponents Unit Test Polynomials and Properties of Exponents Unit Test Theoretical and Experimental Probability Independent and Dependent Events Making Predictions Making Predictions Permutations Combinations Probability Unit Test Probability Unit Test Personal Financial Literacy Lesson 1 © 2015 Connections Education LLC. Lesson 3: Personal Financial Literacy Lesson 3 CE 2015 Algebra Readiness (Pre-Algebra) B Unit 7: Personal Financial Literacy Note: This is not a mandatory unit. If you would like to complete this unit for additional attendance hours or extra credit, please follow the instructions in the attachment tab For today's lesson, refer to the Section Message Boards. Contact your teacher for additional information regarding lesson pacing and assessments. For this lesson, you will submit designated financial literacy activities using a Drop Box. Contact your teacher for additional information regarding the lesson assessment. Complete and submit the Analyzing Financial Situations portfolio item. You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Polynomials and Properties of Exponents Unit Test Polynomials and Properties of Exponents Unit Test Theoretical and Experimental Probability Independent and Dependent Events Making Predictions Making Predictions Permutations Combinations Probability Unit Test Probability Unit Test Personal Financial Literacy Lesson 1 Personal Financial Literacy Lesson 2 © 2015 Connections Education LLC. Lesson 4: Personal Financial Literacy Lesson 4 CE 2015 Algebra Readiness (Pre-Algebra) B Unit 7: Personal Financial Literacy Note: This is not a mandatory unit. If you would like to complete this unit for additional attendance hours or extra credit, please follow the instructions in the attachment tab For today's lesson, refer to the Section Message Boards. Contact your teacher for additional information regarding lesson pacing and assessments. For this lesson, you will submit designated financial literacy activities using a Drop Box. Contact your teacher for additional information regarding the lesson assessment. Complete and submit the Estimating College Costs and Payments portfolio assessment. You must successfully complete the following activities before viewing this item: Using Graphs to Analyze Data Unit Test Using Graphs to Analyze Data Unit Test STAAR Review Lesson 9 Polynomials Adding and Subtracting Polynomials Exponents and Multiplication Exponents and Multiplication Multiplying Polynomials Exponents and Division Polynomials and Properties of Exponents Review Polynomials and Properties of Exponents Unit Test Polynomials and Properties of Exponents Unit Test Theoretical and Experimental Probability Independent and Dependent Events Making Predictions Making Predictions Permutations Combinations Probability Unit Test Probability Unit Test Personal Financial Literacy Lesson 1 Personal Financial Literacy Lesson 2 Personal Financial Literacy Lesson 3 © 2015 Connections Education LLC. Lesson 5: Personal Financial Literacy Lesson 5 CE 2015 Algebra Readiness (Pre-Algebra) B Unit 7: Personal Financial Literacy For today's lesson, refer to the Section Message Boards. Contact your teacher for additional information regarding lesson pacing and assessments. For this lesson, you will submit designated financial literacy activities using a Drop Box. Contact your teacher for additional information regarding the lesson assessment. Complete and submit the Financial Literacy Unit Review portfolio assessment. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. © 2015 Connections Education LLC. Lesson 6: Personal Financial Literacy Lesson 6 CE 2015 Algebra Readiness (Pre-Algebra) B Unit 7: Personal Financial Literacy For today's lesson, refer to the Section Message Boards. Contact your teacher for additional information regarding lesson pacing and assessments. For this lesson, you will submit designated financial literacy activities using a Drop Box. Contact your teacher for additional information regarding the lesson assessment. Complete and submit the Financial Literacy Unit Test portfolio assessment. Your teacher has dropped this assessment and will not count it toward your grade. Please skip it and move on. © 2015 Connections Education LLC.