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.
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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
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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
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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
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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
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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
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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:
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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:
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assignments for this lesson.
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Lesson 2: STAAR Review Lesson 2
Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review
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Lesson 3: STAAR Review Lesson 3
Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review
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Lesson 4: STAAR Review Lesson 4
Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review
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Lesson 5: STAAR Review Lesson 5
Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review
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Lesson 6: STAAR Review Lesson 6
Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review
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Lesson 7: STAAR Review Lesson 7
Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review
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Lesson 8: STAAR Review Lesson 8
Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review
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Lesson 9: STAAR Review Lesson 9
Algebra Readiness (Pre-Algebra) B Unit 4: STAAR Review
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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
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assignments for this lesson.
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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.