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1 Basic Skills/ TAP Test -- Mathematics Boot Camp Study Guide The Basic Skills/ TAP Test of mathematics examines skills required to demonstrate quantitative literacy at the college level through the application of mathematical methods and reasoning to the solution of real-world problems. Comparison Basic Skills to TAP: Mathematics: BS300 - 42 multiple choice; TAP400- 50 multiple choice online TAP Study Guide: http://www.icts.nesinc.com/PDFs/IL_field400_SG.pdf Websites for additional help: Dr. Jim Olsen, WIU: http://faculty.wiu.edu/JR-Olsen/wiu/stu/advising/prep-basic- skills.htm Praxis Pre-professional Skills Test: http://www.ets.org/Media/Tests/PRAXIS/pdf/5730.pdf Math Dictionary: www.mathwords.com Basic Math Help - http://www.basic-mathematics.com ACT practice test (math): http://www.actstudent.org/sampletest/math/math_01.html ACT practice test booklet p. 17-20 (can be printed) (answers at the end of p. 20) http://sat.collegeboard.org/SAT/public/pdf/getting-ready-for-the-sat-subj-tests.pdf ACT practice test booklet explanations for test questions http://sat.collegeboard.org/SAT/public/pdf/SubjectTestsAnswerExplanationsMath.pdf Discovery Education: http://www.webmath.com Free Math Help: http://www.math.com Click “algebra” then “see all 258 skills” Mathway: http://wwwmathway.com Either “enter your problem” or view the “example problems” Purple math: http://www.purplemath.com “tutoring and lessons” or “quizzes and worksheets” 2 Suggested General Strategies for the Math Test: Read the test standards and objectives. This is a source of information about topics covered by the test. This can give a more specific idea of the knowledge and skills you will need to demonstrate on the test. As you are answering the questions, remember to read the directions and the test questions carefully. Read all response options. Remember that the test questions call for the "best answer." Do not choose what appears to be the first reasonable response option; read and evaluate all choices to find the best answer. Draw a quick sketch to illustrate the given information and help you “see” how the pieces of information fit together. If you’re running out of time and cannot quickly determine the best answer to a question, try to eliminate as many of the options as possible. Then, guess among the remaining answer choices. Remember if a problem is long, looks hard or is a topic that gives you trouble, skip it and save for the end. If the problem is more than four lines long, skip it and save it till the end. Always go to the end of a word problem. Read the question. Study the diagram and think about the implications. Answer their question, not your question. When answering, make sure to answer the question asked. If the question asks for “length”, don’t give the value for “width”. Plug-in Technique. Work backwards. If appropriate, try substituting the answers into the problem. Start from the middle choice and work up/down depending on your initial result. Always check your answer for: Does it make sense? Does it answer the question? Are your computations correct? Use any remaining time at the end of the session to check the accuracy of your work. Go back to the questions that gave you difficulty and verify your work on them. PRACTICE, PRACTICE, PRACTICE! There are many problems on websites to work on. You may want to print them out so you can annotate! 3 Key Math Vocabulary: Acute angle -- an angle measuring between zero and 90 degrees Acute triangle -- a triangle with three acute angles Alternate exterior angles -- congruent angles formed by two parallel lines cut by a transversal; located on opposite sides of the transversal outside the parallel lines Alternate interior angles -- congruent angles formed by two parallel lines cut by a transversal; located on opposite sides of the transversal between the parallel lines Bar graph – a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence of bars, or rectangles, corresponding to the possible values, and the length of each is proportional to the frequency. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=50 Box-and-whisker plot – a diagram constructed from a set of numerical data showing a box that indicates the middle 50% of the marked observations together with lines, sometime called ‘whiskers’, that go out from the quartile to the most extreme data value in that direction which is not more than 1.5 times the Inter Quartile Range from the quartile. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=57 http://www.introductorystatistics.com/escout/main/Glossary.htm Circle -- a 2-D figure; the set of all points at a given distance from the center Circle graph – a graph for categorical data. The proportion of elements belonging to each category is proportionally represented as a pie-shaped sector of a circle. Sometimes called a pie chart. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=68 Circumference -- the perimeter of a circle Clusters of data – a portion of high concentration in a data set. For more info: http://www.teachers.ash.org.au/jeather/maths/dictionary.html Complementary angles -- two or more angles that sum to 90 degrees Cone -- a 3-D figure with a circular or non-circular base and rounding sides rising to a point Congruent -- of the same size and shape Corresponding angles -- congruent angles formed by two parallel lines cut by a transversal; positioned on matching "corners" of intersections 4 Cylinder -- a 3-D figure with circular or non-circular parallel congruent bases and curving sides Data – the observations gathered from an experiment, survey or observational study. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=102 Diagonal -- a line segment connecting two non-adjacent vertices of a polygon Equiangular triangle -- a triangle with three congruent angles Equilateral triangle -- a triangle with three congruent sides Histogram – A bar graph presenting the frequencies of occurrence of data points. Sometimes called a frequency histogram For more info: http://www.shodor.org/interactivate/activities/histogram/ Isosceles trapezoid -- a trapezoid with two congruent legs Isosceles triangle -- a triangle with two congruent legs Kite -- a quadrilateral with two pair of adjacent congruent sides Lateral faces -- the polygon shapes making up the sides of a polyhedron Line-of-best-fit – the line that best represents the trend that the points in a scatter plot follow. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=198 Line plot – a line graph that orders the data along a real number line. Also called a dot plot. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=200 Mean – the mean is an appropriate location measure for interval or ratio variables. Suppose there are N individuals in the population and x denotes an interval or ratio variable. Let values for the ith individual be denoted by i x . The mean of x is the number For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=210 Median – suppose the observations in a set of numerical data are ranked in ascending order. The median is the middle observation if there are an odd number of observations, and is the average of the two middlemost observations if there are an even number of observations. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=213 Mode –The mode is the most frequently occurring value in a set of discrete data. There can be more than one mode if two or more values are equally common. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=219 5 Obtuse angle -- an angle measuring between 90 and 180 degrees Obtuse triangle -- a triangle with one obtuse angle Odds – a way of representing the likelihood of an event's occurrence. The odds m:n in favor of an event means we expect the event will occur m times for every n times it does not occur. For more info: http://www.mathwords.com/o/odds.htm Parallel lines -- lines that never intersect in a plane Parallelogram -- a quadrilateral with two pair of parallel sides Perimeter -- the distance around a 2-D figure Perpendicular -- meeting at a 90 degree angle Polygon -- a simple closed two-dimensional shape made of line segments Prism -- a 3-D figure with parallel congruent polygon bases and sides made of parallelograms Probability – the chance/likelihood that a particular event (or set of events) will occur expressed on a scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%. For more info: http://www.mathwords.com/p/probability.htm http://www.teachers.ash.org.au/jeather/maths/dictionary.html Pyramid -- a 3-D figure with a polygon for a base and triangular faces meeting at a point at the top Pythagorean Theorem -- the sum of the squares of the legs of a right triangle equals the square of the hypotenuse; a2 + b2 = c2 where c is always the hypotenuse side Radius -- the distance from the center to the circle Range – the range of a sample (or a data set) is a measure of the spread or the dispersion of the observations. It is the difference between the largest and the smallest observed value. For more info http://www.teachers.ash.org.au/jeather/maths/dictionary.html Rectangle -- a parallelogram with a right angle Regular -- 2-D or 3-D figure having all congruent sides and all congruent angles Rhombus -- a quadrilateral with four congruent sides 6 Right angle -- an angle measuring exactly 90 degrees Right triangle -- a triangle with one right angle Sample – a subset of a population that is obtained through some process, possibly random selection or selection based on a certain set of criteria, for the purposes of investigating the properties of the underlying parent population. For more info: http://www.teachers.ash.org.au/jeather/maths/dictionary.html http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=316 Sphere -- a 3-D figure; the set of all points at a given distance from the center Square -- a quadrilateral with all congruent sides and all congruent angles Stem-and-leaf plot – a semi-graphical method used to represent numerical data, in which the first (leftmost) digit of each data value is a stem and the rest of the digits of the number are the leaves. For more info: http://online.math.uh.edu/MiddleSchool/Modules/Module_5_Prob_Stat/Content/Stat/Ch4_3. pdf Straight angle -- measuring exactly 180 degrees; makes a straight line Supplementary angles -- two or more angles that sum to 180 degrees Transversal -- a line that crosses a pair of parallel lines Trapezoid -- a quadrilateral with one pair of parallel sides Tree diagram – A tree diagram displays all the possible outcomes of an event. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=363 Variable – a quantity that varies. For example, the weight of a randomly chosen member of a football team is such a variable. Variables are usually represented by letters. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=379 Venn diagram – a graphic means of showing intersection and union of sets by representing them as bounded regions. For more info: http://www.teachers.ash.org.au/jeather/maths/dictionary.html http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=380 Vertical angles -- congruent opposite angles formed by two intersecting lines 7 Standard 0014 Solve problems involving integers, fractions, decimals, and units of measurement. The following are examples of content that may be covered under this standard. Solve problems involving integers, fractions, and decimals, including percentages. Solve problems involving ratios and proportions. Solve problems involving units of measurement, including U.S. customary and metric measurements, and conversions, including scientific notation (e.g., 6.05 x 10 8). Use estimation skills to solve problems. Practice Test Samples 1. In a park, the ratio of ducks to geese is 16 to 9. How many of the 300 birds are geese? To solve: The ratio tells me that, of every 16 + 9 = 25 birds, 9 are geese. That is, 9/25 of the birds are geese. Then there are ( 9/25 )(300) = 108 geese 2. Convert 3 gallons to quarts. To solve: Quarts are smaller than gallons; every gallon has four quarts. Since I'm converting from a larger unit (gallons) to a smaller unit (quarts), my answer needs to be a bigger number. So I multiply:(3)(4) = 12 quarts 3. Convert 7920 yards to miles To solve: Miles are bigger than yards; there are 1760 yards in every mile. Since I'm converting from a smaller unit (yards) to a bigger unit (miles), my answer needs to be a smaller number. So I divide: 7920 ÷ 1760 = 4.5 miles 4. The original price of a shirt was $20. It was decreased to $15. What is the percent decrease? To solve: The absolute decrease is 20 - 15 = $5 The percent decrease is the absolute decrease divided by the original price (part/whole). 5 / 20 = 0.25 then convert to a percent = 25% 8 5. Mary has a monthly salary of $2400. She spends $380 per month on food. What percent of her monthly salary does she spend on food? To solve: The part of her salary that is spent on food is $380 out of her monthly salary of $2400… percent = part / whole = 380 / 2400 = 0.16 (rounded to 2 decimal places) then convert to a percent = 16% 6. Solve and write (3.4 x 106) (4.2 x 103) in scientific notation. To solve: (3.4)(4.2) x 10 6+3 = 14.28 x 109 = 1.428 x 1010 7. Solve and write (3 x 104)3 in scientific notation. To solve: 33 x 104x3 = 27 x 1012 = 2.7 x 1013 8. A rectangle has a length of 20 mm and a width of 15 mm. What is the perimeter of the rectangle in cm? To solve: add two widths and two lengths (20 + 20 + 15 +15 = 70mm) then convert the 70 mm into cm (70mm ÷ 10) = 7cm 9. A walking path is 402 m from start to finish. If the path is walked 5 times how far to the nearest kilometer is walked? To solve: multiple 402 x 5 = 2010m then convert the m to km = 2.010 km then round to the nearest km = 2 km 10. Lilly rides 1.3 miles from her house to school then 1.68 miles from school to swim practice and finally she rides 2.2 miles from swim practice back to her house. She does this 5 days a week. Approximately how far does she ride in one week? To solve: round each distance (1.3 →1.0), (1.68 → 2.0), (2.2 →2.0) ….so, in one day she rides 5 miles (1+2+2), 5 miles x 5 days = 25 miles per week. 9 11. Lilly has 24 yards of ribbon for 18 presents she has to wrap. Each present gets the same amount of ribbon. How much ribbon will each present get? To solve: first, convert 24 yards to inches…24 yards x 3 = 72 feet, 72 feet x 12 = 864 inches. Now divide the 864 inches by 18 and that equals 48 inches or the answers might be in only feet (4 feet) or the answer could be in yards and feet (I yard 1 foot) 12. Jessica owns Bubbles pet store, she needs to replace the tubing in the 22 fish tanks at the store. Each tank needs a length of tubing that is 1 foot 8 inches. Plus the owner wants an additional 5 feet of tubing to save for repairs. What is the total length of tubing Jessica needs to buy for his store? To solve: convert 1 foot 8 inches into all inches, 12 + 8 = 20 inches, then multiple 20 inches by 22 tanks and get 440 inches needed for the tank. Next, Convert the extra 5 feet to inches (5 x 12 = 60 inches) now add the inches needed for the tank and the extra inches together: 440 + 60 = 500 inches. BUT the answer probably won’t be in only inches so, divide 500 by 12 to see how many feet can be taken out of the inches: 500 ÷ 12 = 41.66 so we know 41 feet can be taken out. Since 41 feet = 492 inches we know that 500 inches = 41 feet 8 inches if that isn’t a choice then you may have to convert that to yards so 41 feet divide by 3 is 13.66 so we know that 13 yards can be taken out, 13 x 3 is 39 which means we would have 2 feet left over. Another way to write the answer is 13 yards 2 feet 8 inches 10 Standard 0015 Apply mathematical reasoning skills to analyze patterns and solve problems. The following are examples of content that may be covered under this standard. Draw conclusions using inductive reasoning. Draw conclusions using deductive reasoning. Identify errors in mathematical explanations. *To help you understand the difference between inductive and deductive reasoning… Inductive reasoning is from a specific case or cases and deriving a general rule. It draws inferences from observations in order to make generalizations. Inference can be done in four stages: o Observation: collect facts, without bias. o Analysis: classify the facts, identifying patterns o of regularity. o Inference: From the patterns, infer generalizations about the relations between the facts. o Confirmation: Testing the inference through further observation. Inductive reasoning strategies o Derive a general rule in an accepted area and then apply the rule in the area where you want the situation or person to behave. o Give lots of detail, then explain what it all means. o Talk about the benefits of the parts and only get to the overall benefits later. o Take what has happened and give a plausible explanation for why it has happened. Deductive reasoning starts with a general case and deduces specific instances. Deductive reasoning starts with an assumed hypothesis or theory. Deductive reasoning assumes that the basic law from which an argument is made is applicable in all cases. This can let you take a rule and apply it perhaps where it was not really meant to be applied. Deductive conclusions can be valid or invalid. 11 Practice Test Samples 1. Look carefully at the following figures. Then, use inductive reasoning to make a conjecture about the next figure in the pattern. To solve: look carefully at the pattern; you should came up with this figure: 2. Look at the patterns below. Draw the next figure. To solve: The trick is to see that one dot is always placed between and above two dots. Also, the next figure always has one more dot at the very bottom row keeping this in mind; your next figure should look like this: 3. Write the next three numbers in the sequence 7, -7, 14, -42, 168... To solve: Since we are bouncing back and forth between positive and negative numbers, a pattern in addition doesn't look promising. Let's check out multiplication. At first glance, I would say that a negative number is probably what we are looking for here, since it does alternate signs. It doesn't appear to be the same number each time, because 7 times -1 is -7, but -7 times -2 = 14. It looks like we have 7 (-1), -7 (-2), 14 (-3), -42 (-4), 168, ... Aha, we have a pattern in multiplication - we are multiplying by the next negative integer. Since 168(-5) = -840, -840(-6) = 5040, 5040(-7) = -35280. Final Answer: -840, 5040, -35280 12 4. Write the next three numbers in the sequence 3, 5, 9, 15, 23… To solve: see if there is some pattern in addition…we are not adding the same number each time to get to the next number. But, make note of what you do add to each number 3 +2, 5 +4, 9+6, 15 +8, 23, see the pattern? You are always adding the next even number. Since 23+10 = 33, 33 +12 = 45, 45 +14 = 59. Final Answer: 33, 45, 59 5. Use the statements #1-5 below to answer the question that follows: 1. All people wearing hats have blonde hair. 2. Some of the people have red hair. 3. All people who have blonde hair like hamburgers. 4. People who have red hair like pizza. 5. Keith has blonde hair. Which of the following statements MUST be true? a. Keith likes hamburgers. b. Keith has red hair. c. Keith likes pizza. d. Keith is wearing a hat. To solve: with deductive reasoning, you need to be a 100% sure. Since statement 3 says that ALL people who have blonde hair like hamburgers and Keith has blonde hair, then statement a, Keith likes hamburgers, is a 100% guarantee. 6. Jerry, Kevin, Todd and Mark all live on the first floor of an apartment complex. One is a manager, one is a computer programmer, one is a singer, and the other is a teacher. Use the statements below to answer the question Which is the manager? A. Jerry and Todd eat lunch with the singer. B. Kevin and Mark carpool with the manager. C. Todd watches CSI with the manger and the singer. To solve: You can use a process of elimination on this problem. Statement A, Jerry and Todd eat lunch with the singer, doesn’t let us definitively eliminate anyone from being the manager. However, statement B, Kevin and Mark carpool with the manager, eliminates Kevin and Mark from being the manager. And statement C, Todd watches CSI with the manger and the singer, eliminates Todd. The only one that could be (100%, without a doubt) the manager is Jerry. 13 7. Four friends - Suzy, John, Sally, and Tom - each has his or her own hobby. One collect coins, one sews, one cooks, and one plays in a band, not necessarily in that order. Use the statements below to answer: Who is the friend that likes to sew? A. Suzy and John always eat lunch with the friend that plays in the band. B. Sally and Tom carpool with the one who likes to sew. C. John and the friend that likes to cook visited the one who likes to sew. To solve: use a process of elimination on this problem. Statement A, Suzy and John always eat lunch with the friend that plays in the band, doesn't let us definitively eliminate anyone from being the one who likes to sew. However, statement B, Sally and Tom carpool with the one who likes to sew, eliminates Sally and Tom from being the one who likes to sew. Statement C, John and the friend that likes to cook visited the one who likes to sew, eliminates John. The only one that could be (100%, without a doubt) the one who likes to sew is Suzy. 8. Write the next five figures in the pattern. To solve: It looks like several things change throughout this sequence. It starts with one line and then one circle then it has two lines and two circles and then three lines. So, it is alternating between lines and circles and each time it alternates it adds one more of that figure. So the next five figures would be: Note that we stopped at the fifth one, if we would have continued; there would be a total of four lines that follow the three circles. 14 Standard 0016 Solve problems involving algebra and geometry: The following are examples of content that may be covered under this standard. Graph numbers or number relationships. Find the value of the unknown in a given one-variable equation. Express one variable in terms of a second variable in two-variable equation. Solve problems involving lines and angles. Solve problems involving two-and three-dimensional geometric figures (e.g. perimeter and area problems, volume and surface area problems). Practice Test Samples 1. Graph the following ordered pairs of numbers in the coordinate plane: (4, -6), (-3, 5) To solve: Recall: each set of parentheses represents the location of a point in the x-y (coordinate) plane. The first number represents a location to the right or left of the origin (0,0). The second number represents a location above or below the origin. Start at the origin and move accordingly. 2. Find the roots of y = 3x² − 5x – 2 and graph. To solve: Recognize this as a second degree equation. Anticipate a parabola-shaped graph. Generate a table of values by selecting any Real number for an “x” value, substitute it in for x in the equation, and calculate the corresponding value for y. After finding several sets of values, locate the corresponding points on the graph. Sketch the graph, by drawing a smooth curve passing through the points. Remember to select both positive and negative values for x, and using x = 0 is always helpful. **Another way to solve the problem is to factor the equation like this: 3x² − 5x − 2 = (3x + 1)(x − 2). Then set each half equal to 0 to find the roots…if 3x + 1 = 0 then one root is −1/3, if x – 2 = 0 then the other root is 2. The roots are -1/3 and 2 15 3. Solve for x in 4x – 28 = 2x + 14 To solve: Recognize this as a first degree equation; Anticipate no more than 1 solution; Isolate the variable on one side of the equation, combine like terms, and simplify. 4x – 28 = 2x + 14…add 28 to each side and get 4x = 2x + 42, then subtract 2x from each side and get 2x = 42, finally divide each side by 2 and get x = 21 4. Solve for x in x² + 4x – 28 = -7 To solve: Recognize this as a second degree equation; Anticipate the possibility of 2 solutions; Get one side of the equation equal to zero, factor, set each factor to zero and solve for the variable as in the Strategy for a above. or: x2 + 4x – 28 = -7…add 7 to each side and get x2 + 4x – 21, now think you two numbers that when multiplied give you -21 and when added give you 4..yes the two numbers are -3 and 7 so the new expression is (x-3)(x+7), the final set to solve for x is to set each side equal to 0… if x – 3 = 0 then x= 3 and if x + 7 = 0 the x = -7 so the final answer is x = 3 and -7 5. In the diagram below, assume that line l is parallel to line m. The two parallel lines are intersected by a transversal line p One angle measure is given in the diagram. Find the measures of the other angles. To solve: Recall that angles such as that labeled 125⁰ is a vertical angle with c, so it is 125⁰ too. Angle B is supplementary with the angle labeled 125⁰, so together they are 180⁰…(180-125 = 55) Recall that angles such as c and d are called alternate interior angles and will have the same measure. Angles such a and e are corresponding angles and will have the same measure. Remember, this reasoning only works if the lines are parallel. angles a, b, e, f = 55o angles c, d, g = 125o 16 6. A certain rectangle has an area of 28 m². Its length is 2.5 m. Find it width. To solve: Quickly sketch a diagram and label everything you know. Recall that the area of a rectangle is length × width, so if only one dimension is known, divide the area by that dimension to get the other. (hint: sometimes we think of length as being longer than width. In math it’s not a problem if the length turns out to be shorter than the width.) 7. By attaching the edges of the circular sector of paper, a right circular cone is formed. What is the lateral surface area of the cone shown below? (This question adapted from ICTS Study Guide, 2006) A. in² C. in² B. in² D. in² To solve: Recall that a sector is part of a circle with 2 sides that are radii of the circle and that lateral surface area is the area of the region. The formula for area of a circle: A = r². Since our sector is 40⁰ and a full circular area would be 360⁰, we have = of a circle. A full circle with a radius of 8 inches would have an area of ·8², which is 64 in². To answer the question, we would need to find 1/9 of this amount. Notice that this question does not ask for the total surface area. If it were needed, we would have been told on the area of the base of the cone too. The answer is D. in² 17 Standard 0017 Understand concepts and procedures related to data analysis and statistics. The following are examples of content that may be covered under this standard. Interpret information form tables, line graphs, bar graphs, histograms, pictograms, and pie charts. Recognize appropriate representations of various data in graphic form (discrete and continuous). Demonstrate an understanding of fundamental statistical concepts (e.g., mean, correlation, standard deviation). Interpret graphic and nongraphic representations of frequency distributions, percentiles, central tendency, variability, and correlation. Practice Test Samples 1. The scores of a group of graduate students are shown below in a graph. What is the range of the data? a) b) c) d) e) 5 100 40-49 40-50 0-100 To solve: Range is the difference between the largest and smallest data (100-0)= 100. 18 2. The graphs show the growth in the use of cellular phones. Why do they look different? http://www.glencoe.com/sec/math/prealg/prealg05/extra_examples/chapter12/lesson12_5.p df a) b) c) d) The data is different. The vertical scale is different. The horizontal scale is different. The number of people is increasing To solve: Check the scale on each… b) The vertical scale is different 3. According to the graph, approximately how many students needed 20 minutes to locate suitable library materials? http://www.galeschools.com/research_tools/src/create_visual.htm#linegaph2 a) b) c) d) 10 20 40 50 To solve: From 20 on the horizontal axis, go vertically until you hit the graph. Then go horizontally to the vertical axis and read the value… d) 50 19 Use the circle graph to solve problems 4-5. The circle graph shows the percent of the total population of a small town is 9,520 living in the given types of housing. 4. Find the number of people who live in condos and apartments to the nearest whole number. a) 4,094 b) 5,712 c) 5,710 d) 6,000 To solve: Add the percents for condos and apartments (17% + 43% = 60%) and multiply by total population (9,520 x 60%) = b) 5715 5. Find the number of degrees in the sector of the circle that represents people who live in townhomes. Round your answer to the nearest degree. a) 6 b) 21.6 c) 21 d) 216 To solve: Multiply the number of degrees (360°) in a circle by .06 (360° x .06) = b) 21.6 20 Use the stem and leaf plot to solve problems 6 - 7 The weights (in pounds) of a group of high school students are listed below in a stem-and-leaf plot. 6. What is the median of the data? To solve: The median is the data point which is in the center of the data AFTER it is arranged from low to high. Median = 130 7. What is the mode of the data? To solve: The mode is the data point which occurs most frequently. Mode = 142 8. The following scatterplot displays temperatures recorded in various locations at different latitudes on a particular summer day. What can be said about the correlation? a) Negative correlation b) Positive correlation c) No correlation d) no correlation To solve: Since the data is approximately linear, consider the relationship between the values…a) Negative correlation 21 Standard 0018 Solve applied problems using a combination of mathematical skills (including word problems involving one and two variables). The following are examples of content that may be covered under this standard. Apply combinations of mathematical skills to solve a series of related problems. Identify an equation to solve word problems involving one and two variables. Apply number concepts and geometric principles to solve practical problems. Apply statistical principles to analyze patterns and trends in data. Practice Test Samples 1. Solve for x and y in 3x + 2y = 3 and x = 3y – 10 To solve: Replace x in the first equation with its equivalent, (3y - 10) from the second equation. 3x + 2y = 3 3(3y - 10) + 2y = 3 9y - 30 + 2y = 3 11y = 33 y=3 Top equation. Replaced x with (3y - 10). Multiplied out. Simplified. Divide each side by 11 to get answer. Now that y has a value, you can plug that value in either equation and find a value for x. Because the second equation has already been solved for x, it will be easier to plug 3 in for y in that equation. x = 3(3) - 10 x = 9 - 10 x = -1 The solution is the ordered pair (-1,3). 2. Jess has a garden that is 5.2 feet by 3.8 feet. If 3 flowers fit into each square foot, about how many flowers will she be able to plant in the area? To solve: 5.2 → 5 feet, 3.8 → 4, area is L x W so, 5 x 4 = 20 square feet and if 3 flowers can go in each square foot, 20 x 3 = 60 flowers 22 3. Solve for x in 3x = 2x + 1 To solve: By subtracting 2x from each side, the variables are all on the same side of the equation. By combining like terms, the variable is isolated, and the equation is simplified and solved. X = 1 4. Find the value of c. To solve: a2 + b2 = c2 Write the Pythagorean Theorem Then plug in 52 + 122 = c2 169 = c2 Solve for c c = 13 5. A rectangle has a length that is one more than 2 times its width. The perimeter of the rectangle is 26 feet. Find the length of the rectangle and the area of the rectangle. To solve: First draw a picture: X 2X + 1 for the length : the width can be represented as X and then the length would be 2X + 1, next add all the 4 sides together (X) + (X) +(2X + 1) + (2X + 1) = 6X + 2, now set that = to the perimeter of 26, 6X + 2 = 26, simplify by subtracting two from each side 6X = 24, then divide each side by 6 and get X = 4, the width is 4 and then the length is 9 feet. Draw another picture: 4 feet 9 feet for the area: Now that you know the width (4) and the length (9) multiple them together to get the area. Area = Length x Width 9 x 4 = 36, the area is 36 square feet. 23 6. The 24 students in Mr. Madison’s class earn stickers for good work and behavior. During the first week the least amount of stickers earned by a student was 10 and the most amount of stickers earned by a student was 30. Which of the following could be a reasonable estimate of the total number of stickers Mr. Madison gave out during the first week of school? a) 240 b) 480 c) 720 d) 960 To solve: find the average between the least amount and the greatest amount of stickers (10 + 30) ÷ 2 = 20. Then multiple the 20 stickers by the 24 students to get b) 480 5. A rectangular solid measures 3 feet by 4 feet. The volume of the box is 48 cubic feet. If y represents the height of the box, then which equation represents the volume? a) b) c) d) 3 + 4y = 48 7y = 48 12y = 48 12 + y = 48 To solve: remember Volume = height x length x width … c)12y = 48 6. In the figure to the right, AB = BC and the measure of What is the measure of B? a) b) c) d) C is 40 . 40 80 100 140 To solve: remember if AB = BC then angles A and C are equal. And also remember that the sum of interior angles in any triangle is 180 o so, 180o – (40o + 40o) = c) 100o