# Core Academic Skills for Educators: Mathematics

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1 The Praxis Study Companion Core Academic Skills for Educators: Mathematics

3 Welcome to the Praxis Study Companion What should I expect when taking the test on computer? When taking the test on computer, you can expect to be asked to provide proper identification at the test center. Once admitted, you will be given the opportunity to learn how the computer interface works (how to answer questions, how to skip questions, how to go back to questions you skipped, etc.) before the testing time begins. Watch the What to Expect on Test Day video to see what the experience is like. Where and when are the Praxis tests offered? You can select the test center that is most convenient for you. The Praxis tests are administered through an international network of test centers, which includes Prometric Testing Centers, some universities, and other locations throughout the world. Testing schedules may differ, so see the Praxis Web site for more detailed test registration information at www. ets.org/praxis/register. The Praxis Study Companion 3

7 Step 1: Learn About Your Test II. Algebra and Functions A. Seeing Structure in Expressions 1. Apply understanding of arithmetic to algebraic expressions 2. Solve real-life and mathematical problems using numerical and algebraic expressions 3. Use properties of operations to generate equivalent expressions B. Reasoning with Equations and Inequalities 1. Understand the connections between proportional relationships, lines, and linear equations 2. Understand solving equations as a process of reasoning and explain the reasoning 3. Reason about and solve one-variable equations and inequalities 4. Solve equations and inequalities in one variable 5. Analyze and solve linear equations and pairs of simultaneous linear equations 6. Represent and solve equations and inequalities graphically C. Functions 1. Interpreting functions 2. Building functions III. IV. Geometry A. Congruence and Similarity 1. Draw, construct, and describe geometrical figures and describe the relationships between them 2. Experiment with transformations in the plane B. Right Triangles 1. Understand and apply the Pythagorean theorem C. Circles 1. Understand and apply theorems about circles D. Geometric Measurement and Dimension 1. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume 2. Explain volume formulas and use them to solve problems E. Modeling with Geometry 1. Apply geometric concepts in modeling situations Statistics and Probability A. Basic Statistics and Probability 1. Develop understanding of statistical variability 2. Summarize and describe distributions 3. Use random sampling to draw inferences about a population 4. Investigate chance processes and develop, use, and evaluate probability models 5. Investigate patterns of association in bivariate data B. Interpreting Categorical and Quantitative Data 1. Summarize, represent, and interpret data on a single count or measurement variable 2. Interpret linear models C. Making Inferences and Justifying Conclusions 1. Understand and evaluate random processes underlying statistical experiments D. Using Probability to Make Decisions 1. Use probability to evaluate outcomes of decisions The Praxis Study Companion 7

13 Step 3: Practice with Sample Test Questions The answer, therefore, is choice four. The Praxis Study Companion 13

14 Step 3: Practice with Sample Test Questions 2. On the computer screen you will see the following: Since C is the center of the circle, QC and RC are both radii of the circle and therefore have the same length. The answer, therefore, is choice one. The Praxis Study Companion 14

15 Step 3: Practice with Sample Test Questions 3. On the computer screen you will see the following. Solution Strategy 1: Note that all of the answer choices are linear equations. Therefore, the input and output coordinate pairs in the table represent points on a line. With any two coordinate pairs from the table, the slope of the line can be found using Slope Difference in y-values Difference in x-values For example, if (6, 23) and (2, 11) are selected, the slope is Slope = = = The slope of the line is 3. The coordinate pairs in the table indicate that the first pair, (0, 5) is the point at which the line crosses the y-axis. Therefore, the y-intercept is 5. Use the slope-intercept equation, y = mx + b, where m is the slope and b is the y-intercept. When the slope and y-intercept of the line are known, the equation of the line can be found by using substitution. Therefore, the equation y = 3x + 5 expresses the relationship between x and y in the table. The answer, therefore, is choice three. The Praxis Study Companion 15

16 Step 3: Practice with Sample Test Questions Solution Strategy 2: The correct equation must hold when each of the five pairs of values from the table is substituted for x and y. The equation y = x + 5 (1st choice) holds for the pairs of values x = 0, y = 5, but not for the pairs of values x = 2, y = 11. The equations y = x + 6, y = 4x 1 and y = 4x 5 (2nd, 4th, and 5th choices) do not hold for the pairs of values x = 0, y = 5. The equation y = 3x + 5 (3rd choice) holds for all five pairs of values given: if x = 0, then y = 3(0) + 5 = 5, if x = 2, then y = 3(2) + 5 = 11, if x = 6, then y = 3(6) + 5 = 23, and so forth. The answer, therefore, is choice three. The Praxis Study Companion 16

17 Step 3: Practice with Sample Test Questions 4. On the computer screen you will see the following: In the table provided, there are 48 individuals in the group, 18 of whom are women. The probability that the individual selected will be a woman is 18 out of 48, or choice four =. The answer, therefore, is 48 8 The Praxis Study Companion 17

18 Step 3: Practice with Sample Test Questions 5. On the computer screen you will see the following: When dividing a number expressed as a decimal by 10, the decimal point is moved one place to the left, so divided by 10 is The tens place is the second place to the left of the decimal point. In the number , that is the digit Tens Ones Tenths Hundredths Thousandths The Praxis Study Companion 18

19 Step 3: Practice with Sample Test Questions The answer, therefore, is choice four. The Praxis Study Companion 19

20 Step 3: Practice with Sample Test Questions 6. On the computer screen you will see the following: To subtract rational expressions, each rational expression must first be expressed with a common denominator. In this case, a common denominator is 2xy. Multiply each rational expression by an expression equal to 1 so that each rational expression will have the same common denominator 2xy as follows y 2x y 2x y 3y 2x 2xy 2xy 3y 2x 2xy 1 2x y 2x The Praxis Study Companion 20

21 Step 3: Practice with Sample Test Questions The answer, therefore, is choice three. The Praxis Study Companion 21

22 Step 3: Practice with Sample Test Questions 7. On the computer screen you will see the following: For the average of a set of data, the sum of the data entries is equal to the product of the average and the number of entries. Since Average = Sum of data entries Total number of entries, Sum of data entries = Average Total number of entries. Since the average of the absences for the first 4 months is 14, the number of absences for the first 4 months is 14 4= 56. The number of absences in the fourth month is the number of absences in the first 4 months minus the number of absences in the first 3 months, or 56 ( ) = 11. The Praxis Study Companion 22

23 Step 3: Practice with Sample Test Questions The answer, therefore, is choice three. The Praxis Study Companion 23

24 Step 3: Practice with Sample Test Questions 8. On the computer screen you will see the following: The price on Monday was x dollars. A reduction of 25% means that the price on Tuesday is 75% of the price on Monday, or (0.75)x dollars. The price on Tuesday is then reduced by 40%, so the price on Wednesday is 60% of 0.75x dollars, or (0.60)(0.75x) dollars. The answer, therefore, is choice five. The Praxis Study Companion 24

25 Step 3: Practice with Sample Test Questions 9. On the computer screen you will see the following: Since y = 2, substituting the value 2 for y in the equation 4x = y gives the following equations. 4x = (2) 4x = x = 32 Dividing both sides of the equation 4x = 32 by 4 gives the result x = 8. The Praxis Study Companion 25

26 Step 3: Practice with Sample Test Questions The answer, therefore, is choice five. The Praxis Study Companion 26

27 Step 3: Practice with Sample Test Questions 10. On the computer screen you will see the following: A horizontal translation changes the x-coordinate, and a vertical translation changes the y-coordinate. Translating 3 units to the right adds 3 to the x-coordinate. Translating 2 units down subtracts 2 from the y-coordinate. Since the coordinates of vertex A are ( 3, 1), the coordinates corresponding to vertex A after translation are ( 3 + 3, 1 2), or (0, 1). The Praxis Study Companion 27

28 Step 3: Practice with Sample Test Questions The answer, therefore, is choice four. The Praxis Study Companion 28

29 Step 3: Practice with Sample Test Questions 11. On the computer screen you will see the following: The numbers to be placed in the blank spaces must be both greater than 1 4 and less than 3 8. The first pair of numbers listed is 1 2, 0. Although 0 is both greater than 1 4 and less than 3 8, 1 2 is less than 1 4, so this pair is not the correct answer. The second pair of numbers listed is 1 8 and 1 3, both of which are greater than 1 4 and less than 3, so this pair is the correct answer. At least 8 one of the numbers in each of the other pairs is either less than 1 4 or greater than 3 8. The Praxis Study Companion 29

30 Step 3: Practice with Sample Test Questions The answer, therefore, is choice two. The Praxis Study Companion 30

31 Step 3: Practice with Sample Test Questions 12. On the computer screen you will see the following: To answer this question, compute the dollar amount for each of the five categories and compare each dollar amount to \$1,000. Dollar amount for food = \$3,500 x 40% = \$1,400 Dollar amount for lodging = \$3,500 x 35% = \$1,225 Dollar amount for other = \$3,500 x 3% = \$105 Dollar amount for entertainment = \$3,500 x 12% = \$420 Dollar amount for transportation = \$3,500 x 10%= \$350 Of the five dollar amounts, only two are greater than \$1,000. An alternative solution method is to express \$1,000 as a percent of the total budget and determine how many of the percent values given in the circle graph are greater. 1, 000 That is, \$1,000 is 100% 28. 6% of \$3,500. Of the five percent values in the circle graph, 3, 500 only two percent values are greater than 28.6%. The answer is indicated by choice 2. The Praxis Study Companion 31

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