Common Core Coach Mathematics 8 First Edition
Contents Domain 1 The Number System...4 Lesson 1 Understanding Rational and Irrational Numbers...6 Lesson 2 Estimating the Value of Irrational Expressions... 12 Common Core State Standards 8.NS.1 8.NS.2 Domain 1 Review... 16 Domain 2 Expressions and Equations...20 Lesson 3 Applying Properties of Exponents...22 Lesson 4 Understanding Square and Cube Roots...26 Lesson 5 Scientific Notation...30 Lesson 6 Using Scientific Notation...34 8.EE.1 8.EE.2 8.EE.3 8.EE.4 Lesson 7 Lesson 8 Representing and Interpreting Proportional Relationships...40 Relating Slope and y-intercept to Linear Equations...46 8.EE.5 8.EE.6 Lesson 9 Solving Linear Equations in One Variable....52 8.EE.7.a, 8.EE.7.b Lesson 10 Lesson 11 Solving Systems of Two Linear Equations Graphically...56 Solving Systems of Two Linear Equations Algebraically...62 8.EE.8.a 8.EE.8.b Lesson 12 Problem Solving: Using Systems of Equations...... 68 8.EE.8.c Domain 2 Review...72 Domain 3 Functions...76 Lesson 13 Introducing Functions....78 8.F.1 Lesson 14 Comparing Functions Represented in Different Ways...82 Lesson 15 Linear and Nonlinear Functions....86 Lesson 16 Using Functions to Model Relationships....90 Lesson 17 Describing Functional Relationships from Graphs....94 Domain 3 Review...98 8.F.2 8.F.3 8.F.4 8.F.5 2 Problem Solving Performance Task
Common Core State Standards Domain 4 Geometry...102 Lesson 18 Lesson 19 Properties of Rotations, Reflections, and Translations...104 Understanding Congruence of Two-Dimensional Figures (Using Rigid Motions).... 110 8.G.1.a, 8.G.1.b, 8.G.1.c 8.G.2 Lesson 20 Rigid Motion on the Coordinate Plane.... 114 Lesson 21 Dilations on the Coordinate Plane....120 8.G.3 8.G.3 Lesson 22 Understanding Similarity of Two-Dimensional Figures (Using Transformations)....124 8.G.4 Lesson 23 Extending Understanding of Angle Relationships....128 Lesson 24 Angles in Triangles...132 Lesson 25 Explaining the Pythagorean Theorem....136 8.G.5 8.G.5 8.G.6 Lesson 26 Lesson 27 Applying the Pythagorean Theorem in Two and Three Dimensions...142 Applying the Pythagorean Theorem on the Coordinate Plane...146 8.G.7 8.G.8 Lesson 28 Problem Solving: Volume....150 8.G.9 Domain 4 Review...156 Domain 5 Statistics and Probability...160 Lesson 29 Constructing and Interpreting Scatter Plots....162 8.SP.1 Lesson 30 Modeling Relationships in Scatter Plots with Straight Lines...166 8.SP.2 Lesson 31 Using Linear Models to Interpret Data....170 8.SP.3 Lesson 32 Investigating Patterns of Association in Categorical Data... 174 Domain 5 Review...178 Glossary...182 Math Tools...185 8.SP.4 3
1 LESSON Understanding Rational and Irrational Numbers UNDERSTAND All numbers can be written with a decimal point. For example, you can rewrite 2 and 5 with decimal points without changing their values. 2 2.0 or 2.00 or 2.000, and so on 5 5.0 or 5.00 or 5.000, and so on You can expand the decimal places of a number that already has digits to the right of the decimal point. 2.2 2.20 or 2.200, and so on 5. _ 1 5.1 _ 1 or 5.11 _ 1, and so on Each of the numbers above has a decimal expansion that ends either in zeros or in a repeating digit. Any number with a decimal expansion that ends in 0s or in repeating decimal digits is a rational number. UNDERSTAND Some numbers, like the ones below, do not end in 0s or in repeating decimal digits. The three dots, called an ellipsis, mean that digits continue, but not in a repeating pattern. 2 1.41421 5 2.23606 10 3.16227 Any number with a decimal expansion that does not end in 0s or in repeating decimal digits is an irrational number. You have previously worked with a very important number, pi, which is represented by the symbol. 3.14159 The decimal expansion of does not end in 0s or in repeating decimal digits. It is an irrational number. Every real number belongs either to the set of rational numbers or to the set of irrational numbers. Set of all Real Numbers Rational Numbers 2 2.2 5 5.1 Irrational Numbers 2 5 6 Domain 1: The Number System
Connect Is 0.07 rational or irrational? Examine the digits to the right of the decimal point. 0.07 0.070 or 0.0700, and so on 0.07 is rational because its decimal expansion ends in 0s. Is 3. 45 rational or irrational? Examine the digits to the right of the decimal point. 3. 45 3.45 45 or 3.4545 45, and so on 3. 45 is rational because its decimal expansion repeats. Is 10.049846 rational or irrational? Examine the digits to the right of the decimal point. 10.049846 10.049846 is irrational because its decimal expansion does not end in 0s or in repeating decimal digits. Is 8 rational or irrational? Use a calculator to find the decimal form. 8 2.828427125 Examine the digits to the right of the decimal point. 8 is irrational because its decimal expansion does not end in 0s or in repeating decimal digits. DISCUSS How could you show that 4.95271 is a rational number using methods shown above? Lesson 1: Understanding Rational and Irrational Numbers 7
EXAMPLE A Write each of the following rational numbers in fraction form. 3, 0.9, 3.03 1 Express each number as a fraction of the form a b where a and b are integers and b 0. Use the place value of the rightmost digit to determine the value of the denominator. The rightmost digit in 3 is in the ones place, so 3 3 1. The rightmost digit in 0.9 is in the tenths place, so 0.9 9 10. The rightmost digit in 3.03 is in the hundredths place, so 3.03 303 100. 2 3 3 1 0.9 9 10 3.03 303 100 EXAMPLE B Convert the rational number 0. 3 to a fraction. 1 Use algebra. Set the number, 0. 3, equal to n. n 0. 3 There is one repeating digit, so multiply n by the first power of 10, or 10. 10n 3. 3 2 Subtract the number, n, from 10n. 10n 3. 3 n 0. 3 9n 3 3 Solve the equation and simplify the result. 9n 9 3 9 n 3 9 1 3 0. 3 1 3 CHECK How can you work backward from 1 3 to check the answer? 8 Domain 1: The Number System
1 EXAMPLE C Convert 0. 45 to a fraction. Use algebra. Set the number, 0. 45, equal to n. n 0. 45 There are two repeating digits, so multiply n by the second power of 10, or 100. 100n 45. 45 2 Subtract the number, n, from 100n. 100n 45. 45 n 0. 45 99n 45 3 Solve the equation and simplify the result. 99n 45 99 99 n 45 99 5 11 0. 45 5 11 DISCUSS What steps could you use to express the decimal 0.8 3 as a fraction? Lesson 1: Understanding Rational and Irrational Numbers 9
Practice Identify whether the number is rational or irrational. Then explain why it is rational or irrational. 1. 101 2. 8 17 3. 21.192 4. 7 5. 9 6. HINT If a square root has an integer value, is it rational? 7. 50 8. 39. 81 Write three equivalent decimal forms for each number. 9. 19 10. 21.5 11. 44.045 12. 1. _ 1 REMEMBER Adding zeros to the end of a decimal does not change its value. 10 Domain 1: The Number System
Complete each sentence. 13. 11.3 is rational because. 14. 19 is irrational because. 15. 0.08 3 is rational because. 16. 2.1371938 is irrational because. Convert the repeating decimal to a fraction. 17. 0. 6 18. 1. _ 1 19. 4. 4 20. 9. 09 21. 2. 90 22. 4. 54 Choose the best answer. 23. Which is an irrational number? A. 3.3 4 B. 1 C. 20 D. 11.2092 24. Which number is not equivalent to 13.02? A. 13.002 B. 13.020 C. 13.0200 D. 13.020000 Solve. 25. WRITE MATH Convert 3.1 6 to a fraction. Explain your strategy or show the steps you used to convert the number. 26. DESCRIBE Describe two real-life applications of irrational numbers. Lesson 1: Understanding Rational and Irrational Numbers 11
Contents Domain Assessment The Number System...................................... 4 Domain Assessment Expressions and Equations................................ 12 Domain Assessment Functions............................................. 22 Domain Assessment Geometry............................................ 34 Domain Assessment Statistics and Probability................................. 44 Summative Assessment.................................................... 58 3
Domain Assessment The Number System 1. What is the decimal expansion of 1 8? A. 0.1 B. 0.125 C. 0.25 D. 0. 3 4. Which decimal below is the best approximation of the irrational number p? A. 1.41 B. 1.62 C. 2.72 D. 3.14 2. Which of the following is true about the decimal expansion of 1 11? A. ends in 625 B. 3 repeating C. 09 repeating D. 27 repeating 5. Which of the following sets contains only irrational numbers? A. p, 2, 4.238905 B. 1 7, 3.14, 5 C. 2 1 4, 5, 7.717 D. 0. _ 1, 0. 09, 0.1 6 3. Convert the following repeating decimal into a fraction: 0.1111111 A. 1 5 B. 1 7 C. D. 1 8 1 9 Go On 4
6. The figure below is a right triangle. 1 1 Which is the best approximation of the hypotenuse of the triangle? A. 1.21 B. 1.41 C. 1.73 D. 2.24 2 8. Points H, I, J, and K are plotted on the number line below. 0 1 2 H I J K 3 4 5 6 7 8 Which point on the number line represents 7? A. H B. I C. J D. K The Number System 7. Between which two whole numbers is 11? A. 1 and 2 B. 2 and 3 C. 3 and 4 D. 4 and 5 9. Which of the following fractions does not end with a decimal expansion of zeros? A. B. C. D. 1 3 1 4 1 5 1 8 Go On 5
10. A flagpole measures 25 1 11 feet tall. Which repeating decimal represents this height? A. 25.08 3 feet B. 25. 09 feet C. 25. _ 1 feet D. 25.1 6 feet 12. Which of the following sets contains only rational numbers? A. 0. 3, 0.1 6, 3.14 B. 1 9, p, 4 1 6 C. 0. 2, 3 2, 7 D. 0. 8, 0. 87, 0.16075423 11. Which is the best approximation of p 1 9 3? A. 1.27 B. 2.09 C. 3.14 D. 4.05 13. In an art class, Jorge constructs a 2 feet by 4 feet frame for a painting he just finished. He uses the Pythagorean theorem to find the diagonal of the frame, which is 20 feet. He then concludes that the diagonal must be at least 5 feet. Is he correct in his conclusion? A. Yes, because 2 1 4 6. B. Yes, because 20 < 5.48. C. No, because 4 2 5 16 and 5 2 5 25, so 20 must be between 4 and 5. D. No, because 20 5 2 2 1 3 2, so 20 must be between 2 and 3. Go On 6
14. The formula for the circumference of a circle is pd, where d is the diameter. Kim measures the diameter of each of four pools in his neighborhood and uses this formula to find their circumferences, in feet. They are plotted on the number line below. P Q R S 15. Which irrational number below is approximately equal to 2? A. 3 B. 5 C. 7 D. 8 The Number System 0 10 20 30 40 50 Which point best represents the pool with a diameter of 14 feet? A. P B. Q C. R D. S Go On 7
16. This baseball season, Barry gets two hits for every 16 at bats. This statement represents his batting average. A. Write his batting average as a fraction and as a decimal. B. Does Barry s batting average in decimal form end in zeros? 17. Scott cuts his birthday cake into nine equal pieces and eats two of them. A. Use long division to compute in decimal form how much of the cake he ate. (Ensure your answer includes the first four digits of the decimal.) B. Does the decimal appear to repeat indefinitely or end in zeros? Go On 8
18. A middle school with 375 students has 125 students in the eighth grade. Irene says that the eighth-grade class makes up 0. _ 1 of the school. Yolanda says that the eighth-grade class makes up 0. 3 of the school. Fernando says that the eighth-grade class makes up 0. 6 of the school. A. Convert each student s decimal into a fraction. Irene: Yolanda: The Number System Fernando: B. Which student s decimal is correct? C. Which student s decimal represents the population of the rest of the middle school instead of the eighth-grade class? Go On 9
19. Omar has been practicing swimming in his public pool for a swimming race. The farthest he can swim without resting is the diagonal of the pool, which is 200 meters. The three races available are the 10-meter, the 15-meter, and the 20-meter swim. A. Between which two races is the length of the pool diagonal that Omar can swim? B. In which race should Omar compete if he cannot rest during the race? C. If, in the future, Omar can swim the length of a pool diagonal that measures 400 meters, what is the longest race he can swim without resting? Go On 10
20. As part of a homework assignment, Larry uses a ruler to measure the diameter of various circular objects in his home. He then uses the diameter to find their circumferences. Below are his three circumference measurements. soup can lid: 2.5p inches dinner plate: 5p inches clock: 7.75p inches The Number System A. Plot these measurements on the line below, using S for soup can lid, D for dinner plate, and C for clock. 0 5 10 15 20 25 B. The circumference of Larry s wristwatch is 1.5p. Would this measurement be plotted to the left or right of the other household objects on the number line? C. If the circumference of Larry s Frisbee is exactly 18 inches, which household object has the closest measurement? STOP 11
Contents Instructional Overview.......................................... 4 Common Core State Standards Correlation Chart................. 12 Domain 1 The Number System.......................... 16 Lesson 1 Understanding Rational and Irrational Numbers.......... 18 Lesson 2 Estimating the Value of Irrational Expressions............ 20 Common Core State Standards 8.NS.1 8.NS.2 Domain 2 Expressions and Equations................... 22 Lesson 3 Applying Properties of Exponents..................... 24 Lesson 4 Understanding Square and Cube Roots................. 26 Lesson 5 Scientific Notation.................................. 28 Lesson 6 Using Scientific Notation.......................... 30 Lesson 7 Representing and Interpreting Proportional Relationships.. 32 Lesson 8 Relating Slope and y-intercept to Linear Equations.... 34 Lesson 9 Solving Linear Equations in One Variable................ 36 Lesson 10 Solving Systems of Two Linear Equations Graphically...... 38 Lesson 11 Solving Systems of Two Linear Equations Algebraically.... 40 Lesson 12 Problem Solving: Using Systems of Equations........ 42 8.EE.1 8.EE.2 8.EE.3 8.EE.4 8.EE.5 8.EE.6 8.EE.7.a, 8.EE.7.b 8.EE.8.a 8.EE.8.b 8.EE.8.c Domain 3 Functions...................................... 44 Lesson 13 Introducing Functions............................... 46 Lesson 14 Comparing Functions Represented in Different Ways...... 48 Lesson 15 Linear and Nonlinear Functions........................ 50 8.F.1 8.F.2 8.F.3 Lesson 16 Using Functions to Model Relationships................. 52 Lesson 17 Describing Functional Relationships from Graphs......... 54 Domain 4 Geometry...................................... 56 Lesson 18 Properties of Rotations, Reflections, and Translations...... 58 Lesson 19 Understanding Congruence of Two-Dimensional Figures (Using Rigid Motions)................................ 60 Lesson 20 Rigid Motion on the Coordinate Plane............... 62 8.F.4 8.F.5 8.G.1.a, 8.G.1.b, 8.G.1.c 8.G.2 8.G.3 2 Problem Solving Performance Task
Common Core State Standards Lesson 21 Dilations on the Coordinate Plane...................... 64 8.G.3 Lesson 22 Understanding Similarity of Two-Dimensional Figures (Using Transformations).............................. 66 8.G.4 Lesson 23 Extending Understanding of Angle Relationships......... 68 Lesson 24 Angles in Triangles.................................. 70 Lesson 25 Explaining the Pythagorean Theorem.................. 72 8.G.5 8.G.5 8.G.6 Lesson 26 Lesson 27 Applying the Pythagorean Theorem in Two and Three Dimensions................................... 74 Applying the Pythagorean Theorem on the Coordinate Plane................................... 76 8.G.7 8.G.8 Lesson 28 Problem Solving: Volume......................... 78 8.G.9 Domain 5 Statistics and Probability..................... 80 Lesson 29 Constructing and Interpreting Scatter Plots.............. 82 8.SP.1 Lesson 30 Modeling Relationships in Scatter Plots with Straight Lines....................................... 84 8.SP.2 Lesson 31 Using Linear Models to Interpret Data.................. 86 Lesson 32 Investigating Patterns of Association in Categorical Data... 88 8.SP.3 8.SP.4 Answer Key................................................... 90 Math Tools................................................... 118 Appendix A: Fluency Practice.................................... A Appendix B: Standards for Mathematical Practice.................. B 3
LESSON 1 Understanding Rational and Irrational Numbers Learning Objectives Students will understand that rational numbers are numbers that have decimal expansions that terminate in 0s or eventually repeat and that other numbers are irrational numbers. Students will classify a number as rational or irrational and convert rational numbers to fraction form. Vocabulary irrational number a number that cannot be expressed as a terminating or repeating decimal; an irrational number cannot be represented as a b, where a and b are integers and b 0 rational number a number whose decimal form is a terminating or repeating decimal; a rational number can be represented as a b, where a and b are integers and b 0 real number a number with a location on a number line; real numbers are either rational or irrational Common Core State Standard 8.NS.1 Understand informally that every number has a decimal expansion; the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat. Know that other numbers are called irrational. Before the Lesson Provide students with an opportunity to review the relationship between the set of whole numbers and the set of integers. Ask: What is the opposite of 4? (24) How are opposites located on a number line? (They are the same distance but in different directions from 0.) Extend the discussion to include the rational numbers. Additionally, review how to convert common fractions, such as 3 4 or 2 5, to decimals; and common decimals, such as 0.6 or 0.25, to fractions. You might want to use Fluency Practice page A3 to help students review Solving Equations with Decimals: px 1 q 5 r. Understand Connect Review the definition of a rational number as a number that can be written as the ratio of two integers. Discuss why this definition is the same as a number with a decimal expansion that ends in 0s or in repeating decimal digits. Connect the definitions by considering 1 3, whose decimal expansion is 0. 3. Point out that all terminating decimals and all repeating decimals are rational. Check that students understand why all terminating decimals are rational and end in zeros. Ask: Why can you expand 4.283 to end in zero? (4.283 is equivalent to 4.2830. You can insert zeros to the right of the last digit in a decimal number without changing its value.) Extend the discussion to include irrational numbers. Point out that the square root of a non-perfect square number is always irrational. Ask: Why is 4 rational and not 5? ( 4 5 2.0 but 5 cannot be 18
represented with a decimal expansion that ends in 0.) Make sure students recognize that the ellipsis at the end of a decimal number means the digits continue but do not repeat in a pattern. To connect the concept to procedural understanding, have students refer back to the definitions of rational and irrational numbers. Emphasize the importance of using the definition to classify a real number as rational or irrational. Remind students to use a calculator to find decimal approximations of the square root of a non-perfect square number, such as 8. Ask: Why is it important to find the decimal form of 8 to classify the number as rational or irrational? (to verify that it does not have a decimal expansion that ends in 0s or in repeating decimal digits) DISCUSS MP6 MP8 Discuss with students how to use the given methods to verify that the number is rational. Answers may vary. Possible answer: 4.95271 5 4.952710 or 4.9527100, and so on. Because the decimal expansion ends in 0, the number 4.95271 is rational. Domain 1 Examples EXAMPLE A Remind students that all rational numbers can be expressed as fractions. Point out that some of these fractions are improper fractions. Discuss how to use place value to write each decimal number as a fraction. EXAMPLE B This example involves using algebra to write a repeating decimal as a fraction. Discuss why n and the original decimal are both multiplied by 10. Emphasize that the goal is to subtract so that the repeating decimal is eliminated from the equation. CHECK MP4 MP6 Discuss why writing the fraction as a decimal is working backward. If needed, review how to write a fraction as a decimal by computing 1 4 3. Divide the numerator by the denominator. EXAMPLE C In this example, a decimal with two repeating digits is converted to a fraction. Emphasize that the decimal is multiplied by 100 so that the repeating part of the decimal can be eliminated. Otherwise, the steps are the same as for converting a decimal with a single repeating digit. DISCUSS MP4 MP7 Discuss with students how to recognize the power of ten needed to multiply to convert a repeating decimal to a fraction. Have students explain the reasoning behind the order of the steps used to convert a repeating decimal to a fraction. Answers may vary. Possible answer: Let n 5 0.8 3. Then 10n 5 8. 3 and 100n 5 83. 3. Subtract 10n from 100n to get 90n. Subtract 8. 3 from 83. 3 to get 75. Set 90n equal to 75 and solve to get n 5 5 6. Practice As students are working, pay special attention to problems 23 and 24, which provide an opportunity for students to distinguish rational and irrational numbers from a list of choices. For answers, see page 90. Common Errors When converting a repeating decimal to a fraction, students may not multiply by the correct power of ten. When students make this error, discuss why the repeating decimal is multiplied by a power of ten. Point out that if they multiply by a power of ten that is too great, the solution will be harder to simplify. If the power of ten is not great enough, the repeating part of the decimal will not be subtracted. Ask: Why do you set the original repeating decimal equal to n and then multiply both sides of that equation by a power of ten? (When you multiply n by a power of ten, you can subtract n to remove the repeating part of the decimal. Then solve the new equation for n to write the fraction.) 19
LESSON 2 Estimating the Value of Irrational Expressions Learning Objectives Students will understand that all rational and irrational numbers can be represented by points on a number line. Students will find a rational approximation of an irrational number. Before the Lesson Review how to find the two integers that a non-perfect square root lies between. Ask: What two integers does 7 lie between? How do you know? (2 and 3 because 4 5 2, 9 5 3, and 4, 7, 9. So, 2, 7, 3.) Repeat with other examples. Common Core State Standard 8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p 2 ). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. Understand Connect Number lines can help students understand how to refine the rational approximation of an irrational number. Begin by discussing why the location is an approximation. Point out that getting a better approximation requires looking more and more closely at square numbers to squeeze the approximation between. After identifying that 12 lies between 3 and 4, ask: Why compute the squares of 3.4 and 3.5 to approximate 12? (Because 12 is about halfway between 9 and 16 but closer to 9.) Point out that other squares may be selected, but they may require more calculations to zero in on the approximation. For example, 3.3 2 5 10.89, and 12 does not lie between 10.89 and 11.56. Discuss the similar situation with 3.46 and 3.47. To connect the concept to procedural understanding, remind students that the process requires getting closer and closer to 8 when decimals between 2 and 3 are squared. Start with the whole-number approximation, move to tenths, then hundredths, and finally thousandths. Remind students that 8 should always be between the two approximations. Check that they understand how the closer approximation is selected. For example ask: Why is 8 closer to 8.0089 than to 7.9524? (8 is 0.0089 units from 8.0089 while it is 0.0476 units from 7.9524.) TRY MP6 MP7 Review how to find a square root. Remind students that they can square the result to check the answer. To the nearest tenth, 7 is 2.6. 20
Practice Point out that answers for problems 10 12 should be in fraction form. For problem 18, review the meanings of 2 3 and 3 2 as products: 2 3 5 2 3 3, and 3 2 5 3 3 2. For answers, see page 90. Common Errors Students may confuse the square roots and the squares. Encourage students to use a number line to help them distinguish between the two forms of numbers and to keep track of the approximations. Remind them to carefully check differences as they refine estimates to a given place. Domain 1 21
LESSON 3 Applying Properties of Exponents Learning Objectives Students will understand the meaning of integer exponents and the rules for computing with exponential expressions that have the same base. Students will apply the properties of integer exponents to generate equivalent numerical expressions. base exponent Vocabulary in a power, the number that is used as a factor the number of times indicated by the exponent in a power, the number that indicates how many times the base is used as a factor Common Core State Standard 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3 2 3 3 25 5 3 23 5 1 5 1 3 3 27. Before the Lesson Discuss how to write a repeated addition expression as a multiplication expression. For example, 3 1 3 1 3 1 3 5 4 3 3. Point out that the multiplication expression is a more concise way to write the repeated addition. Likewise, exponents provide a more concise way to write a repeated product. Review how to write 3 3 3 3 3 3 3 as an expression with an exponent (3 4 ). Then evaluate the expression with students (81). Understand Connect Review how to evaluate an expression when the exponents are 0, 1, and 21. For example, 4 0 5 1, 4 1 5 4, and 4 21 5 1 4. Discuss all the equivalent forms of 4 27 : 4 27 5 1 4 7 5 1 5 1 4 7 16,384. Emphasize that an exponent means repeated multiplication of the base. The exponent tells how many times to multiply the base by itself. To help develop conceptual understanding, discuss why each property works by working through the steps that illustrate how to simplify each of the exponential expressions. To connect the concept to procedural understanding, point out that the properties make evaluating an exponential expression easier. For example, in the first problem it is much easier to evaluate 7 2 5 49 than 7 6 5 117,649 7 4 2,401 5 49. Caution students to pay attention to the signs of the exponents. In the fourth problem, remind students that a fraction bar denotes division and that to divide fractions, the first term is multiplied by the reciprocal of the second term. 24
DISCUSS MP2 Discuss with students how to simplify the expression. Encourage them to work in steps by first writing an equivalent expression for the numerator with a positive exponent and then writing an equivalent expression for the denominator with a positive exponent. When both exponents are positive, it is easier to simplify. Answers may vary. Possible answer: Rewrite 2 25 as 1 and 7 22 as 1. Then 225 becomes 1 4 1 5 2 5 7 2 7 22 2 5 7 2 1 3 72 2 5 1 5 72. 2 5 Practice Remind students that using the properties of exponents should simplify the computations in problems 13 24. For answers, see pages 91 and 92. Common Errors Students may incorrectly apply properties of exponents. Encourage students having difficulty to write out the meaning of the exponential expression, then to simplify the expression, and finally to evaluate the expression. Caution them to be especially careful about confusing multiplication (where the exponents are added) and exponentiation (where the exponents are multiplied). Domain 2 25