Numbers & Operations 8 th Grade Chapter Questions 1. What are the properties of a square? 2. What does taking the square root have to do with the area of a square? 3. Why is it helpful to memorize perfect squares? 4. What can be helpful when finding the square roots of numbers greater than 400? 5. Explain how to take the square root of a fraction or a decimal. 6. Explain how to approximate a square root. 7. What is the difference between an irrational and rational number? 8. Explain how to convert between the different forms of a rational number. Numbers & Operations Chapter Problems Converting Repeating Decimals to Fractions and Decimal Expansion Classwork 1. Write each repeating decimal as a fraction in simplest form * 2. Find the decimal expansion of the following * : NJ Center for Teaching and Learning ~ 1 ~ www.njctl.org
Homework 3. Write each repeating decimal as a fraction in simplest form * 4. Find the decimal expansion of the following * : Squares, Square Roots & Perfect Squares Classwork 5. A square has an area of 9 units 2. What is the side length of a square of this area? Draw a square with an area of 9 units 2. What is the square root of 9? Explain why your answers in parts (a) and (c) are the sam * From Engage NY NJ Center for Teaching and Learning ~ 2 ~ www.njctl.org
6. Fill in the following table: Side Length of a square (units) Area of the square (units 2 ) 1 2 3 4 5 6 7 8 9 10 11 12 13 7. Explain how the table above helps you find the square root of 121? 8. Simplify each square root. Homework 9. A square has an area of 36 units 2. What is the side length of a square of this area? NJ Center for Teaching and Learning ~ 3 ~ www.njctl.org
Draw a square with an area of 36 units 2. What is the square root of 36? Explain why your answers in parts (a) and (c) are the sam 10. Fill in the following table: Side Length of a square (units) Area of the square (units 2 ) 14 15 16 17 18 19 20 11. Simplify each square root. NJ Center for Teaching and Learning ~ 4 ~ www.njctl.org
Squares of Numbers Greater Than 20 Classwork 12. Fill in the following table: Side Length of a square (units) Area of the square (units 2 ) 10 20 30 40 50 60 70 80 90 100 13. If you compare that to the table of side lengths from 1-10, what pattern do you notice? 14. Simplify each square root. g. h. i. j. NJ Center for Teaching and Learning ~ 5 ~ www.njctl.org
Homework 15. Simplify each square root. g. h. i. j. Simplifying Perfect Square Radical Expressions Classwork 16. Simplify each square root. g. h. i. j. k. l. m. n. o. Homework 17. Simplify each square root. - NJ Center for Teaching and Learning ~ 6 ~ www.njctl.org
g. h. i. - j. k. l. m. n. o. Approximating Square Roots Classwork 18. What two integers do the following square roots fall between? 19. Draw and label a number line from 0 to 10. Place the following square roots on the number lin 20. Estimate the following square roots. g. h. NJ Center for Teaching and Learning ~ 7 ~ www.njctl.org
i. j. 21. Approximate the square root to the nearest integer 22. For what integer x is closest to 7.42? * 23. For what integer x is closest to 5.1? * 24. For what integer x is closest to 3.9? * Homework 25. What two integers do the following square roots fall between? 26. Draw and label a number line from 0 to 10. Place the following square roots on the number lin 27. Estimate the following square roots. g. * From Engage NY NJ Center for Teaching and Learning ~ 8 ~ www.njctl.org
h. i. j. 28. Approximate the square root to the nearest integer 29. For what integer x is closest to 5.2? * 30. For what integer x is closest to 6.1? * 31. For what integer x is closest to 6.9? * Rational & Irrational Numbers Classwork. 32. Circle the numbers below that are rational 3.5 π g. h. 0.25 i. j. 0. 33. Given the statement If x is a rational number then is irrational. Which values of x make the statement false? * A) 15 B) 24 C) 4 D) 20 34. Given the statement If x is a rational number then is irrational. Which values of x make the statement false? * A) 25 B) 49 C) 5 D) 59 * From Engage NY NJ Center for Teaching and Learning ~ 9 ~ www.njctl.org
Homework 35. Circle the numbers below that are irrational. 6.75 g. h. π i. j. 0. 36. Given the statement If x is a rational number then is irrational. Which values of x make the statement false? * A) 36 B) 8 C) 16 D) 121 37. Given the statement If x is a rational number then is irrational. Which values of x make the statement false? * A) 144 B) 12 C) 48 D)30 Properties of Exponents Classwork 38. Complete each equation for the missing value: (5 2 )(5 5 ) = 5? (12 7 )(12 3 ) = 12? (3-2 )(3 5 ) = 3? (4 9 )(4-3 ) = 4? (5 4 )(5? ) = 5 12 (10 7 )(10? )(10-6 ) = 10 3 g. 3 4 3 2 = 3? 9 5 h. 6 = 5? 5 5 9 i. 8 = 9? 9 j. 12 4 12 6 = 12? * From Engage NY NJ Center for Teaching and Learning ~ 10 ~ www.njctl.org
k. 10 8 10? = 10 3? 2 l. 3 = 2 4 2 39. A rectangle has a length of mm and a width of mm. Write an expression for the area of the rectangle as a power of 5. * 40. Express the volume of a cube with a side length of inches as a power of 7. * 41. a) Write an exponential expression for the area of a rectangle with a length of meters and a width of meters. b) Evaluate the expression to find the area of the rectangl * Homework 42. Complete each equation for the missing value: (12 2 )(12 7 ) = 12? (2 5 )(2 2 ) = 2? (5-3 )(5 5 ) = 5? (15 8 )(15-5 ) = 15? (6 7 )(6? ) = 6 15 (11-6 )(11? )(11 8 ) = 11 5 g. 7 7 7 3 = 7? 10 11 h. 6 = 11? 11 i. 3 7 3 9 = 3? 2 2 6 j. 10 = 2? 6 13 k.? = 13 2 13 l. 5? 5 6 = 5 3 43. A rectangle has a length of mm and a width of mm. Write an expression for the area of the rectangle as a power of 4. * 44. Express the volume of a cube with a side length of inches as a power of 2. * * From Engage NY * From Engage NY NJ Center for Teaching and Learning ~ 11 ~ www.njctl.org
45. a) Write an exponential expression for the area of a rectangle with a length of meters and a width of meters. b) Evaluate the expression to find the area of the rectangl * Roots & Radicals Review Determine whether the given numbers are perfect squares. Circle your answer. 1) 1 Yes No 2) 8 Yes No 3) 16 Yes No 4) 25 Yes No 5) 82 Yes No Circle the simplified version of each square root: 6) 14 12 72 21 7) 10 6 0.6 18 8) -7 0.7 0.07-0.07 NJ Center for Teaching and Learning ~ 12 ~ www.njctl.org
Circle whether the given number is rational or irrational 9) π rational irrational 10) 0.875 rational irrational 11) ) rational irrational 12) Between what two integers does the following square root fall? 4 & 5 6 & 7 7 & 8 5 & 6 13) (4 7 )(4 3 ) = 4? 10 24 4 5 14) Approximate 15) Find the missing value 11 4 11 6 = 11? 16) 17) (6 7 )(6-2 ) = 6? 18) A rectangle has a length of cm and a width of cm. What is the area of the rectangle written as a power of 4? 19) Draw and label a number line from 0-10. Place the following numbers on the number line: *,, * From Engage NY NJ Center for Teaching and Learning ~ 13 ~ www.njctl.org
20) Write as a fraction in simplest form. * 21) Write as a fraction in simplest form. * 22) Write two exponential expressions with like bases. Leave all answers in simplified exponential form. Expression 1: Expression 2: Multiply your expressions. Divide your expressions. Raise your first expression to the 5 th power. NJ Center for Teaching and Learning ~ 14 ~ www.njctl.org
Answer Key 1. 5. 3 units 3 units 2. 3. 7 6. 3 units 3 Area of a Square = Side 2 and 9 = 3 2 Side Length of a Square (units) Area of the square (units 2 ) 1 1 2 4 4. 4 2 4 2 2x y z 10 3 3 3 x y z yz xz 2 3 2 2x y z 6 3 2 3 x y z 65y 2x y z 14x x y z 2 2 3 10yz 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 2 2 g. 2x yz 7 yz 12 144 h. i. 4 x y z 22z 4 3 3x y z 3 xy 13 169 7. Since Area of a square = side 2, the square root of the area = sid So, 121 = 11. NJ Center for Teaching and Learning ~ 15 ~ www.njctl.org
8. 9. 5 8 9 7 4 6 units 12. Side Length of a square (units) 10 100 20 400 30 900 Area of the square (units 2 ) 6 units 40 1600 50 2500 6 units 6 Area = Side 2 and 36 = 6 2 60 3600 70 4900 80 6400 10. Side Length of a square Area of the square 90 8100 100 10,000 11. 17 20 14 19 12 (units) (units 2 ) 14 196 15 225 16 256 17 289 18 324 19 361 20 400 13. Each answer in this table is 100 times greater than the corresponding answer in the other tabl (or 10 2 times greater). 14. 53 89 22 80 45 15 g. 29 h. 97 i. 31 j. 66 15. 71 36 92 55 61 83 g. 24 h. 49 i. 50 NJ Center for Teaching and Learning ~ 16 ~ www.njctl.org
j. 17 16. 5 8-9 No real solution 7 5 7 g. No real solution h. ½ 5 i. 7 j. ½ k. 0.8 l. 0.09 m. -0.5 n. 0.04 o. No real solution 17. 17-20 8 19 No real solution 1/3 g. ½ h. No real solution i. -3/4 j. 1/10 k. No real solution l. -0.14 m. 0.7 n. 0.19 o. 0.5 18. 19. 20. 2.45 8.37 7.42 3.74 10.29 6.4 g. 8.94 h. 8.06 i. 2.83 j. 15.26 21. 7 6 8 3 9 22. 55 23. 26 24. 15 NJ Center for Teaching and Learning ~ 17 ~ www.njctl.org
25. 26. 27. 8.83 2.65 7.94 5.39 6.48 11.75 g. 17.32 h. 12.17 i. 4.58 j. 7.21 28. 4 6 4 6 7 29. 27 30. 37 31. 48 32. Rational Irrational Irrational Rational Irrational Rational g. Irrational h. Rational i. Rational j. Rational 33. c 34. a,b 35. Rational Irrational Rational Rational Rational Rational g. Irrational h. Irrational i. Rational j. Rational 36. a,c,d 37. a 38. 7 10 3 6 8 2 g. 2 h. 3 i. -3 j. -2 k. 5 l. 7 39. mm 2 40. in 3 NJ Center for Teaching and Learning ~ 18 ~ www.njctl.org
41. x m 2 42. 9 7 2 3 8 3 g. 4 h. 4 i. -2 j. -4 k. 4 l. 9 43. mm 2 44. in 3 45. x in 2 Review answers 1. Yes 2. No 3. Yes 4. Yes 5. No 6. B 7. C 8. D 9. Irrational 10. rational 11. irrational 12. b 13. a 14. ~7 15. -2 16. 4 17. 5 18. cm 2 19. =5.29 20. 21. =5.48 =9.1 22. Expressions will vary NJ Center for Teaching and Learning ~ 19 ~ www.njctl.org