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Dear Family, Content Overview The goal of this unit of Math Expressions is for your child to become fluent with dividing whole numbers and with comparing, adding, subtracting, multiplying, and dividing fractions and decimals. Below is a summary of the topics in this unit. Addition and Subtraction of Fractions and Decimals 3 10 1.40 \\ - 0.25 1.15 4 = ___ 10 + ___ 12 = ___ 22 = 1___ 7 2 + __ __ 5 3 15 15 15 15 Equivalent Fractions Multiply the numerator and denominator by the same number. Divide the numerator and denominator by the same number. 5 • 2 = ___ 10 5 = _____ __ 10 = 10 ÷ 2 = __ 5 ___ _______ 6 6•2 12 12 12 ÷ 2 6 Multiplication of Fractions and Decimals Multiply numerators and denominators. 4= 2 8 2 • __ • 4 = ___ __ _____ 3 5 3• 5 15 Count decimal places in the factors to place the decimal point in the product. 0.12 • 0.4 = 0.048 2 places 1 place 3 places Division of Fractions and Decimals Multiply by the reciprocal. 3 = __ 14 2 • __ 7 = ___ 2 ÷ __ __ 7 5 5 3 15 Sometimes you can divide numerators and denominators. 2 = _______ 8 ÷ 2 = __ 8 ÷ __ 4 ___ 15 3 15 ÷ 3 5 Make the divisor a whole number by multiplying it and the dividend by the same number. Here we multiply both by 10. 0.4 _____ 0.2 0.08 If you have any questions or comments, please call or write to me. Sincerely, Your child’s teacher Unit 3 addresses the following standards from the Common Core State Standards for Mathematics with California Additions: 6.NS.1, 6.NS.2, 6.NS.3, 6.NS.4, and all Mathematical Practices. UNIT 3 LESSON 1 Place Value and Whole Number Division 57 Estimada familia, Un vistazo general al contenido El objetivo de esta unidad de Expresiones en matemáticas es que su hijo domine la división de números enteros y que compare, sume, reste, multiplique y divida correctamente fracciones y decimales. Debajo hay un resumen de algunos de los temas de esta unidad. Suma y resta de fracciones y decimales 3 10 1.40 \\ - 0.25 1.15 4 = ___ 10 + ___ 12 = ___ 22 = 1___ 7 2 + __ __ 5 15 15 15 15 3 Fracciones equivalentes Multiplicar el numerador y el denominador por el mismo número. Dividir el numerador y el denominador entre el mismo número. 5 • 2 = ___ 10 5 = _____ __ 10 = 10 ÷ 2 = __ 5 ___ _______ 6 6•2 12 12 12 ÷ 2 6 Multiplicación de fracciones y decimales Multiplicar los numeradores y denominadores. 4= 2 8 2 • __ • 4 = ___ __ _____ 3 5 3• 5 15 Contar los lugares decimales en los factores para colocar el punto decimal en la respuesta. 0.12 • 0.4 = 0.048 2 lugares 1 lugar 3 lugares División de fracciones y decimales Multiplicar por el recíproco. 3 = __ 2 ÷ __ 2 • __ 7 = ___ 14 __ 7 5 5 3 15 Algunas veces se pueden dividir los numeradores y denominadores. 2 = _______ 8 ÷ 2 = __ 8 ÷ __ 4 ___ 3 15 5 15 ÷ 3 Dividir el divisor y el dividendo entre el mismo número para convertir el divisor en un número entero. Aquí se multiplican ambos por 10. 0.4 _____ 0.2 0.08 Si tiene preguntas, por favor comuníquese conmigo. Atentamente, El maestro de su hijo En la Unidad 3 se aplican los siguientes estándares auxiliares, contenidos en los Estándares estatales comunes de matemáticas con adiciones para California: 6.NS.1, 6.NS.2, 6.NS.3, 6.NS.4, y todos los de prácticas matemáticas. 58 UNIT 3 LESSON 1 Place Value and Whole Number Division 3–1 ► Decimal and Whole Number Secret Code Cards 2, 0 0 0 0 .1 3 0 0 0 .0 3 3 0 1 0 .0 1 2 9 0 .4 .0 0 6 0 0 .0 0 2 7 0 0 .6 2,000 0 .1 300 0 .03 1 30 2 9 0 .01 0 .4 60 70 UNIT 3 LESSON 1 0 .002 0 .6 Place Value and Whole Number Division 59 3–1 ► Decimal and Whole Number Secret Code Cards $1,000 $1,000 $100 $100 $100 $1 $10 $10 $10 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $10 $10 $10 $10 $10 $10 $10 $10 $10 $10 $10 $10 $10 60 UNIT 3 LESSON 1 Place Value and Whole Number Division 3–1 Content Standards 6.NS.2 Mathematical Practices MP.2, MP.3, MP.6, MP.7, MP.8 ► Discuss and Summarize Fill in the blanks and discuss how the parts of each problem are related. Place Value 3 10 1 10 1 10 1 10 1 10 1 10 1 TNBMMFS 10 5IPVTBOET )VOESFET 5FOT 0/&4 5FOUIT )VOESFEUIT 5IPVTBOEUIT 1,000. 100. 10. 1. 0. 1 0.01 0.001 1000 1 100 1 10 1 1 1 1 10 1 100 1 1000 $1,000.00 $100.00 $10.00 $1.00 $0.10 $0.01 $0.001 2,000 300 60 $1 000 $1,000 $1,000 $100 $100 $100 $10 $10 $10 $10 $10 1 0 .6 0 .0 3 0.002 2, 03 06 001000..60.030 2 1 a. r(SBEF1MBDF7BMVF1PTUFSr¥CZ)PVHIUPO.JGGMJO)BSDPVSU1VCMJTIJOH$PNQBOZr*4#/ CJMM 5JN.BJOJFSP"MBNZ b. c. 4 10 MBSHFS 2,361.632 = 2,000 + + $10 $1 + 2 a. + 0 .6 + 0.002 3 2 + ____ + ______ 10 100 1,000 + ______ + ______ + ______ 1,000 1,000 1,000 + b. c. d. UNIT 3 LESSON 1 + ___ 0 .6 0 0 + 0 .0 3 0 + 0 .0 0 2 0 .6 3 2 Place Value and Whole Number Division 61 3–1 Vocabulary dividend divisor quotient remainder ► Discuss Division Meanings The 49 sixth graders raised $2,361 toward their class trip. How much is that for each student? When $2,361 is split 49 ways, each student gets $40. Write 4 in the tens place. Take away the $1,960 that has been shared out, leaving $401 to be shared next. divisor 4 __________ 4 __________ 49 $2,361.00 49 $2,361.00 - 1 96 401 dividend Take away the 49 dimes shared out from the 90 dimes, leaving 41 dimes. Change the 41 dimes to 410 pennies. 48.1 __________ 49 $2,361.00 1 96 ______ 401 - 392 90 -49 41 3. Each student gets $ 62 UNIT 3 LESSON 1 48.1 __________ 49 $2,361.00 - 1 96 401 - 392 90 -49 4 10 Each student gets 8 more dollars. Take away the $392 shared out. 48 __________ 49 $2,361.00 - 1 96 401 - 392 9 The multiplier 8, which was used before, works again. Each student gets 8 pennies. 48.18 __________ 49 $2,361.00 1 96 401 - 392 90 -49 4 10 and there are There are 9 dollars left, which is 90 dimes. Each student gets 1 dime, so write 1 in the tenths place. 48.1 __________ 49 $2,361.00 - 1 96 401 - 392 90 That makes 392 pennies shared out of the 410 pennies, leaving 18 pennies. quotient__________ 48.18 49 $2,361.00 - 1 96 401 - 392 90 -49 4 10 - 3 92 18 remainder cents left over. Place Value and Whole Number Division 3–3 ► Rule About Multiplying Decimals Example 0.4 • 0.02 Ignore the decimal point and multiply: 4 • 2 = 8 Place decimal point so there are three decimal places: 0.4 • 0.02 = 0.008 We can state a rule about multiplying with decimal numbers. Rule: Ignore the decimal points and multiply. Then place the decimal point so the number of decimal places in the product is equal to the total number of decimal places in the factors. 1 place 2 places 3 places 3. Find each product in two ways: 1) Use the rule above; 2) Use the method of dividing into 10 or 100 parts and then multiplying by the number of tenths or hundredths. Show or explain your work. The first one is done for you. Use the Rule a. 0.2 • 0.3 Divide into Parts and Multiply (0.3 ÷ 10) • 2 = 0.03 • 2 = 0.06 0.06; two decimal places b. 0.02 • 0.3 c. 0.2 • 3 d. 0.02 • 3 e. 2 • 0.3 f. 2 • 0.03 4. Explain why the two methods give the same result. Use either method to solve these problems. 5. 0.04 • 2 = 6. 0.4 • 2 = 7. 0.3 • 0.3 = 8. 3 • 0.3 = 9. 0.04 • 6 = 10. 0.5 • 7 = 11. 0.3 • 0.8 = 12. 7 • 0.6 = UNIT 3 LESSON 3 Multiplying by a Decimal 63 64 UNIT 3 LESSON 3 Multiplying by a Decimal 3–5 ► How Many Decimal Places? Use the fact that 39 • 74 = 2,886 to solve each problem. 20. 0.39 • 7,400 = _______ 21. 0.39 • 740 = _______ 22. 0.39 • 74 = _______ 23. 39 • 7.4 = _______ 24. 0.74 2,886 25. 0.74 288.6 26. 0.74 28.86 27. 74 288.6 28. 3.9 • 740 = 29. 3.9 • 74 = 30. 3.9 • 7.4 = 31. 39 • 0.74 = _______ 32. 7.4 2,886 PATH to FLUENCY _______ 33. 7.4 288.6 _______ 34. 7.4 28.86 _______ 35. 74 28.86 Solve. 36. 0.7 • 0.3 = ___ 37. 0.06 • 7 = ____ 38. 8 • 0.4 = ___ _____ 39. 0.12 42 40. 3.2 2.4 41. 0.06 54 42. 0.27 1.35 0.09 43. × 52 8.5 44. × 4.2 7.2 45. × 0.25 18 46. × 0.6 _____ 47. 3.02 90.6 UNIT 3 LESSON 5 _____ 48. 7.5 0.06 ____ 49. 0.8 6.8 _____ 50. 0.2 0.95 Multiplication or Division 65 3–9 ► Strategies for Comparing Look for patterns in the fractions and decimals below. Below are general and special cases for comparing fractions and decimals. General Cases 1= 2 = 4 = 8 2 4 8 7 8 3=6 4 8 5 8 1=2=4 2 4 8 1.000 = 1.0 0.9 0.875 0.83 ≈ 5 6 0.8 = 4 5 0.750 = 0.75 0.7 0.67 ≈ 4 = 2 0.625 3 0.6 = 6 3 5 0.500 = 0.50 = 0.5 = 3 6 3 8 1=2 4 8 1 8 0.4 = 2 0.375 5 0.33 ≈ 2 = 1 6 3 0.3 0.250 1 = 0.25 0.2 = 5 0.17 ≈ 1 0.125 0.1 6 Case 1: Same denominator or number of places Fraction with greater 3 4 __ __ numerator is greater. 5 5 Ignore decimal point. 0.7 > 0.5 Greater number is greater. Case 2: Same numerator or digits Fraction with lesser 6 __ denominator is greater. 9 Decimal with leftmost non-zero 0.3 digit is greater. Case 3: Different denominators Find equivalent fractions with a common denominator to change fractions to Case 1. 3 __ 6 __ 8 0.03 2 __ 4 3 Case 4: Mixed numbers Number with greater whole 9 1 2___ 5__ 8 10 number is greater. If whole numbers are the same, compare 4.7 > 4.07 fractions using Cases 1–3. Special Cases for Fractions Case 5: Denominators are factors of 10 or 100. Compare the decimal 3 4 __ ___ equivalents. 5 10 1 1 . One fraction < __ . Case 6: One fraction > __ 1 Fraction > __ is greater. 2 2 5 __ 2 3 __ 7 8 Use strategies to compare. 0= 0=0 2 4 8 0.000 28. 0.76 5 31. __ 8 66 UNIT 3 LESSON 9 0.67 5 __ 6 2 29. __ 3 32. 0.09 7 ___ 1 30. 1__ 12 2 1__ 5 2 0.1 4 33. __ 9 3 __ 5 Mixed Problem Solving Unit 3 Select all that apply. _____ ______ 2. 25 2,515 1. 2 629 A 213 A 55 B 314 R1 B 95.6 C 314.5 C 100.6 D 345 D 100 R15 3. Is the problem below a multiplication or a division problem? Explain your answer. The rectangular floor of a playpen has an area of 14 square feet. 1 feet. What is the other measure? One measure of the floor is 3__ 2 4. Select True or False for each statement. 4a. 0.9 ÷ 7.5 = 0.12 True False 4b. 63.2 ÷ 8 = 7.8 True False 4c. 72.8 ÷ 5.2 = 1.4 True False 4d. 7 ÷ 0.35 = 20 True False UNIT 3 TEST 67 Unit 3 Name Date 5. Without doing any computation, predict if the quotient 73 ÷ 0.42 will be greater than or less than the product 0.42 • 73. Explain your reasoning in terms of the size of 0.42. Use the numbers and the decimal point (if needed) to write the product. Each tile may be used more than once. 0 1 2 4 6 7 8 . 0.06 7. ×12 0.4 6. ×0.3 Fraction Pair 2 __ __ ,1 5 __ ___ ,3 7 3 12 4 LCM Equivalent Fraction Pair Multiply. Write the product. 9. 5.8 × 2.5 Product: 68 UNIT 3 TEST 10. 2.9 × 0.81 Product: 7 ___ ___ , 4 10 15 © Houghton Mifflin Harcourt Publishing Company 8. The fraction pairs in the table are not equivalent. Find the LCM for each fraction pair. Rewrite each pair as an equivalent fraction pair using the LCM as the denominator. Unit 3 Find the sum or difference. Select one number from each column to make the equation true. 3 7 + __ = 11. __ 8 5 7 12. __ + __ = 8 whole number 0 1 2 3 9 fraction 5 __ 8 4 __ 5 1 __ 4 3 __ 4 6 whole number fraction 0 3 __ 1 7 __ 2 11 ___ 3 11 ___ 7 9 13 18 Write a number in the box to make the equation true. 2 13. 3__ 5 1 = 1__ 2 - 1.78 = 0.82 14. 15 . 15. George calculated a sum of ___ 24 15 ? Choose all that apply. Which fractions are equivalent to ___ 24 A 10 ___ D 5 __ B 5 __ E 3 __ C 7 ___ F 30 ___ 16 6 12 UNIT 3 TEST 8 8 48 69 Unit 3 16. Read the division problem below. Then select True or False for each statement. _3_ ÷ 7 = 4 16a. To find the quotient, first find a common denominator. True False 16b. The quotient will be less than _34_. True False 16c. You will need to simplify the quotient. True False Draw a line to match each missing numerator or denominator to its value. You will not use all of the choices. • • • • 3 6 • ________ = ___ 17. __ 5 3 1 18. __ • ________ = ___ 4 35 • • • • • • 2 3 4 5 6 7 Choose the expressions that make the equation true. Select all that apply. 8 4 19. ___ ÷ __ = 15 5 1 A __ 3 70 1 5 20. 1__ ÷ 1__ = 8 A 2 7 2___ 16 B 2 __ B 39 __ 16 C 20 ___ C 26 __ 24 D 40 ___ D 1 1__ 3 60 60 UNIT 3 TEST 12 20 Unit 3 Name Date 21. Seth wrote an expression that simplifies to 0.36. Choose Yes or No to indicate if the expression has a value of 0.36. 21a. 36 • 0.01 Yes No 21b. 3.6 • 0.1 Yes No 21c. 3.6 ÷ 0.1 Yes No 21d. 0.036 ÷ 0.01 Yes No 22. Darnell has two scraps of cloth. The first scrap is 3.75 cm 1 as long as the length of the first. in length. The second scrap is __ 5 Part A What is the total length of the scraps? Show your work. cm Part B Answer the following question without making a calculation. Explain how you know the answer. 1 as long as the first, If Darnell’s second scrap was __ 3 would the total length of the scraps be greater than or less than the answer you found in Part A? UNIT 3 TEST 71 Unit 3 Name Date 1 23. Lydia has 8__ cups of juice to serve. 2 Part A 3 -cup, how many full servings can she make? If each serving is __ 4 What amount of juice, if any, will be left over? Show your work. servings leftover Part B If she needs 15 servings, how much more juice does she need? A 1 2__ cups 4 B 3 cups C 3 2__ cups 4 D 4 cups Part C 5 -cup. Without calculating, Lydia decides to make each serving of juice __ 8 predict if she can make more full servings or fewer full servings using 1 cups of juice. Explain your choice. Then confirm your prediction. the 8__ 2 Predict: © Houghton Mifflin Harcourt Publishing Company Confirm: 24. Dana’s dog Skippy weighs 20.4 pounds. In comparison, her cat Bruiser weighs 0.75 as much as Skippy. 24a. Does Bruiser weigh more or less than Skippy? Explain. 24b.How much does Bruiser weigh? pounds 72 UNIT 3 TEST