ELH Env. G6 T1 PDF
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This document contains math content covering topics like adding, subtracting, multiplying, and dividing decimals and fractions, including examples and practice problems.
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TOPIC USE POSITIVE 1 RATIONAL NUMBERS How can you fluently add, subtract, ? Topic Essential Question multiply, and divide decimals? How...
TOPIC USE POSITIVE 1 RATIONAL NUMBERS How can you fluently add, subtract, ? Topic Essential Question multiply, and divide decimals? How can you multiply and divide fractions? Topic Overview Topic Vocabulary 1-1 Fluently Add, Subtract, and Multiply Decimals reciprocal 1-2 Fluently Divide Whole Numbers and Decimals 1-3 Multiply Fractions 3-Act Mathematical Modeling: Stocking Up 1-4 Understand Division with Fractions 1-5 Divide Fractions by Fractions 1-6 Divide Mixed Numbers 1-7 Solve Problems with Rational Numbers Lesson Digital Resources INTERACTIVE STUDENT EDITION Access online or offline. VISUAL LEARNING ANIMATION Interact with visual learning animations. ACTIVITY Use with Solve & Discuss It, Explore It, and Explain It activities, and to explore Examples. VIDEOS Watch clips to support 3-Act Mathematical Modeling Lessons and STEM Projects. Go online 2 Topic 1 Use Positive Rational Numbers 3-ACT MATH Stocking Up When shopping for groceries, it is useful to set a budget and stick to it. Otherwise, you may buy items you do not need and spend more money than you should. Some people avoid overspending by bringing cash to pay for their groceries. If you bring $50 in cash, you cannot spend $54. Think about this during the 3-Act Mathematical Modeling lesson. PRACTICE Practice what KEY CONCEPT Review important you’ve learned. lesson content. TUTORIALS Get help from Virtual Nerd, GLOSSARY Read and listen to right when you need it. English/Spanish definitions. MATH TOOLS Explore math ASSESSMENT Show what with digital tools. you’ve learned. GAMES Play Math Games to help you learn. Topic 1 Use Positive Rational Numbers 3 TOPIC 1 Project VIDEO Did You Know? Engineers design equipment to make you safer. Engineering is the application of math and science to solve problems. Engineers solve problems by designing and building products, materials, machinery, structures, transportation vehicles, and so many other things. Engineers work in nearly every area from chemical and electrical engineering to biomedical and oceanographic engineering. Engineers find ways to Engineers help improve and enhance keep you healthy. performance of all kinds of products. Your Task: Improve Your School Think like an engineer! Take a walk around the inside and the outside of your school building. Make a list of specific things or areas that need improvement. Then choose one idea and do some background research to gain an understanding of factors that might impact improvement efforts. In the next topic, you and your classmates will learn about and implement the engineering design process to propose possible ways to make the improvements. 4 Topic 1 enVision STEM Project TOPIC Review What You Know! GET READY! 1 Vocabulary compatible numbers Choose the best term from the box to complete each definition. decimal 1. Numbers that are easy to compute mentally are. divisor estimate 2. The number used to divide is the. quotient 3. A(n) is an approximate answer. 4. The result of a division problem is a(n). Whole Number Operations Calculate each value. ____ 5. 4⟌348 6. 9,007 − 3,128 7. 35 × 17 ______ 8. 7,964 + 3,872 9. 22⟌4,638 10. 181 × 42 Mixed Numbers and Fractions Write each mixed number as a fraction. Write each fraction as a mixed number. 1 11. 8__ 3 12. 5 __ 5 13. 2 __ 4 14. 3 __ 3 5 8 9 24 15. ___ 43 16. ___ 59 17. ___ 32 18. ___ 7 9 8 5 Verbal Expressions 1 of 12” and “12 divided by 4” related? 19. How are the expressions “ __ 4 Decimals 20. What decimal does this model represent? Explain. Topic 1 Use Positive Rational Numbers 5 Language Development Fill in the boxes with terms and phrases related to Decimals and Fractions from the given bank. Include illustrations or examples. annex zeroes (1–1) algorithm (1–2) compatible numbers (1–1) denominator (1–3) decimal point (1–1) mixed number (1–3) estimate (1–1) numerator (1–3) hundredths (1–1) unit fraction (1–3) line up place value (1–1) reciprocal (1–4) rounding (1–1) rewrite (1–4) tenths (1–1) multistep problems (1–7) Decimals Fractions 6 Topic 1 Use Positive Rational Numbers TOPIC PICK A PROJECT 1 PROJECT 1A What is the most challenging board game you have ever played? PROJECT: MAKE YOUR OWN BOARD GAME PROJECT 1B What is your favorite party food? PROJECT: PLAN THE MENU FOR A SCHOOL FUNDRAISER Topic 1 Pick a Project 7 PROJECT 1C If you planted a garden, what would be in it? PROJECT: DESIGN A VEGETABLE AND HERB GARDEN PROJECT 1D How much food does a tiger eat? PROJECT: PRESENT A PROPOSAL FOR A TIGER EXHIBIT 8 Topic 1 Pick a Project Lesson 1-1 Fluently Add, Solve & Discuss It! ACTIVITY Subtract, and Maxine is making a model windmill Multiply Decimals for a science fair. She is connecting 4 cardboard tubes together vertically. Go Online Each tube is 0.28 meter in length. What is the combined measure of the connected tubes? I can… add, subtract, and multiply Use Appropriate Tools decimals. You can use decimal grids to calculate with decimals. 0.28 m Focus on math practices Look for Relationships Suppose that Maxine made another windmill model by connecting 4 cardboard tubes that are each 2.8 meters long. What is the combined measure of this model? What relationships do you see in the factors you used here and above? Explain how this helps you solve the problem. 9 ? Essential Question How can you add, subtract, VISUAL LEARNING ASSESS and multiply with decimals? Scan for EXAMPLE 1 Add Decimals Multimedia Kim and Martin swam 50 meters. Martin took 0.26 second longer than Kim. What Find 50.9 + 0.26. was Martin’s time in the race? t Be Precise Why is precision important 50.9 when working with decimals? 0.26 Estimate first by rounding each addend. Kim’s time: 50.9 rounds to 51. 0.26 rounds to 0.3. 50.9 seconds 60 55 5 51 + 0.3 = 51.3 50 10 Martin’s time: 45 15 0.26 second longer 40 20 35 25 Find the sum. 30 Annex a zero so 50.90 each place has + 0.26 _ a digit. Remember to line up the place values to add. Add each place. 1 You can regroup the 5 0 .90 sum of nine tenths _ + 0.26 and two tenths. 51.16 Martin swam the race in 51.16 seconds. The sum 51.16 is close to the estimate, 51.3. Try It! Suppose that Martin finished the race 0.47 second after Kim. What was Martin’s time in the race? Use an estimate to check that your answer is reasonable. Convince Me! If Martin finished the race 0.267 second after Kim, you would need to add 0.267 to 50.9 to solve the problem. How is adding 0.267 to 50.9 different from adding 0.26 to 50.9? 10 1-1 Fluently Add, Subtract, and Multiply Decimals EXAMPLE 2 Subtract Decimals ACTIVITY ASSESS Amy ran a race in 20.7 seconds. Katie finished the race 0.258 second before Amy. How long did it take Katie to run the race? Find 20.7 − 0.258. To find the difference, line up the place values. 20.700 Annex zeros as placeholders. 20.7 − 0.258 _ s Subtract each place. Regroup as needed. 9 6 1010 0.258 0 0 20. 7 – 0. 2 5 8 _ Estimate the difference by rounding. 20. 4 4 2 20.7 − 0.3 = 20.4 Katie ran the race in 20.442 seconds. 20.442 is close to the estimate, 20.4, so the answer is 0.258 rounds to 0.3. reasonable. Try It! Suppose that Katie finished the race 0.13 second before Amy. What was Katie’s time in the race? Use an estimate to check that your answer is reasonable. EXAMPLE 3 Multiply Decimals What is the area of this antique map? Use the formula A = ℓ wto find the area of the map. North Asia Multiply as you would with whole numbers. Then America place the decimal point in the product. Annex zeros if 2.5 ft Africa needed. The number of decimal places in the product South is the sum of the number of decimal places in the factors. America Australia 3.25 2 decimal places (hundredths) × 2.5 _ 1 decimal place (tenths) 1625 _+ 6500 3.25 ft 8.125 3 decimal places (tenths times hundredths equals thousandths) The area of the antique map is 8.125 ft 2. Try It! 0.43 decimal place(s) How do you determine where to place the decimal point in × 0.2 _ decimal place(s) the product? 0.086 decimal place(s) Annex zeros if needed. 1-1 Fluently Add, Subtract, and Multiply Decimals 11 KEY CONCEPT KEY CONCEPT 9 1 6 1010 To add decimals, line up 5 0 .90 To subtract decimals, line up 0 0 20. 7 place values and add. _ + 0.26 place values and subtract. – 0. 2 5 8 _ Regroup as needed. 51.16 Regroup as needed. 20. 4 4 2 1.35 To multiply decimals, multiply as you × 4.6 _ would with whole numbers. Then use the 810 number of decimal places in the factors to place the decimal point in the product. + 5400 _ 6.210 Do You Understand? Do You Know How? 1. Essential Question How can you add, In 5–10, find each sum or difference. subtract, and multiply with decimals? 5. 5.9 + 2.7 6. 4.01 − 2.95 7. 6.8 − 1.45 8. 9.62 − 0.3 2. Generalize How is adding and subtracting 9. 2.57 + 7.706 10. 15 − 6.108 decimals similar to and different from adding and subtracting whole numbers? In 11–16, place the decimal point in the correct location in the product. 11. 4 × 0.94 = 376 12. 5 × 0.487 = 2435 3. What can you do if a decimal product has final zeros to the right of the decimal point? 13. 3.4 × 6.8 = 2312 14. 3.9 × 0.08 = 312 15. 0.9 × 0.22 = 198 16. 9 × 1.2 = 108 4. Critique Reasoning Diego says that the product of 0.51 × 2.427 will have five decimal places. Is Diego correct? Explain. In 17 and 18, find each product. 17. 5.3 × 2.7 18. 8 × 4.09 12 1-1 Fluently Add, Subtract, and Multiply Decimals PRACTICE TUTORIAL Name: Practice & Problem Solving Scan for Multimedia In 19–27, find each sum or difference. 19. 2.17 − 0.8 20. 4.3 + 4.16 21. 46.91 − 28.7 22. 4.815 + 2.17 23. 5.1 − 0.48 24. 27 + 0.185 25. 9.501 − 9.45 26. 14 + 9.8 27. 12.65 + 14.24 In 28–33, find each product. 28. 7 × 0.5 29. 12 × 0.08 30. 24 × 0.17 31. 0.4 × 0.17 32. 1.9 × 0.46 33. 3.42 × 5.15 34. Write an equation that illustrates the following: 35. The Bright-O Shampoo Factory includes A number with two decimal places multiplied by 1.078 ounces of vanilla oil in a 6.35-ounce a number with one decimal place. The product bottle of shampoo. How much of the bottle has only two nonzero digits. of shampoo is NOT vanilla oil? In 36–38, use the graph to solve. 36. The fastest speed a table tennis ball has been Fastest Sporting Speeds hit is about 13.07 times as fast as the speed for 110 106 the fastest swimming. What is the speed for the 100 95.69 table tennis ball? 90 80 Miles per Hour 70 60 50 37. Look for Relationships How fast would 40 1.5 times the fastest rowing speed be? Before 27.79 30 you solve, tell the number of decimal places in 20 13.99 your answer. 10 5.35 0 Fastest Fastest Fastest Fastest Fastest Swimming Running Rowing Luge Thrown Baseball 38. Which activity has a recorded speed about Human Activity 7 times as fast as the fastest rowing speed? 1-1 Fluently Add, Subtract, and Multiply Decimals 13 39. Matthew bought a jersey, 40. Anna’s running time for a race was a pennant, and a hat. 23.1 seconds. Another runner’s time was He paid with a $50 bill 5.86 seconds faster. Find the other and some money he runner’s time. borrowed from his friend. If Matthew got $6.01 in change from the cashier, how much did he borrow from his friend to pay for all the items? 41. Higher Order Thinking Explain why 0.25 ×0.4 42. The wings of some hummingbirds beat has only one decimal place in the product. 52 times per second when hovering. If a hummingbird hovers for 35.5 seconds, how many times do its wings beat? 43. The students at Walden Middle School are selling tins of popcorn to raise money for new uniforms. They sold 42 tins in the first week. How much money did they make in the first week? Assessment Practice 44. Use the information in the table to solve each problem. Trails in Everglades National Park Trail Length (kilometers) Bayshore Loop 3.2 Coastal Prairie 12.1 Rowdy Bend 4.2 Snake Bight 2.6 PART A PART B What is the combined length in kilometers of the How many kilometers longer is the Coastal Prairie Bayshore Loop trail and the Rowdy Bend trail? trail than the Snake Bight trail? 14 1-1 Fluently Add, Subtract, and Multiply Decimals Lesson 1-2 Fluently Divide Solve & Discuss It! ACTIVITY Whole Numbers Some friends went to lunch and and Decimals split the bill equally. If each person paid $6.75, how many Go Online people went to lunch? Use a diagram or equation to explain your thinking. I can… divide whole numbers and Reasoning How can decimals. you use reasoning to create a representation of the problem? Focus on math practices Reasoning Suppose $7.00 was added to the bill for a dessert that everyone shared. How much more does each person have to pay? 15 ? Essential Question How can you divide whole numbers VISUAL LEARNING ASSESS and decimals? Scan for Divide Whole Numbers EXAMPLE 1 Multimedia by Whole Numbers A tortilla bakery makes 863 packages of tortillas to sell to restaurants. Each restaurant receives the same number of packages as a complete order. How many restaurants can receive a complete order? Use Structure How can you use structure to divide 863 by 18? Find 863 ÷ 18 = n. Start by dividing the tens. A bar diagram can be used to represent the problem. 4 ____ Step 1 Divide packages of 863 18⟌ 863 Step 2 Multiply tortillas −72 _ Step 3 Subtract packages n Step 4 Compare 18 n, completely 14 per box filled boxes Next, bring down the ones. Repeat the steps as Use compatible numbers to estimate 863 ÷ 18. needed to complete the division. 47 R17 ____ 900 ÷ 20 = 45 18⟌863 The answer is reasonable since _ −72 47 is close to the The quotient of 863 ÷ 18is about 45, so the first 143 estimate, 45. digit of the quotient will be in the tens place. _ −126 17 The bakery can sell complete orders to 47 restaurants. Try It! Workers at an electronics company pack 2,610 smart phones in boxes. Each box holds 9 smart phones. How many boxes do they fill? 9 2, 6 1 0 Convince Me! Why is the first digit of the quotient in the - 1 8 Try It! not in the same place as the first digit of the quotient in Example 1? - 8 1 0 - 0 0 16 1-2 Fluently Divide Whole Numbers and Decimals Divide Whole Numbers by Whole EXAMPLE 2 ACTIVITY ASSESS Numbers with Decimal Quotients How can you write a decimal quotient when dividing whole numbers? Find 180 ÷ 8. 22 ____ Write the remainder as a 22.5 ______ 8⟌180 decimal. Place the decimal 8⟌180.0 Estimate. Because 180 ÷ 10 = 18, − 16 point and annex a 0 in the − 16 start dividing in the tens place. _ _ 20 tenths place. 20 Divide the tens and ones. _− 16 Then complete the division. − 16 _ 4 40 − 40 _ 0 EXAMPLE 3 Divide Decimals Use the division algorithm to divide with decimals. A. Find $809.40 ÷ 12. B. Find $4.20 ÷ $1.40. $809.40 $4.20 n n n n n n n n n n n n n $1.40 Use compatible numbers to estimate, and Multiply both the divisor and the dividend by then divide to solve. the same power of 10 that will make the divisor a whole number. 809.40 is close to 840, and 840 ÷ 12 = 70. Multiply 1.40 and 4.20 by 10 2 or 100. Place the decimal point in the quotient above the decimal point in the dividend. _____ 3____ Divide. Place a decimal 1.40⟌4.20 140⟌420 point in the quotient 67.45 _______ _ – 420 if needed. 12⟌ 809.40 The quotient 67.45 is close 0 –72 _ to the estimate of 70, so 89 the answer is reasonable. $4.20 ÷ $1.40 = 3 –84 _ 54 –_ 48 60 –_ 60 0 $809.40 ÷ 12 = $67.45 Try It! Divide. a. 65 ÷ 8 b. 14.4 ÷ 8 c. 128.8 ÷ 1.4 1-2 Fluently Divide Whole Numbers and Decimals 17 KEY CONCEPT KEY CONCEPT To divide by a decimal, rewrite the decimal so that 35.2 ÷ 0.16 you are dividing by a whole number. Multiply both 220 ______ the divisor and the dividend by the same power of 100 100 16⟌3,520 10. Then divide as you would with whole numbers. × 35.2 _ × 0.16 _ – 32 _ 200 600 32 5000 + 1000 _ _ – 32 + 30000 _ 16.00 0 3,520.0 Do You Understand? Do You Know How? 1. Essential Question How can you divide R whole numbers and decimals? 5. 48 9, 8 5 3 - - 2. When dividing with decimals, why is it necessary to multiply both the divisor and the dividend by - the same power of 10? In 6 and 7, divide. Record remainders. ____ 6. 2,789 ÷ 36 7. 18⟌153 3. Use Structure Explain how you can decide where to place the first digit of the quotient for 6,139 ÷ 153. In 8 and 9, divide. Write remainders as decimals. ____ 8. 4⟌139 9. 215 ÷ 2 In 10 and 11, divide. ____ 10. 5⟌34.75 11. 215.25 ÷ 5 4. Use Structure How do you know where to place the decimal point in the quotient when dividing a decimal by a whole number? In 12 and 13, divide. Annex zeros if needed to write remainders as decimals. ____ 12. 5.3 ÷ 0.2 13. 0.4⟌8.9 18 1-2 Fluently Divide Whole Numbers and Decimals PRACTICE TUTORIAL Name: Practice & Problem Solving Scan for Multimedia Leveled Practice In 14 and 15, divide. R 8. 14. 62 5, 8 4 1 15. 4 3 5 0. - - 3 - 4 - 1 3 2 - In 16–19, divide. Record remainders. ____ ____ 16. 2,593 ÷ 21 17. 19⟌6,927 18. 9⟌2,483 19. 968 ÷ 38 In 20–23, divide. Write remainders as decimals. ____ ____ 20. 5⟌56 21. 232 ÷ 40 22. 44 ÷ 10 23. 4⟌2,626 In 24–27, divide. ____ ____ 24. 6⟌$54.18 25. 187.2 ÷ 8 26. 7⟌6.3 27. 137.5 ÷ 5 In 28–31, divide. Annex zeros if needed to write remainders as decimals. ____ ____ 28. 6.4 ÷ 0.8 29. 0.6⟌0.2430 30. 52.056 ÷ 7.23 31. 0.745⟌9.089 32. Ants are one of the Thorny Devil lizard’s favorite foods. It can eat 45 ants per minute. How long would it take this lizard to eat 1,080 ants? Express your answer in minutes. 1-2 Fluently Divide Whole Numbers and Decimals 19 33. Critique Reasoning Henrieta divided 0.80 by 20 34. Which brand of fruit as shown. Is her work correct? If not, explain why snacks costs less per and give a correct response. pound? How much less? Brand A Brand B _____ 0.40 15 lb 25 lb 20⟌0.80 $16.20 $22.25 _ – 80 0 35. Be Precise How many times as much does each item cost in 2010 as in 1960? 1960 2010 Movie Ticket Item Cost Cost Regular Popcorn Movie Ticket $0.75 $9.75 Regular Popcorn $0.25 $4.10 Regular Drink Regular Drink $0.35 $3.08 36. Higher Order Thinking Kendra has 5.5 pounds 37. You and a friend are paid $38.25 for doing yard of popcorn and wants to package it equally in work. You worked 2.5 hours and your friend 50 bags. How can she use place-value reasoning to worked 2 hours. You split the money according find the amount of popcorn to put in each bag? to the amount of time each of you worked. How much is your share of the money? Explain. Assessment Practice 38. What is the value of the expression 1,248 ÷ 25? 39. Which expression has the same solution as 3,157 ÷ 41? 𝖠 49 𝖠 1,852 ÷ 24 𝖡 49 R 9 𝖡 1,928 ÷ 25 𝖢 49.9 𝖢 2,079 ÷ 27 𝖣 49 R 23 𝖣 2,184 ÷ 28 20 1-2 Fluently Divide Whole Numbers and Decimals Lesson 1-3 Multiply Fractions Solve & Discuss It! ACTIVITY Go Online The art teacher gave each student half of a sheet of paper. Then she asked the students to color one fourth of their pieces of paper. What part of the original sheet did the students color? I can… use models and equations to multiply fractions and mixed Model with Math How can you use numbers. a picture to represent the problem? Focus on math practices Reasoning Should your answer be less than or greater than 1? Explain. 21 ? Essential Question How can you multiply fractions VISUAL LEARNING ASSESS and mixed numbers? Scan for EXAMPLE 1 Multiply Unit Fractions Multimedia There was __ 14 of a pan of lasagna left. Tom ate 1 __ 3 of this amount. What fraction of a whole pan of lasagna did Tom eat? To find a part of a whole, multiply to solve the problem. 13 × __ Find __ 14 . ONE WAY Divide one whole into fourths. ANOTHER WAY 1 into Divide __ Divide each of the other Shade 1 of the 3 rows 4 3 equal parts. 14 s into 3 equal parts. __ 31 . yellow to represent __ Shade 1 of the 4 columns red 14 . to represent __ 1 12 parts make one whole, so one part is __ 12 . 1 × __ __ 1 × 1 = ___ 1 = _____ 1 31 × __ The orange overlap shows the product __ 14 . 3 4 3×4 12 1 out of 12 parts are shaded orange. 1 of a whole pan of lasagna. Tom ate ___ 12 1 × __ __ 1 × 1 = ___ 1 = _____ 1 3 4 3×4 12 1 of a whole pan of lasagna. Tom ate ___ 12 Try It! 14 × __ Find __ 15 using the area model. Explain. 1 __ 1 of rows 4 1 __ 1 of columns 5 Convince Me! 14 × __ Why is the product of __ 15 less than each factor? 22 1-3 Multiply Fractions EXAMPLE 2 Multiply Fractions ACTIVITY ASSESS Find __ 3 using a number line. 2 × __ 1 1 3 4 3 3 __13 means 1 of 3 equal parts, so __ 34 is __ 13 of __ 14 . __ 23 means 2 of 3 equal parts, so __ 34 is 2 times __ 23 of __ 14 . 1 2 3 0 4 4 4 1 23 × __ __ 34 = __ 6 12 12 or __ Try It! 34 × __ Find __ 46 using the number line. Explain. 1 2 3 4 5 0 6 6 6 6 6 1 EXAMPLE 3 Multiply Mixed Numbers 12 × 2 __34 . Find 7__ 34 is about 8 times 3. Estimate first. 7 __12 times 2__ So, the answer should be about 24. ONE WAY You can use an area model to find ANOTHER WAY You can use an equation to find the partial products. Then add to find the final the product. Rename the mixed numbers and product. then multiply. 1 7 __ 3 = ___ 1 × 2 __ 15 × ___ 11 7 2 2 4 4 2 165 = ____ 1 8 2 2 × 7 = 14 2× 2 =1 = 20 __ 5 8 3 3 21 3 1 3 or 5 1 × 2 = 8 Because 20 __58 is close to the estimate of 24, the 4 4 ×7= 4 4 4 answer is reasonable. 3 = 1 + __ 14 + 1 + 5 __ 4 8 3 = 20 __ 2 + __ 14 + 1 + 5 __ 5 8 8 8 14 is renamed 5__ 5__ 28 . Try It! A clothing factory makes T-shirts. If each machine makes 13 T-shirts per hour, how many T-shirts does one machine 3__ make in 4__ 12 hours? Write and solve an equation. 1-3 Multiply Fractions 23 KEY CONCEPT KEY CONCEPT You can find the product of fractions or mixed numbers. Multiply the numerators. 2 × __ __ 2 × 3 = ___ 3 = _____ 6 or ___ 3 1 × 1 __ 3__ 10 × __ 1 = ___ 10 × 3 = ___ 3 = ______ 30 or 5 5 4 5×4 20 10 3 2 3 2 3×2 6 Multiply the denominators. Rename mixed numbers as fractions. Do You Understand? Do You Know How? 1. Essential Question How can you 56 × __ 7. Find __ 12 . Use the model multiply fractions and mixed numbers? to help solve. 3 × __ 8. Find __ 4 . 4 9 2. Reasoning Is the product of __ 36 × __ 54 equal to the 34 × __ product of __ 56 ? Explain. In 9–16, find each product. 2 × __ 9. __ 1 5 × __ 10. __ 1 3 2 9 9 3. Construct Arguments Why is adding __ 39 and __ 69 different from multiplying the two fractions? 11. ___ 3 7 × __ 1 × __ 12. __ 1 10 4 3 4 5 × __ 13. __ 3 3 × ___ 14. __ 11 6 7 5 12 12 of a pan of cornbread left from a 4. Tina has __ 4 × __ 15. ___ 2 3 × __ 16. __ 2 dinner party. She eats __ 12 of the leftover part the 10 5 4 9 next night. How much of the whole pan does Tina eat? Write and solve an equation. In 17 and 18, estimate the product. Then complete the multiplication. 5. Construct Arguments Explain how you would multiply 5 × 2 __12 . 3 × 8 = ______ 17. 2 __ 8 = × __ 4 4 1 6. In Example 1, find the fraction of a whole pan 78 of of lasagna that Tom ate if he started with __ 1 × 1 __ 1 = _____ 18. 4 __ × _____ = a pan. 2 4 2 4 24 1-3 Multiply Fractions PRACTICE TUTORIAL Name: Practice & Problem Solving Scan for Multimedia In 19 and 20, find each product. Shade the model to help solve. 5 1 × __ 19. __ 3 6 2 × ___ 20. __ 1 3 12 In 21–28, find each product. 7 × __ 21. __ 1 2 × ___ 22. __ 1 5 × __ 23. __ 7 3 1 × __ 24. __ 8 2 5 12 7 9 2 4 1 × __ 25. __ 7 5 × ___ 26. __ 9 1 × __ 27. __ 1 3 1 × __ 28. __ 4 8 6 10 4 8 3 7 In 29–36, estimate the product. Then find each product. 1 × 4 __ 29. 2 __ 1 3 × 8 __ 30. __ 1 1 × 3 __ 31. 1 __ 1 1 × __ 32. 3 __ 2 6 2 4 2 8 3 5 3 1 × 6 33. 3 __ 1 × 3 34. 5 __ 3 × 4 35. 2 __ 1 × 5 __ 36. 4 __ 1 4 3 8 8 2 In 37 and 38, use the diagram at the right. Tremont Trail 37. Linda walked __34 of the length of the Tremont Trail before 1 Seton Trail 3 2 miles 1 stopping for a rest. How far had Linda walked on the trail? 14 miles Wildflower Trail 3 2 8 miles 38. The city plans to extend the Wildflower Trail to make it 12 times its current length in the next 5 years. How 2__ long will the Wildflower Trail be at the end of 5 years? 1-3 Multiply Fractions 25 39. The world’s smallest gecko is __ 34 inch long. 40. Higher Order Thinking In Ms. Barclay’s An adult male Western Banded Gecko is classroom, __25 of the students play chess. Of the 13 times as long. How long is an adult male 7__ students who play chess, __ 56 also play sudoku. If there are 30 students in Ms. Barclay‘s class, Western Banded Gecko? how many play chess and sudoku? 41. The Boca Grande Causeway in Florida is about 78 is multiplied by __ 42. If __ 45 , will the product be 49 times as long as the Golden Gate Bridge 1__ greater than either of the two factors? Explain. in San Francisco. The Golden Gate Bridge is about 9,000 feet long. About how long is the Boca Grande Causeway? 34 43. Be Precise To amend the U.S. Constitution, __ 34 of a bottle of 44. A scientist had __ of the 50 states must approve the amendment. a solution. She used __ 16 of the If 35 states approve an amendment, will the solution in an experiment. Constitution be amended? How much of the bottle did she use? 12 of all eligible voters cast 45. In the voting for City Council Precinct 5, only __ Fraction of votes. What fraction of all eligible voters voted for Shelley? Morgan? Candidate Votes Received Who received the most votes? 3 __ Shelley 10 5 __ Morgan 8 Assessment Practice 46. Which of these equations is equivalent 47. Which of these equations is equivalent to 12 × 3 __15 = 4 __12 ? to 1__ __34 × 8 __15 = 6 __ 3 20 ? Select all that apply. 𝖠 4 __ 12 ÷ 3 __15 = 1 __12 3 ÷ 8 __ __ 3 1 = 6 ___ 4 5 20 3 ÷ __ 6 ___ 3 = 8 __ 1 𝖡 1 __ 12 ÷ 4 __12 = 3 __15 20 4 5 3 ÷ 8 __ 6 ___ 3 1 = __ 𝖢 1 __ 12 ÷ 3 __15 = 4 __12 20 5 4 3 ÷ 6 ___ __ 3 = 8 __ 1 𝖣 3 __ 15 ÷ 4 __12 = 1 __12 4 20 5 3 = __ 1 ÷ 6 ___ 8 __ 3 5 20 4 26 1-3 Multiply Fractions TOPIC 1 Name: MID-TOPIC 1. Vocabulary How can you use a compatible number to estimate a CHECKPOINT quotient when dividing a decimal by a whole number? Lesson 1-2 2. Keaton is building a rectangular tabletop and wants to put a metal border around the edge. The length of the tabletop is 1.83 meters and the width is 0.74 meter. Use the formula P = 2ℓ + 2wto find the perimeter of the tabletop. Lesson 1-1 Norbert’s Nursery 3. Norbert’s Nursery is having a sale. Flats of flowers are priced as marked, Flower Price per Flat including tax. Jake buys 2 flats of petunias, 3 flats of daisies, and 1 flat Petunia $5.25 of begonias. If he pays with a $50 bill, how much change should Jake Daisy $7.65 receive? Lesson 1-1 Begonia $8.40 4. Marguerite is selling space in an advertisement book for a community fund-raising event. Each __ 14 page in the book costs $15.50. What is the cost 34 page? Lesson 1-1 for __ 𝖠 $62.00 𝖡 $46.50 𝖢 $20.67 𝖣 $11.63 5. What is the value of 170 ÷ (4 × 5)? Lesson 1-2 6. Lucia walks 2 __34 miles on Monday. On Monday, she walks 1 __1 times farther 2 than on Tuesday. Which equation can be used to find how far Lucia walks on Tuesday? Lesson 1-3 𝖠 2 __34 × 1 __12 = 4 __18 𝖡 2 __ 34 + 1__ 12 = 4__ 14 𝖢 2 __34 ÷ 1 __12 = 1 __56 𝖣 1 __12 ÷ 2 __34 = ___ 6 11 How well did you do on the mid-topic checkpoint? Fill in the stars. Topic 1 Use Positive Rational Numbers 27 TOPIC 1 MID-TOPIC PERFORMANCE TASK Nyan Robotics Team received their challenge for the year and Parts List has to buy parts to build their robot for competitions. Part Cost per Part Beam $5.95 Channel $8.50 PART A Motor controller $99.75 Team members Eric and Natalia secure a grant for $75.00 to buy beams Motor mount $17.55 and channels. If the team needs 3 beams and 6 channels, will the grant Gear $12.15 cover the cost? If so, how much of the grant will remain? Sprocket $3.00 Wheel $18.90 Axle $4.35 PART B Team members Corinne, Kevin, and Tomas decide to share the cost of 2 motor controllers and 4 wheels equally. How much does each member need to contribute? PART C Nyan Robotics has a budget of $99 to buy sprockets, axles, and gears. 23 of the budget on sprockets, how much money from the If they spend __ budget remains to buy axles and gears? 28 Topic 1 Use Positive Rational Numbers 3-ACT MATH 3-Act Mathematical Modeling: Stocking Up Go Online ACT 1 1. After watching the video, what is the first question that comes to mind? 2. Write the Main Question you will answer. 3. Construct Arguments Predict an answer to this Main Question. Explain your prediction. 4. On the number line below, write a number that is too small to be the answer. Write a number that is too large. Too small Too large 5. Plot your prediction on the same number line. Topic 1 3-Act Mathematical Modeling 29 ACT 2 6. What information in this situation would be helpful to know? How would you use that information? 7. Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically. 8. Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question. 9. What is your answer to the Main Question? Is it higher or lower than your prediction? Explain why. 30 Topic 1 3-Act Mathematical Modeling ACT 3 10. Write the answer you saw in the video. 11. Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference? 12. Make Sense and Persevere Would you change your model now that you know the answer? Explain. Topic 1 3-Act Mathematical Modeling 31 ACT 3 Extension Reflect 13. Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question? 14. Reasoning How did you represent the situation using symbols? How did you use those symbols to solve the problem? SEQUEL 15. Model with Math The store purchases boxes of pasta for $0.82 and cans of sauce for $1.62. How much profit does the store make from this purchase? 32 Topic 1 3-Act Mathematical Modeling Lesson 1-4 Explore It! ACTIVITY Understand Students are competing in a 4-kilometer Division with relay race. There are 10 runners. Fractions Go Online I can… use models and equations to represent fraction division. Start: 0 km Each runner runs 2 km. 5 Finish: 4 km A. Use the number line to represent the data for the race. 0 1 2 3 4 B. Use multiplication or division to describe your work on the number line. Focus on math practices Model with Math Describe what a number line would look like if there 12 kilometer in a 5-kilometer race. were 10 runners each running __ 33 ? Essential Question division of fractions? How can you represent VISUAL LEARNING ASSESS Scan for EXAMPLE 1 Divide Whole Numbers by Fractions Multimedia Mr. Roberts has a board that is 3 feet long. He plans to cut the 34 foot long to build a set of shelves. board into pieces that are each __ Use Structure How How many shelves can he make? 34 s are in 3? many __ 0 ft 1 ft 2 ft 3 ft ONE WAY Write 3 as a fraction with a denominator ANOTHER WAY Use a number line to show 12 of 4, __ 4 . Think of division as repeated subtraction. 3 feet. Divide it into __34 -foot parts. 12 ft 4 Board 0 1 2 3 3 s shelves Each shelf ft 4 3 = 4. So, 3 ÷ __ 4 9 12 __ ___ 6 __ 3 __ 4 4 4 4 When the divisor is less than 1, the __3 − − __ 3 − __ 3 3 − __ 4 4 4 4 quotient is greater than the dividend. 9 __ 6 __ 3 __ 0 4 4 4 Mr. Roberts can make 4 shelves. Mr. Roberts can make 4 shelves. Try It! 23 -foot-long pieces can be cut from A board is 6 feet long. How many __ the board? Use the number line to show your work. Convince Me! Why is the number of pieces that can be cut from the board greater than the number of feet in the length of the board? 34 1-4 Understand Division with Fractions Divide Fractions by EXAMPLE 2 Whole Numbers ACTIVITY ASSESS How much cake will each person get if 3 friends decide 12 ÷ 3. to share half a cake equally? Find __ 1 . Draw a picture to show __ 1 into 3 equal parts. Divide __ 1 of the whole. Each part is __ 2 2 6 1 __ 1 ÷ 3 __ 1 1 __ ÷ 3 = __ 2 2 2 6 1 2 Each person will get 16 of the cake. Try It! 2 ÷ 4. Make a diagram to find __ 3 Use Relationships to Divide EXAMPLE 3 Whole Numbers by Fractions You can use what you know about dividing fractions to find and use 4 = 28 × __ 8 ÷ __ 1 = 2 a pattern. Look at the division and multiplication sentences at the 1 4 23 . right. Find and use a pattern to solve 4 ÷ __ 1 = 105 × __ 2 = 10 5 ÷ __ 2 1 The pattern shows that when you divide by a fraction, you get the same result as when you multiply by its reciprocal. 3 3 = 43 × __ ÷ __ 4 = 4 4 3 Rewrite the problem as a Two numbers whose product is 4 ÷ __ 3 2 = 4 × __ multiplication problem using 3 2 1 are called reciprocals of each the reciprocal of the divisor. other. If a nonzero number is = __ 3 4 × __ named as a fraction __ a , then its 1 2 b b __ 12 or 6 reciprocal is a . = ___ 2 Try It! 3 . Use the pattern in the table above to find 8 ÷ __ 4 3 = 8 × 8 ÷ __ = 4 1-4 Understand Division with Fractions 35 KEY CONCEPT KEY CONCEPT To divide a whole number by a fraction: Write the whole 4 = ___ 14 ÷ __ 14 ÷ __ 4 number as a fraction. 7 1 7 Multiply the whole number by
