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STUDY LINK 4 1 Equivalent Fractions 105 73 74 Name Date Time Copyright © Wright Group/McGraw-Hill Find an equivalent fraction by multiplying. 1. 4 5 2. 17 0 3. 1 4 4. 2 3 5. 5 4 6. 2 2 Find an equivalent fraction by dividing. 7. 19 2 8. 12 00 0 9. 14 6 10. 3 10 2 11. 1 50 0 12. 1 26 4 Write 3 equivalent fractions for each number. 13. 1 3 14. 17 05 0 15. 6 16. 1 52 Write each fraction in simplest form. 17. 18 6 18. 6 9 19. 13 5 20. 1 20 5 21. 16 6 22. 1 44 9 Find the missing numbers. 23. 1 5 1x 5 24. 2 3 1y 8 25. 1 25 5 5m 0 x ym Practice Divide. Express the remainder as a fraction in simplest form. 26. 246 54 27. 257 30 28. 144 10LESSON 4 1 Name Date Time Factor Rainbows 106 Copyright © Wright Group/McGraw-Hill When listing the factors of a number, you need to be certain that you have included all the factors in your list. Creating a factor rainbowis one way to do this. A factor rainbow is an organized list of factor pairs. Example:factor rainbow for 24 Every number is divisible by 1. Because 1 º 24 24, use an arc to show that 1 and 24 are paired. Now try dividing by 2. Because 2 º 12 24, use an arc to pair 2 and 12. Continue your divisibility tests by moving to 3, which is the next factor greater than 2. 24 is divisible by 3, so 3 º 8 24. Pair 3 and 8. From the arcs you have drawn, you can see that all remaining factors must be between 3 and 8. Try dividing 24 by 4. Because 4 6 24, use an arc to pair 4 and 6. Any remaining factors must be between 4 and 6. The only whole number between 4 and 6 is 5. Notice that 5 does not divide into 24 evenly, so your rainbow is complete. Use the example above when completing the factor rainbow for 36. Because 36 is a square number, one of the factors (6) is paired with itself. 124 1 2 12 24 123 81224 1234 681224 1234 681224 24 2 16691236
LESSON 4 1 Name Date Time Factor Rainbows continued 107 Copyright © Wright Group/McGraw-Hill Use the examples on page 106 to help you complete the factor rainbow for each number. 1. factor rainbow for 18 2. factor rainbow for 48 3. factor rainbow for 12 4. factor rainbow for 32 5. factor rainbow for 40 6. factor rainbow for 64 13 918 2 146 16848 48
LESSON 4 1 Name Date Time Applications of the GCF 108 Copyright © Wright Group/McGraw-Hill Some real-world problems involve finding the greatest common factor (GCF) of a set of numbers. Solve. 1. Tyrone is preparing snack packs for the class field trip. He has 60 bags of chips and 90 bottles of fruit juice. Each pack should have the same number of bags of chips and the same number of bottles of fruit juice. What is the greatest number of snack packs that Tyrone can make with no bags or bottles left over? The greatest number of snack packs that he can make is . Each snack pack will have bags of chips and bottles of fruit juice. 2. Carla has 30 blue beads, 60 red beads, and 72 white beads. What is the maximum number of friends to whom Carla can give the same number of beads and have no beads left over? The maximum number of friends to get beads is . Each friend will get blue beads, red beads, and white beads. 3. Ms. Mendis wants to split her class into groups for a bridge-building contest. There are 32 students in her class. She has 16 bottles of wood glue, 1,200 craft sticks, and 24 jars of paint. What is the greatest number of groups that Ms. Mendis can make so each group gets the same number of supplies and no supplies are left over? The greatest number of groups that she can make is . Each group will get bottles of wood glue, craft sticks, and jars of paint.
STUDY LINK 4 2 Comparing and Ordering Fractions 109 75–77 Name Date Time Copyright © Wright Group/McGraw-Hill Write , , or to make a true number sentence. For each problem that you did not solve mentally, show how you got the answer. 1. 4 5 2 5 2. 3 8 1 3 3. 3 4 1 27 0 4. 1 29 0 19 09 0 5. 4 7 14 0 6. 2 3 7 9 7. Circle each fraction that is less than 1 2. 3 6 16 0 1 3 2 5 16 1 1 22 5 8. Write the fractions in order from smallest to largest. 1 3 1 15 6 17 4 1 5 2 5 4 59 0 16 0 11 2 Practice 9. 13.987 4.09 10. 5.9 2.068 11. 0.9 0.077 12. 8 3.643
LESSON 4 2 Name Date Time Benchmark Fractions 110 Copyright © Wright Group/McGraw-Hill Cut out the fraction cards. Remove the benchmark cards 0, 1 2, and 1. Arrange these benchmark cards in order from left to right, leaving enough space to place fraction cards between them. Sort the remaining 9 cards into 3 piles—fractions closest to 0, fractions closest to 1 2, and fractions closest to 1. Fill the space between 0, 1 2, and 1 with your sorted cards, positioning them in order from smallest to largest. Record the order in which you placed the cards on the number line. 1 2 48 100 1 5 3 4 1 4 1 3 2 3 1 20 6 10 9 10 01 01 1 2
111 Copyright © Wright Group/McGraw-Hill LESSON 4 3 Name Date Time Fractions of a Square 1. What fraction of the large square is … a. Square A? b. Triangle B? c. Triangle E? d. Parallelogram G? 2. What fraction of the large square are the following pieces, when put together? Write a number sentence to show your answer. a. Triangles B and C b. Triangles E and F c. Square A and Triangle C d. Square A and Triangle E e. Triangles E and B f. Square A and Parallelogram G g. Triangles D, E, and F and Parallelogram G B DCA E F G Math Message
LESSON 4 3 Name Date Time Fractions of a Square continued 112 Copyright © Wright Group/McGraw-Hill A A B B B B E E E E E E E E A A B B B B E E E E E E E E Cut out each shape.
LESSON 4 3 Name Date Time Fractions of a Square continued 113 Copyright © Wright Group/McGraw-Hill Cut out each shape. G G G G
STUDY LINK 4 3 Adding and Subtracting Fractions Copyright © Wright Group/McGraw-Hill 114 Name Date Time Add or subtract. Write each answer in simplest form. If possible, rename answers as mixed numbers or whole numbers. 1. 1 3 1 6 2. 3 4 15 6 3. 9 4 2 5 4. 2 9 4 9 5. 1 6 3 4 6. 15 2 3 4 7. 7 9 2 5 8. 5 4 3 4 9. 7 8 2 4 10. 5 3 2 5 11. 1 11 2 17 2 12. 4 5 13 0 13. 1 85 23 4 14. 3 5 1 2 Practice Solve mentally. 15. 3 0.30 16. 0.60 0.02 17. 2 0.02 83
You can use fraction strips to find common denominators, sums, and differences. Study the examples below and then solve the problems. Example 1:Solve. 1 2 1 3 2 6 Step 1Use fraction strips to model both fractions. Step 2Rename each fraction using a common denominator. Step 3Write the sum. 3 6 2 6 5 6 Use your fraction strips to find each of the following sums. 1. 1 4 2 3 17 2 2. 17 2 1 3 17 2 3. 3 4 1 8 Example 2:Solve. 2 3 1 4 17 2 Step 1Use fraction strips to model both fractions. Rename each fraction using a common denominator. Step 2Remove fraction strips to find the difference. Step 3Write the difference. 2 3 1 4 15 2 Use fraction strips to find each of the following differences. 4. 1 2 1 3 2 6 5. 7 8 1 4 1 8 6. 1 3 1 4 LESSON 4 3 Name Date Time Using Fraction Strips to Add and Subtract 115 Copyright © Wright Group/McGraw-Hill 1 2 1 3 1 6 1 6 1 6 1 6 1 6 1 2 1 3 1 3 1 3 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 4 1 12 1 12 1 12 1 3 1 3 1 12 1 4 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12
LESSON 4 3 Name Date Time Paper Pool 116 Copyright © Wright Group/McGraw-Hill Paper Pool is played on a rectangular grid. An imaginary ball is hit from the lower left pocket of the rectangular grid at a 45angle. The ball travels at a 45angle along the diagonals of the squares making up the grid. When the ball hits a side of the grid, it bounces off that side at a 45angle and continues to travel along a diagonal that has not already been crossed. Play ends when the ball lands in any of the corner pockets. The length of the ball’s path is found by counting the number of diagonals of the individual squares that the ball crosses before landing in a pocket. 1. Draw the path of the ball on each rectangular grid below. Then record the length of the ball’s path. a. b. Length of ball’s path: units Length of ball’s path: units c. d. Length of ball’s path: units Length of ball’s path: units 2. For each grid in Problem 1, compare the length of the ball’s path to the dimensions of the grid. For example, the length of the ball’s path for a 4 6 grid is 12. Describe any patterns you notice. 1.Ball is hit from lower left pocket and travels at a 45° angle across the diagonals of the grid squares. 4.Because the ball crossed the diagonals of 12 squares before landing in a pocket, the length of its path is 12 units. 3.Ball hits a pocket and play ends. 2.Ball hits side(s) and bounces off at a 45° angle. Length of ball’s path:12 units Dimensions of grid:4 units by 6 units
LESSON 4 4 Name Date Time Math Message 117 Copyright © Wright Group/McGraw-Hill Cut out the ruler below. Use it to measure line segment ABto the nearest 11 6inch. length of AB— 0123 4 5 inches AB LESSON 4 4 Name Date Time Math Message Cut out the ruler below. Use it to measure line segment ABto the nearest 11 6inch. length of AB— 0123 4 5 inches AB Copyright © Wright Group/McGraw-Hill
STUDY LINK 4 4 ,Fractions and Mixed Numbers Copyright © Wright Group/McGraw-Hill 118 Name Date Time 1. In a national test, eighth-grade students answered the problem shown in the top of the table at the right. Also shown are the 5 possible answers they were given and the percent of students who chose each answer. a. What mistake do you think the students who chose C made? b. Explain why B is the best estimate. 2. A board is 6 3 8inches long. Verna wants to cut enough so that it will be 5 1 8inches long. How much should she cut? (unit) 3. Tim is making papier-mâché. The recipe calls for 1 3 4cups of paste. Using only 1 2-cup, 1 4-cup, and 1 3-cup measures, how can he measure the correct amount? Add or subtract. Write your answers as mixed numbers in simplest form. Show your work on the back of the page. Use number sense to check whether each answer is reasonable. 4. 31 41 1 4 5. 4 2 1 4 6. 12 3 2 3 7. Circle the numbers that are equivalent to 2 3 4. 1 7 4 6 4 3 7 1 41 Practice Solve mentally. 8. 5 º 18 9. 6 º 41 10. 9 º 48 11. 7 º 45 Estimate the answer to 1 12 3 7 8. You will not have enough time to solve the problem using paper and pencil. Possible Percent Who Chose Answers This Answer A.17% B.2 24% C.19 28% D.21 27% E.I don’t know. 14%
LESSON 4 4 Name Date Time Fraction Counts and Conversions 119 Copyright © Wright Group/McGraw-Hill Most calculators have a function that lets you repeat an operation, such as adding 1 4to a number. This is called the constant function. To use the constant function of your calculator to count by 1 4s, follow one of the key sequences below, depending on the calculator you are using. Calculator A Press: Display: 5 1 1 4 Press: Display: 5 4 Calculator B Press: Display: 5 4 Press: Display: 1 1 4 K 0 + + 4 1 Op1 Op1 Op1 Op1 Op1 Op1 d 4 n 1 + Op1 1. Using a calculator, start at 0 and count by 1 4s to answer the following questions. a. How many counts of 1 4are needed to display 6 4? b. How many counts of 1 4are needed to display 1 1 2? 2. Use a calculator to convert mixed numbers to improper fractions or whole numbers. a. 23 4 b. 17 4 c. 24 4 d. 31 42 3. How many 1 4s are between the following numbers? a. 3 4and 2 b. 6 4and 2 3 4 c. 13 4and 4 d. 3 and 4 1 2
LESSON 4 5 Name Date Time Math Message 120 Copyright © Wright Group/McGraw-Hill Add or subtract. Be ready to explain your solution strategies for Problems 6 and 8. 1. 2. 3. 4. 5. 6. 7. 8. 217 2 1 1 2 217 2 1 16 2 21 4 3 1 8 22 8 3 1 8 51 4 3 3 4 10 7 9 4 1 9 63 8 5 7 8 21 5 3 3 5 LESSON 4 5 Name Date Time Math Message Add or subtract. Be ready to explain your solution strategies for Problems 6 and 8. 1. 2. 3. 4. 5. 6. 7. 8. 217 2 1 1 2 217 2 1 16 2 21 4 3 1 8 22 8 3 1 8 51 4 3 3 4 10 7 9 4 1 9 63 8 5 7 8 21 5 3 3 5 Copyright © Wright Group/McGraw-Hill
STUDY LINK 4 5 Mixed-Number Practice 121 Name Date Time Copyright © Wright Group/McGraw-Hill Solve mentally. 12. 11 24 2 32 1 25 1 3 13. 4.5 3.4 7.5 2.5 14. $2.35$9.60$8.05$1.99 15. 55 83 3 42 1 48 3 8 1. Answer the following questions about the rectangle shown at the right. Include units in your answers. a. What is the perimeter? b. If you were to trim this rectangle so that it was a square measuring 1 1 4inches on a side, how much would you cut from the base? from the height? 2. Michael bought 1 peck of Empire apples, 1 peck of Golden Delicious apples, a 1 2-bushel of Red Delicious apples, and 1 1 2bushels of McIntosh apples. a. How many bushels of apples did he buy in all? b. Michael estimates that he can make about 12 quarts of applesauce per bushel of apples. About how many quarts of applesauce can he make from the apples he bought? Add or subtract. Show your work and estimates on the back of the page. 3. 21 31 2 3 4. 61 35 2 3 5. 41 2 2 3 6. 6 5 4 9 7. 43 82 3 4 8. 31 42 3 4 9. 9 2 2 5 10. 41 42 5 6 11. 51 42 17 0 1 21 in. 3 42 in. Practice1 peck 1 4bushel 84–86
LESSON 4 5 Name Date Time Representing Mixed Numbers 122 Copyright © Wright Group/McGraw-Hill Study the example row. Then complete the table. Use Row 5 to represent a mixed number of your choice. Whole As As Mixed As Number Improper As Picture Number Sum as Fraction Fraction Quotient Example: 1 1 3 1 1 3 3 3 1 3 4 3 4 3 1. 6 6 5 6 11 6 2. 3. 4. 1 42 3 4 15 4 5.
STUDY LINK 4 6 Fraction Multiplication 123 88 89 Name Date Time Copyright © Wright Group/McGraw-Hill Use the fraction multiplication algorithm below to solve the following problems. Fraction Multiplication Algorithm a bº dc ba ºº dc 1. 3 5º 2 4 2. 3 7º 5 9 3. 5 º 3 8 4. 1 11 2º 1 4 5. 5 6º 7 8 6. 13 0º 17 0 7. 2 5º 7 9 8. 4 7º 8 9. 12 º 18 1 10. South High beat North High in basketball, scoring 4 5of the total points. Rachel scored 1 4of South High’s points. What fraction of the total points did Rachel score? 11. Josh was making raisin muffins for a party. He needed to triple the recipe, which called for 3 4cup raisins. How many cups of raisins did he need? 12. At Long Middle School, 7 8of the sixth graders live within 1 mile of the school. About 2 3of those sixth graders walk to school. None who live a mile or more away walk to school. About what fraction of the sixth graders walk to school? 13. a. For Calista’s 12th birthday party, her mom will order pizza. 3 4of the girls invited like vegetables on their pizza. However, 1 3of those girls won’t eat green peppers. What fraction ofall the girls will eat a green-pepper-and- onion pizza? b. If 12 girls are at the party (including Calista), how many girls will not eat a green-pepper-and-onion pizza? Practice Solve. 14. 12 º 0.75 15. 0.2 º 0.5 16. 0.4 º 0.25
124 Copyright © Wright Group/McGraw-Hill LESSON 4 6 Name Date Time Modeling Fraction Multiplication You can use an area model to find a product. Example: 1 4º 1 3 Shade 1 4of Shade 1 3of The product is the area the grid this way: the grid this way: that is double-shaded. Since 11 2of the grid is double-shaded, 1 4º 1 3 11 2. Shade each factor and then find the product. 1. 2. 1 6º 4 5 3 4 0 3 4º 7 8 2 31 2 3. 4. 2 3º 4 7 2 8 1 5 8º 1 2 15 6 5. Which of the following represents a general pattern for the special cases in Problems 1–4? Circle the best answer. A a bº dc a cºº db B a bº dc a cºº dd C a bº dc ba ºº dc D a bº dc ba dc 1 4 1 3 1 4 2 3 4 7 5 8 1 2 4 5 1 6 3 4 7 8 89
LESSON 4 6 Name Date Time A Nature Hike Problem 125 Copyright © Wright Group/McGraw-Hill Pavan, Jonathan, Nisha, and Gloria walked on the sixth-grade nature hike. They finished 1 2the length of the trail in the first hour. Then they slowed down. In the second hour, they walked 1 2the distance that was left. In the third hour, they moved even more slowly and walked only 1 2the remaining distance. If they continue slowing down at this rate, what fraction of the trail will they walk in their fourth hour of hiking? Explain or show how you solved the problem. LESSON 4 6 Name Date Time A Nature Hike Problem Pavan, Jonathan, Nisha, and Gloria walked on the sixth-grade nature hike. They finished 1 2the length of the trail in the first hour. Then they slowed down. In the second hour, they walked 1 2the distance that was left. In the third hour, they moved even more slowly and walked only 1 2the remaining distance. If they continue slowing down at this rate, what fraction of the trail will they walk in their fourth hour of hiking? Explain or show how you solved the problem. Copyright © Wright Group/McGraw-Hill
LESSON 4 7 Name Date Time Math Message 126 Copyright © Wright Group/McGraw-Hill Write the following mixed numbers as fractions. 1. 31 4 2. 23 8 3. 57 8 4. 35 6 5. 413 0 6. 213 6 Write the following fractions as mixed numbers. 7. 1 29 8. 1 37 9. 2 47 10. 2 81 11. 2 54 12. 3 94 13. Multiply. Show your work on the back of the page. Be prepared to explain how you found your answer. 3 3 8º 1 2 5 LESSON 4 7 Name Date Time Math Message Write the following mixed numbers as fractions. 1. 31 4 2. 23 8 3. 57 8 4. 35 6 5. 413 0 6. 213 6 Write the following fractions as mixed numbers. 7. 1 29 8. 1 37 9. 2 47 10. 2 81 11. 2 54 12. 3 94 13. Multiply. Show your work on the back of the page. Be prepared to explain how you found your answer. 3 3 8º 1 2 5 Copyright © Wright Group/McGraw-Hill
LESSON 4 7 Name Date Time A Photo Album Page 127 Copyright © Wright Group/McGraw-Hill
LESSON 4 7 Name Date Time Album Photos 128 Copyright © Wright Group/McGraw-Hill 21 2in. 4 1 8in.2 1 8in. 2 1 8in. 2 1 8in. 4 in. 3 5 8in. 2 3 4in.
STUDY LINK 4 7 Multiplying Mixed Numbers 129 Name Date Time Copyright © Wright Group/McGraw-Hill Rename each mixed number as a fraction. 1. 14 5 2. 26 6 3. 52 3 4. 31 2 Rename each fraction as a mixed number or whole number. 5. 1 42 6. 3 83 7. 1 65 8. 2 30 Multiply. Write each answer in simplest form. If possible, write answers as mixed numbers or whole numbers. 9. 5 º 3 5 10. 21 3º 1 4 5 11. 5 6º 2 1 2 12. 11 6º 4 2 3 13. 33 4º 2 1 8 14. 71 2º 2 2 3 Solve mentally. 15. 8 º 3.5 16. 12 º 5.25 17. 4.2 º 15 Practice 71 72 90
LESSON 4 7 Name Date Time Modeling Multiplication 130 Copyright © Wright Group/McGraw-Hill An area model can help you keep track of partial products. The area of each smaller rectangle represents a partial product. Example:2 º 4 3 4 Find the area of each smaller rectangle. 2 º (4 3 4) (2 º 4) (2 º 3 4) 2 º 4 82 º 3 41 1 2 Then add the two areas to find the area of the largest rectangle. 8 1 1 29 1 2 So, 2 º 4 3 49 1 2 Find the area of each smaller rectangle. Then add the areas. 1. 21 4º 5 (2 1 4) º 5 2 º 5 1 4º 5 So, 2 1 4º 5 2. 22 3º 3 3 4(2 2 3) º (3 3 4) 2 º 3 2 º 3 4 2 3º 3 2 3º 3 4 So, 2 2 3º 3 3 4 3. 17 8º 3 1 2(1 7 8) º (3 1 2) 1 º 3 1 º 1 2 7 8º 3 7 8º 1 2 So, 1 7 8º 3 1 2 2 º 4 = 8 3 4 º = 1 1 2 3 4 24 2 25 1 4 23 2 3 3 4 3 1 7 8 1 2
LESSON 4 7 Name Date Time Buying an Aquarium 131 Copyright © Wright Group/McGraw-Hill Robert wants to buy an aquarium for his bedroom. Use the dimensions below to figure out whether Robert has enough floor space for a free-standing aquarium after his furniture is in the room. Ignore doors and windows and work with only total floor space. Robert will need enough space to walk around his furniture. Drawing a floor plan might help. How much floor space is available after Robert places the furniture but before he buys the aquarium? Does Robert have enough space for the aquarium? Length (ft) Width (ft) Area (ft 2) Room9 1 2 93 4 Desk3 1 4 21 2 Bed6 1 4 33 4 Dresser3 1 4 21 4 Bookcase1 1 4 31 2 Aquarium 2 1
STUDY LINK 48 Fractions, Decimals, and Percents Copyright © Wright Group/McGraw-Hill 132 53 54 59 60 Name Date Time Fill in the missing numbers below. Then shade each large square to represent all three of the equivalent numbers below it. Each large square is worth 1. 1. 2. 3. 4 5 —10 —— % 6 8 —100 —— % 30% —100 —10 Rename the fractions as decimals. 4. 17 4 5. 6 8 6. 25 0 7. 14 5 Rename the decimals as fractions in simplest form. 8. 0.4 9. 0.10 10. 0.68 11. 0.25 Rename the fractions as percents. 12. 2 55 0 13. 26 4 14. 1 38 0 15. 1 29 0 Rename the percents as fractions in simplest form. 16. 50% —100 17. 40% —100 18. 100% —100 19. 180% —100 Experiment People often don’t realize that fractions, decimals, and percents are numbers. To them, numbers are whole numbers like 1, 5, or 100. Try the following experiment: Ask several adults to name four numbers between 1 and 10. Then ask several children. Keep a record of all responses on the back of this page. How many named fractions, decimals, or percents? Now ask the same people to name four numbers between 1 and 3. Report your findings.
LESSON 48 Name Date Time Renaming Fractions 133 Copyright © Wright Group/McGraw-Hill Fraction: —100 Decimal: Percent:% Fraction: —100 Decimal: Percent:% Fraction: —100 Decimal: Percent:% Fraction: —100 Decimal: Percent:%
STUDY LINK 49 Copyright © Wright Group/McGraw-Hill 134 Name Date Time Decimals, Percents, and Fractions Rename each decimal as a percent. 1. 0.65 2. 0.334 3. 0.02 4. 0.4 5. 2.7 6. 3.09 Rename each percent as a decimal. 7. 27% 8. 53.9% 9. 8% 10. 60% 11. 180% 12. 115% Use division to rename each fraction as a decimal to the nearest hundredth. Then rename the decimal as a percent. 13. 7 80.% 14. 15 20.% 59 60
LESSON 49 Name Date Time Fractions, Decimals, and Percents 135 Copyright © Wright Group/McGraw-Hill Write each fraction or decimal as a percent. 1. 1 51 0 2. 3 5 3. 7 8 4. 0.45 5. 0.745 6. 0.0925 Write each percent as a decimal. 7. 65% 8. 4% 9. 9.2% 10. 15 1 2% 11. 20% 12. 2% 13. Enter your results from Problems 11 and 12 on the appropriate lines below. Then complete the pattern. 200% 20% 2% 0.2% 0.02% 0.002% Percents Greater Than and Less Than 1% You can apply the meaning of percentand a power-of-10 strategy to rename percents greater than or less than 1 percent as equivalent decimals. Example: Write 125% as a decimal. 125% means 125 per hundred. or 1 12 05 0 125 100 1.25.Example: Write 0.15% as a decimal. 0.15% means 0.15 per hundred. or 0 1. 01 05 0.15 100 0.00.15 Write each percent as a decimal. 14. 375% 15. 278% 16. 400 1 2% 17. 0.165% 18. 0.03% 19. 0.005%
LESSON 4 10 Name Date Time Math Message 136 LESSON 4 10 Name Date Time Math Message 1. Find the equivalent decimal and percent for each of the fractions in the table below. Use a mental math and/or paper-and-pencil strategy to complete the table. 2. There were 90 questions on the math final exam. Max correctly answered 72 questions. What percent of the questions did he answer correctly? Fraction Decimal Percent 11 0 1 4 1 23 5 36 0 5 71 5 Copyright © Wright Group/McGraw-Hill Copyright © Wright Group/McGraw-Hill 1. Find the equivalent decimal and percent for each of the fractions in the table below. Use a mental math and/or paper-and-pencil strategy to complete the table. 2. There were 90 questions on the math final exam. Max correctly answered 72 questions. What percent of the questions did he answer correctly? Fraction Decimal Percent 11 0 1 4 1 23 5 36 0 5 71 5
STUDY LINK 4 10 Circle Graphs 137 Name Date Time Use estimation to make a circle graph displaying the data in each problem. (Hint:For each percent, think of a simple fraction that is close to the value of the percent. Then estimate the size of the sector for each percent.) Remember to graph the smallest sector first. 1. According to the 2000 Census, 21.2% 2. In 2004, NASA’s total budget was of the U.S. population was under the $15.4 billion. 51% was spent on Science, age of 15, 12.6% was age 65 or older,Aeronautics, and Exploration. 48.8% was and 66.2% was between the ages of spent on Space Flight Capabilities, and 15 and 64.0.2% was spent on the Inspector General. 3. 98.3% of households in the United States 4. The projected school enrollment for have at least one television. the United States in 2009 is 72 million students. 23.2% will be in college, 22.9% will be in high school, and 53.9% will be in Grades Pre-K–8. Age of U.S. Population Households with TV U.S. School Enrollment NASA Budget Copyright © Wright Group/McGraw-Hill
138 Copyright © Wright Group/McGraw-Hill LESSON 4 10 Name Date Time Estimating and Measuring Sector Size Shade the circles below to represent your estimate of the given percent or fraction. Then check your estimates using the Percent Circle on the Geometry Template. 55% 22% 2 3 1 6 Starting with the smallest sector, use the Percent Circle to create a circle graph that shows 10% blue, 4% red, 60% green, and 26% yellow. Label each sector with the percent it represents.
139 Copyright © Wright Group/McGraw-Hill STUDY LINK 4 11 Name Date Time Percent Problems The results of a survey about children’s weekly allowances are shown at the right. 1. Lincoln School has about 500 students. Use the survey results to complete this table. 2. The sixth grade at Lincoln has about 60 students. Use the survey results to complete this table. A rule of thumb for changing a number of meters to yards is to add the number of meters to 10% of the number of meters. Examples:5 m is about 5 (10% of 5), or 5.5, yd. 10 m is about 10 (10% of 10), or 11, yd. 3. Use this rule of thumb to estimate how many yards are in the following numbers of meters. a. 3 m is about 3 (10% of 3), or , yd. b. 8 m is about 8 (10% of 8), or , yd. c. 20 m is about 20 (10% of 20), or , yd. Amount of Predicted Number of Allowance Students at Lincoln $0 $1–$4 $5 $6 or more Amount of Predicted Number of Sixth- Allowance Grade Students at Lincoln $0 $1–$4 $5 $6 or more Amount of Allowance Percent of Children $0 30% $1–$4 20% $5 25% $6 or more 25% 49 50
140 LESSON 4 11 Name Date Time Modeling Fractional Parts of a Number You can use a diagram to model fractional parts of a number. Example:Find 4 5of $150. First, think about $150 being divided equally among 5 people. One Way Another Way 1 5of $150 $30 5 5of $150 $150 4 5of $150 1 5of $150 $30 4 ( 1 5of $150) 5 5 1 5 4 5 4 $30 $120 $150 $30 $120 4 5of $150 $120 1. Use the diagram to find the amounts. 13 2of $30 19 2of $30 1 3of $30 2 3of $30 1 6of $30 15 2of $30 17 2of $30 1 4of $30 5 6of $30 2. Complete the diagram below. Then find the amounts. 1 8of $48 7 8of $48 3 8of $48 1 4of $48 1 2of $48 3 4of $48 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 $48 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 $2.50 $2.50 $2.50 $2.50 $2.50 $2.50 $2.50 $2.50 $2.50 $2.50 $2.50 $2.50 $30 $30 1 5 1 5 1 5 1 5 1 5 1 5 $30 $30 $30 $30 $150 Total amount Amounts for each Fractional parts 87 Copyright © Wright Group/McGraw-Hill
141 Geometry: Congruence, Constructions, and Parallel Lines In Fourthand Fifth Grade Everyday Mathematics,students used a compass and straightedge to construct basic shapes and create geometric designs. In Unit 5 of Sixth Grade Everyday Mathematics, students will review some basic construction techniques and then devise their own methods for copying triangles and quadrilaterals and for constructing parallelograms. The term congruent will be applied to their copies of line segments, angles, and 2-dimensional figures. Two figures are congruent if they have the same sizeand the same shape. Another approach to congruent figures in Unit 5 is through isometry transformations. These are rigid motions that take a figure from one place to another while preserving its size and shape. Reflections (flips), translations (slides), and rotations (turns) are basic isometry transformations (also known as rigid motions). A figure produced by an isometry transformation (the image) is congruent to the original figure (the preimage). Students will continue to work with the Geometry Template, a tool that was introduced in Fifth Grade Everyday Mathematics. The Geometry Template contains protractors and rulers for measuring and cutouts for drawing geometric figures. Students will review how to measure and draw angles using the full-circle and half-circle protractors. Students will also use a protractor to construct circle graphs that represent data collections. This involves converting the data to percents of a total, finding the corresponding degree measures around a circle, and drawing sectors of the appropriate size. Measures often can be determined without use of a measuring tool. Students will apply properties of angles and sums of angles to find unknown measures in figures similar to those at the right. One lesson in Unit 5 is a review and extension of work with the coordinate grid. Students will plot and name points on a 4-quadrant coordinate grid and use the grid for further study of geometric shapes. Please keep this Family Letter for reference as your child works through Unit 5. slide P turn flip ab transversal parallel lines cd ef gh If the measure of any one angle is given, the measures of all the others can be found without measuring. The sum of the angles in a triangle is 180°. Angles a and b have the same measure, 70°. 40° ab 2 in. 2 in. STUDY LINK 4 12 Unit 5: Family Letter Name Date Time Copyright © Wright Group/McGraw-Hill
Copyright © Wright Group/McGraw-Hill 142 Math Tools Your child will use a compass and a straightedge to construct geometric figures. A compass is used to draw a circle, or part of a circle, called an arc. A straightedge is used only to draw straight lines, not for measuring. The primary difference between a compass-and-straightedge construction and a drawing or sketch of a geometric figure is that measuring is not allowed in constructions. Vocabulary Important terms in Unit 5: adjacent angles Two angles with a common side and vertex that do not otherwise overlap. In the diagram, angles aand bare adjacent angles. So are angles band c, dand a, and cand d. congruent Figures that have exactly the same size and shape are said to be congruent to each other. The symbol means “is congruent to.” line of reflection (mirror line) A line halfway between a figure (preimage) and its reflected image. In a reflection, a figure is flipped over the line of reflection. ordered pair Two numbers, or coordinates, used to locate a point on a rectangular coordinate grid. The first coordinate xgives the position along the horizontal axis of the grid, and the second coordinate ygives the position along the vertical axis. The pair is written (x,y). reflection (flip) The flipping of a figure over a line (line of reflection) so its image is the mirror image of the original (preimage). reflex angle An angle measuring between 180° and 360°. rotation (turn) A movement of a figure around a fixed point or an axis; a turn. supplementary angles Two angles whose measures add to 180º. Supplementary angles do not need to be adjacent. translation (slide) A transformation in which every point in the image of a figure is at the same distance in the same direction from its corresponding point in the figure. Informally called a slide. vertical (opposite) angles The angles made by intersecting lines that do not share a common side. Same as opposite angles. Vertical angles have equal measures. In the diagram, angles 1 and 3 are vertical angles. They have no sides in common. Similarly, angles 4 and 2 are vertical angles. 1 2 4 3 preimageimage C D AB C' D'A' B' line of reflection a b d c Unit 5: Family Letter cont. STUDY LINK 412
143 Copyright © Wright Group/McGraw-Hill Do-Anytime Activities To work with your child on the concepts taught in this unit, try these interesting and engaging activities: 1.While you are driving in the car together, ask your child to look for congruent figures, for example, windows in office buildings, circles on stoplights, or wheels on cars and trucks. 2.Look for apparent right angles or any other type of angles: acute (less than 90º) or obtuse (between 90º and 180º). Guide your child to look particularly at bridge supports to find a variety of angles. 3.Triangulation lends strength to furniture. Encourage your child to find corner triangular braces in furniture throughout your home. Look under tables, under chairs, inside cabinets, or under bed frames. Have your child count how many examples of triangulation he or she can find in your home. Building Skills through Games In Unit 5, students will work on their understanding of geometry concepts by playing games such as those described below. Angle TangleSeeStudent Reference Book,page 306 Two players need a protractor, straightedge, and blank paper to play Angle Tangle. Skills practiced include estimating angle measures as well as measuring angles. Polygon CaptureSeeStudent Reference Book, page 330 Players capture polygons that match both the angle property and the side property drawn. Properties include measures of angles, lengths of sides, and number of pairs of parallel sides. Students will review concepts from previous units by playing games such as: 2-4-8 and 3-6-9 Frac-Tac-Toe(Decimal Versions) SeeStudent Reference Book, pages 314–316 Two players need a deck of number cards with four each of the numbers 0–10; a gameboard; a 5 5 grid that resembles a bingo card; a Frac-Tac- To eNumber-Card board; markers or counters in two different colors; and a calculator. The two versions, 2-4-8 Frac-Tac-Toeand3-6-9 Frac-Tac-Toe, help students practice conversions between fractions and decimals. Unit 5: Family Letter cont. STUDY LINK 412
Copyright © Wright Group/McGraw-Hill 144 As You Help Your Child with Homework As your child brings assignments home, you may want to go over the instr\ uctions together, clarifying them as necessary. The answers listed below will guide you through some of the Unit 5 Stu\ dy Links. Study Link 5 1 2. a.çH b.çJ c. çD d.çABC, çGFE, çL 3b. 180 3c.360 Study Link 5 2 1.mç y 2.mç x 3. mç c mç a mç t 4. mç q mç r mç s 5. mç a mç b mç c m ç d mç e mç f m ç g mç h mç i 6. mç w mç a mç t m ç c mç h 7. 12 8.30 9.110 Study Link 5 3 2. a. 1,920,000 adults b. 3,760,000 adults 3.7, 0, 0.07, 0.7, 7 4. 0.06, 11 0, 0.18, 0.2, 0.25, 0.75, 4 5,4 4 Study Link 5 4 Sample answers for 1–3: 1. Vertex C:( 1,2 ) 2. Vertex F:( 5,10 ) Vertex G:( 3,7 ) 3. Vertex J:( 2,1 ) 4.Vertex M:( 2, 3) 5. Vertex Q:(8, 3) Study Link 5 5 1. 2. 3. 4. 64 5.243 6.1 7.64 Study Link 5 7 1.mç r 47° m çs 133° m çt 47° 2. mç NKO 10° 3. mç a 120° m çb 120° m çc 60° 4. mç a 57° m çc 114° m çt 57° 5. mç x 45° m çy 45° m çz 135° 6. mç p 54° 7. 0.0027 8.0.12 9.0.0049 10.0.225 Study Link 5 8 2.A': ( 2, 7) B': ( 6, 6) C ': ( 8, 4) D': ( 5, 1) 3. A'': ( 2,1 ) B'': ( 6,2 ) C '': ( 8,4 ) D'': ( 5,7 ) 4. A''' :( 1, 2) B''' :( 2, 6) C ''':( 4, 8) D''':( 7, 5) 5. 0.3 6.0.143 7.0.0359 Study Link 5 9 3.Sample answers: All of the vertical angles have the same measure; all of the angles along the transversal and on the same side are supplementary; opposite angles along the transversal are equal in measure. Study Link 5 10 1. a. 50°;çYZW plus the 130° angle equals 180°, so çYZW 50°. Because opposite angles in a parallelogram are equal, çX also equals 50°. b. 130°; m çYZW 50° and çY and çZ are consecutive angles. Because consecutive angles of parallelograms are supplementary, ç Y 130°. 2. Opposite sides of a parallelogram are congruent. 3. 110°; Adjacent angles that form a straight angle are supplementary. 4. square 5.rhombus 105° 75° 105° 75° 90° 100° 100° 80° 140° 140° 40° 120° 60° 120° 70° 80° 120° 135° 45° 135° 115° 120° J K L M X Unit 5: Family Letter cont. STUDY LINK 412