Objective To add and subtract mixed numbers with like denominators. 1 materials Teaching the Lesson Key Activities Students practice adding and subtracting mixed numbers that have fractions with like denominators. Math Journal 1, pp. 132 and 133 Student Reference Book, pp. 84–86 Study Link 4 3 Key Concepts and Skills • Convert between fractions and mixed numbers. [Number and Numeration Goal 5] • Use multiplication and division facts to find equivalent fractions and to simplify fractions. [Operations and Computation Goal 2] • Add and subtract mixed numbers with like denominators. [Operations and Computation Goal 3] Teaching Master (Math Masters, p. 117) scissors See Advance Preparation Key Vocabulary mixed number • proper fraction • improper fraction • simplest form Ongoing Assessment: Recognizing Student Achievement Use journal page 133. [Operations and Computation Goal 3] 2 materials Ongoing Learning & Practice Students practice estimating sums of fractions by playing Fraction Action, Fraction Friction. Students practice and maintain skills through Math Boxes and Study Link activities. 3 materials Differentiation Options READINESS Students use a calculator to practice counting by fractions and converting between improper fractions and mixed numbers. ENRICHMENT Students read a poem about fractions. EXTRA PRACTICE Students use bills and coins to model and simplify mixed numbers. Additional Information Advance Preparation For the Math Message in Part 1, make one copy of Math Masters, page 117 for every two students. 272 Unit 4 Rational Number Uses and Operations Math Journal 1, p. 134 Student Reference Book, p. 317 Study Link Master (Math Masters, p. 118) Game Master (Math Masters, p. 446) Geometry Template; calculator Teaching Master (Math Masters, p. 119) Math Talk: Mathematical Ideas in Poems for Two Voices calculator; coins and bills of various denominations Technology Assessment Management System Journal page 133, Problems 9, 12, 14, and 15 See the iTLG. Getting Started Mental Math and Reflexes Math Message Students rename improper fractions as mixed numbers. Suggestions: Complete a copy of the Math Message problem. 3 1 1 2 2 5 2 1 3 3 75 3 18 4 4 5 1 1 4 4 13 1 2 6 6 108 3 21 5 5 Study Link 4 3 Follow-Up Go over the answers. 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION NOTE If students are fairly skilled at finding (Math Masters, p. 117) sums and differences of fractions, this lesson may take less than one day. Ask a volunteer to demonstrate and explain how to use the paper ruler to measure the line segment. Sample answer: Line up the right end of A 苶B 苶 with the mark for 4 in. on the ruler. The left end 5 of 苶 AB 苶 is aligned with the mark for 16 in. The total length of A 苶B 苶 is 5 5 4 in. 16 in., or 4 16 in. This ruler shows the two parts of a mixed number: the whole number and the fraction. A mixed number can be viewed as the sum of a whole number and a fraction. Discuss why fractions greater than 1 are easier to interpret when 3 written as mixed numbers. For example, 24 clearly represents a number greater than 2 but less than 3. This is not as obvious 3 11 when 24 is written as the improper fraction 4. Writing Mixed Numbers Teaching Master Name WHOLE-CLASS DISCUSSION in Simplest Form Date LESSON Time Math Message 4 4 Cut out the ruler below. Use it to measure line segment AB to the 1 nearest inch. 16 A Review the meanings of proper fraction, improper fraction, and simplest form with the class. B — length of AB A fraction in which the numerator is less than the denominator is called a proper fraction. A proper fraction names a number that is less than 1. 3 9 5 16 4 in. 0 1 2 3 4 5 inches 0 Examples: 8, 10, 4 A fraction in which the numerator is equal to or greater than the denominator is called an improper fraction. An improper fraction names a number that is greater than or equal to 1. 5 7 9 Examples: 5, 2, 3 Name Date LESSON Time Math Message 4 4 Cut out the ruler below. Use it to measure line segment AB to the 1 nearest inch. 16 A B — length of AB 0 1 2 3 4 5 inches Math Masters, p. 117 Lesson 4 4 273 Student Page Date Time LESSON A mixed number is in simplest form if the fraction part is a proper fraction in simplest form. Adding and Subtracting Mixed Numbers 4 4 2 5 4 5 Example 1: 1 2 ? 84–86 Step 1 Add the fractions. Then add the whole numbers. 6 3 65 3 5 2 15 5 1 3 5 5 4 2 5 1 3 1 5 1 4 5 6 3 5 Add. Write your answers in simplest form. 1 4 5 1. 2 14 2. 2 3 5 3 1 4 414 7 735 5 4 8 Example 2: 1 2 8 1 5 4 3. 3 2 4 2 Step 2 If necessary, rename the difference. 5 4 8 5 4 8 1 2 8 1 2 8 4 4 Example 3: 2 13 1 2 8 2 2 2 8 1 5 3 9 be written using proper fractions, the fraction part is not required to be in simplest form. Students should know how to name fractions in simplest form for standardized tests. However, they may often find it helpful to work with fractions or mixed numbers that are not in simplest form when they are computing with fractions. 2 18 3 4 ? Step 1 Subtract the fractions. Then subtract the whole numbers. 8 NOTE While Everyday Mathematics requests that mixed-number answers 4 18 4. 4 Examples: 27 is in simplest form. 45 and 39 are not in simplest form because they contain an improper fraction. 4 4 38 is not in simplest form because 8 is not in simplest form. Step 2 If necessary, rename the sum. 5 Write 34 on the board and ask a volunteer to rename it in simplest form. Use pictures similar to those below to show the procedure. ? Notice that the fraction in the first mixed number is less than the fraction in the second 2 1 1 mixed number. Because you can’t subtract 3 from 3, you need to rename 5 3. Step 1 Rename the first mixed number. 1 1 5 3 4 1 3 3 Step 2 Subtract the fractions. Then subtract the whole numbers. 1 4 3 3 4 5 3 1 4 3 2 13 1 4 4 1 2 13 4 4 3 4 3 1 1 1 4 1 4 1 4 1 4 2 3 3 132 Math Journal 1, p. 132 5 34 1 1 1 1 Time 4 4 5 7 88 7 8 5 3 8 6. 1 4 5 2 4 3 2 5 1 2 4 5 5 7. 2 10. 1 2 1 3 5 5 11. 1 5 3 14 3 1 3 3 5 1 4 3 1 3 7 6 8 1 5 42 52 2 9 23 33 10 A U D I T O R Y 1 4 7 12 inches 14. Mr. Ventrelli is making bread. He adds cups 1 of white flour and 14 cups of wheat flour. The recipe calls for the same number of cups of water as cups of flour. How much water should he add? 2 12 cups of water 15. Evelyn’s house is between Robert’s and Elizabeth’s. How far is Robert’s house from Elizabeth’s? 1 2 3 4 mi 3 1 4 mi 2 miles Robert’s Evelyn’s 1 16 22 ELL 3 3 8 1 14 Elizabeth’s 133 Math Journal 1, p. 133 274 1 44 Have students model mixed numbers using bills and quarters. To 5 model 34, use $3 and 5 quarters. Five quarters is equal to $1 and 1 quarter. Therefore, $3 and 5 quarters $4 and 1 quarter. 1 3 4 3 4 12. 2 6 6 7 8. 3 13. Joe has a board that is 8 4 inches long. He cuts 1 off 14 inches. How long is the remaining piece? 1 4 Adjusting the Activity 2 2 3 1 3 1 1 3 Add or subtract. 4 6 1 4 5 4 8 9. 1 4 8 8 3 8 8 7 8 3 75 8 Step 2 Subtract the fractions. Then subtract the whole numbers. 871 1 84–86 Step 1 Rename the whole number. 3 4 1 1 4 Write several such mixed numbers on the board. Have students rename them in simplest form. Suggestions: 5 Example 4: 8 3 8 ? 5. 4 4 1 4 354 can be renamed as 414. Adding and Subtracting Mixed Numbers cont. 1 4 1 4 Student Page LESSON 1 4 Date 1 4 1 3 1 5 4 3 Unit 4 Rational Number Uses and Operations K I N E S T H E T I C T A C T I L E V I S U A L Student Page Adding Mixed Numbers INDEPENDENT ACTIVITY with Like Denominators Fractions Addition of Mixed Numbers One way to add mixed numbers is to add the fractions and the whole numbers separately. This may require renaming the sum. (Math Journal 1, p. 132; Student Reference Book, p. 84) 5 Example 7 Find 48 28. Step 1: Add the fractions. Step 2: Add the whole Step 3: Rename the sum. numbers. 2 4 Write the following problem on the board: 15 25 ? Ask students to solve and then share their strategies. Go over the steps on journal page 132 before having students complete Problems 1–4 on their own. Bring the class together to share solutions. If necessary, provide more practice, particularly with problems that require renaming. Refer students to page 84 in the Student Reference Book. 5 4 5 8 4 2 7 8 12 8 2 7 5 8 6 12 8 7 8 12 6 8 6 8 8 61 4 8 1 2 7 7 1 48 28 72 4 8 4 8 If the fractions do not have the same denominator, first rename the fractions so they have a common denominator. 3 Example 2 Find 34 53. Step 1: Rename and add 5 3 3 4 2 3 3 5 2 Step 3: Rename the sum. Step 2: Add the whole numbers. the fractions. 3 9 12 8 12 17 12 3 5 8 9 12 8 12 17 12 8 17 12 8 12 12 8 1 9 5 12 5 12 5 12 5 34 53 912 Subtracting Mixed Numbers PARTNER ACTIVITY Some calculators have special keys for entering mixed numbers. 3 Example 2 Solve 34 53 on a calculator. On Calculator A: Key in 3 with Like Denominators Unit On Calculator B: Key in 3 (Math Journal 1, pp. 132 and 133; Student Reference Book, p. 85) 3 3 4 n d 4 5 + 5 Unit 2 2 n 3 Enter = d 3 Check Your Understanding Solve Problems 1–3 without a calculator. Solve Problem 4 with a calculator. 1 Go over the three subtraction examples on journal pages 132 and 133 (Examples 2–4). Ask students to solve Problem 5. If successful, they should continue and complete the page. You may need to provide additional practice before continuing or refer students to page 85 in the Student Reference Book. Ongoing Assessment: Recognizing Student Achievement 7 4 1. 28 78 1 2 2. 35 22 3 4 3. 63 34 Ch k 6 4. 149 87 417 Student Reference Book, p. 84 Journal Page 133 Problems 9, 12, 14, and 15 Use journal page 133, Problems 9, 12, 14, and 15 to assess students’ ability to add mixed numbers with like denominators. Students are making adequate progress if they can calculate the sums in Problems 9, 12, 14, and 15. Some students may be able to calculate the differences in Problems 5–8, 10, 11, and 13. [Operations and Computation Goal 3] Student Page Fractions Subtraction of Mixed Numbers If the fractions do not have the same denominator, first rename them as fractions with a common denominator. 7 3 Find 38 14. Example Step 1: Rename the fractions. 3 7 8 1 3 4 7 8 1 6 8 3 Step 2: Subtract the Step 3: Subtract the whole fractions. 3 7 1 38 14 28 numbers. 3 8 7 3 7 1 6 1 6 8 8 1 8 8 2 1 8 To subtract a mixed number from a whole number, first rename the whole number as the sum of a whole number and a fraction that is equivalent to 1. 2 Find 5 23. Example Step 1: Rename the whole number. 2 2 3 2 3 2 2 3 Step 3: Subtract the whole fractions. 4 3 5 Step 2: Subtract the 1 5 23 23 numbers. 4 3 4 3 3 2 2 3 1 3 3 2 2 3 1 3 2 When subtracting mixed numbers, rename the larger mixed number if it contains a fraction that is less than the fraction in the smaller mixed number. Example 1 3 Find 75 35. Step 1: Rename the larger mixed number. 7 1 5 3 3 5 1 3 Step 2: Subtract the Step 3: Subtract the whole fractions. numbers. 6 6 5 6 6 5 6 6 3 3 3 3 3 3 5 3 75 35 35 5 5 3 5 5 3 3 5 Student Reference Book, p. 85 Lesson 4 4 275 Game Master Name Date Time Fraction Action, Fraction Friction Card Deck 1 2 1 3 2 3 1 2 4 3 2 Ongoing Learning & Practice 1 4 Playing Fraction Action, PARTNER ACTIVITY Fraction Friction 3 4 1 6 1 6 5 6 1 12 1 12 5 12 5 12 7 12 7 12 11 12 11 12 (Student Reference Book, p. 317; Math Masters, p. 446) Distribute one set of 16 Fraction Action, Fraction Friction cards and one or more calculators to each group of two or three players. Review game directions on page 317 of the Student Reference Book. Play a few practice rounds with the class. Math Boxes 4 4 INDEPENDENT ACTIVITY (Math Journal 1, p. 134) Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 4-2. The skill in Problem 5 previews Unit 5 content. Students will need the Percent Circle on the Geometry Template to complete Problem 5. Math Masters, p. 446 Study Link 4 4 INDEPENDENT ACTIVITY (Math Masters, p. 118) Home Connection Students practice addition and subtraction of mixed numbers. Study Link Master Student Page Date Time LESSON Name 4 4 18 a. 45 2 5 26 b. 39 2 3 1 3 a. 10 5 7 10 56 c. 80 7 10 25 d. 625 1 25 5 1 b. 12 3 3 4 7 4 c. 9 9 1 3 6 d. 8 0 9 4 Write 2 fractions equivalent to . e. 27 12 18 8 3 4 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. Sample answers: a. 4. Thomas Jefferson was born in 1743. George Washington was born m years earlier. In what year was Washington born? Choose the best answer. following set of numbers. 1.5, 2.8, 3.4, 4.5, 2.2, 8.4 3.1 b. mean 3.8 m 1743 The mean would double. 1743 m 240 5. The table below shows the results of a survey in which people were asked which winter Olympic sport they most enjoyed watching. Use a Percent Circle to make a circle graph of the results. Favorite Sport Percent of People Surveyed Luge 35% Ice hockey 15% Figure skating 40% Other 10% Winter Olympic Sports Preferences 28% D. 21 27% 14% 1 4 1 inches (unit) 3 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. 1 3 1 1 1 1 2 2 2 4 3 4. 3 1 5. 4 2 6. 1 4 4 4 1 4 3 3 2 3 Circle the numbers that are equivalent to 24. 7 6 4 3 7 11 4 Practice Solve mentally. 8. 5 º 18 90 9. 6 º 41 246 145 146 Unit 4 Rational Number Uses and Operations C. 19 Tim is making papier-mâché. The recipe calls for 14 cups of paste. Using only 1 1 1 -cup, -cup, and -cup measures, how can he measure the correct amount? 2 4 3 1 Sample answer: He can use three 2-cup measures and 1 one 4-cup measure. 134 276 24% 3 14 Math Journal 1, p. 134 7% B. 2 3. Luge Other Figure skating A. 1 A board is 68 inches long. Verna wants to cut enough 1 so that it will be 58 inches long. How much should she cut? 7. Ice hockey Explain why B is the best estimate. 2. 1743 m 136 137 7 Percent Who Chose This Answer E. I don’t know. m 1743 Suppose you multiplied each data value by 2. What would happen to the mean? Possible Answers Both fractions are close to 1, so their sum should be close to 2. 83 74 12 Estimate the answer to 13 8. You will not have enough time to solve the problem using paper and pencil. What mistake do you think the students who chose C made? They may have added only the numerators. b. 3. Find the median and mean for the a. median 1. 2. Add or subtract. Then simplify. Time ⴙ, ⴚ Fractions and Mixed Numbers 4 4 1. Write each fraction in simplest form. Sample answers: Date STUDY LINK Math Boxes Math Masters, p. 118 10. 9 º 48 432 11. 7 º 45 315 Teaching Master Name 3 Differentiation Options Date LESSON Time Fraction Counts and Conversions 4 4 Most calculators have a function that lets you repeat an operation, such 1 as adding to a number. This is called the constant function. To use 4 1 the constant function of your calculator to count by s, follow one of the 4 key sequences below, depending on the calculator you are using. PARTNER ACTIVITY READINESS Using a Calculator for Calculator A Calculator B Op1 + Press: 1 n 4 d Op1 1 14 Display: 5 Fraction Counts Display: 1. 1 a. b. 2. 1 4 6 4 1 4 1 1? 2 6 6 How many counts of are needed to display ? How many counts of are needed to display Use a calculator to convert mixed numbers to improper fractions or whole numbers. 11 11 3 7 4 4 a. 2 b. 1 4 3. 5 4 K Using a calculator, start at 0 and count by 4s to answer the following questions. c. Reading about Fractions 0 Display: 14 Display: PARTNER ACTIVITY + + 1 5 4 (Math Masters, p. 119) ENRICHMENT 4 Press: Press: To provide experience with fractions and mixed numbers, have students use the constant function on a calculator to define and generate fraction counts. They also study and apply patterns involving unit fractions, improper fractions, and mixed numbers. 1 Press: Op1 Op1 Op1 Op1 Op1 5–15 Min 4 3 4 4 2 d. 6 12 4 3 1 4 How many s are between the following numbers? 3 a. 4 c. and 2 3 4 1 and 4 5 9 6 b. 4 d. 3 4 and 2 1 2 3 and 4 5 6 5–15 Min in Poetry Math Masters, p. 119 Literature Link To further explore fractions, have students read the poem “Proper Fractions” in Math Talk: Mathematical Ideas in Poems for Two Voices. Suggest that students recite the poem in their spare time and present it to the class. PARTNER ACTIVITY EXTRA PRACTICE Modeling Mixed Numbers 5–15 Min with Bills and Coins To provide extra practice simplifying mixed numbers, have students use bills and coins to model renaming procedures. Suggestions: 4 13 24 $2 and 4 quarters; 3 9 3 21 0 $2 and 13 dimes; 3 10 1 14 $1 and 9 quarters; 34 31 11 32 0 $3 and 31 nickels; 4 20 Lesson 4 4 277