Subtracting Mixed Numbers


 Kathleen Hicks
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1 Subtracting Mixed Numbers Objective To develop subtraction concepts related to mixed numbers. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Find equivalent names for mixed numbers. [Number and Numeration Goal ] Convert between fractions and mixed numbers. [Number and Numeration Goal ] Subtract mixed numbers. [Operations and Computation Goal ] Use benchmarks to estimate differences. [Operations and Computation Goal ] Key Activities Students subtract mixed numbers with like denominators by renaming the minuend. Students use benchmarks to estimate differences. Informing Instruction See page. Materials Math Journal, p. Math Masters, p. 1 Study Link slate 1 Playing MixedNumber Spin Math Journal, p. Math Masters, pp. and 9 per partnership: large paper clip Students practice adding and subtracting fractions and mixed numbers with like and unlike denominators. Recognizing Student Achievement Use journal page. [Operations and Computation Goals and ] Math Boxes Math Journal, p. Geometry Template Students practice and maintain skills through Math Box problems. Study Link Math Masters, p. Students practice and maintain skills through Study Link activities. READINESS Subtracting Mixed Numbers Students use a countingup algorithm to explore mixednumber subtraction. ENRICHMENT Exploring a Pattern for Fraction Addition and Subtraction Math Masters, p. Students explore patterns that result from adding and subtracting unit fractions. EXTRA PRACTICE Minute Math Minute Math, pp. 1 and 1 Students add mixed numbers. Advance Preparation Teacher s Reference Manual, Grades pp. 1, 1 0 Unit Fractions and Ratios
2 Getting Started Mental Math and Reflexes Have students name a common denominator for the following: Sample answers: halves and thirds sixths fifths and thirds fifteenths fourths and eighths eighths fifths and eighths fortieths halves, thirds, and fourths twelfths halves, fifths, and sixths thirtieths Mathematical Practices SMP1, SMP, SMP, SMP Content Standards.NF.1,.NF.,.G.,.G. Math Message Solve Problems 1 at the top of journal page. Use benchmarks to estimate the differences and to check the reasonableness of your solutions. Study Link FollowUp Have partners compare answers and resolve differences. Ask volunteers how they know their answers are in simplest form. A fraction is in simplest form if the numerator and denominator have no common factors except 1. 1 Teaching the Lesson Math Message FollowUp (Math Journal, p. ) WHOLECLASS DISCUSSION Ask students to share their estimating strategies for Problem using benchmarks. Sample answer: I know that _ is close to and =. I estimated my answer to be about. Ask volunteers to write their solutions on the board and explain their strategies. Expect that most students will have subtracted the wholenumber and fraction parts separately and renamed the differences in Problems 1 and in simplest form. Subtracting Mixed Numbers with Renaming Write this problem on the board.  1 _ WHOLECLASS Ask: How does this problem differ from the Math Message problems? The fraction being subtracted from,, is smaller than the fraction that is being subtracted, _. Ask volunteers to suggest solution strategies. Sample answer: Rename to make the fraction part larger. _  1_ = 1_ Remind students that when they add mixed numbers, some sums might have to be renamed to write them in simplest form. For example, a sum of 7_ could be renamed: 7_ = + _ + _, or _ = _. Emphasize that the two mixed numbers are equivalent; 7_ is another name for _. Date Math Message Subtract. Subtracting Mixed Numbers 1. _. _. 7 _ 1 Renaming and Subtracting Mixed Numbers Fill in the missing numbers.. = _. = _. _ = _ 7. 7_ 9 = 1_ 9 Subtract. Write your answers in simplest form. Show your work. 7. = 9. _ = _ = 11. _ 7_ = 1. Isaac would like to practice the violin 7 hours this week. So far, he has practiced _ hours. How many more hours does he need to practice this week? 1 _ hours Student Page 7 _ Math Journal, p. 7_EMCS_S_MJ_U0_7.indd /1/11 : PM Lesson 1
3 Have students rename to an equivalent mixed number with a larger fraction part. 1 7_, or _ Write their responses on the board or a transparency, and illustrate the mixed numbers with pictures. = 1 7_ = _ Informing Instruction Watch for students who have difficulty renaming mixed numbers for subtraction. Have students practice the following steps: 1. Write the wholenumber part of the mixed number as an equivalent addition expression that adds 1. _ = _. Use the denominator to rename the 1 as a fraction. _ = + _ + _. Combine the fraction parts. _ = + 7_ Ask students to use each of the responses to solve the problem. Circulate and assist. When most students have finished, have volunteers explain which mixednumber name was more efficient for solving the subtraction problem. _, because with 17_ you need to rename the fraction again in order to express the difference as a mixed number. Explain that deciding how to rename a fraction depends on the problem. Sometimes it might be more efficient to work with mixed numbers, other times it might be more efficient to work with improper fractions. In this case, the more efficient mixednumber name is a matter of personal preference. Pose several problems. Encourage students to estimate each difference using what they know about benchmarks with fractions. They should use their estimates to check the reasonableness of their solutions. In each case, ask students first how they would rename the minuend and then how they would solve the problem. Suggestions:  _ = 7_ ; solution:  = _ ; solution: _ _  1 = _ ; solution: _  _ = 7_ ; solution: _ _ 1 = 17_ ; solution: 1 Carlie purchased 10 feet of ribbon to make bows. She used 7_ feet to make one bow. How much ribbon remains to make additional bows? 7 feet Sammy needs cups of sugar to make sugar cookies. He only has 1 _ cups of sugar. How much more sugar does he need for the recipe? 1 cups Unit Fractions and Ratios
4 Write this problem on the board.  1 Ask: How would you solve this problem? Expect that students will likely recognize is smaller than and might suggest renaming the fraction parts with common denominators. Ask volunteers to model their strategies on the board. Sample answer: Use as a common denominator. Rename the mixed numbers. _  1_ Rename the minuend. _ = _. Subtract. _  1_ =. Pose a few mixednumber subtraction number stories. Suggestions: Daniel has a DVD set that provides hours of viewing. So far, he has watched _ of an hour. How many more hours of the DVD set does he still have to watch? _, or _ hours Serena measured out _ feet of rope to make a jump rope. Her friends told her that it needed to be feet long. How much more rope did she need to measure out? _ ft more Subtracting Mixed Numbers (Math Journal, p. ; Math Masters, p. 1) PARTNER Date MixedNumber Spin Materials Math Masters, p. large paper clip Players Directions 1. Each player writes his or her name in one of the boxes below.. Take turns spinning. When it is your turn, write the fraction or mixed number you spin in one of the blanks below your name.. The first player to complete 10 true sentences is the winner. 7_EMCS_S_MJ_U0_7.indd Student Page Name Name + < + < + > + >  < 1  < 1  <  < + > 1 + > 1 + < 1 + < 1 + < + <  =  =  > 1  > 1 + > + > + < + < + > + > Math Journal, p. /1/11 1:01 PM Ask students to complete the journal page. Remind them to use benchmarks to estimate the difference. Ask students to use an Exit Slip (Math Masters, page 1) to explain their strategy for solving Problem 10 on Math Journal, page. Links to the Future Subtraction of mixed numbers with unlike denominators continues in Sixth Grade Everyday Mathematics. Ongoing Learning & Practice Playing MixedNumber Spin (Math Journal, p. ; Math Masters, pp. and 9) PARTNER Algebraic Thinking Students practice estimating sums and differences of mixed numbers with like and unlike denominators by playing MixedNumber Spin. Players use benchmarks to estimate sums and differences as they record number expressions that fit the parameters given on the MixedNumber Spin record sheet, Math Journal, page. Each partnership makes a spinner using a large paper clip anchored by a pencil point to the center of the spinner on Math Masters, page. They use the numbers they spin to complete the number sentences on the record sheet. Ask students to save their spinners for future use. Copy Math Masters, page 9 to provide additional record sheets. Recognizing Student Journal Achievement Page Use the MixedNumber Spin record sheet on journal page to assess students ability to make estimates for adding and subtracting mixed numbers. Students are making adequate progress if they correctly complete number sentences using the benchmarks and 1. Some students may be able to make the estimates and comparisons using and as benchmarks. [Operations and Computation Goals and ] Lesson
5 Date Math Boxes 1. Make a circle graph of the survey results. Spent on Homework Percent of Students 0 9 minutes % 0 9 minutes % 0 9 minutes 10% minutes 1% hours or more %. Write each numeral in numberandword notation. a.,000,000 million b.,000 thousand c. 1,000,000,000 1 billion d. 9,00,000 9 hundredthousand. Complete the What s My Rule? table and state the rule. in out Rule out = in Student Page Spent on Homework 1% % 10% %. True or False? Write T or F. a. All rectangles are parallelograms. b. A kite is not a parallelogram. c. Some parallelograms are squares. d. Some trapezoids are not quadrangles.. Find the area of the rectangle. m Area: 7 m (unit) title 1 m % T T T F Math Journal, p. Area = b h 19 Math Boxes (Math Journal, p. ) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 1. The skill in Problem previews Unit 9 content. Writing/Reasoning Have students write a response to the following: Explain one advantage and one disadvantage to using number and word notation in Problem. Sample answer: One advantage is that the number is written the same way that it is said aloud. One disadvantage is that you cannot perform operations on them as easily. Writing/Reasoning Have students write a response to the following: Explain how you determined your answer to Problem c. Sample answer: The answer is true because a parallelogram has two pairs of parallel sides with opposite sides congruent. When all sides of a parallelogram are the same length and form right angles, the parallelogram is also a square. 7_EMCS_S_MJ_G_U0_7.indd /1/11 :9 PM Study Link (Math Masters, p. ) INDEPENDENT Home Connection Students practice renaming and subtracting mixed numbers. Differentiation Options Name Date STUDY LINK Subtracting Mixed Numbers Fill in the missing numbers _ = = 1 9. _ = Subtract. Write your answers in simplest form.. _ = 1. _ 7 = 10_ 7. 7 _ = 7. _. _ 9. _ _ 7_ 10 1 _ _ = _ 9 = 1.  _ 10 = _ = 1. _  1 _ = 1. _  7_ = Practice Study Link Master 1, * 0 = 17. * 0 = 1. 1,00 * 0 = 19. * 0 = Math Masters, p.,0 0, READINESS Subtracting Mixed Numbers SMALLGROUP 1 0 Min To explore mixednumber subtraction, have students use a countingup algorithm. Remind students that another way to subtract is to count up. With whole numbers, they could solve  by thinking plus what equals? Discuss how they would count up to solve this problem: _  _. Expect that students will suggest _ plus what equals _? Explain that counting up is one way to answer plus what? Demonstrate how to count up from _ to _. 1. Count up, keeping track of the amounts used. Count up : _ + = Count up 1: + 1 = 1 Count up _ : + _ = _. Add _ = 1_ So the difference between _ and _ is 1_. _ 1 EMCS_B_MM_G_U0_797.indd //11 :0 PM Unit Fractions and Ratios
6 Pose problems for students to solve by counting up. Suggestions: _ _ 10 _  1_ 1_  _ ENRICHMENT PARTNER Exploring a Pattern for Fraction Addition and Subtraction (Math Masters, p. ) 1 0 Min PROBLEM SOLVING Name Date Add. Addition and Subtraction Patterns 1, or _ 9_ 1. a. 1 + = b. + = c. + = 0 d. + = e. + = Teaching Master Sample answer: In each problem, the answer is the sum of the denominators over the product of the denominators.. What pattern do you notice in Problems 1a 1e?. Use the pattern above to solve these problems. _ a = b = c =. Do you think this pattern also works for problems like +? Explain. _ 199 9,900 Sample answer: Yes. This pattern will work whenever two unit fractions are added. _ 1 0 7_ 1 Algebraic Thinking To apply students understanding of addition and subtraction of fractions, have them find patterns in unit fraction addition and subtraction problems. When most students have completed the problems, discuss the pattern in Problem 1. In each problem, the answer is the sum of the denominators over the product of the denominators. Problem asks whether this pattern works for the sum of any two unit fractions a +. It does. Add b a and using the QCD method: b a + b = _ 1 b a b + _ 1 a b a = b_ a b + a_ b a = _ b + a a b Have students verify that this pattern holds true for other unit fractions, for example: +. The pattern for Problem is similar. The denominator of the result is the product of the denominators of the unit fractions being subtracted. The numerator is always 1. Problems a e involve the subtraction of two unit fractions where the difference between the denominators is 1. The pattern described in Problem f reflects that type of problem. You may want to extend the Math Master activity by having students find the results of the following subtraction problems (where the difference between the denominators of the unit fractions is greater than 1):  _ 10  _  1 Ask students to describe a pattern for subtracting any two unit fractions. Sample answer: The numerator of the result is found by subtracting the denominator of the first fraction from the denominator of the second fraction. The denominator of the result is the product of the denominators of the two unit fractions. 7_ 0. The plus signs in Problem 1 have been replaced with minus signs. Find each answer. a. 1  = b.  = c.  = 1 d.  = 0 e.  = 0 1 EMCS_B_MM_G_U0_797.indd Sample answer: The numerator is always 1. The denominator of the result is the product of the denominators of the subtracted fractions. f. Describe the pattern. Math Masters, p. //11 :0 PM EXTRA PRACTICE SMALLGROUP 1 Min Minute Math To offer students more experience with adding mixed numbers, see Minute Math, pages 1 and 1. Lesson
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