Lesson 5

Subtract Mixed Numbers

Est. Class Sessions: 2–3

Developing the Lesson

Part 2: Estimate with Closest To Game

Introduce Closest To Game. Students play a game in which they use benchmarks and other methods to estimate the difference of two numbers. Students need to be organized into groups of two to four to play the game. Prior to play, each player needs to cut out the set of four Number Squares on the Closest To pages in the Student Activity Book. In addition, one set of cards from the Closest To Game Cards Master needs to be cut out per game.

Display the Closest To pages from the Student Activity Book to introduce the game. Review the directions with the class. For each round, players will estimate the difference to a mixed number subtraction problem and predict whether the difference is closest to 0, 1, 2, or 3. Players will then find the exact difference and determine whether it is closest to 0, 1, 2, or 3. Each correct estimate earns one point. The player with the most points at the end of the game wins.

In Game 1, students will start with the minuend 33/4 and will subtract the mixed number displayed on a game card. Use the example problem on the recording sheet, 33/4 − 21/8 to demonstrate a round of the game.

Ask:

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  • Explain how you would estimate the difference between 33/4 and 21/8 . (Possible response: 21/8 is a little more than 2. 33/4 − 2 is 13/4 . I think the difference is about 13/4 .)
  • Do you predict the difference will be closest to 0, 1, 2, or 3? (Possible response: closest to 2)
  • Show how you solve 33/4 − 21/8 . (Possible response: I noticed there were fourths and eighths in the problem so I can use a ruler like a number line. First I deal with the whole numbers, 3 − 2. I start at the 33/4 inch mark and take 2 away. I am at the 13/4 inch mark. Then I deal with the fractions, 3/4 − 1/8 . From 13/4 , I take 1/8 away and the answer is 15/8 .)
  • Who would like to show another way to solve it? (Possible response: First I turn each fraction into improper fractions 15/4 − 17/8 . Then I find a common denominator of 8. 30/8 − 17/8 = 13/8 . Then I change 13/8 to 15/8 .)
  • Is the exact difference closest to 0, 1, 2, or 3? How do you know? (closest to 2; Possible response: 14/8 is 11/2 . 15/8 is more than that so it is closer to 2 than 1.)

Play Closest To Game 1. Allow students time to play the game. Encourage students to practice paper-and-pencil methods, but provide access to rulers and fraction circle pieces. Remind students they can use the Fraction Chart, Fractions on Number Lines Chart, or Multiplication and Division Facts page in the Student Guide Reference section to help them find equivalent fractions and common denominators.

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Closest To Game. If a student is struggling with using paper-and-pencil methods to subtract fractions, he or she can be paired to play with a partner who can use a paper-and-pencil strategy confidently. Both students can estimate the differences, and one student can find the exact difference. Being able to estimate fraction sums and differences is very important, especially for students who struggle with using paper-and-pencil algorithms.

Another alternative is to create a deck of Closest To Game Cards only with fractions that can be modeled with fraction circle pieces (e.g., 21/2 , 13/8 , 33/4 , 21/3 , 23/12 , 12/5 , etc.) Pull the game cards displaying other fractions out (e.g., 21/7 , 25/11 , 33/16 , 15/9 , etc.) Encourage students to use fraction circle pieces during the game.

Read What Do I Do. After students finish Game 1 of Closest To, ask them to work with a partner to solve 22/5 − 17/10 . Allow students to invent their own ways to deal with the regrouping situation and solve the problem.

Discuss students' strategies.

Ask:

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  • Who would like to show how they solved 22/5 − 17/10 ? (Possible response: I looked at the fractions and noticed that 10 is a common denominator. 22/5 is the same as 24/10 . I thought of 24/10 as 1 + 10/10 + 4/10 or 114/10 and then I could solve 114/10 − 17/10 = 7/10 . See Figure 4.)
  • Did anyone solve this problem a different way? Show us.

After some discussion, direct students' attention back to the Subtract Mixed Numbers pages in the Student Guide. Read the vignette in the What Do I Do section and discuss how Mr. Moreno's students solve 22/5 − 17/10 . Discuss the common misconception presented in the vignette. When solving a problem such as 22/5 − 17/10 , some students subtract the whole numbers and then want to subtract the smaller fraction from the larger fraction ( 7/10 − 2/5 ). Ask students to use fraction circle pieces to explain why this doesn't work. The fractions are part of the mixed number, not a separate number.

Another common difficulty occurs when students are asked to solve a problem like 2 − 3/4 . When one number is not a fraction, students may incorrectly rewrite the problem so that the whole number has a denominator common with that of the fraction, for example 2/4 − 3/4 .

Display the problem 2 − 3/4 and ask:

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  • Who can explain how to use mental math to solve 2 − 3/4 ? (Possible response: Each whole is made of four fourths. I start at 3/4 and add 1/4 to make 1, and then add on 1 more to make 2. The difference is 11/4 .)
  • Who can show how to use the fraction circle pieces to justify this answer? (Start with 2 red wholes. In order to take 3/4 away, I have to exchange a red whole for 4 yellow fourths. Take 3 yellow fourths away and you have 11/4 left.)
  • Who would like to show us a paper-and-pencil solution? (Possible response: 14/4 − 3/4 = 11/4 or 8/4 − 3/4 = 5/4 )

Play Closest To Game 2. Students will encounter subtraction problems that require regrouping when they play Game 2 of Closest To. Encourage students to use any strategy that makes sense to them when they solve these problems.

After students have played Game 2, ask them to save the game materials, or collect the pages to save for the Workshop in Lesson 6. They will have another opportunity to play Closest To during the Workshop.

Student Guide: 477
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Student Activity Book: 405 407 408
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Master:
Solving 22/5 − 17/10
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