Just as with addition, students can use invented strategies and mental math to solve mixed number subtraction problems. Finding ways to compose and decompose fractions in ways that make sense to them is important. They can work with models and then build on their invented strategies and understanding of equivalence to develop paper-and-pencil strategies.
Present the following problem to students:
It is 21/2 miles to Ruby Falls from John's tent. He has already hiked 11/8 miles. How much farther does he need to hike?
- What is a number sentence that describes this problem? (21/2 − 11/8 )
- Explain how you would estimate the difference. (Possible response: 11/8 is close to 1, so I think 21/2 − 1 = 11/2 . I think the difference will be a little less than 11/2 .)
- Try to use fraction circle pieces to solve 21/2 − 11/8 . Is your solution close to the estimated difference?
- Who would like to show us how to solve the problem with circle pieces? (See Figure 1. Possible response: I have 2 red wholes and 1 pink half. I can take 1 red away for 2 − 1, but I can't take 1/8 from 1/2 unless I trade the pink piece for 4 blues. Then I can take 1 blue, or 1/8 , away and I have 1 red and 3 blues left, 13/8 .)
- Can someone show another way to solve the problem? (Possible response: First I subtract the whole numbers to get 1. Then I thought about common denominators. I know that 1/2 is equivalent to 4/8 , and 4/8 − 1/8 is 3/8 . He has to walk 13/8 miles more.)
- Is this solution reasonable? How do you know?
Continue to discuss many strategies for solving the problem by directing students to the Subtract Mixed Numbers pages in the Student Guide. Several strategies for solving 21/2 − 11/8 are presented on the pages. Providing solution strategy options to students will ensure that more students understand and are able to solve the problems correctly. Ask students to review the strategies with a partner.
After students have discussed the variety of strategies, ask:
- Do you always need to find a common denominator to subtract mixed numbers? (No; Maya and Keenya used other strategies to subtract 21/2 − 11/8 .)
Students can use invented strategies to solve fraction problems without finding common denominators. However, finding common denominators is helpful. Explain to students that in order to add or subtract, you need to have equal-sized parts. When you find common denominators, the parts being added or subtracted are the same size.
Display Luis's first number sentence:
21/2 − 11/8
Then rewrite the equation with the equivalent fraction 24/8 in place of 21/2 :
24/8 − 11/8
- Does 21/2 − 11/8 = 24/8 − 11/8 ? Is this still the same equation? (yes; 21/2 and 24/8 are equivalent.)
- Why do you think Luis renamed the fraction? (Luis wanted to find a common denominator so that equal-sized parts could be combined. Then he could take eighths away from eighths.)
- Use Luis's paper-and-pencil method to solve 22/3 − 13/5 . (See Figure 2. Possible response: I multiply the denominators together to find a common denominator, 15. I subtract the whole numbers and then I subtract the fractions.)
- Ana changes the mixed numbers to improper fractions. Use Ana's paper-and-pencil method to solve 22/3 − 13/5 . (See Figure 3. Possible response: I change the mixed numbers to improper fractions, 8/3 − 8/5 . Then I find a common denominator by multiplying the denominators together. 40/15 − 24/15 = 16/15 . I simplify 16/15 to 11/15 .)
- Who would like to show how to check these answers with addition? (I add the answer 11/15 back to what I took away, 13/5 , and should get 22/3 . 11/15 + 13/5 = 11/15 + 19/15 = 210/15 . I can simplify that to 22/3 , so the answer is correct.)
If some students still have trouble understanding equivalency, encourage them to continue to use models like fraction circle pieces and tools such as the Fractions on Number Lines Chart or Fraction Chart in the Student Guide Reference section to find and gain experience with equivalent fractions.
Assign Questions 1–11 in the Explore section. Students will solve problems using fraction circle pieces, mental math, or paper and pencil. Encourage students to refer to the Fractions on Number Lines Chart, Fraction Chart, and Multiplication and Division Facts pages in the Student Guide Reference section if they need help finding equivalent fractions or common denominators.
Assign Home Practice Parts 3–4 after Part 1.
Use Check-In: Questions 8–11 on the Subtract Mixed Numbers pages in the Student Guide and the corresponding Feedback Box in the Teacher Guide to assess students' abilities to subtract mixed numbers including those with unlike denominators using fraction circle pieces, mental math, and paper-and-pencil methods [E8]; find common denominators and use them to subtract fractions [E10]; find equivalent fractions using tools and strategies [E1]; estimate differences of mixed numbers [E9]; represent the simplest form of a fraction [E2]; find a strategy [MPE2]; and check for reasonableness [MPE3].
