Lesson 11

Different Ways to Subtract

Warm-up: Number Talk: Mixed Number Addition and Subtraction (10 minutes)

Narrative

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for adding and subtracting a fraction and a whole number. For these problems, students do not need to focus on a common denominator as the numbers either have the same denominator or one of the numbers in the sum is a whole number. Their strategies for thinking about the sums and differences will be helpful throughout the lesson as they calculate more complex differences involving mixed numbers. 

Launch

  • Display one problem.
  • “Give me a signal when you have an answer and can explain how you got it.”
  • 1 minute: quiet think time

Activity

  • Record answers and strategy.
  • Keep problems and work displayed.
  • Repeat with each problem.

Student Facing

Find the value of each expression mentally.

  • \(3+\frac{7}{8}\)
  • \(3-\frac{7}{8}\)
  • \(1\frac{5}{8}+\frac{6}{8}\)
  • \(1\frac{5}{8}-\frac{6}{8}\)

Student Response

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Activity Synthesis

  • “How did you find the value of \(1\frac{5}{8} + \frac{6}{8}\)?” (I made a fraction from the mixed number and then added the numerators. I added on to get 2 and then added the rest of the eighths.)

Activity 1: Challenging Differences (20 minutes)

Narrative

The purpose of this activity is for students to subtract a fraction or mixed number from a mixed number. There are multiple strategies available and the differences are selected in order to highlight these strategies:

  • subtracting the whole number and fraction parts of the numbers separately 
  • rewriting the mixed number to facilitate subtraction
  • adding on to find the difference

In each case, students will need to choose a common denominator in their calculation. Students first identify expressions that are equivalent to the mixed number that appears in all of the differences they calculate. These expressions are deliberately chosen to support the listed techniques to find the value of the subtraction expressions. The goal of the activity synthesis is to compare and connect several different strategies and consider the benefits and challenges of each strategy.

When students adapt their subtraction strategy to the numbers, they look for and make use of structure (MP7).

This activity uses MLR7 Compare and Connect. Advances: Conversing.

Launch

  • Groups of 2

Activity

  • 10 minutes: independent or group work 
  • monitor for students who:
    • add on to find the value of \(3\frac{5}{8}-1\frac{15}{16}\)
    • use an equivalent expression that has a fraction greater than 1, such as \(2\frac{13}{8}\), to find the value of \(3\frac{5}{8}-1\frac{12}{16}\)
MLR7 Compare and Connect
  • “Create a visual display that shows your thinking about \(3\frac{5}{8}-1\frac{15}{16}\). You may want to include details such as notes, diagrams or drawings to help others understand your thinking.”
  • 2 minutes: independent or group work
  • 5 minutes: gallery walk 

Student Facing

  1. Circle all of the expressions that are equivalent to \(3\frac{5}{8}\). Explain or show your reasoning.

    • \(\frac{20}{8}\)
    • \(2\frac{13}{8}\)
    • \(3\frac{10}{16}\)
  2. Find the value of each expression. Explain or show your reasoning.

    • \(3\frac{5}{8}-\frac{3}{16}\)
    • \(3\frac{5}{8}-1\frac{15}{16}\)
    • \(3\frac{5}{8}-1\frac{12}{16}\)

Student Response

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Activity Synthesis

  • “What is the same and what is different between the strategies?” (Some people added on or used a number line. Some people changed the mixed number to a whole number plus a fraction greater than one. Some people changed the mixed number to a fraction. Some people got different, equivalent answers.)
  • Display: \(3\frac{5}{8}=2\frac{13}{8}\)
  • “How do we know this is true?” (There are 8 eighths in 1 so I can break up the 3 as 2 and 1 and put the 1 with the \(\frac{5}{8}\) to make \(\frac{13}{8}\).)
  • Invite previously selected students to share how they found the value of \(3\frac{5}{8} - 1\frac{12}{16}\).
  • “How was rewriting \(3\frac{5}{8}\) as \(2\frac{13}{8}\) helpful in this calculation?” (I could take 1 from 2 and take \(\frac{12}{16}\) from \(\frac{13}{8}\) after rewriting it as \(\frac{6}{8}\).)
  • Invite previously selected students to share how they found the value of \(3\frac{5}{8}-1\frac{15}{16}\).
  • “Why did you decide to use that strategy?” (I noticed that \(\frac{15}{16}\) was really close to 1 so it was easy for me count up.) 
  • “How was rewriting \(3\frac{5}{8}\) as \(3\frac{10}{16}\) helpful in this calculation?” (I needed to add a sixteenth to get to 2 and it’s easy to combine sixteenths and sixteenths.)
  • “We are going to find the values of more differences of mixed numbers and fractions in the next activity.”
  • Keep displays available for students to refer to during activity 2.

Activity 2: Find the Difference (15 minutes)

Narrative

The purpose of this activity is for students to find the value of differences of mixed numbers. The numbers are chosen to encourage a variety of strategies that were highlighted in the previous activity. Students should be encouraged to find the differences in a way that makes sense to them. This may mean choosing a different strategy depending on the problem but it could also mean writing each difference as a difference of fractions and then finding a common denominator. 

Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for finding the difference of each expression before they begin. Students can speak quietly to themselves, or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language

Launch

  • Groups of 2

Activity

  • 5 minutes: independent think time
  • 5 minutes: small-group work time
  • Monitor for students who:
    • add on to find the value of \(9\frac{1}{8}-8\frac{8}{9}\)
    • rewrite \(\frac{10}{4}\) as 2\(\frac{1}{2}\) to find the value of 3\(\frac{1}{2}-\frac{10}{4}\)
    • use an equivalent expression with a fraction greater than one, such as \(3\frac{24}{15}-1\frac{10}{15}\), to find the value of 4\(\frac{3}{5}-1\)\(\frac{2}{3}\)

Student Facing

Find the value of each difference. Explain or show your reasoning.

  1. \(9\frac{1}{8}-8\frac{8}{9}\)
  2. \(3\frac{1}{2}-\frac{10}{4}\)
  3. \(4\frac{3}{5}-1\frac{2}{3}\)

Student Response

For access, consult one of our IM Certified Partners.

Advancing Student Thinking

If students do not have a strategy to find the value of 4\(\frac{3}{5}-1\)\(\frac{2}{3}\), write some expressions that are equivalent to \(4\frac{3}{5}\), such as \(4\frac{9}{15}\) and \(3\frac{24}{15}\) and ask, “How do you know these expressions are equivalent?”

Activity Synthesis

  • Ask previously selected students to share in the given order.
  • “Why did you decide to add on to \(8\frac{8}{9}\)?” (\(\frac{8}{9}\) is really close to 1.)
  • “Can you also subtract in 2 steps to find the value of \(9\frac{1}{8} - 8\frac{8}{9}\)?” (Yes, I can subtract \(\frac{1}{8}\) from \(9\frac{1}{8}\) to get 9 and then take away \(\frac{1}{9}\) more to get \(8\frac{8}{9}\). That’s the same idea as adding on.)
  • “Why did you decide to use \(2\frac{1}{2}\) instead of \(\frac{10}{4}\)?” (It is easy to subtract \(3\frac{1}{2}-2\frac{1}{2}\).)

Lesson Synthesis

Lesson Synthesis

“Today we found differences of mixed numbers and fractions.”

Display the differences.

  • \(3\frac{5}{8}-1\frac{15}{16}\)  
  • \(3\frac{5}{8}-\frac{3}{16}\)  
  • \(3\frac{5}{8}-1\frac{12}{16}\)
  • \(9\frac{1}{8}-8\frac{8}{9}\)
  • 3\(\frac{1}{2}-\frac{10}{4}\)
  • 4\(\frac{3}{5}-1\)\(\frac{2}{3}\)

“How can we sort these expressions based on the strategies we used?” Sample responses:

  • We could put \(9\frac{1}{8}-8\frac{8}{9}\) and \(3\frac{5}{8}-1\frac{15}{16}\) together because they both have fractions that are close to 1 and adding on was a good strategy to find these differences.
  • We could put \(3\frac{1}{2}-\frac{10}{4}\)  and \(3\frac{5}{8}-\frac{3}{16}\)  together because we just had to find a common denominator for the fractional part and write \(\frac{10}{4}\) as a mixed number to find the values. We could take the whole number from the whole number and the fraction from the fraction.
  • We could put  \(3\frac{5}{8}-1\frac{12}{16}\) and  4\(\frac{3}{5}-1\)\(\frac{2}{3}\) together because they were the most challenging or they required the most steps.

“How do you decide which strategy to use when finding the difference of mixed numbers or a mixed number and a fraction?” (I look at the numbers and think about which strategy would be easy to use and accurate.)

Cool-down: Mixed Differences (5 minutes)

Cool-Down

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