In this activity, students solve contextual problems that involve subtracting fractions of which at least one value is a mixed number and it is necessary to decompose one or both numbers. Students find differences in any way that makes sense to them. They may use number-line diagrams, reason in terms of addition, or perform repeated partial subtractions (without necessarily writing expressions or equations). For instance, to find the difference between and , they may:

• Draw a number line, partition it into eighths, locate , and count back 7 spaces to the left to represent subtraction of .
• Reason: “What number plus gives or ?”
• First subtract from , and then subtract another .
• First subtract 1 from , and then add back .

Students also may use the insights they gained previously, about rewriting and decomposing whole numbers, to facilitate subtraction and consider whether they could be applied to subtraction of a fraction from a mixed number. When students recognize the mathematical features of objects in the real world and ask questions that arise from a presented situation, they reason abstractly and quantitatively (MP2).

• Select 2–3 students who use different methods to briefly share their responses. Record subtraction and addition expressions or equations that students wrote or mentioned.
• Students will take a closer look at the same numbers in the next activity.

Previously, students found differences of two fractions, including mixed numbers, using any way that made sense to them. This activity formalizes and makes explicit how such differences can be found by writing equivalent fractions and decomposing a whole number or a mixed number. When students share their responses with a partner and revise them based on the feedback they receive, they construct viable arguments and critique the reasoning of others (MP3).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing.

MLR1 Stronger and Clearer Each Time

• Ask each student to find a new partner.
• “Explain your response to the last part of the second problem to your new partner.”
• “Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–4 minutes: structured partner discussion.
• Repeat with at least one other partner.
• “Revise your initial response based on the feedback you got from your partners.”
• 2 minutes: independent work time