In this activity, students use any strategy that makes sense to them to reason about subtraction of a fraction from a whole number. They begin by using an image to support their reasoning. Later, when no image is given, students may use a variety of ways to find differences. In the Activity Synthesis students share, explain, and relate different approaches for solving the problem (MP3).

At this point, students are not expected to decompose or rewrite numbers, using expressions and equations. They may perform the reasoning intuitively and informally. Later, they will formalize the different ways of recording the decomposition of numbers for subtraction. During the Activity Synthesis, the teacher records students’ thinking as drawings, equivalent fractions, or expressions, so it is visible to all students.

Monitor for and select to share in the Activity Synthesis students with the following approaches to solve the second problem about pouring water out of a 4-cup pitcher:

• Remove or count back  at a time, using a discrete drawing (for example, 1 square for each cup) or a number line. Sample response:
  • A number line from 0 to 4, partitioned into thirds, with 5 jumps of to the left of 4
    Teacher records above each whole on the representation, rewrites as , and records: .
• Remove or count back whole numbers first, and then thirds, using discrete drawings or a number line. Sample responses:
  • A number line from 0 to 4, partitioned into thirds between 2 and 3, with a jump from 4 to 3 and then another jump of to the left
    Teacher rewrites as or and records: and Rewrites 3 as and records: .

• Invite previously selected students to share in the given order. Record or display their work for all to see.
• Connect students’ approaches by asking:
  •  “What were different ways to think about 4 whole cups when subtracting?” (3 cups and cup, 2 cups and  cups,  cups) 
• Connect students’ approaches to the learning goal by asking:
  • “How did the strategies use equivalent fractions?” (We could think about each whole as or , or all the wholes completely partitioned into fourths or thirds on a diagram or number line or while explaining our thinking.)

This activity makes explicit what students may have noticed in earlier activities, namely, that we can subtract a fraction from a whole number if we:

• Write an equivalent fraction for the whole number.
• Decompose the whole number into a sum of smaller numbers, which could be fractions, or a whole number and a fraction.

This sorting task gives students opportunities to analyze fractions and expressions closely and make connections (MP7). Each card has a value equivalent to either or . Though students may sort the cards in a few valid ways, the discussion should focus on the values on the cards. Students then apply their insights from the sorting work to subtract a fraction from whole numbers.

This activity uses MLR2 Collect and Display. Advances: conversing, reading, writing.

• See the Lesson Synthesis.