Previously, students considered non-unit fractions in terms of equal groups of unit fractions or as a product of a unit fraction and a whole number. This activity prompts students to think about non-unit fractions as sums of other fractions. The given context—about measuring fractional amounts, using measuring cups of certain sizes—allows students to continue thinking in terms of equal groups, but also invites them to consider a fractional quantity as a sum of two or more fractions with the same denominator. For instance, students may see as 5 groups of or as , but they also may see that is equal to . Students record such a decomposition as an equation. When students connect the quantities in the story problem to an equation, they reason abstractly and quantitatively (MP2).

Students may not be familiar with the use of measuring cups. Consider demonstrating how to use a 1-cup measure to obtain different whole numbers of cups. 

• Select students to share their responses in the order shown in the activity notes (from informal or concrete to formal or symbolic, from multiplication to addition).
• Record the different ways of expressing the quantities in the recipe.
• Consider drawing diagrams and annotating them to help students relate the expressions and the quantities.
• Highlight that each quantity can be written as a product of a whole number and a unit fraction, but it also can be written as a sum of smaller fractions. For example:
  • Barley: , or

    

  • Broth: or , or

    

In the previous activity, students saw that a fraction can be decomposed into a sum of fractions with the same denominator and that it can be done in more than one way. In this activity, they record such decompositions as equations. The last question prompts students to consider whether any fraction can be written as a sum of smaller fractions with the same denominator. Students see that only non-unit fractions (with a numerator greater than 1) can be decomposed that way. Students observe regularity in repeated reasoning as they decompose the numerator, 9, into different parts while the denominator in all cases is 5 (MP8).

• Invite students to share their equations. Display or record them for all to see.
• Next discuss students' responses to the last question. Select students with different explanations to share their reasoning.
• If not mentioned by students, highlight that fractions with a numerator of 1 (unit fractions) cannot be further decomposed into smaller fractions with the same denominator because it is already the smallest fractional part. Other fractions, with a numerator other than 1 (non-unit fractions), can be decomposed into fractions with the same denominator.