The purpose of this activity is for students to identify equivalent sums of fractions and use them to find the values of sums of fractions with different denominators. In a previous grade, students learned to use factors and multiples to generate and identify equivalent fractions. They recall that technique here, and then use those equivalent fractions to find sums. This helps reinforce the idea that it is helpful, when adding two fractions, for the fractions to have the same denominator, while also recalling how to find an equivalent fraction with a different denominator.

When students identify that equivalent fractions with the same denominator help to find the value of a sum, they notice and take advantage of the meaning and the structure of fractions (MP7).

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

• “How can you use number lines to help you find fractions that are equivalent to ?”
• Draw 2 different number lines to represent . “How can you adapt the diagrams to show =?”

• Invite previously selected students to share how they know the expressions and have the same value as .
• “How do you know that ?” (I can divide each into 4 equal parts. Those parts are s, and there are 8 of them.)
• “Why is the expression  helpful for finding the sum?” (It’s all twelfths. I have 8 of them and 10 more, so that's .)
• “Which expression did you choose to find the sum?” (I used , because the numbers were smaller.)

This activity builds on the previous activity in which students saw how equivalent expressions can be a valuable tool to add or subtract fractions. The purpose of this activity is for students to generate an equivalent expression in order to find the value of a difference of fractions. Monitor for students who:

• Find equivalent fractions with smaller numerators and denominators than the given fractions.
• Find equivalent fractions with greater numerators and denominators than the given fractions.

• Invite previously selected students to share how they found the value of .
• “How are the strategies for finding the value of the expression the same?” (Both change one of the fractions to an equivalent fraction so the fractions have the same denominator.)
• “How are the strategies for finding the value of the expression different?” (To make the denominator bigger I multiply by a whole number. To make the denominator smaller I divide by a whole number.)
• “Why is it important to have the same denominator?” (I can add or subtract the number of parts when they are the same size.)

The purpose of this activity is for students to solve a problem that involves finding the difference of fractions. Students may use addition or subtraction to solve the problem. Either way, they will need to find a common denominator of the fractions. One of the numbers is a mixed number, so students may:

• Convert the mixed number to a fraction.
• Find the difference in steps, adding on or subtracting.

When students recognize mathematical features of objects in the real world, they model with mathematics (MP4).

• Continue to the Lesson Synthesis.