The purpose of this activity is for students to sort expressions showing sums and differences of fractions. Students likely will notice that some cards include expressions with mixed numbers, some include fractions that have the same denominator, and others include fractions with different denominators. Students are not expected to find the values of the expressions as that will be the work of the next activity. One way of sorting, however, may be based on how to find the value of the expression.

• Invite previously selected groups to share their categories and how they sorted their cards.
• Choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between expressions that have the same denominator and expressions that have different denominators.
• Display:
• “How could you find the value of this expression?” (I can just take 1 from 2 since both are thirds.)
• Display:
• “Why is finding the value of this expression different?” (It’s thirds and sixths, so I can’t just take away the sixth.)
• “In the next activity, we will find the values of expressions such as these.”

The purpose of this activity is for students to add and subtract fractions in ways that make sense to them. Students may use strategies such as drawing tape diagrams or number lines, or they may use computations to find a common denominator. 

Monitor for, and select to share in the Activity Synthesis, students who use:

• The meaning of fractions to explain why .
• A diagram or a number line to find the value of and .
• Equivalent fractions and arithmetic to find the value of  and .

The approaches are sequenced from more concrete to more abstract to help students connect a variety of different, but familiar, representations as they make sense of adding and subtracting fractions with unlike denominators. Students, who choose to draw number lines or tape diagrams, use appropriate tools strategically (MP5). Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently. For an example for each approach, look at the Student Responses.

• Invite previously selected students to share in the given order. Record or display their work for all to see.
• Connect students’ approaches by asking:
  • “How was different than the other 2 sums?” (It was thirds and thirds, so I could just add them. I did not need to find any equivalent fractions to make the denominators the same.)
  • “How was the way you found the value of  different than the other expressions?” (For the first one, I had thirds and thirds, so I could just add them. For the second one, it was thirds and sixths, so I just had to change the thirds to sixths. Here I had to change both the thirds and the half to sixths to get parts of the same size.)
• Connect students’ approaches to the learning goal by asking:
  • “Why is having a common denominator helpful when adding or subtracting fractions?” (When all parts have the same size, I can just add or subtract the number of parts.)