Lesson 8

The purpose of this Which One Doesn’t Belong is for students to recall representations of fractions they have seen in an earlier course. Two of the representations are fraction strips and the other two are number lines. These representations will be useful to students in this and future lessons as they think about representing equivalent fractions.

• Groups of 2
• Display the image.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “How do the diagrams in B and C help us see the relationship between thirds, sixths, and twelfths?” (We can see that \(\frac{1}{3}=\frac{4}{12}\) and \(\frac{1}{3}=\frac{2}{6}\).)

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

• Create a set of cards from the blackline master for each group of 2.

• Groups of 2 or 4
• Distribute one set of pre-cut cards to each group of students.
• “In this activity, you will sort some cards into categories of your choosing. When you sort the expressions, you should work with your partner to come up with categories.”

• 8 minutes: partner work time
• Monitor for students who sort the expressions according to whether the denominators of the fractions are the same or different.

• Select 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: \(\frac{2}{3} - \frac{1}{3}\)
• “How could you find the value of this expression?” (I can just take 1 from 2 since they are both thirds.)
• Display: \(\frac{2}{3} - \frac{1}{6}\)
• “Why is finding the value of this expression different?” (It’s thirds and sixths so I can’t just take the sixth away.)
• “In the next activity we will find the values of expressions like these.”

The purpose of this activity is for students to add and subtract fractions in a way that makes 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 students with the following strategies to share in the synthesis: 

• use the meaning of fractions to explain why \(\frac{2}{3}+\frac{2}{3}=\frac{4}{3}\)
• use a diagram like a number line to find the value of \(\frac{2}{3} - \frac{1}{6}\) and \(\frac{2}{3}+\frac{1}{2}\)
• use equivalent fractions and arithmetic to find the value of \(\frac{2}{3} - \frac{1}{6}\) and \(\frac{2}{3}+\frac{1}{2}\)

Students who choose to use the number line or tape diagrams use appropriate tools strategically (MP5). 

• Groups of 4

• 5 minutes: independent work time
• 5 minutes: small-group discussion
• As students work, consider asking:
  • “What is the same about these expressions? What is different?”
  • “How did you decide which strategy to use?”

If students do not have a strategy to add or subtract fractions with unlike denominators, refer to one of the expressions and ask, “How could you use one number line to show both of these fractions?”

• Ask selected students to share their response for how to add or subtract fractions with the same denominator.
• “How was this sum 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.)
• Ask selected students to display their responses how to add and subtract fractions with unlike denominators, such as \(\frac{2}{3}+\frac{1}{2}\).  
• “How was your strategy for finding this sum different than the other problems?” (For the first one, I had thirds and thirds so 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.)
• “Why is having a common denominator helpful when adding or subtracting fractions?” (When the parts all have the same size I can just add or subtract the number of parts.)