The purpose of this activity is for students to apply what they have learned about using common denominators to add and subtract fractions with unlike denominators. In a previous lesson, students added two fractions where neither denominator was a multiple of the other, , using a strategy that made sense to them. In this activity students see more complex examples. Having built an understanding that they need to find equivalent fractions with a common denominator students will develop strategies for finding a common denominator (MP7, MP8).

Monitor for students who:

• Look at multiples of the denominators and pick a common denominator. 
• Notice that the product of the denominators is a common denominator of the two fractions.

Representation: Internalize Comprehension. Begin by asking students: “Do these expressions remind anyone of something we have done before?”

• Ask previously selected students to share their responses.
• “How did you decide which common denominator to use?” (I know that both 4 and 6 are factors of 12, or I know that 4 and 6 are factors of 24 because 24 is .)
• “How did you use the common denominator to find the sum?” (I found equivalent expressions with 12 or 24 as a denominator, and then I could add the fractions since they had the same denominator.)
• “In the next activity, we are going to see a general strategy to find a common denominator of two fractions.”

The purpose of this activity is for students to explain why the product of the denominators of two fractions is always a common denominator of the two fractions. Students noticed in the previous activity that there are several possible common denominators. Sometimes it is possible to just see a common denominator. For example, for students might notice that 9 is a common denominator because it is a multiple of 3. It can be convenient, however, to have a strategy that always works, especially for more challenging denominators. After explaining why the product of two denominators is always a common denominator of a pair of fractions (MP3), students practice finding sums and differences of fractions in any way that makes sense to them. This may include:

• Using the product of the denominators.
• Thinking about each pair of fractions individually.

Both strategies are important. For example,  since  is equivalent to . The number is probably easier to grasp mentally than the number  which is the result when using the product of the denominators.

• Invite students to share how they found the value of
• “How did your strategy compare to Lin’s method?” (I used the product of the denominators for a common denominator, just like Lin.)
• Invite students to share how they found the value of .
• “Does Lin’s strategy work here, too?” (Yes, I used 30, which is .)
• Invite selected students to share their responses for .
• Display: and
• “Are these fractions equivalent? How do you know?” (Yes, the numerator and the denominator of the second fraction are 10 times the numerator and the denominator of the first.)
• “Which of these denominators do you prefer?” (I like using hundredths because I’m used to them. I like using thousandths because I did not have to think about finding a common denominator. I just took the product of 20 and 50.)