Lesson 10

The purpose of this How Many do You See is for students to visualize a common denominator for two fractions. The diagram can be seen as showing \(\frac{8}{12} + \frac{3}{12}\) but it can also be seen as showing \(\frac{2}{3} + \frac{1}{4}\). The area diagram provides a way to visualize why the product of two denominators works as a common denominator for two fractions. 

• Groups of 2
• “How many do you see? How do you see them?”  
• Display the image.
• 1 minute: quiet think time

• Display the image. 
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “How does the diagram show \(\frac{2}{3} + \frac{1}{4}\)?” (There are \(\frac{2}{3}\) of the left square and \(\frac{1}{4}\) of the right square.)
• “What is the value of \(\frac{2}{3} + \frac{1}{4}\)? How do you know?” (\(\frac{11}{12}\) because there are 11 shaded pieces and each one is \(\frac{1}{12}\).)

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, \(\frac{1}{2} + \frac{1}{3}\), 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 these students who:

• look at multiples of the denominators and pick a common one 
• notice that the product of the denominators is a common denominator for the two fractions

• Groups of 2

• 5 minutes: individual work time
• 5 minutes: partner discussion
• monitor for students who:
  • use twelfths as a common denominator to find the value of \(\frac{3}{4}+\frac{4}{6}\)
  • use twenty-fourths as a common denominator to find the value of \(\frac{3}{4}+\frac{4}{6}\)

If students don’t use equivalent fractions to find the value of the expressions, ask, “How can you write an equivalent sum (or difference) of fractions with the same denominator?”

• Ask previously selected students to share their responses.
• “How did you decide which common denominator to use?” (I know that 4 and 6 are both factors of 12 or I know that 4 and 6 are factors of 24 because 24 is \(4 \times 6\).)
• “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 for 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 for 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 \(\frac{2}{3} + \frac{5}{9}\) 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 for 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, \(\frac{3}{50} + \frac{11}{100} = \frac{17}{100}\)  since \(\frac{3}{50}\) is equivalent to \(\frac{6}{100}\). The number\( \frac{17}{100}\) is probably easier to grasp mentally than the number \(\frac{850}{50,000}\)which is what you get if you use the product of the denominators.

• Groups of 2

• 5 minutes: individual work time
• 5 minutes: partner work time
• Monitor for students who use different denominators for the last two sums and differences.

• Invite students to share how they found the value of \(\frac{3}{8} + \frac{1}{5}\)
• “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 \(\frac{7}{10} - \frac{2}{3}\).
• “Does Lin’s strategy work here too?” (Yes, I used 30 which is \(10 \times 3\).)
• Invite selected students to share their responses for \(\frac{7}{20} + \frac{41}{50}\).
• Display: \(\frac{117}{100}\) and \(\frac{1,170}{1,000}\)
• “Are these fractions equivalent? How do you know?” (Yes, the numerator and denominator for the second one are 10 times the numerator and 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 the common denominator. I just took the product of 20 and 50.)