Lesson 19

Compare to 1

Warm-up: What Do You Know About $\frac{15}{14}\times\frac{23}{30}$? (10 minutes)

Narrative

The purpose of this What Do You Know About _____ is for students to share what they know about and how they can represent the product \(\frac{15}{14}\times\frac{23}{30}\). The numbers were intentionally chosen to make finding the exact value of the product challenging.

Launch

  • Display the expresion.
  • “What do you know about \(\frac{15}{14}\times\frac{23}{30}\)?”
  • 1 minute: quiet think time

Activity

  • Record responses.
  • “How could we find the value of the product \(\frac{15}{14}\times\frac{23}{30}\)?” (Find the product of the numerators and the product of the denominators.)

Student Facing

What do you know about \(\frac{15}{14}\times\frac{23}{30}\)?

Student Response

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Activity Synthesis

  • “Is \(\frac{15}{14}\times\frac{23}{30}\) less than, equal to, or greater than \(\frac{23}{30}\)? Why?” (It is greater since \(\frac{15}{14}\) is greater than 1.)

Activity 1: Compare Fraction Products on the Number Line (15 minutes)

Narrative

The goal of this activity is to continue to compare the size of a product of fractions to the size of the second factor. In addition to the number line representation which students have worked with in the last few lessons, they also see a different expression that represents the product. In the next activity, this expression will be combined with the distributive property to explain in all cases why multiplying a number by a fraction less than one results in a smaller number while multiplying by a fraction greater than one results in a larger number (MP8).

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed _____, so I matched . . . .” Encourage students to challenge each other when they disagree.
Advances: Representing, Conversing

Launch

  • Groups of 2

Activity

  • 1–2 minutes: quiet think time
  • 6–8 minutes: partner work time

Student Facing

  1. Match the expressions and number lines that show the same value.

    • \(\frac{2}{5} \times \frac{4}{3}\)
    • \(\frac{3}{4} \times \frac{5}{2}\)
    • \(\frac{4}{3} \times \frac{5}{2}\)

    • \(\left(1+\frac{1}{3}\right) \times \frac{5}{2}\)
    • \(\left(1-\frac{3}{5}\right) \times \frac{4}{3}\)
    • \(\left(1-\frac{1}{4}\right) \times \frac{5}{2}\)

    ANumber line. 

    BNumber line.

    CNumber line. 

  2. Choose one of the expressions from each set and explain whether the value is greater than or less than the second factor.

Student Response

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Activity Synthesis

  • Invite students to share their matches.
  • “How did you find the matching number line for \(\frac{3}{4} \times \frac{5}{2}\)?” (I saw that two of the number lines have \(\frac{5}{2}\) on them and looked for the one that showed \(\frac{3}{4}\) of \(\frac{5}{2}\). I knew which one it was because \(\frac{3}{4}\) of \(\frac{5}{2}\) is less than \(\frac{5}{2}\).)
  • “How did you find the matching expression for \(\frac{3}{4} \times \frac{5}{2}\)?” (I looked for an expression with \(\frac{5}{2}\) and only one of them had another factor with the value \(\frac{3}{4}\).)
  • “How did you know whether the value of \(\frac{3}{4} \times \frac{5}{2}\) was greater than or less than \(\frac{5}{2}\)?” (I knew it was less because \(\frac{3}{4}\) is less than 1. That was what helped me find the right number line.)

Activity 2: True Statement (20 minutes)

Narrative

The goal of this activity is to use the distributive property to explain why multiplying a number by a fraction greater than one increases the size of the number while multiplying by a fraction less than one decreases the size of the number. Expressions are particularly useful here because they show explicitly how the size of the number relates to the product. For example writing \(\frac{3}{5}\) as \(1 - \frac{2}{5}\) and then multiplying by \(\frac{4}{7}\) gives: \(\displaystyle \left(1 - \frac{2}{5}\right) \times \frac{4}{7} = \frac{4}{7} - \left(\frac{2}{5} \times \frac{4}{7}\right)\)

The revealing part of this calculation is that the structure of the right hand side shows that it is less than \(\frac{4}{7}\) without calculating the exact value (MP7). It must be less than \(\frac{4}{7}\) because it is \(\frac{4}{7}\) minus some other number.

Engagement: Internalize Self-Regulation. Provide students an opportunity to self-assess and reflect on their own progress. For example, provide students with questions that relate to the size of the factors for them to reflect on once the activity is complete.
Supports accessibility for: Conceptual Processing, Attention, Memory

Launch

  • Groups of 2

Activity

  • 1–2 minutes: quiet think time
  • 8–10 minutes: partner work time
  • Monitor for students who use the expressions in the first problem to make the comparisons and then generalize about what happens when you multiply a number by any fraction greater than 1 or less than 1.

Student Facing

  1. Rewrite each expression as a sum or difference of 2 products.

    1. \(\left(1 - \frac{2}{5}\right) \times \frac{4}{7}\)

    2. \(\left(1 + \frac{1}{5}\right)\times \frac{4}{7}\)

    3. \(\left(1 - \frac{3}{8}\right)\times \frac{4}{7}\)

    4. \(\left(1 + \frac{1}{8}\right)\times \frac{4}{7}\)

  2. Fill in each blank with \(<\) or \(>\) to make the inequality true.

    1. \(\left(1 - \frac{2}{5}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}} \,\frac{4}{7}\)
    2. \(\left(1 + \frac{1}{5}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}}\, \frac{4}{7}\)
    3. \(\left(1 - \frac{3}{8}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}} \,\frac{4}{7}\)
    4. \(\left(1 + \frac{1}{8}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}} \,\frac{4}{7}\)
  3. Describe the value of the product when \(\frac{4}{7}\) is multiplied by a fraction greater than 1. Explain your reasoning.
  4. Describe the value of the product when \(\frac{4}{7}\) is multiplied by a fraction less than 1. Explain your reasoning.

Student Response

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Activity Synthesis

  • Invite students to share their expressions for the products in the first problem.
  • Display the equation: \(\left(1 - \frac{2}{5}\right)\times \frac{4}{7} = \frac{4}{7} - \left(\frac{2}{5} \times \frac{4}{7}\right)\)
  • “How can you see that the value of the expression is less than \(\frac{4}{7}\)?” (It’s \(\frac{4}{7}\) minus something.)
  • “Does this reasoning also work for \(\left(1 - \frac{3}{8}\right)\times \frac{4}{7}\)?” (Yes, it’s again \(\frac{4}{7}\) minus some other number.)
  • “Will this reasoning work whenever you multiply a number less than 1 by \(\frac{4}{7}\)?” (Yes, I’ll always get \(\frac{4}{7}\) minus an amount so that’s less than \(\frac{4}{7}\).)

Lesson Synthesis

Lesson Synthesis

“Today we compared the value of a product of fractions to the value of one of the factors without calculating the product.”

Display product: \(\frac{7}{9} \times \frac{15}{13}\).

“What are some ways you can compare the value of the product with \(\frac{15}{13}\)?” (I can calculate the value, but the numbers are complicated. I can make a number line diagram and see that it is to the left of \(\frac{15}{13}\). I can rewrite \(\frac{7}{9}\) as \(1-\frac{2}{9}\) and see that it is less.)

“What are some ways you can compare the value of the product with \(\frac{7}{9}\)?” (I can calculate the value. I can make a number line diagram and see that it is to the right of \(\frac{7}{9}\). I can rewrite \(\frac{15}{13}\) as \(1+\frac{2}{13}\) and see that it is more.)

Cool-down: Compare without Calculating (5 minutes)

Cool-Down

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Student Section Summary

Student Facing

Number line. 2 tick marks, labeled 0, 4 sevenths. Point between the tick marks, labeled 3 fifths times 4 sevenths. 

In this section, we learned how to compare the size of a product to the size of the factors. To compare \(\frac{3}{5} \times \frac{4}{7}\) with \(\frac{4}{7}\), for example, we can put them on a number line.
Since \(\frac{3}{5}\) is 3 equal parts with 5 parts in the whole, it is to the left of \(\frac{4}{7}\), only part of the way there. We can also see this by writing \(\frac{3}{5}\) as \(1 - \frac{2}{5}\).

\(\left(1 - \frac{2}{5}\right)\times \frac{4}{7} = \frac{4}{7} - \left(\frac{2}{5} \times \frac{4}{7}\right)\)

The product is less than \(\frac{4}{7}\) because it is \(\frac{4}{7}\) minus a fraction.