Lesson 18

Compare Without Multiplying

Warm-up: Notice and Wonder: Expressions and Number Lines (10 minutes)

Narrative

The purpose of this warm-up is for students to interpret a multiplication expression as a location on the number line. This builds on work students did in the previous lesson with an emphasis now on precisely locating the expression using the meaning of multiplication. Students will build on this idea and locate the value of more complex expressions throughout the lesson.

Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

Activity

• 1 minute: partner discussion
• Share and record responses

Student Facing

What do you notice? What do you wonder?

$$\frac{2}{3} \times 5$$

Activity Synthesis

• “How can you find the approximate location of $$\frac{2}{3} \times 5$$ on the number line?” (I can find the value and add more tick marks to find it exactly. I can divide the number line between 0 and 5 into 3 equal parts and $$\frac{2}{3} \times 5$$ will be the second of those tick marks.)

Activity 1: Approximate Location (15 minutes)

Narrative

The purpose of this activity is for students to understand, using complex numbers and no context, the relationship between the size of a product and the size of one of the factors. They begin by using a number line to locate such products and then choose the numerator or denominator of a fraction in order to make a product smaller, the same, or greater. To choose the number correctly students need to understand both:

• the relationship between the numerator and denominator of a fraction and the size of the fraction
• the relationship between the size of a factor and the size of the product

When students locate the expressions on the number line they use their understanding of multiplication, fractions, and the structure of the number line (MP7).

Representation: Access for Perception. Synthesis: Use gestures during the discussion to emphasize if the product was greater than, less than, or equal to twelve.
Supports accessibility for: Conceptual Processing, Fine Motor Skills

• Groups of 2

Activity

• 3–5 minutes: independent work time
• 3–5 minutes: partner discussion
• Monitor for students who notice patterns as they solve the last problem. For example, they notice that to make $$\frac{\boxed{\phantom{100}}}{11} \times 12 > 12$$ true, the numerator in the fraction must be larger than the denominator.

Student Facing

1. Label each expression at its approximate location on the number line.

Partner A

1. $$\frac{2}{5} \times 12$$
2. $$\frac{5}{3} \times 12$$
3. $$\frac{7}{7} \times 12$$

Partner B

1. $$\frac{4}{7} \times 12$$
2. $$\frac{8}{5} \times 12$$
3. $$\frac{9}{9} \times 12$$
2. Choose a number to put in each box to make the statement true.

1. $$\frac{\boxed{\phantom{\frac{0}{000}}}}{\Large{11}} \times 12 > 12$$
2. $$\frac{\boxed{\phantom{\frac{0}{000}}}}{\Large{15}} \times 12 = 12$$
3. $$\frac{\Large{13}}{\boxed{\phantom{\frac{0}{000}}}} \times 12 < 12$$

Student Response

If students don’t use the number line to consider the relationship between the factors, ask them to consider where $$\frac{1}{2}$$ of 12 would be located. What about $$\frac{1}{4}$$ of 12? Ask them to explain how they can identify the location of the value of these expressions without multiplying.

Activity Synthesis

• Display the equation: $$\frac{\boxed{\phantom{100}}}{15} \times 12 = 12$$
• “What solution(s) did you find for this statement?” (Just 15.)
• “Why is there only one solution?” (Because the only multiple of 12 that’s 12 is $$1 \times 12$$.)
• Display the inequality: $$\frac{13}{\boxed{\phantom{100}}} \times 12 < 12$$
• “What solutions did you find for this statement?” (14, 15, 16, and so on.)
• “What do the solutions all have in common?” (They are all more than 13.)
• “Why?” (Because the product will only be less than 12 if the fraction is less than 1. That means the numerator has to be smaller than the denominator.)
• “How do the number lines help us understand the comparison?” (They show the relationship between the size of the fraction and the value of the product.)

Activity 2: An Unknown Number (20 minutes)

Narrative

In the previous activity students located numerical expressions on the number line, noticing that $$\frac{2}{5} \times 12$$, for example, is less than 12 because it is only 2 out of 5 equal parts making 12. The goal of this activity is for students to extend this reasoning to all numbers, including 12 but also including fractions which is new. Students continue to use a number line to support their reasoning and the reasoning is identical to what students did in the previous lesson comparing different distances students ran to Priya's (unknown) distance. If P is how far Priya ran in miles then $$\frac{1}{2} \times {\rm P}$$ is halfway between 0 and P on the number line whether P is a whole number or a fraction.

MLR7 Compare and Connect. Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “What did the approaches have in common?”, “How were they different?”, and “Did anyone solve the problem the same way, but would explain it differently?”

• Groups of 2

Activity

• 6–8 minutes: independent work time
• 3–5 minutes: partner discussion
• Monitor for students who:
• label the number line with tick marks to show the location of the value of each expression in relation to A
• refer to $$\frac{8}{8}$$ to explain why $$\frac{13}{8}\times \frac{11}{39}$$ is greater than $$\frac{11}{39}$$
• draw a number line or use the given number line to show the relationship between $$\frac{17}{53}$$ and $$\frac{2}{3}\times \frac{17}{53}$$

Student Facing

1. The number A is shown on the number line. Label the approximate location of the value of each expression. Explain or show your reasoning.

• $$\frac{1}{4} \times A$$
• $$2 \times A$$
• $$\frac{13}{8} \times A$$
• $$\frac{2}{3} \times A$$
2. Is $$\frac{13}{8}\times \frac{11}{39}$$ less than, greater than, or equal to $$\frac{11}{39}$$? Explain or show your reasoning.
3. Is $$\frac{2}{3}\times \frac{17}{53}$$ less than, greater than, or equal to $$\frac{17}{53}$$? Explain or show your reasoning.

Activity Synthesis

• Invite students to share how they found the location of $$\frac{2}{3} \times \text{A}$$ on the number line.
• “Why is $$\frac{2}{3} \times \text{A}$$ to the left of A?” (It’s less than A since it’s missing $$\frac{1}{3}$$ of A. So it’s to the left.)
• “How do you know $$\frac{13}{8} \times \text{A}$$ is to the right of A on the number line?” (Because $$\frac{13}{8}$$ is more than 1. It's an extra $$\frac{5}{8}$$ of A.)
• Invite students to share how they compared $$\frac{13}{8} \times \frac{11}{39}$$ with $$\frac{11}{39}$$.
• I know $$\frac{13}{8}$$ is more than 1 so that means the product is bigger.
• I can use the number line and imagine that A is $$\frac{11}{39}$$.

Lesson Synthesis

Lesson Synthesis

“Today we compared products to the size of one factor when both factors were fractions.”

Display: $$\underline{\hspace{0.7cm}}\times 6<6$$

“What numbers make this statement true?” ($$\frac{1}{2}$$, $$\frac{2}{3}$$, $$\frac{5}{8}$$, any number that’s less than 1)
Display: $$\underline{\hspace{0.7cm}}\times 6>6$$
“What numbers make this statement true?” (2, 5, 10, $$\frac{8}{5}$$, any number that’s greater than 1)

Record the numbers so students see them.

“What if I replace 6 with $$\frac{3}{8}$$? Do your numbers still make the statements true?” (Yes, half of $$\frac{3}{8}$$ is still less than $$\frac{3}{8}$$ and twice $$\frac{3}{8}$$ is still more than $$\frac{3}{8}$$.)

Record student explanations and keep a copy to refer to during future lessons.