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 its factors. They begin by using a number line to locate such products and then choose the numerator or the denominator of a fraction in order to make a product smaller, the same, or greater. To choose the number correctly, students need to understand the relationship between:

• The numerator and the denominator of a fraction and the size of the fraction.
• The size of one 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).

• “How are the expressions the same? How are they different?”
• “Where would of 12 be located? What about of 12? How might you identify the approximate location of these expressions, without multiplying?”

• Display the equation:
• “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 .)
• Display the inequality:
• “What solutions did you find for this statement?” (14, 15, 16, and so on.)
• “What do all the solutions have in common?” (All are greater than 13.)
• “Why?” (Because the product will be less than 12 only if the fraction is less than 1. That means the numerator has to be less 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.)

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

• Invite students to share how they found the location of  on the number line.
• “Why is to the left of A?” (It’s less than A since it’s missing  of A. So it’s to the left.)
• “How do you know is to the right of A on the number line?” (Because  is greater than 1. It's an extra of A.)
• Invite students to share how they compared with .
  • I know is greater than 1 so that means the product is bigger.
  • I can use the number line and imagine that A is .