Compare Products by Scaling
Compare Products by Scaling is a Grade 5 math skill from Illustrative Mathematics Chapter 6 (Place Value Patterns and Decimal Operations) that teaches students to compare multiplication products without computing by examining the fraction multiplier. Multiplying by a fraction greater than 1 scales up; by a fraction less than 1 scales down; by a fraction equal to 1 leaves the number unchanged. This reasoning also compares two products by comparing their fraction multipliers.
Key Concepts
Property For any positive number $a$: Multiplying $a$ by a fraction greater than 1 results in a product greater than $a$. ($a \times \frac{b}{c} a$ if $\frac{b}{c} 1$) Multiplying $a$ by a fraction less than 1 results in a product less than $a$. ($a \times \frac{b}{c} < a$ if $\frac{b}{c} < 1$) Multiplying $a$ by a fraction equal to 1 results in a product equal to $a$. ($a \times \frac{b}{c} = a$ if $\frac{b}{c} = 1$).
Examples Is $8 \times \frac{5}{4}$ greater than, less than, or equal to 8? Since $\frac{5}{4} 1$, the product is greater than 8. Is $\frac{7}{8} \times \frac{1}{2}$ greater than, less than, or equal to $\frac{7}{8}$? Since $\frac{1}{2} < 1$, the product is less than $\frac{7}{8}$. Which is greater: $15 \times \frac{2}{3}$ or $15 \times \frac{3}{4}$? Since $\frac{3}{4} \frac{2}{3}$, multiplying 15 by the larger fraction results in a larger product. So, $15 \times \frac{3}{4}$ is greater.
Explanation This skill involves reasoning about the size of a product without performing the actual multiplication. This is also known as scaling. When you multiply a number by a fraction greater than 1, you are scaling it up, so the result is larger. Conversely, multiplying by a fraction less than 1 scales the number down, making the result smaller. This allows you to quickly compare products by analyzing the value of the fraction you are multiplying by.
Common Questions
How do you compare products by scaling in Grade 5?
Compare the fraction multiplier to 1. If the fraction > 1, the product is greater than the original number. If the fraction < 1, the product is less. If the fraction = 1, the product equals the original. Also: larger fraction multiplier → larger product (for the same base number).
How do you compare two products with the same starting number?
Compare the two fractions you are multiplying by. The larger fraction produces the larger product. For example, 15 × (3/4) vs. 15 × (2/3): since 3/4 > 2/3, the first product is larger.
What chapter covers compare products by scaling in Illustrative Mathematics Grade 5?
Compare Products by Scaling is covered in Chapter 6 of Illustrative Mathematics Grade 5, titled Place Value Patterns and Decimal Operations.
What does it mean to scale a number with a fraction?
Scaling means changing the size of a number by multiplying it by a scale factor. Fractions > 1 scale up (enlarge), fractions < 1 scale down (reduce), and fractions = 1 preserve the original value.
What is an example of comparing products by scaling?
Is (7/8) × (1/2) greater or less than 7/8? Since 1/2 < 1, the product is less than 7/8. Is 8 × (5/4) greater or less than 8? Since 5/4 > 1, the product is greater than 8.