Comparing Products to the Original Number
Comparing Products to the Original Number is a Grade 5 math skill from Illustrative Mathematics Chapter 6 (Place Value Patterns and Decimal Operations) that teaches students to compare n × (a/b) to n without computing: if a/b < 1, the product is less than n; if a/b = 1, the product equals n; if a/b > 1, the product is greater than n. This scaling concept builds fraction multiplication number sense.
Key Concepts
Property For any positive number $n$: $n \times \frac{a}{b} < n$ if $\frac{a}{b} < 1$ $n \times \frac{a}{b} = n$ if $\frac{a}{b} = 1$ $n \times \frac{a}{b} n$ if $\frac{a}{b} 1$.
Examples Is $5 \times \frac{3}{4}$ greater than or less than 5? Since $\frac{3}{4} < 1$, the product is less than 5. Is $12 \times \frac{5}{5}$ greater than or less than 12? Since $\frac{5}{5} = 1$, the product is equal to 12. Is $8 \times \frac{7}{6}$ greater than or less than 8? Since $\frac{7}{6} 1$, the product is greater than 8.
Explanation This skill involves comparing the size of a product to the original number without actually calculating the product. When you multiply a positive number by a fraction, you are scaling it. If you multiply by a fraction less than 1, the number gets smaller. If you multiply by a fraction greater than 1, the number gets larger.
Common Questions
How do you compare a product to the original number without calculating?
Look at the fraction you're multiplying by. If it's less than 1, the product is smaller than the original number. If it's equal to 1, the product equals the original. If it's greater than 1, the product is larger.
What is an example of comparing a fraction product to the original number?
Is 5 × (3/4) greater or less than 5? Since 3/4 < 1, the product is less than 5. Is 8 × (7/6) greater or less than 8? Since 7/6 > 1, the product is greater than 8.
What chapter covers comparing fraction products in Illustrative Mathematics Grade 5?
Comparing products to the original number is covered in Chapter 6 of Illustrative Mathematics Grade 5, titled Place Value Patterns and Decimal Operations.
Why does multiplying by a fraction greater than 1 make a number larger?
A fraction greater than 1 represents more than one whole, so multiplying by it scales the number up. It's like multiplying by a number greater than 1 — the result is always larger than the original.
Why does multiplying by a fraction less than 1 make a number smaller?
A fraction less than 1 represents a part of a whole, so it scales the number down. You're taking only a portion of the original quantity, resulting in a product smaller than what you started with.