Learn on PengiIllustrative Mathematics, Grade 5Chapter 6: Place Value Patterns and Decimal Operations

Lesson 13: Compare to 1

In this Grade 5 Illustrative Mathematics lesson from Chapter 6: Place Value Patterns and Decimal Operations, students learn to compare decimal numbers to the benchmark value of 1, determining whether a given decimal is greater than, less than, or equal to 1. Students apply place value reasoning to make and justify these comparisons using mathematical language and notation.

Section 1

Comparing Products to the Original Number

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.

Section 2

Comparing Fraction Products to 1

Property

To compare the product of two fractions to 1 without calculating, examine each fraction.
If both fractions are less than 1, their product will also be less than 1.
If one fraction is greater than 1 and the other is less than 1, their product could be greater than, less than, or equal to 1.

Examples

Is $\frac{3}{4} \times \frac{7}{8}$ greater than or less than 1? Since $\frac{3}{4} < 1$ and $\frac{7}{8} < 1$ , their product is less than 1.
Is $\frac{2}{5} \times \frac{4}{3}$ greater than or less than 1? We cannot tell without calculating because $\frac{2}{5} < 1$ and $\frac{4}{3} > 1$ . In this case, $\frac{2}{5} \times \frac{4}{3} = \frac{8}{15}$ , which is less than 1.
Is $\frac{5}{2} \times \frac{3}{4}$ greater than or less than 1? We cannot tell without calculating because $\frac{5}{2} > 1$ and $\frac{3}{4} < 1$ . In this case, $\frac{5}{2} \times \frac{3}{4} = \frac{15}{8}$ , which is greater than 1.

Explanation

This skill helps you estimate the size of a product involving fractions by comparing it to the benchmark of 1. When you multiply a number by a fraction less than 1, the result is smaller than the original number. Therefore, multiplying two fractions that are both less than 1 will always result in a product that is even smaller and thus less than 1. However, if one factor is greater than 1 and the other is less than 1, you must perform the calculation to determine how the product compares to 1.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Comparing Products to the Original Number

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

Section 2

Comparing Fraction Products to 1

Property

Examples

Is $\frac{3}{4} \times \frac{7}{8}$ greater than or less than 1? Since $\frac{3}{4} < 1$ and $\frac{7}{8} < 1$ , their product is less than 1.
Is $\frac{2}{5} \times \frac{4}{3}$ greater than or less than 1? We cannot tell without calculating because $\frac{2}{5} < 1$ and $\frac{4}{3} > 1$ . In this case, $\frac{2}{5} \times \frac{4}{3} = \frac{8}{15}$ , which is less than 1.
Is $\frac{5}{2} \times \frac{3}{4}$ greater than or less than 1? We cannot tell without calculating because $\frac{5}{2} > 1$ and $\frac{3}{4} < 1$ . In this case, $\frac{5}{2} \times \frac{3}{4} = \frac{15}{8}$ , which is greater than 1.

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Comparing Products to the Original Number

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

Section 2

Comparing Fraction Products to 1

Property

Examples

Is $\frac{3}{4} \times \frac{7}{8}$ greater than or less than 1? Since $\frac{3}{4} < 1$ and $\frac{7}{8} < 1$ , their product is less than 1.
Is $\frac{2}{5} \times \frac{4}{3}$ greater than or less than 1? We cannot tell without calculating because $\frac{2}{5} < 1$ and $\frac{4}{3} > 1$ . In this case, $\frac{2}{5} \times \frac{4}{3} = \frac{8}{15}$ , which is less than 1.
Is $\frac{5}{2} \times \frac{3}{4}$ greater than or less than 1? We cannot tell without calculating because $\frac{5}{2} > 1$ and $\frac{3}{4} < 1$ . In this case, $\frac{5}{2} \times \frac{3}{4} = \frac{15}{8}$ , which is greater than 1.