Lesson 11: Compare Products Using Diagrams

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.

Property

When a number is multiplied by a fraction, the product can be compared to the original number.
An area model or tape diagram can be used to visualize this comparison.
Shading a fraction of a whole shows whether the product is less than, equal to, or greater than the original number.
For a 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

• To compare $4 \times \frac{2}{3}$ to $4$, draw 4 wholes. Divide each whole into thirds and shade $\frac{2}{3}$ of each whole. The total shaded area is less than the original 4 wholes, so $4 \times \frac{2}{3} < 4$.
• To compare $2 \times \frac{5}{4}$ to $2$, draw 2 wholes. Divide each whole into fourths. To show $\frac{5}{4}$ of each whole, you need more than one whole for each, so the total product will be greater than the original 2 wholes. Thus, $2 \times \frac{5}{4} > 2$.

Explanation

Using a diagram helps you see why multiplying by a fraction changes a number. When you multiply a number by a fraction less than one, you are taking only a part of that number, so the result is smaller. Multiplying by a fraction greater than one means you are taking more than the original number, so the result is larger. These diagrams provide a visual proof for the rules of scaling.