Compare Products Using Diagrams
Students learn to use area models or tape diagrams to compare the product of a number multiplied by a fraction to the original number, determining whether the product is less than, equal to, or greater than the original based on whether the fraction is less than, equal to, or greater than 1, as covered in Illustrative Mathematics Grade 5, Chapter 6: Place Value Patterns and Decimal Operations. If the fraction is less than 1, the product is smaller; if greater than 1, the product is larger.
Key Concepts
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.
Common Questions
How does multiplying by a fraction affect a number?
If the fraction is less than 1, the product is smaller than the original; if the fraction equals 1, the product equals the original; if the fraction is greater than 1, the product is larger.
How do you use a diagram to compare n x (a/b) to n?
Draw n whole units; shade a/b of each unit; if less than all of each unit is shaded, the product is less than n; if more than each unit is shaded, the product is greater than n.
What does multiplying by 2/3 do to a number?
Multiplying by 2/3 (less than 1) makes the result smaller than the original number; for example, 4 x 2/3 = 8/3 which is less than 4.
What does multiplying by 5/4 do to a number?
Multiplying by 5/4 (greater than 1) makes the result larger than the original number; for example, 2 x 5/4 = 10/4 = 2.5 which is greater than 2.
Why are diagrams helpful for comparing fraction products?
Diagrams provide a visual proof of how multiplying by a fraction scales a number; seeing the shaded portion directly compares the product to the original whole, making the abstract rule concrete.