5th Grade Multiplication as Scaling Lesson (Free PDF)
Multiplication as scaling means that multiplying a number by a factor stretches or shrinks it. A factor greater than 1 makes the product larger than the other factor. A factor equal to 1 keeps it the same. A factor less than 1, such as 3/4, makes the product smaller.
Grade 5 · Common Core State Standards: 5.NF.B.5a, 5.NF.B.5b
- A scale factor greater than 1 makes the product larger.
- A scale factor equal to 1 leaves the product the same.
- A scale factor less than 1 makes the product smaller.
- A fraction times a fraction less than 1 is smaller than both factors.
- Compare the factor to 1 to predict the size before you calculate.
What does it mean to scale a number by multiplying?
Multiplying can make a number larger, smaller, or keep it the same. This is called scaling. A scale factor is the number you multiply by. Look at \(12 \times 2 = 24\), \(12 \times 1 = 12\), and \(12 \times \frac{1}{2} = 6\). The bars show how 12 stretches to 24 or shrinks to 6.
- If f is greater than 1, the product is greater than n.
- If f equals 1, the product equals n.
- If f is less than 1, the product is less than n.
How can you tell if a product is larger or smaller without calculating?
Compare the scale factor to 1. A fraction with a numerator greater than its denominator is greater than 1, such as \(\frac{5}{4}\). A fraction with a numerator less than its denominator is less than 1, such as \(\frac{3}{4}\). A fraction with equal numerator and denominator equals 1.
| Scale factor | Compared with 1 | Product compared with 8 |
|---|---|---|
| \(\frac{5}{4}\) | greater | \(8 \times \frac{5}{4} = 10\) is larger |
| \(\frac{4}{4}\) | equal | \(8 \times \frac{4}{4} = 8\) is the same |
| \(\frac{3}{4}\) | less | \(8 \times \frac{3}{4} = 6\) is smaller |
What happens when you multiply two fractions less than 1?
The product is smaller than both factors. You take a part of a part. For example, \(\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}\), and \(\frac{1}{2}\) is smaller than \(\frac{3}{4}\) and smaller than \(\frac{2}{3}\).
What are the steps for predicting and checking?
- Find the scale factor.
- Compare it with 1.
- Predict whether the product is greater than, less than, or equal to the other factor.
- Multiply.
- Check that the answer matches the prediction.
Worked example: Is \(\frac{4}{3} \times 9\) greater than 9? The factor \(\frac{4}{3}\) is greater than 1, so yes. Check: \(\frac{4}{3} \times 9 = \frac{36}{3} = 12\), and 12 is greater than 9.
Questions about multiplication as scaling
Does multiplication always make a number bigger?
No. Multiplying by a number greater than 1 makes it bigger, but multiplying by a fraction less than 1 makes it smaller. For example, 8 times 3/4 is 6, which is less than 8.
How do I know if a fraction is greater than 1?
Compare the numerator and denominator. If the numerator is greater than the denominator, such as 5/4, the fraction is greater than 1. If it is smaller, such as 3/4, the fraction is less than 1.
Why is a fraction times a fraction smaller than both factors?
You take a part of a part. One half of one third is only one sixth, which is smaller than both one half and one third.
What happens when I multiply by 1?
The number stays the same. Any fraction that equals 1, such as 4/4 or 7/7, also keeps the number the same.
Why predict the size before calculating?
A prediction helps catch mistakes. If a factor is less than 1 but the product is larger than the other factor, there must be an error in the calculation.
All Grade 5 chapters: Grade 5 math.
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