Working with Fractions · Lesson 4 of 8
Understanding the Size of a Product
“Predict how multiplication changes positive quantities and explain why factor order does not change the answer.”
• Predict whether multiplying by a positive number shrinks, preserves, or enlarges a quantity. • Compare a product with both its factors without unnecessary calculation. • Explain the order-of-multiplication property using rectangle area. • Use estimates and comparisons to detect calculation errors.
Is the Product Always Greater than the Numbers Multiplied?
Multiplication often first appears as repeated addition of whole numbers. That experience can suggest that every product must be larger than its factors. Fractions show why we need a more precise idea. Multiplying by a number between 0 and 1 takes only part of a quantity; multiplying by a number greater than 1 takes more than one copy.
Throughout this lesson, the quantities being compared are positive. This matters: the statements about increasing and decreasing are being made for positive quantities. Also keep 1 separate from numbers above and below it, because one copy leaves the quantity unchanged.
Problem
Compare 1/4 × 8 with both factors.
- 1.Taking one quarter of 8 gives 8 ÷ 4 = 2.
- 2.The product 2 is less than 8 because only one quarter of that amount was taken.
- 3.But 2 is greater than 1/4: the same multiplication can be read as eight copies of 1/4. Thus the product lies between the two factors.
Both Factors Between Zero and One
If both factors are smaller than 1, each represents less than one whole copy. Taking part of an already fractional amount makes the result smaller than that amount. Because the order can be interchanged, the result is smaller than either positive factor.
Problem
Compare 3/4 × 2/5 with both factors.
- 1.Multiply to get 6/20 = 3/10.
- 2.Use denominator 20 to compare: 3/4 = 15/20 and 2/5 = 8/20.
- 3.Since 6/20 is less than both 15/20 and 8/20, the product is smaller than both factors. It is three quarters of 2/5, so that makes sense.
Both Factors Greater Than One
A factor does not have to be a whole number to enlarge a quantity. A factor such as 4/3 represents one complete copy and another third. Multiplying any positive amount by it gives more than the original amount. If both factors exceed 1, this explanation applies when either factor is treated as the original quantity.
Problem
Compare 4/3 × 4 with both factors.
- 1.The product is 16/3 = 5 1/3.
- 2.This exceeds 4, because multiplying by 4/3 takes one and one-third copies of 4.
- 3.It also exceeds 4/3, because multiplying that amount by 4 takes four copies. Both factors are greater than 1.
| Factors | Example | Position of the product |
|---|---|---|
| Both greater than 1 | 4/3 × 4 = 16/3 | Greater than both factors |
| Both between 0 and 1 | 3/4 × 2/5 = 3/10 | Less than both factors |
| One below 1 and one above 1 | 3/4 × 5 = 15/4 | Between the two factors |
| One factor equals 1 | 1 × 7/9 = 7/9 | Equal to the other factor |
When one factor is below 1 and the other is above 1, make both comparisons separately. For 3/4 × 5, taking three quarters makes the answer smaller than 5. Taking five copies makes it larger than 3/4. These two explanations establish the answer’s position even before calculating 15/4.
A numerator greater than 1 does not tell you whether its fraction is greater than 1. Compare numerator with denominator: 7/9 is below 1, while 9/7 is above 1. Do not decide that a product must increase simply because its calculation uses multiplication.
Order of Multiplication
A rectangle’s area does not change when we describe its width first instead of its height. A rectangle with sides 1/2 and 1/4 has the same area as one with those sides interchanged. This visual fact agrees with the arithmetic rule: the products of the two numerators and of the two denominators do not depend on their order.
Problem
Explain why 1/2 × 1/4 and 1/4 × 1/2 have the same value.
- 1.A unit square divided into 2 columns and 4 rows has 8 equal cells. One selected cell has area 1/8.
- 2.If rows and columns are interchanged, a 4-column, 2-row square still has 8 equal cells.
- 3.Both products therefore equal 1/8. The interpretation of which fraction is taken first changes, but the resulting quantity does not.
Create two examples for each row of the comparison table. Predict the position of each product before calculating. If a result disagrees with the prediction, check whether the fractions were multiplied or simplified incorrectly.
Quiz
For positive x, what happens to x when it is multiplied by 2/7?
Without multiplying, compare 5/6 × 3/8 with its factors.
Where does 7/5 × 2/3 lie?
Which factor leaves a positive quantity unchanged?
Why is 3/7 × 5/4 equal to 5/4 × 3/7?
Practice Problems
- Predict the position of each product relative to both factors, then calculate: 2/3 × 3/5; 7/4 × 6/5; 5 × 2/7.
- Without calculating the exact product, decide which statements are true for 565/465 × 707/676: it is greater than the first factor; greater than the second factor; greater than 1. Explain.
- Find one example in which a product is below both factors and one in which it is between them. Justify the predictions.
- A student claims 3/8 × 5/6 = 15/14. Explain why comparison with the factors shows that this cannot be right, then calculate correctly.
- Use a rectangle to explain why 2/3 × 3/4 = 3/4 × 2/3.
- Compare multiplying a positive amount by 0, by 1, by 4/5, and by 5/4. Which comparison case needs separate treatment?
Products: 2/5, below both; 21/10, above both; 10/7, between 2/7 and 5. Both 565/465 and 707/676 exceed 1, so all three statements are true. Sample examples: 1/2 × 1/3 = 1/6; 1/2 × 3 = 3/2. The claimed 15/14 exceeds 1 while both factors are below 1; the correct result is 15/48 = 5/16. The swapped rectangle products both equal 1/2. Multiplication by 0 gives 0 and needs a boundary case; by 1 preserves the amount; by 4/5 decreases it; by 5/4 increases it.
Key Takeaways
• Multiplying a positive quantity by a factor between 0 and 1 makes it smaller. • Multiplying a positive quantity by a factor greater than 1 makes it larger. • Two factors below 1 give a product below both; two factors above 1 give a product above both. • If one positive factor is below 1 and the other is above 1, the product lies between them. • Swapping the factors leaves a product unchanged, as seen through rectangle area or the multiplication rule.