5.NF.5: Multiplication as Scaling
Also written as 5.NF.B.5 and CCSS.MATH.CONTENT.5.NF.B.5 · Cluster: Apply and extend previous understandings of multiplication and division.
The Official Wording
Interpret multiplication as scaling (resizing), by:
Common Core State Standards, Grade 5, Number & Operations: Fractions · Read it at the source · Checked October 14, 2026
What 5.NF.5 Means in Plain English
Fifth graders predict the size of a product before multiplying. For 3/4 × 8 they say less than 8, because 3/4 is less than 1. For 5/4 × 8 they say more than 8. Multiplying by 4/4 is multiplying by 1, which is why 3/4 = 12/16.
This is reasoning, not computing: students judge the product’s size before any arithmetic. It explains why 2/3 of a 5/6-yard bed is smaller than the bed (5.NF.4), and it seeds the ratio reasoning of 6th grade (6.RP.3) and scale drawings in 7th (7.G.1).
5.NF.5 I Can Statements
- I can predict whether a product is bigger or smaller than a factor.
- I can explain why multiplying by a fraction less than 1 shrinks a number.
- I can explain why multiplying by 4/4 does not change a fraction.
5.NF.5 Worked Example
Three products of the same number, put in order before anyone multiplies. That ordering is the whole skill.
Put these in order from smallest to largest without working them out: 2/3 × 12, 5/5 × 12, 7/4 × 12. Explain each.
- All three start with 12, so compare the other factor to 1. 2/3 is less than 1, so 2/3 × 12 is less than 12: two thirds of twelve is only part of it.2/3 < 1, so 2/3 × 12 < 12
- 5/5 equals 1. Multiplying by 1 changes nothing, so 5/5 × 12 is exactly 12. (This is also why 2/3 × 5/5 = 10/15 is still 2/3.)5/5 = 1, so 5/5 × 12 = 12
- 7/4 is more than 1 (it is 1 3/4), so 7/4 × 12 is more than 12: one whole 12 plus three quarters of another.7/4 > 1, so 7/4 × 12 > 12
- Rank them by the factors alone: 2/3 < 5/5 < 7/4, so the products come in the same order.2/3 × 12 < 5/5 × 12 < 7/4 × 12
Check it: Now compute to confirm: 8, 12 and 21. The order matches, and 8 < 12 < 21 is exactly what the factors predicted.
Common Mistakes with 5.NF.5
Believing multiplication always makes bigger
“2/3 × 12 must be more than 12 because we multiplied”
Try this: Reframe the product as “of”: 2/3 of 12 is part of 12. Only whole numbers above 1 built the habit, and fractions break it.
Comparing to the wrong factor
“3/4 × 8 is bigger than 3/4, so multiplying by 3/4 made it bigger”
Try this: Name which factor is being scaled before judging. 8 is scaled by 3/4, so compare to 8, and 6 is less than 8.
Thinking equivalent fractions grow
“6/8 is bigger than 3/4 because 6 and 8 are bigger”
Try this: Show the scaling: 3/4 × 2/2 = 6/8, and 2/2 is 1. Multiplying by 1 cannot change the size, only the piece count.
Where 5.NF.5 Fits
Practice 5.NF.5
Quick Check
- Without multiplying: is 4/5 × 30 more or less than 30?
- Without multiplying: is 9/8 × 16 more or less than 16?
- Why is 3/5 × 4/4 still equal to 3/5?