Cogito
Grade 6 Math · Chapter 1 · Lesson 3
Why You Multiply and Not Add
The most common mistake in all of middle school.
10 problems · about 18 minutes · 6.RP.A.1, 6.RP.A.3
What this lesson teaches
The child explains why scaling a ratio requires multiplying both quantities rather than adding to both.
- Scaling a ratio multiplies both quantities.
- Adding the same amount to both changes the comparison.
- Draw it if you are unsure. Only the right answer keeps the shape.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Equivalent Ratios: 4 : 5. With 16 of the first, how many of the second?
Answer 20
Why 16 : 20.
Review — What a Ratio Says: 6 pens and 9 pencils. Pens to total: what is the second number?
Answer 15
Why 6 : 15.
Warm Up
Straightforward practice. Get the method working first.
4 problems6 : 9 scaled so the first is 12. What is the second?
Answer 18
Why 12 : 18.
How does a ratio compare two numbers?
Answer By how many times bigger one is.
Why A ratio compares by division, not by difference.
2 : 5 scaled so the first is 6. What is the second?
Answer 15
Why Scaled by 3.
3 : 4 scaled so the first is 12. What is the second?
Answer 16
Why Scaled by 4, so not 13.
Build It Up
The same ideas with more to keep track of.
2 problems5 : 8 scaled so the first is 10. What is the second?
Answer 16
Why Doubled.
Is 3 : 5 the same comparison as 6 : 8?
Answer No. 6 : 10 would be.
Why 6 : 8 came from adding 3 to each.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Right Answer: 4 : 6 scaled so the first is 6. What is the second?
Answer 9
Why 6 : 9, because both were multiplied by the same factor.
The Squash: 1 : 4 squash to water. With 3 squash, how much water?
Answer 12 water
Why 3 : 12. Adding 2 to each would give 3 : 6, which tastes much stronger.