Question 1

Do more workers always take more time?

Accepted Answer

At the same rate per worker, more workers can finish a fixed task sooner. (Tempting mistake: If workers double, the time for a fixed task doubles.. Why it feels right: I expect both amounts to rise together.)

Question 2

If your child says: “The ratios match whenever their differences match.”

Accepted Answer

Ask: For 6:3 and 5:2, do 6 times 2 and 3 times 5 match? What’s true: Equal cross-products show equal ratios, even when the differences differ.

Question 3

If your child says: “The same side ratio means the triangles have the same size.”

Accepted Answer

Ask: How can the sizes match when the longest sides measure 5 cm and 10 cm? What’s true: A valid side ratio fixes shape, while the chosen lengths determine size. Any two sides together must be longer than the third.

Question 4

If your child says: “There are three angles, so I divide the whole by three.”

Accepted Answer

Ask: Does the largest angle contain the same number of equal parts as a smaller angle? What’s true: Add the ratio numbers because they count equal parts, not groups.

Question 5

If your child says: “I measure every slice from the first line.”

Accepted Answer

Ask: Where would the next slice fit if you measured over the pencils’ slice again? What’s true: Each new slice begins at the boundary where the previous slice ends.

Question 6

If your child says: “Working together takes six plus three intervals.”

Accepted Answer

Ask: If three cupfuls arrive each interval, how could six cupfuls need nine intervals? What’s true: The amounts added during each interval combine, so the separate filling times do not add.

Question 7

If your child says: “Whenever one quantity increases, the other must decrease.”

Accepted Answer

Ask: Does adding another identical coin reduce the pile’s value? What’s true: At a fixed value per coin, increasing the count increases the total value.

Question 8

If your child says: “I compare the sleeping angle with the largest band.”

Accepted Answer

Ask: Does the largest band contain the whole day, or only part of it? What’s true: The complete turn represents the whole day, including every band.

Question 9

If your child says: “Twice as many bowls means twice as many chana in each.”

Accepted Answer

Ask: Where would the extra chana come from if both the bowls and their portions doubled? What’s true: Doubling the number of equal portions halves each portion when the total stays fixed.

Proportional Reasoning-2: common misconceptions

The tempting mistake & worked visual

Do more workers always take more time?

If your child says… (8 concept checks in this chapter)

Use this chapter three ways

Common misconceptions

How to teach Proportional Reasoning-2 →

15 practice questions with answers →

The tempting mistake & worked visual

Do more workers always take more time?

If your child says… (8 concept checks in this chapter)

Use this chapter three ways

Common misconceptions

How to teach Proportional Reasoning-2 →

15 practice questions with answers →

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