Multiplicative Comparison

Activities to try

1. 1

   Build the bar, place the unknown, decide × or ÷

   Why it works: The position of the unknown in the comparison — not the presence of the word 'times' — decides whether you multiply or divide. A bar model (or number line) makes that position visible.

   1. Read a comparison story, e.g. '36 is 4 times a number — what is the number?'
   2. Draw a referent bar (the quantity being compared to) and an 'n-times' bar made of that many equal copies; a double number line or a ×n/÷n scale-factor arrow represents the same scaling relationship if you prefer.
   3. Place a '?' marker on whichever bar is unknown.
   4. Decide the operation from the position of the '?': if the whole n-times bar is unknown, multiply; if the referent or the number of copies is unknown, divide. For '36 is 4 times a number', the referent is unknown, so 36 ÷ 4 = 9.
   5. Write the matching equation with a symbol for the unknown, e.g. 36 = 4 × n, then n = 36 ÷ 4 = 9.
2. 2

   Matched pair: 'more than' versus 'times as many'

   Why it works: 'More than' means add a fixed amount; 'times as many' means repeat the quantity. The same numbers give different answers, so you must read the comparison structure, not just spot a keyword — and which quantity is unknown decides whether you multiply or divide.

   1. Solve two near-identical stories built on the same numbers: (a) 'Maya has 3 stickers. Tom has 5 MORE than Maya — how many does Tom have?' and (b) 'Maya has 3 stickers. Tom has 5 TIMES as many as Maya — how many does Tom have?'
   2. Build or draw both: for (a), Maya's bar plus 5 more; for (b), 5 copies of Maya's bar.
   3. Compute both and compare: (a) 3 + 5 = 8; (b) 3 × 5 = 15.
   4. State in your own words why the two differ, then make up your own matched 'more'/'times' pair and solve both.
   5. Try a reverse problem: 'A blue ribbon is 12 cm. That is 3 times as long as the red ribbon. How long is the red ribbon?' Build blue as 3 equal parts; one part is red, so 12 ÷ 3 = 4 cm.
3. 3

   Sort, match, and solve mixed problems

   Why it works: Deciding the comparison type and the unknown's position up front determines the operation; this is checkable and should be practised across varied contexts.

   1. Given a mixed set of comparison stories, sort each into 'multiplicative' or 'additive' before solving.
   2. Match each story to an equation written with a symbol for the unknown.
   3. Solve for the unknown wherever it sits — compared quantity, referent, or scale factor.
   4. Check targeted traps: for '36 is 4 times a number, the number is…', confirm the answer is smaller than 36 (9), not larger (144).