Multiplicative Comparison

Activities to try

1. 1

   Scaling bars: times as many vs more than

   Why it works: 'n times as many' is a scaling relationship — n copies of the referent — not n added on; and the same pair of bars can be read two ways: the multiplicative comparison '35 is 5 times as many as 7' and its reciprocal '7 is one-fifth of 35'.

   1. Start with a short referent bar labelled red (e.g. red = 7). Build a second bar, blue, that is 'n times as long' by stacking n copies of the red bar end to end; for ×5 this makes blue = 35.
   2. Alongside the multiplicative bar, build an ADDITIVE comparison bar: red plus a fixed amount (red + 5 = 12). Place the two blue results side by side so '5 times as many' (35) visibly towers over '5 more' (12).
   3. Write the equation that matches the multiplicative bars: blue = n × red (e.g. blue = 5 × red, so blue = 35).
   4. Re-read the SAME pair of bars (7 and 35) two ways: the multiplicative-comparison statement '35 is 5 times as many as 7', and its reciprocal '7 is one-fifth of 35 / 7 is one of the five equal copies that make 35'. State both out loud. (The matching equation 35 = 5 × 7 also carries two multiplicative comparisons — '35 is 5 times as many as 7' and '35 is 7 times as many as 5' — which you will write out in the sort-and-translate activity.)
   5. Take verbal cards such as 'Sarah has 3 times as many stickers as Tom, who has 8' and build the correct bars, then write the equation (3 × 8 = 24).
2. 2

   Tower comparison: how many copies fit?

   Why it works: 'Times as tall' compares by how many COPIES of the referent fit, not by how many more; the multiplier tells you how many copies.

   1. The red tower is 4 cubes tall; the blue tower is 4 TIMES as tall. Build blue by stacking four separate red towers end to end, then count: 4 × 4 = 16 cubes.
   2. Explain in words what '4 times as tall' means: blue is four copies of red's height.
   3. Now build a '4 MORE than red' tower (4 + 4 = 8) beside it and describe how it differs from the '4 times' tower.
   4. Given both towers (red = 4, blue = 16), work out how many times as tall blue is, and say how you know (how many red towers fit into blue).
   5. Try a variant: if red were 5 cubes and blue is 4 times as tall, how tall is blue? (20)
3. 3

   Sort and translate: multiplicative or additive?

   Why it works: 'Times as many' describes a multiplicative relationship (larger = n × smaller) and 'more than' describes an additive relationship (larger = smaller + n); the operation you actually carry out depends on which quantity is unknown (e.g. finding the smaller quantity from 'Sam has 3 times as many as Jo, and Sam has 24' means dividing). Translation must run reliably in both directions, and one equation can encode two multiplicative comparisons.

   1. Sort word-problem statements into two piles: 'times as many' (multiplicative) and 'more than' (additive). Example multiplicative: 'Sam has 5 times as many as Jo.' Example additive: 'Sam has 5 more than Jo.'
   2. Match each verbal statement to its equation, e.g. 'Sam has 5 times as many' → ×5 (not +5).
   3. Translate both ways: given an equation like 6 × 4 = 24, write a 'times as many' sentence; given a sentence, write the equation.
   4. For 35 = 5 × 7, write both multiplicative comparisons it supports: '35 is 5 times as many as 7' and '35 is 7 times as many as 5'.