Multiplying fractions

Activities to try

1. 1

   Jump-counter number line for n × a/b

   Why it works: Because the unit part stays the same size, multiplying by a whole number only multiplies the COUNT of parts: n × a/b = (n×a)/b, with the denominator unchanged. A mixed number × whole requires multiplying every part.

   1. Set the jump size to 3/8 and dial the number of jumps to 4; watch four identical hops march along the line and land on 12/8.
   2. Re-group the landing point into a mixed number: 12/8 = 1 4/8 (= 1 1/2).
   3. Keep the jump size fixed and change only how many jumps there are for n × 1/4 with n = 3, 5, 8, each time reading off the total count of quarters and converting past-one results to mixed or whole numbers (e.g. 5 × 1/4 = 5/4 = 1 1/4; 8 × 1/4 = 8/4 = 2).
   4. Switch to a mixed-number mode: lay three whole 2 1/3 bars end to end so every part is tripled, and read that three 2s and three thirds make 6 + 3/3 = 7.
   5. Say aloud what changed and what stayed the same after each build: the unit (the eighth, the third) never changed, only how many of them there were.
2. 2

   Build from repeated addition before stating the rule

   Why it works: Multiplication of a unit or non-unit fraction by a whole number is accumulation of equal parts; the numerators accumulate and the denominator is the unchanging unit. This is distinct from adding a whole number to a fraction.

   1. Write 4 × 3/8 as 3/8 + 3/8 + 3/8 + 3/8 and add the numerators while keeping the eighth as the unit, reaching 12/8.
   2. Contrast this with the additive slip of adding the whole number to the fraction (e.g. 4 + 3/8) and explain in words how multiplication differs from addition, so the operations are not blurred.
   3. Convert the improper product to a mixed number (12/8 = 1 1/2; 6/5 = 1 1/5) and back, naming how many whole ones and what is left over.
3. 3

   Targeted practice that baits the multiply-denominator error

   Why it works: Fraction × whole and mixed × whole can be computed reliably by scaling the count and keeping the unit fixed; the multiply-denominator slip gives an answer equal to the original fraction, which therefore cannot be four lots of it — a whole-number multiple greater than one must be larger than the original.

   1. Complete fill-in equations such as '4 × 3/8 = _/8 = 1 _/8', writing the missing count and the mixed number.
   2. Choose the correct value of 4 × 3/8 from options that include 12/32 (denominator wrongly multiplied), then justify why 12/32 cannot be right.
   3. Order n × 1/4 for n = 3, 5, 8 from smallest to largest and, where it exceeds one, state each as a mixed number or whole number (5/4 = 1 1/4; 8/4 = 2).
   4. Work 2 1/3 × 3, showing both the part-by-part method and the improper-fraction method (7/3 × 3 = 21/3 = 7).