Multiplying fractions (age 10+)

Activities to try

1. 1

   Two-way area shader: a fraction of a fraction

   Why it works: Multiplying two proper fractions means taking a part of a part; the crosshatched overlap sits plainly INSIDE both striped regions, so the product is smaller than either proper factor, and the b×d grid gives the denominator automatically (no common denominator needed). The whole-number case is different: 2/3 × 4 is four copies of 2/3, so 8/3 is larger than 2/3.

   1. Shade 4/5 of a unit square with vertical stripes.
   2. Shade 2/3 of the square with horizontal stripes, over the same region.
   3. Watch the square rule itself into a 3×5 grid of fifteenths; count the crosshatched overlap (8 tiles) and read the product as 8/15.
   4. Repeat for 3/4 × 2/3: see the overlap as 6 tiles out of 12, then use a simplify step to collapse 6/12 to 1/2 on the same picture.
   5. Say each product in words using 'of': '2/3 × 4/5 is two-thirds OF four-fifths.'
2. 2

   Contrast multiplication with the addition rule

   Why it works: Multiplication and addition of fractions run different procedures: addition needs a common denominator to combine like-sized parts; multiplication multiplies across because you are partitioning the unit in two directions.

   1. Work out 2/3 × 1/3 on the area grid: shade 1/3 one way, take 2/3 of it the other way, and count the overlap — 2 tiles out of 9, i.e. 2/9.
   2. State explicitly: the answer is NOT 2/3; two-thirds OF one-third is smaller than one-third.
   3. Compare side by side with an addition problem that needs re-partitioning (e.g. 2/3 + 4/5): notice addition needs same-sized parts (a common denominator) before you can combine, while multiplication splits the unit BOTH ways and the b×d grid supplies the denominator.
   4. Explain in words why you do not need a common denominator to multiply.
3. 3

   Compute and simplify (practice and self-check)

   Why it works: Fluency: multiply numerators and denominators directly, always put the result in simplest form, and expect the product of two proper fractions to be smaller than each factor.

   1. Complete fill-in-the-blank items of the form '2/3 × 4/5 = __/__ = __ (simplest form)'.
   2. Answer a multiple-choice item for 2/3 × 1/3 where '2/3' (denominator left unchanged) is offered as a tempting wrong answer, and choose/justify the correct 2/9.
   3. Finish a problem that yields 6/12 and reduce it to 1/2 as a required step.
   4. Estimate before computing: predict whether each product is bigger or smaller than its factors.