L5 · Multiplying and Dividing Fractions as Rational Numbers

Learning goal

“I can multiply and divide any two fractions or mixed numbers - positive or negative - using the area model for multiplication and reciprocal reasoning for division, and I can say WHY the procedures work.”

Reciprocal — the fraction flipped so the product is 1: the reciprocal of 2/5 is 5/2, and 2/5 x 5/2 = 1 (review from L4, now the engine of division).

Area model — multiplication drawn as a rectangle: a 2/3-by-4/5 rectangle inside a unit square has area 8/15 - the grid shows WHY numerators multiply and denominators multiply.

Simplest form — a fraction whose numerator and denominator share no common factor (review); cancelling a common factor BEFORE multiplying keeps numbers small.

Worked example

2/3 x 4/5. Step 1: Draw the unit square. Cut it vertically into 3 equal columns and shade 2 of them (2/3 of the width). Cut it horizontally into 5 equal rows and shade 4 of them (4/5 of the height). The overlap - the part shaded twice - is a rectangle of 2 columns by 4 rows. Step 2: Count: the whole square is 3 x 5 = 15 small cells; the overlap is 2 x 4 = 8 cells. So 2/3 x 4/5 = 8/15. The procedure "multiply numerators, multiply denominators" is just this counting, written down. Step 3: Signs ride along separately (L3): a negative factor flips the sign of the area, and cancelling common factors first (divide a numerator and a denominator by the same number) keeps the arithmetic small - it never changes the value.

Check yourself — explain it

In the area model for 2/3 x 4/5, why is the denominator of the answer 15?

Paired problem — example still visible

-3/4 x 2/9 -> different signs -> negative; cancel 3 with 9 first: -1/4 x 2/3 = -1/6

Faded completion — last step is yours

"5/6 x 3/10: cancel 3 with 6 and 5 with 10 first - the product becomes 1/2 x 1/2 = ___." -> 1/4

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

Start practice →Take the unit quiz