Working with Fractions · Lesson 2 of 8
Multiplying Two Fractions
“See a fraction of a fraction using equal parts, grids, and rectangle areas.”
• Interpret multiplication of fractions as taking a fraction of a fractional quantity. • Represent products using equal rows and columns within a fixed unit square. • Connect the product to rectangle area. • Extend the visual method to fractions greater than one.
Multiplying Two Fractions
A fraction can multiply another fraction just as it can multiply a whole number. The central question is still: what amount are we taking, and what fraction of that amount do we need? A diagram helps us keep the original whole fixed while a smaller region is divided again.
For a tortoise that travels 1/4 kilometre in one hour, half an hour gives half of that quarter kilometre. We are finding 1/2 of 1/4, not adding 1/2 kilometre and 1/4 kilometre. Represent one kilometre by one whole square and first shade its quarter.
A square with side length 1 unit and area 1 square unit. Its entire area can represent one whole quantity in a fraction model.
Problem
A tortoise travels 1/4 km each hour. Find its distance in 1/2 hour.
- 1.Take half of the distance travelled in one hour: 1/2 of 1/4 km.
- 2.Divide the whole square into 4 rows, then split every row into 2 columns. There are 8 equal cells.
- 3.Half of the quarter is 1 cell out of those 8, so 1/2 × 1/4 = 1/8. The distance is 1/8 km.
Taking Several Parts of a Fraction
When the multiplier is 3/4, first take one quarter of the quantity, then take three such parts. Suppose the quantity is 2/5. Divide the whole square into five rows and shade two rows. Four columns split that shaded amount into four equal portions. Three of those portions form the required product.
Problem
A tortoise covers 2/5 km each hour. How far does it go in 3/4 hour?
- 1.Find one quarter of its one-hour distance. Two shaded rows split into four columns give 2 cells out of 20: 1/4 of 2/5 is 2/20.
- 2.Three quarters of the hour gives three copies of that distance: 3 × 2/20 = 6/20 km.
- 3.Simplify by dividing numerator and denominator by 2: 6/20 = 3/10. The tortoise travels 3/10 km.
Equal parts are essential. Counting three shaded pieces among ten pieces proves a fraction of 3/10 only if all ten pieces have equal area. The row-column grid guarantees this equality. The denominator counts every equal cell in the whole, not just cells inside the earlier shaded region.
Connection between the Area of a Rectangle and Fraction Multiplication
A rectangle can have fractional side lengths. Within a unit square, a rectangle of width 1/2 unit and height 1/4 unit occupies one of eight equal small rectangles. Its area is therefore 1/8 square unit. This is also the product of its two side lengths.
Problem
Find the area of a rectangle with sides 2/3 unit and 4/5 unit.
- 1.Divide the unit square into 3 equal columns and 5 equal rows. It contains 15 equal small rectangles.
- 2.The required rectangle covers 2 columns and 4 rows, so it contains 2 × 4 = 8 cells.
- 3.Each cell has area 1/15 square unit. The rectangle has area 8/15 square unit, giving 2/3 × 4/5 = 8/15.
Fractions Greater Than One
A unit square can also help when a quantity exceeds one whole. Represent 3/2 using one complete unit square and half of another. To take one quarter of this amount, divide its three half-unit pieces into four equal pieces each. One quarter of 3/2 is three eighths of the original unit.
Problem
Find 5/4 × 3/2 by first taking one quarter of 3/2.
- 1.The quantity 3/2 consists of three half-unit pieces. Divide each half into four equal portions: each resulting piece is 1/8 of one unit.
- 2.One quarter of the entire 3/2 amount contains three such pieces, giving 3/8.
- 3.Five quarters of that amount means five copies of 3/8: 5 × 3/8 = 15/8 = 1 7/8.
Draw 1/3 of 1/5 using a unit square. Then sketch 1/4 of 2/3. Explain why the first diagram needs 15 equal cells and the second needs 12.
Do not replace the original whole with the smaller shaded region halfway through a calculation. Half of a quarter is half of the quarter-region but one eighth of the original whole. For area answers, use square units, not ordinary length units.
Quiz
Half of one quarter of the same whole is:
A unit square is divided into 3 columns and 5 rows. What fraction is one cell?
Three quarters of two fifths is:
A rectangle has sides 1/3 m and 1/5 m. Its area is:
In a fraction grid, why must all counted cells have equal area?
Practice Problems
- Use unit squares to show 1/3 × 1/5, 1/4 × 1/3, 1/5 × 1/2, and 1/6 × 1/5.
- Draw and calculate 2/3 × 4/5, 1/4 × 2/3, 3/5 × 1/2, and 4/6 × 3/5.
- Without drawing all the cells, predict how many equal cells a grid for 1/12 × 1/18 would contain. What is the product?
- A rectangle is 3/4 m long and 2/5 m wide. Calculate its area and describe the selected cells in a unit-square model.
- Explain one quarter of 3/2 using more than one unit square. Then find three quarters of 3/2.
- A student shades 2 cells out of 8 unequal pieces and claims the shaded area is 1/4. Explain why the cell count alone does not prove the claim.
Unit-fraction products: 1/15, 1/12, 1/10, 1/30. Other products: 8/15, 1/6, 3/10, 2/5. A 12-column, 18-row grid has 216 equal cells; the product is 1/216. Rectangle area: 6/20 = 3/10 m². One quarter of 3/2 is 3/8; three quarters is 9/8 = 1 1/8. Unequal pieces can represent different amounts, so count is not enough to determine area fraction.
Key Takeaways
• A fraction times a fraction can be understood as a fraction of a fraction. • Keep the original whole unchanged throughout the visual model. • Rows and columns produce equal cells that can be counted reliably. • The product of fractional side lengths gives the area of a rectangle. • Fractions greater than one can be represented using more than one unit square.