Find fraction products with partitions and rectangular area

Learning goal: Interpret a fraction of a fraction and connect fraction multiplication with the area of a rectangle. Define the original whole, partition it into equal regions and explain why numerator and denominator products appear in the calculation.

Scope: This lesson teaches a fraction times a fraction (5.NF.4a) and fractional rectangle area by tiling (5.NF.4b), with real-world problems (5.NF.6). A fraction times a whole-number quantity, such as (2/3) × 4, and mixed-number products are taught in the companion lesson on scaling and mixed products.

A garden model assigns 4/5 of a unit strip to flowers. Two thirds of that flower section will contain blue flowers. The blue amount is 2/3 of 4/5, represented by (2/3) × (4/5). The word of identifies a part of the already selected quantity, not another amount added to it.

Draw one whole rectangle and divide it into five equal columns. Select four columns to represent 4/5 of the whole. Now divide the entire rectangle into three equal rows and select two of those rows. The overlap between the four selected columns and two selected rows contains eight small rectangles. The entire original whole contains fifteen equal rectangles, so the overlap is 8/15 of that whole.

The denominator fifteen comes from five columns times three rows. The numerator eight comes from four selected columns times two selected rows. Therefore (2/3) × (4/5) = (2 × 4)/(3 × 5) = 8/15. The unselected cells remain part of the denominator because the denominator describes the complete reference whole, not just the flower section.

The same drawing can model area if the original rectangle is a one-meter by one-meter square. A smaller rectangle measuring 4/5 meter by 2/3 meter has area 8/15 square meter. Each grid cell is 1/5 meter wide and 1/3 meter tall, so its area is 1/15 square meter. Eight such equal cells cover the smaller rectangle without gaps or overlaps.

Unit-fraction squares give another tiling method. A rectangle 1/2 meter by 3/4 meter can be tiled by squares with side length 1/4 meter. Two such sides fit along the half-meter dimension, and three fit along the three-quarter-meter dimension. Six small squares cover the rectangle. Each has area (1/4) × (1/4) = 1/16 square meter, so the total is 6/16 = 3/8 square meter.

Direct multiplication gives (1/2) × (3/4) = 3/8, agreeing with the square tiling. Do not call the answer 3/8 meter: area uses square units because it measures the two-dimensional region covered. A fractional side length is not an area by itself, just as the side of a whole-number square is not its square-unit count.

Check size with the meaning of taking a part. Two thirds of four fifths must be less than four fifths because two thirds selects only part of that amount. Eight fifteenths meets the condition since four fifths equals twelve fifteenths. Adding 2/3 + 4/5 would produce a different quantity and could not represent the overlap.

A picture is reliable only if the partitions are equal. Five uneven columns cannot all represent fifths of the same whole. Label the complete unit boundary, the selected rows and columns, and the overlap before writing the product. This keeps the equation tied to the model instead of treating the grid as decoration.

Worked product: (3/4) × (2/3) = 6/12 = 1/2. A whole grid with four rows and three columns has twelve equal cells, and the three-by-two overlap contains six. Both fractional names describe the same area.

Worked context: A rectangular card is 1/2 foot by 3/4 foot. Its area is 3/8 square foot. Six quarter-foot-side squares cover it, and each square contributes 1/16 square foot. The counts and units support the multiplication.

Guided practice: Use transparent row and column overlays on a whole rectangle. Count both the complete partition and overlap. Then tile a fractional rectangle with equal unit-fraction squares and connect the count to the product of side lengths.

Independent performance: Solve one “part of a part” story and one area problem with fractional side lengths. Show a labeled whole, equal partitions, overlap or tiling, equation and unit. Estimate whether the product should exceed or fall below each relevant positive factor. Assess geometric accuracy and interpretation separately from symbolic multiplication.