Interpret multiplication as scaling and solve mixed-number products

Learning goal: Interpret multiplication by a positive fraction as resizing a quantity and predict a product's size before calculating. Extend the reasoning to whole numbers and mixed numbers, using partitions or equivalent fractions to justify the computation.

Scope: This lesson teaches a fraction times a whole number (5.NF.4a), scaling (5.NF.5) and mixed-number products (5.NF.6). Tiling a rectangle with unit-fraction squares (5.NF.4b) is taught in the companion lesson on fraction products and area; the rectangle below reuses that idea.

Take two thirds of a four-meter length. Partition the four-meter quantity into three equal shares, then select two shares. Each share is 4/3 meter, and two shares total 8/3 meters, or 2 2/3 meters. Thus (2/3) × 4 = 2 × 4 ÷ 3. The original four meters is the quantity being partitioned; the answer is measured in meters.

The same product can be seen as four groups of 2/3 meter. By the commutative property, switching the order of the factors preserves the number. One interpretation takes a fraction of a whole-number quantity; another repeats a fractional quantity a whole number of times. Explain which story you are representing even when the result is the same.

For a positive starting amount, multiplying by a fraction less than one makes a smaller amount. Multiplying by one preserves it. Multiplying by a fraction greater than one makes a larger amount. For a six-meter model, factors 1/2, 1 and 3/2 give three meters, six meters and nine meters. The bars use the same scale so the relative lengths show the scaling faithfully.

These comparisons do not require calculating the exact product. Multiplying by a whole number greater than one is the familiar case: 3 × 12 is three copies of 12, so it is more than 12. A fraction greater than one works the same way. Since 9/8 = 1 + 1/8, (9/8) × 12 is one whole copy of 12 plus an extra eighth of 12, so it is greater than 12. A fraction less than one takes only part of the amount. Since 7/8 = 1 − 1/8, (7/8) × 12 is 12 with an eighth of 12 missing, so it is less than 12. Compare the multiplier with one, then state what happens to the other positive factor. The condition about a positive amount matters: zero multiplied by any of these factors stays zero.

Equivalent fractions connect with multiplying by one. To rename 2/3 as 8/12, multiply by 4/4. The factor 4/4 equals one, so it changes the fraction's representation without changing its value. Multiplying by four alone would quadruple the amount. Both numerator and denominator change in the equivalence calculation because the multiplier is a ratio equal to one.

A mixed-number factor can represent more than one copy. Suppose a recipe needs 2 1/2 cups of a dry ingredient for one batch, and a class prepares 1 1/2 batches, scaling every ingredient by the same factor. The required amount is (2 1/2) × (1 1/2). One batch contributes 2 1/2 cups; another half batch contributes 1 1/4 cups. Together they require 3 3/4 cups.

Renaming gives the same product: (5/2) × (3/2) = 15/4 = 3 3/4. A rectangle with side lengths 2 1/2 and 1 1/2 can also be partitioned into four parts: 2 × 1, (1/2) × 1, 2 × (1/2), and (1/2) × (1/2). Their sum is 2 + 1/2 + 1 + 1/4 = 3 3/4. The last quarter is essential; multiplying whole parts and fractional parts separately would omit the cross products.

Worked prediction: Without evaluating, (11/10) × 8 is greater than 8, and (9/10) × 8 is less than 8. The multipliers lie above and below one. Explain the comparison before computing 8.8 or 7.2.

Worked context: Three boards each measure 1 3/4 feet. Total length is 3 × 7/4 = 21/4 = 5 1/4 feet, assuming no overlap. It is between three and six feet because each board is between one and two feet.

Guided practice: Draw bars for one positive amount scaled by less than one, exactly one and greater than one. Keep the same unit scale across all bars. Explain why equivalent-fraction multiplication by n/n leaves the original length unchanged.

Independent performance: Predict the size of several products before computing them, then solve one whole-by-fraction and one mixed-by-mixed context. Use a partition or decomposition and retain units. Check a solution missing cross products and explain the missing region or partial batch. Evaluate the prediction, model and calculation separately.