Multiply a fraction by a whole number

Learning goal: Explain a fraction as a multiple of a unit fraction, multiply fractional amounts by whole numbers and solve equal-group measurement stories. Multiplication by a whole number counts equal groups. If each group is a fractional amount, the product counts how many fractional pieces appear altogether. The size of the whole-unit reference must stay fixed.

Begin with five fourths. This is five copies of one fourth: 5/4 = 5 × (1/4). Draw a whole strip partitioned into fourths and one fourth of another matching strip. Count the five selected quarter pieces. Their total passes one whole and reaches one and one fourth. A fraction with numerator greater than denominator is a useful description of repeated unit pieces.

Now consider three groups of two fifths. Each group contains two one-fifth pieces, and there are three groups. The illustration uses three separate bars, each showing two fifths of one whole unit. In total there are six selected one-fifth pieces: 3 × (2/5) = 6 × (1/5) = 6/5. Arrange five of these pieces into one whole, with one fifth left, to get 1 1/5.

The denominator stays five because the pieces remain fifths. Three groups of two fifths is not six fifteenths. Multiplying both numerator and denominator by three would produce an equivalent name for just one group's amount, 2/5 = 6/15. Multiplying the fraction by three must triple the amount. Distinguish renaming pieces from accumulating groups.

In general, n groups of a/b contain n × a copies of 1/b. Therefore n × (a/b) = (n × a)/b. This statement summarizes the model; it should not replace understanding it. Identify what n counts, what a counts in each group and which whole b partitions. All three quantities have different roles.

A measurement story gives the product a unit. Four banners each require 3/8 meter of trim. The combined length is 4 × (3/8) = 12/8 meters. Eight eighths make a meter, leaving four eighths, so the group needs 1 4/8 meters. This lies between one and two meters. The product is larger than one banner's trim length because four positive equal lengths are joined.

Estimate with nearby whole numbers and benchmarks. Three groups of two fifths must total more than two fifths, the amount in one group, and less than three wholes, because each group is less than one whole. A sharper check uses halves: two fifths is less than one half, so three groups total less than one and one half. The exact answer, one and one fifth, fits both checks. A result of six fifteenths would still equal one group and fail the repeated-group check.

A product need not exceed one. Two portions of 1/6 liter total 2/6 liter. Count the unit pieces first, then decide whether enough are present to form a whole. If a product equals exactly one, such as four groups of one fourth, say so; do not force a mixed-number form with a nonzero fraction.

Worked story: Five small flags each use 3/8 meter of ribbon. Let r be the total: r = 5 × (3/8) = 15/8 meters. Since 8/8 < 15/8 < 16/8, the length lies between one and two meters. Its mixed-number name is 1 7/8 meters. Draw five groups of three eighth pieces to account for all fifteen.

Worked error check: A learner writes 3 × (2/5) = 6/15. That is still two fifths, so it cannot represent three such groups. Keep fifths as the unit and count six of them: 6/5.

Guided practice: Build several equal groups of fraction pieces and rearrange the selected pieces into wholes. Record repeated addition, unit-fraction multiplication and the final product for the same model. Keep unselected regions visible in each initial whole so each portion is defined clearly.

Independent evidence: Solve a new equal-group fraction story, label the unit, and locate the product between neighboring whole numbers or exactly at a whole. Explain why the denominator is unchanged. Include one product below one and another above one to test whether the learner reasons about amount rather than expecting one answer form.