Multiply whole numbers by decimal amounts

Learning goal: Multiply a whole number by a decimal through hundredths using groups, place value and a written method. Explain what the product counts. This lesson focuses on whole-number factors times decimal amounts, including factors through the hundreds place.

Scope: Ohio's 5.NBT.7 covers all four decimal operations in three parts. This lesson teaches part b, whole numbers times decimals. The companion lessons on decimal addition and subtraction (part a) and decimal division models (part c) complete the standard.

Four labels each require 0.35 meter of tape. The total is 4 × 0.35 meter. Each amount is thirty-five hundredths of a meter, so four groups contain 4 × 35 = 140 hundredths. That equals 1.40 meters, or 1.4 meters. The product is not 140 meters because the unit being counted during the whole-number calculation was a hundredth of a meter.

Use a bar model with four separate groups of 0.35. Within each group, 0.30 + 0.05 separates tenths from hundredths. Across all four groups, 4 × 0.30 = 1.20 and 4 × 0.05 = 0.20. The sum 1.20 + 0.20 = 1.40 matches the hundredths calculation. Both methods account for all the tape without changing the size of the unit.

A written calculation can first multiply 35 × 4 to obtain 140, then interpret that count as hundredths because 0.35 = 35/100. This explanation is stronger than inserting a decimal point by guess. If the original amount were 3.5 meters, the same whole-number digits would count tenths and the result would be 14 meters. The place value of the original factor is essential.

For 24 notebooks priced at a fictional $1.25 each, use 24 × 125 cents. Twenty-four groups of one dollar contribute $24, and twenty-four groups of a quarter dollar contribute $6. The total is $30. A partial-product calculation gives 1.25 × 20 = 25.00 and 1.25 × 4 = 5.00; adding yields 30.00. These are equivalent decompositions of the same repeated amount.

A three-digit whole factor works in the same way. If 125 markers each use 0.08 meter of ribbon, the total is 125 × 8 hundredths = 1,000 hundredths = 10 meters. Decompose 125 into 100 + 20 + 5 if useful: 8.00 + 1.60 + 0.40 = 10.00 meters. The larger number of groups changes the total count, not the value of one hundredth.

Estimate and compare to bounds. Four groups of 0.35 must be more than four groups of 0.30 and less than four groups of 0.40, so the answer lies between 1.20 and 1.60. A product of 0.14 or 14 falls outside those bounds. For the notebook example, 24 amounts a little greater than one dollar should cost a little more than $24.

The commutative property allows 4 × 0.35 and 0.35 × 4 to have the same numerical product. In the tape story, explain the representation as four groups of 0.35 meter. A context gives meaning to the factors even when switching their written order preserves the product. Keep the measurement or money unit attached to the repeated amount.

Worked decomposition: 36 × 2.45 = (30 × 2.45) + (6 × 2.45) = 73.50 + 14.70 = 88.20. Each partial product counts the same decimal unit across a different number of groups.

Worked error analysis: A learner writes 125 × 0.08 = 100. The whole-number multiplication 125 × 8 gives 1,000 hundredths, not 1,000 tenths. Dividing the count by 100 gives 10. An estimate using 125 × 0.1 = 12.5 also shows that 100 is too large.

Guided practice: Draw repeated decimal amounts using bars or equal grids. Rename each amount as tenths or hundredths before multiplying, and connect that unit to the final written decimal.

Independent performance: Solve one measurement and one fictional price problem, including a three-digit number of groups. Provide an estimate or upper/lower bound, show a decomposition or model, and explain the product's unit. Check by repeated addition for a small case and by another decomposition for a larger one. Assess unit reasoning alongside arithmetic.