Compute decimal sums, differences and products with place value

Learning goal: Use standard algorithms to add, subtract and multiply multi-digit decimals, explaining why decimal places are aligned or counted. Estimate and check the numerical size of each result. The companion division lesson completes the four-operation scope.

For 18.475 + 6.82, write 6.82 as 6.820 so corresponding place values align. Thousandths add to five thousandths. Seven hundredths plus two hundredths give nine hundredths. Four tenths plus eight tenths give twelve tenths, renamed as one whole and two tenths. Including that regrouped whole, the result is 25.295.

The standard columns represent like units: ones with ones, tenths with tenths, hundredths with hundredths and thousandths with thousandths. Right-aligning the final digits without checking decimal points could put two hundredths under five thousandths, changing the quantities. Adding a trailing zero to 6.82 preserves its value; moving the decimal point does not.

For 20.000 − 7.586, regroup across the zeros. Rename one of the twenty wholes as ten tenths, then one tenth as ten hundredths, then one hundredth as ten thousandths. The amount is nineteen wholes, nine tenths, nine hundredths and ten thousandths. Subtract seven wholes, five tenths, eight hundredths and six thousandths to obtain 12.414.

Check with addition: 12.414 + 7.586 = 20.000. The thousandths combine to ten, the hundredths including the new hundredth combine to ten, and the tenths including the new tenth combine to ten. This reverses the exchanges and restores twenty wholes.

Multiplication uses a different place-value relationship. For 3.24 × 1.6, compute the whole-number product 324 × 16 = 5,184. The original factors are 324 hundredths and 16 tenths, so the product is 5,184 thousandths, or 5.184. Two decimal places in the first factor and one in the second produce thousandths because hundredths multiplied by tenths are thousandths.

In the standard whole-number multiplication, 324 × 6 = 1,944 and 324 × 10 = 3,240. Their sum is 5,184. The second partial product is shifted because the 1 in 16 represents ten, not one. The decimal interpretation is applied to the complete product, preserving the factors' place values.

An area model confirms 3.24 × 1.6. Split 1.6 into 1 + 0.6: 3.24 × 1 = 3.24 and 3.24 × 0.6 = 1.944. Their sum is 5.184. The two subareas cover the complete rectangle without overlap. The model explains the distributive relationship behind the algorithm.

Small decimal factors need leading placeholder zeros in the product. 0.24 × 0.03 uses 24 × 3 = 72, but hundredths times hundredths gives ten-thousandths. The product is 72/10,000 = 0.0072, not 0.72. Both factors are positive and below one, so the product is smaller than either factor. That magnitude check rejects the misplaced decimal.

Keep the operation's alignment rule distinct. Addition and subtraction require matching like places. Multiplication combines places through products; aligning decimal points in the partial-product calculation is not the governing rule. Explain the quantities rather than transferring a remembered instruction from a different operation.

Worked estimate: 3.24 × 1.6 is a little more than 3 × 1.6 = 4.8 and must exceed 3.24 because the multiplier exceeds one. The exact 5.184 is plausible; 0.5184 would fail the second comparison.

Worked subtraction: 5.03 − 0.875 becomes 5.030 − 0.875 = 4.155. Adding 4.155 and 0.875 returns 5.030. Equal written lengths make the places visible but do not eliminate the need to regroup.

Guided calculation: Annotate every regrouping in the sum and difference with its place value. For the product, name the units in each original factor and in the result. Compare the standard algorithm with the rectangle's two partial products.

Independent performance: Compute multi-digit decimal sums, differences and products, including unequal decimal lengths, zeros requiring regrouping and a product below one hundredth. Use standard algorithms, estimate, and provide inverse checks for sums or differences and a distributive check for a product. Explain why the decimal placement follows from units. Continue with the companion division lesson before claiming all decimal operations have been demonstrated.