Combine and compare decimal amounts using like units

Learning goal: Add and subtract whole numbers and decimals through hundredths in real problems, connecting a model with a written method. Decimal addition combines like units: ones with ones, tenths with tenths and hundredths with hundredths. The decimal point helps keep those places aligned.

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

A class has 12.75 meters of cord and receives another 3.6 meters. Write 3.6 as 3.60 to display the hundredths place without changing its value. Adding 12.75 + 3.60 gives 16.35 meters. The five hundredths remain five hundredths. Seven tenths plus six tenths make thirteen tenths, renamed as one whole and three tenths. The whole amounts, including that new whole, total sixteen.

You can express the entire calculation in hundredths. 12.75 meters is 1,275 hundredths of a meter, and 3.60 meters is 360 hundredths. Their sum is 1,635 hundredths, or 16.35 meters. Renaming the quantities shows why the digits in the written columns can be added with familiar whole-number methods while retaining their units.

A small model illustrates the regrouping without requiring thousands of objects. Combine 0.75 meter and 0.60 meter on two equal hundred grids, each representing one meter. There are 75 + 60 = 135 selected hundredths. Regroup one hundred of them as a whole meter, leaving 35 hundredths. The total is 1.35 meters. Unselected regions still belong to each defined whole; a partly shaded grid is not a smaller whole.

For subtraction, suppose a 15-meter length of cord has 8.67 meters cut from it. Write 15.00 − 8.67. Rename one of the fifteen wholes as ten tenths, then rename one of those tenths as ten hundredths. The amount is now fourteen wholes, nine tenths and ten hundredths. Subtract eight wholes, six tenths and seven hundredths, leaving six wholes, three tenths and three hundredths: 6.33 meters.

Check subtraction with addition: 8.67 + 6.33 = 15.00. Seven hundredths plus three hundredths make ten hundredths, one tenth. That tenth with six tenths and three tenths makes ten tenths, one whole. Eight ones, six ones and that whole make fifteen. The inverse check tests both the digits and their places. A result of 7.33 would fail because the sum would exceed the original fifteen meters.

Zeros and unequal written lengths often cause errors. In 5 + 0.48, write 5.00 + 0.48, not 0.05 + 0.48. The whole number five belongs in the ones place. The same holds for a whole number with hundreds: 125 + 3.08 is 125.00 + 3.08 = 128.08, with the five in the ones place. In 23.4 − 2.86, rename 23.4 as 23.40 and obtain 20.54. Right-aligning the final written digits without considering place would match tenths with hundredths and change the quantities.

Estimate before calculating. 12.75 + 3.6 is roughly 13 + 4, so a sum near seventeen is sensible; 1.635 is not. The exact sum can differ from the estimate because rounding changes the inputs. Keep the unit in the final answer and decide whether addition represents combining amounts or subtraction represents a difference or what remains.

Worked money context: A fictional supply order costs $27.85 plus $6.40. Rename the amounts as 2,785 cents and 640 cents. The sum is 3,425 cents, or $34.25. This price is an invented arithmetic example, not a current shop quotation.

Worked correction: For 15.00 − 8.67, a learner subtracts the smaller digit from the larger in each column, regardless of which number it belongs to, and obtains 13.67. That method does not preserve the original amount: 8.67 + 13.67 = 22.34, not 15. Exchange units before subtracting and check the resulting 6.33 by addition.

Guided practice: Model a sum crossing one whole with equal hundred grids and a subtraction requiring exchange. Label the entire grid as one unit. Explain how each exchange appears in the written method.

Independent performance: Solve a cord problem and a fictional money problem, including a whole number combined with a decimal. Show a model or expanded-unit explanation, an estimate and an inverse check. Continue with the connected multiplication and division lessons for the other decimal operations. Assess choice of operation, like-unit reasoning and accuracy separately.