Add and subtract money and time with models

Learning goal: Model addition and subtraction of money, express monetary answers in decimal notation and use number lines to calculate intervals in hours and minutes. Money and time both require attention to units, but their exchanges differ. One dollar equals one hundred cents; one hour equals sixty minutes. A decimal fraction of a dollar must not be interpreted as clock minutes.

A price of $2.75 means two dollars and seventy-five cents. Represent it with two dollar units and seventy-five one-cent units, or with equivalent labeled value groups. The illustration uses value bars rather than pictures of particular coins. Their purpose is to show that every one-dollar bar can be exchanged for one hundred cents without changing the amount of money.

To add $2.75 and $1.60, first add dollars and cents. Two dollars plus one dollar gives three dollars. Seventy-five cents plus sixty cents gives 135 cents. Exchange one hundred of those cents for one dollar, leaving thirty-five cents. The sum is four dollars and thirty-five cents, written $4.35. An all-cents model agrees: 275 + 160 = 435 cents.

For change from five dollars after spending $3.68, rename five dollars as five hundred cents. Subtract 500 − 368 = 132 cents, or $1.32. Count up to check: $3.68 plus $0.32 reaches $4.00, and one more dollar reaches $5.00. The two increases total $1.32. A money answer written as $1.3 is numerically meaningful, but $1.30 is the customary way to write one dollar thirty cents, with two digits after the decimal point.

Clock intervals use sixty minutes per hour. To find elapsed time from 9:35 a.m. to 11:10 a.m. on the same morning, draw a time line. From 9:35 to 10:00 is twenty-five minutes. From 10:00 to 11:00 is sixty minutes. From 11:00 to 11:10 is ten minutes. The total is ninety-five minutes, or one hour thirty-five minutes.

A proportional timeline should show the sixty-minute jump longer than the twenty-five-minute and ten-minute jumps. Label all endpoints so the jumps can be checked against the clock readings. Subtracting 935 from 1110 as whole numbers would ignore the sixty-minute exchange and the meaning of the colon in clock notation.

To add an interval, move forward from the starting time. Begin at 2:45 p.m. and add one hour twenty-five minutes. One hour reaches 3:45. Fifteen minutes reaches 4:00, and ten more reaches 4:10 p.m. To find a starting time from an ending time, move backward. Ending at 4:10 after a forty-minute activity means starting at 3:30 p.m.

Adding or subtracting durations also requires exchanging sixty minutes. One hour forty minutes plus fifty minutes is one hour ninety minutes, which equals two hours thirty minutes. Two hours ten minutes minus forty-five minutes can be renamed one hour seventy minutes minus forty-five, leaving one hour twenty-five minutes. Check by adding the difference and the removed interval.

Worked purchase: In this made-up purchase, a notebook costs $1.85 and a folder $0.90. Their combined cost is 185 + 90 = 275 cents, or $2.75. Paying $3.00 leaves 300 − 275 = 25 cents, written $0.25. Draw three dollar bars, exchange a bar into cents, and label the cost and remaining value.

Worked clock check: A session begins at 10:50 a.m. and lasts thirty-five minutes. Ten minutes reaches 11:00 and twenty-five more reaches 11:25 a.m. The jumps total thirty-five minutes and the ending time stays before noon.

Guided practice: Build a purchase and change model using labeled paper value units. Then draw a timeline crossing an hour boundary. State the exchange used in each model and explain why one hundred applies to cents while sixty applies to minutes.

Independent evidence: Solve a two-item purchase with change, an elapsed-time problem and a missing-start-time problem. Show models, label a.m. or p.m. and state whether the times are on the same day. Record monetary results with decimal notation and time results with hours and minutes. Use the metric companion lesson for distance, mass and volume problems.