Add and subtract mixed numbers with like denominators

Learning goal: Add and subtract mixed numbers with like denominators, explain regrouping and check answers in context. A mixed number combines whole units and a fraction of one more unit. Two and three fourths means 2 + 3/4. In a strip model, each complete strip represents one whole. All strips must use the same whole-unit length.

To add 2 3/4 and 1 2/4, combine the whole units and the quarter units. Two plus one is three wholes. Three fourths plus two fourths is five fourths. Since four fourths make one whole, five fourths are one and one fourth. Therefore the complete sum is 4 1/4. Regrouping four quarter pieces into a whole preserves the total amount.

Another strategy renames everything in fourths. Two wholes are eight fourths, so 2 3/4 = 11/4. One whole is four fourths, so 1 2/4 = 6/4. Add 11/4 + 6/4 = 17/4. Four sets of four fourths account for sixteen fourths, leaving one fourth: 17/4 = 4 1/4. Both methods describe the same pieces.

Subtracting can require exchanging a whole for fractional pieces. Consider 3 1/4 − 1 3/4. There is only one loose quarter in the starting mixed-number form, but three quarters must be removed along with a whole. Rename one of the three wholes as four fourths. The start becomes 2 5/4: two intact wholes plus five quarters. Now subtract one whole and three quarters to leave 1 2/4.

The renamed start has the same value: 3 1/4 = 2 5/4. As sums of fourths, 13/4 = 4/4 + 4/4 + 4/4 + 1/4 = 8/4 + 5/4, two decompositions of the same amount. The whole count decreased by one while the quarter count increased by four. Simply changing 1/4 to 5/4 without reducing the whole count would add an extra whole. Draw the exchanged strip partitioned into four quarters to show exactly where the new pieces came from.

The all-fraction method checks this subtraction. Three and one fourth is 13/4, and one and three fourths is 7/4. Subtract to get 6/4, which is 1 2/4. Add the difference to the removed amount: 1 2/4 + 1 3/4 = 2 5/4 = 3 1/4. The check rebuilds the original quantity.

Do not subtract the smaller numerator from the larger regardless of its position. For 3 1/4 − 1 3/4, subtracting whole parts to get two and using 3 − 1 to get two quarters would give 2 2/4. That answer fails the addition check. The order of the amounts matters; regroup the starting quantity instead of reversing the subtraction.

Choose an efficient strategy for the numbers. For 4 5/6 − 2 1/6, subtracting whole and fractional parts directly gives 2 4/6 with no exchange. For 4 − 1 3/4, rename four as 3 4/4, then remove 1 3/4 to get 2 1/4. Whole numbers can be written with fractional parts when it helps represent the action.

Worked measuring story: A spool has 3 1/4 meters of cord. A project uses 1 3/4 meters. Let r be the remaining length: r = 3 1/4 − 1 3/4 = 13/4 − 7/4 = 6/4 = 1 2/4 meters. The result, one and a half meters, is reasonable: the spool holds between 3 and 3 1/2 meters and the project uses between 1 1/2 and 2 meters, so more than 3 − 2 = 1 meter and less than 3 1/2 − 1 1/2 = 2 meters should remain.

Worked addition: Two containers hold 1 5/8 liters and 2 6/8 liters of water. Their combined volume of water is 3 11/8 = 4 3/8 liters. The eleven eighths contain one whole liter and three more eighths.

Guided practice: Use strips to solve an addition crossing a whole and a subtraction requiring an exchange. Describe each renaming before computing. Compare the mixed-number and all-fraction methods and show that their answers occupy the same total strip length.

Independent evidence: Solve a measurement story with a mixed-number sum and one with a difference. Include a labeled model, equation and inverse-operation check. Ask the learner to explain where regrouped pieces came from and why 2 5/4 still names 3 1/4. These demonstrations reveal more than a correct final numeral alone.