Explain efficient addition and subtraction within one hundred

Learning goal: Add and subtract within 100 with increasingly fluent place-value strategies, add up to four two-digit numbers and explain why the chosen method preserves value. A written numerical method should record thinking rather than replace it with unexplained digit moves.

For 47 + 28, decompose 28 into 20 and 8. Add two tens: 47 + 20 = 67. Then add eight ones: 67 + 8 = 75. Another method combines forty and twenty to make sixty, and seven and eight to make fifteen. Sixty plus fifteen is seventy-five. Both methods add all the original tens and ones exactly once.

The model begins after the tens and ones of 47 and 28 have been put together: four tens and two tens make six tens, and seven ones and eight ones make fifteen ones. Fifteen ones can be composed into one ten and five ones, making seven tens and five ones overall. The exchange changes the grouping, not the quantity. “Carry one” is unclear unless the learner explains that the one represents a composed ten.

Subtraction can decompose the amount removed. For 75 − 28, remove twenty to reach 55, then remove eight to reach 47. Or use the related addition fact 47 + 28 = 75. Counting upward from 28 to 75 can also find the difference: two to reach 30, forty to reach 70 and five to reach 75, for a total difference of 47.

Compensation can make addition efficient if the adjustment is tracked. For 49 + 26, move one from 26 to 49, giving 50 + 25 = 75. The total is unchanged because one addend increased by one while the other decreased by one. Increasing both addends would change the total and would need a different correction.

Add four two-digit numbers by grouping conveniently. For 18 + 27 + 12 + 23, combine 18 + 12 = 30 and 27 + 23 = 50. Then 30 + 50 = 80. Addition permits reordering and regrouping the addends; none is omitted or counted twice. A running total of 18, 45, 57, 80 is another valid method.

When adding up to four two-digit numbers, the total may exceed 100 even though each addend has two digits. For example, 35 + 25 + 40 + 20 = 120. This is different from the separate fluency goal of adding and subtracting within 100. State the scope of each task instead of treating a two-digit addend as a guarantee of a two-digit sum.

Fluency develops through repeated varied practice. Encourage a learner to choose a suitable strategy and explain it, then become more efficient with familiar relationships. The purpose is not to force the same decomposition for every problem or to reward an unexplained answer solely because it was quick.

Worked example 1: Regroup a sum. Seven ones and eight ones give fifteen ones. Exchange ten of them for a ten, leaving five ones. Together with the original six tens, the result is seven tens and five ones: 75.

Worked example 2: Keep all four addends. In 18 + 27 + 12 + 23, mark each addend as it enters a pair. The convenient pairs total 30 and 50. Their sum is 80. The marking checks that reordering did not lose an amount.

Worked example 3: Check subtraction. For 75 − 28 = 47, add 47 + 28. Getting 75 confirms that the removed amount and remaining amount reconstruct the starting total.

Guided practice: Compare two methods for the same two-digit sum and explain the equality of their results. Solve subtraction by removal and by missing-addend reasoning. Add three and four two-digit numbers using a running total and convenient groups.

Independent tasks: Solve unfamiliar within-100 addition and subtraction efficiently, record a written method and explain at least one place-value step. Add a four-addend problem, including a case whose sum exceeds 100, and verify that every addend was included once.

Feedback: If digits are combined without place values, return to tens and ones. If compensation changes both numbers in the same direction, compare the original and adjusted total with objects. Credit a method when its numerical record and explanation preserve the original quantities.