Compute with tens and ones and explain the regrouping

Learning goal: Add within one hundred using place-value models, find ten more or ten less mentally with an explanation, and subtract multiples of ten from multiples of ten when the difference is zero or positive. Include two-digit plus one-digit and two-digit plus a multiple of ten cases. Written methods should record understood grouping, not replace it with an unexplained carrying rule.

For 34 + 5, keep three tens and combine four ones with five ones. The new ones total is nine, so the result is three tens and nine ones: 39. Tens are combined with tens and ones with ones because those units represent different amounts. Five single counters do not become five tens merely because they are placed near the tens drawing.

For 34 + 20, add two tens to the three existing tens. The four ones remain unchanged. The result is five tens and four ones, or 54. Compare this with adding five ones so the learner can explain why different digits change in different examples.

Sometimes ten ones must be composed into a new ten. For 28 + 6, the eight ones and six ones make fourteen ones. Group ten of those ones as another ten, leaving four ones. With the two original tens, there are now three tens and four ones: 34. No value disappears in regrouping; ten individual ones are renamed as one ten.

The organizer records the same 28 + 6 calculation in stages. It shows parts and equivalent regroupings, not additional collections that should all be added together. Ask where all six added ones can be found after the regrouping. Two helped complete the new ten with the original eight; four remain as loose ones.

Ten more than 52 is 62 because adding one ten changes five tens to six tens while two ones remain. Ten less is 42 for the corresponding removal. Explain mentally without counting ten individual steps. Include a boundary example such as ten more than 95 being 105 using the established count-through-120 range; the principle is still an added group of ten.

For 70 − 30, work with seven tens minus three tens. Four tens remain, so the difference is forty. Keep the word tens attached to the small-number fact 7 − 3 = 4. Otherwise a child may report four individual ones. The relationship between addition and subtraction gives a check: three tens plus four tens make seven tens, so 30 + 40 = 70 confirms 70 − 30 = 40. Restrict these multiple-of-ten subtraction examples to the stated 10–90 starting/removal range and nonnegative differences.

Worked example 1: Compose a ten. Solve 47 + 6. Seven ones and six ones make thirteen ones. Regroup ten, leaving three ones, and combine the new ten with the original four tens. The answer is 53. Check by counting on or another known relationship.

Worked example 2: Two-digit addition model. For 28 + 16, combine two tens and one ten, then eight ones and six ones. Three tens and fourteen ones regroup to four tens and four ones: 44. This explains why tens add to tens and why sometimes another ten is formed.

Worked example 3: Zero difference. For 60 − 60, remove all six tens. No tens or ones remain, so the result is zero. Zero is a legitimate result in the specified nonnegative range.

Guided practice: Build examples for adding ones, adding tens and composing a new ten. Record each with drawings or equations that match the model. Ask for mental ten-more/ten-less explanations using several two-digit starts, and subtract tens using both objects and a written relation.

Independent tasks: Solve 36 + 3, 36 + 20, 58 + 7 and 37 + 25 with reasoning; find ten more and less than 68 without counting individual steps; and solve 80 − 50 and 40 − 40. Include one drawing showing where regrouped ones went. Accuracy alone does not establish that the learner understands the regrouping.

Assessment and feedback: Score the units, the computation and the explanation. If the child reports 80 − 50 as three, restate three tens and convert that to thirty ones. If a written method moves a digit without a model explanation, rebuild the quantities. Extend by comparing two different correct written strategies for the same within-one-hundred sum.