Adding and subtracting (age 8+)

Activities to try

1. 1

   Cascade the thousand: borrow across two zeros with base-ten quantities

   Why it works: Borrowing across zero is not a special rule: when an adjacent column is empty you continue the exchange one column further left, and a single thousand can cascade down through the zeros to free up ones. The written digits are a record of quantities actually regrouped.

   1. Set up 4002 - 1367 in place-value columns with Th/H/T/O headers, aligning from the ones column on the right.
   2. Build 4002 physically (or on a screen model): four thousand-blocks and two ones, with empty hundreds and tens.
   3. Try to subtract 7 ones from 2 ones; notice there is nothing in the hundreds or tens to exchange from, so the exchange must come from the thousands.
   4. Exchange one thousand-block into ten hundred-blocks, then one hundred into ten tens, then one ten into ten ones — watching the single thousand ripple down through the two empty columns until the ones column holds 12.
   5. Record each changed digit in the written algorithm in step with the blocks (4002 becomes 3 thousands, 9 hundreds, 9 tens, 12 ones) and finish the subtraction to reach 2635.
2. 2

   Align and compose: mixed-length columnar addition and subtraction

   Why it works: Digits are added or subtracted by matching place value, anchored from the ones column on the right regardless of how many digits each number has; a column result of ten or more is carried, and a column that is too small to subtract requires exchanging from the next column left.

   1. Set out 3456 + 878 using place-value headers or grid paper, placing the 8 of 878 under the ones column (not under the left of 3456).
   2. Add column by column; when the ones make 14, write 4 and carry one ten to the tens column rather than writing a two-digit entry in the ones.
   3. Set out 6250 - 483, anchoring the 3 under the ones column, and subtract, exchanging from the next column left wherever the top digit is too small.
   4. For each answer, round both numbers to estimate first, then check the exact result with the inverse operation.
3. 3

   Check it two ways: estimate and inverse

   Why it works: Checking is not redoing the same subtraction: estimation catches place-value-scale slips, and the inverse addition independently confirms the exact answer. They are complementary, not repetition.

   1. After solving 4002 - 1367, round both numbers (about 4000 - 1400 = 2600) on a number line or strip and say whether 2635 is reasonable.
   2. Check the same answer a second, independent way by adding the result to the number subtracted: 2635 + 1367 should return 4002.
   3. Answer: 'What if it had been 4000 - 1367 instead — does that change how you borrow?' and explain.
   4. Compare the two forms of checking and say what each one catches that the other might miss.