Addition and subtraction

Combine separate parts into a whole

Addition combines counts. Eight stones in one bag and five different stones in another give 8 + 5 = 13 stones. Eight and five are the addends; thirteen is their sum. Both parts count stones, and no stone belongs to both parts. Keep these conditions visible.

Overlapping lists count shared stones twice when their lengths are added. Five stones and four grams also cannot form a stone total. Define the counted unit and check that the numerical sum fits the story.

Subtract to remove, compare or find a missing part

Taking five stones away from thirteen leaves eight: 13 − 5 = 8. Thirteen is the starting whole; five is the removed part; eight is the remaining part. Subtraction’s result is called a difference. Identify what each number counts rather than choosing an operation from one isolated word.

Subtraction also compares without physically removing anything: nine seeds versus six gives a difference of three seeds. A missing-part question asks what completes a whole: thirteen stones include five in one part, so the other part contains eight.

Use the parts to check the whole

Let A and B count separate parts and T their whole. A + B = T connects with T − B = A and T − A = B. These inverse relationships recover one part by removing the other. Letters name counts, not individual objects.

Check 13 − 5 = 8 by rebuilding 8 + 5 = 13. Check addition by removing a known part. The inverse tests the numerical relationship; also check the unit and that each object was counted once.

Keep zero and the order meaningful

Adding an empty part changes nothing: 7 + 0 = 7 and 0 + 7 = 7. Removing nothing gives 7 − 0 = 7; removing all seven gives 7 − 7 = 0. Zero records a real empty count. Adding or removing zero means no movement on the number line.

Addition allows 8 + 5 = 5 + 8. Subtraction’s order generally matters: eight minus three gives five; three minus eight has a negative result, studied later. State the starting count and the removed count in a take-away story.

Use ten as an intermediate point

For 8 + 5, split five into two and three. Add two to reach ten, then the remaining three to reach thirteen. On the number line, cross five unit intervals right from eight. The starting mark is not a move.

For 13 − 5, remove three to reach ten, then the remaining two to reach eight. The total removed remains five. Crossing five intervals left gives the same result. This plan changes no part of the question.

Regroup the value, rather than changing it

For 28 + 17, combine two tens and eight ones with one ten and seven ones: three tens and fifteen ones. Exchange ten ones for one ten to get four tens and five ones, or 45.

For 45 − 17, exchange one ten for ten ones: three tens and fifteen ones still equal 45. Remove one ten and seven ones, leaving 28. Changing five ones to fifteen while keeping four tens would increase the value, rather than regroup it.

Connect the parts, equation and line move

Choose parts A and B from 0–10 in the explorer. Each circle or square represents one counter; shapes distinguish the parts, not different units. Combine shows A + B = T. Remove B shows T − B = A; Remove A shows T − A = B. In this activity, the whole never exceeds twenty.

Marked removals track the original whole; count remaining counters as the result. Match the equation to unit intervals on the fixed 0–20 segment and inspect the inverse check. Explain an empty part’s zero count. Practise fresh cases; next is “Multiplication and division.”

Worked examples

Try it yourself

1. One tray contains nine seeds and another six. How many more seeds does the first tray contain? Explain why neither tray needs to be emptied.

   Show the answer and reasoning

   Three more seeds: 9 − 6 = 3.

   Pair six seeds in the larger collection with the six in the smaller; three remain unmatched. This compares quantities. Check 6 + 3 = 9.

2. Sixteen beads are split into two parts with no bead shared between them. One part contains nine. Find the other part and check the whole.

   Show the answer and reasoning

   Seven beads: 16 − 9 = 7.

   Nine plus the missing part must make sixteen. Check 9 + 7 = 16; the answer counts beads, not parts.

3. Part A has zero counters and part B has six. Give the results for Combine, Remove B and Remove A, and explain the empty part.

   Show the answer and reasoning

   Combine: 0 + 6 = 6. Remove B: 6 − 6 = 0. Remove A: 6 − 0 = 6.

   A is a named part with no counters. Removing the whole six leaves none; removing the empty part changes neither count nor line position.

4. Find 72 − 38 by regrouping tens and ones. State an equal-value representation of 72 before removing anything, then check your answer.

   Show the answer and reasoning

   Thirty-four. Regroup 72 as six tens and twelve ones, then remove three tens and eight ones.

   60 + 12 = 72 preserves the whole. Three tens and four ones remain. The inverse check is 34 + 38 = 72.

What to carry forward

• Combine separate parts counted in the same unit; subtraction can remove, compare or find a missing part.
• Use inverse checks, give zero its meaning, and keep subtraction’s order clear.
• Count intervals, keep track of each part, and regroup without changing the represented value.