Mathematics K–6 · Years 3–4
Balancing number sentences: finding the missing value
Number and algebra
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The idea
The equals sign means both sides balance, so a missing value is whatever makes the two pans level, and the same balance shows addition and subtraction as a pair.
What you need
- an equal-arm balance and a set of identical masses (for example 40 metal washers of the same size)
- 8 small identical opaque bags (an empty bag goes in the other pan for each bag in use, so the bags' own mass cancels)
- number sentence cards with a missing value shown as a box: 8 + □ = 15, □ - 6 = 9, 3 x □ = 12, 20 = 4 x □
- a recording sheet
How to do it
- Put 15 washers in the right pan. Put 8 in the left pan and add washers one at a time until the pans level. Count the added washers and write 8 + 7 = 15.
- Put 9 washers and a separate group of 6 washers in the right pan, with an empty bag. Fill a bag in the left pan one washer at a time until the pans level, and count the bag: that is the missing value in □ - 6 = 9. Take the group of 6 off the right pan and 6 washers out of the bag, and see the pans stay level with 9 against 9.
- Put 12 washers and three empty bags in the right pan and three identical bags in the left pan. Add washers to the bags equally until the pans level, then open one bag and count.
- Take washers off both pans two at a time and watch the balance stay level. Then take from one pan only and watch it tip.
- Write each card's answer and a second sentence that says the same thing the other way (for example 15 - 8 = 7).
What you should see
8 + □ = 15 balances at 7; □ - 6 = 9 balances when the bag holds 15; 3 x □ = 12 balances with 4 washers in each bag; 20 = 4 x □ gives 5. Removing the same number from both pans keeps the balance level, and removing from one pan tips it, which is why the same change must be made to both sides. Each addition sentence pairs with a subtraction sentence that the same balance shows.
What changes
This activity lists no variables to change, measure and keep the same.
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The equals sign means the answer comes next.
- In 20 = 4 x □ the missing number is 80 because the numbers are multiplied.
Safety card
Hazards
- washers dropped on feet
- pinching at the pivot
Controls
- work over the table; keep fingers clear of the pivot
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package is not needed; ordinary classroom supervision under the school's usual risk management.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
Sources
The pages the author read to write this activity.