What it shows
To multiply 97 by 96, write how far each is below 100 (3 and 4). Subtract crosswise, 97 minus 4 = 93, and multiply the deficits, 3 times 4 = 12, to get 9,312. The slider lines up the two numbers and their deficits so the answer appears in two steps.
The science
Write the numbers as 100 minus a and 100 minus b. Their product is 10,000 minus 100a minus 100b + ab, which is 100(100 minus a minus b) + ab. The bracket is the first number minus the other number's deficit, and ab is the product of the deficits, written as two digits. For numbers above 100 the surpluses are added crosswise instead: 103 times 104 = 107 followed by 12, giving 10,712. If the product of deficits has three digits, the hundreds carry across: 88 times 89 is 77 and 132, giving 7,832.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A card strip 50 cm long with two sliding windows
- A number strip from 90 to 110
- A marker
- A calculator for checking
- Glue and a cutter
Safety
For this project: An adult must cut the windows in the card.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Draw the number strip from 90 to 110 and mark the deficit or surplus from 100 under every number: 10 under 90, 0 under 100, 10 over 110.
Cut two windows in the outer strip so each shows a chosen number with its deficit.
Set 97 and 96 in the windows, read the deficits 3 and 4, and write the crosswise subtraction 97 minus 4 = 93 (or 96 minus 3, which is the same).
Multiply the deficits, 3 times 4 = 12, and write the answer 93 followed by 12: 9,312. Check on a calculator.
Try 98 times 95, 92 times 94 and 99 times 99, writing each as two steps on a record card.
Set 103 and 104 to show the above-100 case: 103 + 4 = 107 and 3 times 4 = 12, so 10,712.
Set 88 and 89 to show the carry: 88 minus 11 = 77 and 12 times 11 = 132, so 7,700 + 132 = 7,832.
Write the algebra on a side card so the trick is seen as a theorem, not magic.
What you should see
The slider gives 9,312 for 97 times 96, 10,712 for 103 times 104 and 7,832 for 88 times 89, all matching the calculator.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What is 98 times 97?
9,506: deficits 2 and 3, 98 minus 3 = 95, 2 times 3 = 06.
2Why is the crosswise subtraction the same whichever way you do it?
Because 100 minus a minus b is symmetric in a and b: 97 minus 4 and 96 minus 3 both give 93.
3What base would you use for 48 times 47?
Base 50, with the left part multiplied by 50 instead of 100: 48 minus 3 = 45, 45 times 50 = 2,250, plus 2 times 3 = 6, so 2,256.
4When do you need a carry?
When the product of the deficits is 100 or more, as in 88 times 89.
Take it further
Make a base-1,000 strip for numbers like 997 times 994, where the deficit product is written with three digits: 991,018.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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