What it shows
To square a number ending in 5, multiply the tens digit by the next number up and write 25 after it: 35 squared is 3 times 4 = 12 followed by 25, so 1,225. The flip board shows each step and the algebra behind it: (10a + 5) squared = 100a(a + 1) + 25.
The science
Write the number as 10a + 5. Squaring gives 100a squared + 100a + 25, which is 100a(a + 1) + 25. The part a(a + 1) lands in the hundreds place, so the answer is the product of the tens digit and its successor, followed by 25. The same idea squares 105 as 10 times 11 = 110 followed by 25, giving 11,025, and it also multiplies two numbers with the same tens digit whose units add to 10, such as 43 times 47 = 4 times 5 followed by 3 times 7, which is 2,021.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A cardboard base 40 cm by 20 cm
- Flip cards on rings or tape hinges for the tens digit and the answer
- A marker
- A calculator for checking
- A small whiteboard sheet for visitors to try
Safety
This project uses safe, everyday materials. Still, read the rules below with a parent before you start.
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.
Step by step
7 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.
Make a row of flip cards 1 to 9 for the tens digit a, and a fixed card showing 5 beside it.
Below, make a flip set for a times (a + 1) with the values 2, 6, 12, 20, 30, 42, 56, 72, 90, and a fixed card showing 25.
Flip to a = 3: the board shows 35 above and 12 then 25 below, reading 1,225.
Flip through every digit and check each answer on a calculator: 15 squared is 225, 25 squared is 625, up to 95 squared is 9,025.
Write the algebra proof on the side panel in three lines: expand (10a + 5) squared, factor 100a(a + 1), add 25.
Add a second panel for the related trick 43 times 47 = 2,021 and 62 times 68 = 4,216, and show the condition: same tens digit, units adding to 10.
Time a visitor squaring 85 on the board against a friend on a calculator.
What you should see
All nine squares from 15 to 95 read correctly on the board and match the calculator, and the side panel proves why the trick works.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
01What is 65 squared?
4,225, from 6 times 7 = 42 followed by 25.
02Why does the answer always end in 25?
Because 5 squared is 25 and the other terms are multiples of 100.
03Does the trick work for 115 squared?
Yes: 11 times 12 = 132 followed by 25, giving 13,225.
04What is 43 times 47 by the related rule?
2,021, from 4 times 5 = 20 and 3 times 7 = 21.
Take it further
Make a board for squaring numbers near 50: 50 + d squared is 2,500 + 100d + d squared, so 53 squared is 2,809.
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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