Two of our upper elementary children work out the formula for a trinomial square using a Montessori peg board. That is  (a + b + c)²  or (a + b + c)  x  (a + b + c).

They start by building the square of 9 i.e. 9 x 9 on the peg board. Shown below with 81 green pegs. Next 9 is broken down as the trinomial  (2 + 3 + 4)  and the 81 beads are divided as shown below.

Trinomial square of 9

Then 2, 3 & 4 are replaced by a, b & c as shown below.

Trinomial square of (a + b + c)

Finally, we can read all the sections on the peg board and derive the formula  (a + b + c)² =  a² + b² + c² + 2ab + 2ac + 2bc

Part 1 over – now lets work out 234² to see if our formula is correct.

First, 234 is expressed as the trinomial (200 + 30 + 4)² on the peg board using 5 unit (green), tens (blue) and hundred (red) beads where a = 200, b= 30, c = 4. NB thousands revert back to green pegs in the two (ab) rectangles and ten thousands to blue pegs in the single a² square (bottom right).

Trinomial square of 234

This provides the values below.

234² = 200² + 30² + 4² + 2(200 x 30) + 2(200 x 4) + 2(30 x 4)

234² = 40000 + 900 + 16 + 2(6000) + 2(800) + 2(120) as shown below

Trinomial square of 234

Which add up to 54756

Trinomial square of 234

For a better explanation just ask your own 11 or 12 year old using these pictures.