# Counters

Counters are a versatile manipulative that have many uses in the maths classroom.  I have demonstrated a selection in this section.

MathsBot has a great selection of interactive counters.

Counters

Double sided counters

Directed counters

Place value counters

Algebra discs

The examples below demonstrate the connections that can be made between number and algebra using counters.

## Four operations with base 10 and base x counters

### Multiplication

152

152 x 2

exchange ten 10’s for 1 hundred

x2 + 5x + 2

2(x2 + 5x + 2) = 2x2 + 10x + 4

152

152 x 10

Each counter moves one place to the left (10 times bigger)

x2 + 5x + 2

x(x2 + 5x + 2) = x3 + 5x2 + 2x

Each counter moves one place to the left (x times bigger)

13 x 12 = 13x2 + 13x10

(x + 3)(x + 2) ≡ 2(x+3) + x(x+3) ≡ x2 + 5x + 6

Each "counter" moves one place to the left (x times bigger)

## Standard Index Form

Standard index form is all about base 10.

### Non-standard standard form problems

Write 23 x 102 in standard index form.

Write 0.23 x 10-2 in standard index form.