Money is really good for learning places and decimal fractions.

Metric rule(r)s are excellent. We had a meter rule that had millimeter increments, so
a meter was 10 decimeters (this unit should not be overlooked), 100 centimeters, and 1000 millimeters. Once kids learn the dollar system, with cents being decimal fractions,
it’s easy to teach 1.000 meter, .100 m, 0.010 m, and .001 m units, and assign problems using their rules.

The English-unit rule is superb for teaching 1/(2^n) fractions and doing early work converting bases.

I mainly want to talk about pennies. Get 2000 pennies. (This is a free manipulative, you can exchange it back at the bank for a $20 bill when you’re done.)

Let’s say you want to teach basic multiplication. Let’s say we want to do 9s to 90.
The child counts up 90 pennies—watch this to make sure that the counting is correct– and puts them on the table. Then she makes a stack of 9 pennies. She uses this as a measure, by first counting and stacking another 9 pennies next to the first, and maybe a third counted stack next to the first two. Then have her create more stacks by visual inspection to get equal heights, rather than by counting. If there is a penny left over, or one-short, have her examine all the stacks. You do this too.

Ordered in a row, we call this 9 x 10.

Now, we take a penny from the last stack, place it on the first stack, then take another to place on the second, and so forth, until we run out of pennies for the last stack. We now examine the stacks, see they are all equal height. We count the pennies in the first stack, 10, and conclude that this is what all the stacks have. But we now have 9 stacks. We call this 10 x 9. The commutative property. (We start with smaller numbers of pennies, this being just an example of a sequence of exercises.)

We want to teach mathematics language. So we have our student write 9 x 10 = 10 x 9.

We can take each stack and break it into two stacks of 5 pennies. We can see that 2 x 5 x 9= 90. The child can count the pennies by fives to 90. The child can see there are 18 stacks, so 5 x 18 = 90.

We teach an order of operations, using parentheses in an equation to be an instruction to first add the quantities, here 5+5 within the parens, then multiplying the sum by the number of stack pairs. Then we can create other stack values, 1+9, 2 + 8, … and show that by adding these number pairs, the sum is always 10, and we have 9 of these pairs, and there are still 90 pennies on the table.

We then separate the stacks into same-height groups. So, for example, if we have 3 + 7 pennies, we separate all the 3-penny stacks (9 of them) from 7-penny stacks (9 of them),
and we multiply 3 x 9 and 7 x 9, and sum 27 + 63 = 90. We can even count all the pennies in the 3-stack and the 7-stack to verify that 3 x 9 indeed equals 27, and 7 x 9 indeed equals 63.

So we write an equation, (3 + 7) x 9 = 3 x 9 + 7 x 9. We give this a name, distributive property. (We can apply this to the earlier exercise of transferring one penny to the 9-penny stacks 9 p/s x 10s = (9 +1) p/s x 9s (distributive and commutative properties.)

We then have the child do other stack combinations, similarly, and write her own equations. Writing mathematical sentences is a crucial exercise. It is the fundamental building block of later multi-sentence mathematical compositions.

Is learning the distributive property hard? It sure seems to be. Pennies make it easy.

The associative property is a piece of cake with pennies.

With 90 pennies (or a smaller number at the outset), you can do a lot. You can make three groups of 3 10-penny stacks, break the thirty pennies in each group into various sum combinations, and reinforce the distributive property. You can group 5 pennies, 15, and 45.

After this, you can scale up to pretty large numbers with 2000 pennies. You can, for example create square arrays, and rectangular arrays. So, for example, if you have a 4×4 array of single pennies, the child writes an equation 4 x 4 = 16. You place a penny on top of the singles, now the child can write 2 x 4 x 4 = 32. Keep adding pennies.

You can teach cubics: 1 penny = 1. 2 pennies per stack in two rows and columns = 2 x 2 x 2 =8 pennies, 3-penny stacks in 3 rows and columns = 3 x 3 x 3 = 27, and so on.

You can set up penny problems such as laying out 1 penny and to its right two stacks of 8 pennies and to the far right four stacks of 7 pennies. Teach the child to write an equation 1 + 2×8 + 4×7 = , and solve the equation. (This BTW teaches WHY multiplication precedes addition in the order of operations. It’s not an arbitrary rule.)

You can teach quadratic equations to 10-11 year olds with pennies.

In every exercise, the pennies can be hand-counted, whenever the child wants a verification that she is doing the multiplication right.

Instead of worksheets that have fill in the blanks like 2 + 2 x 9 = ___, the child writes her own equations. Instead of pushing students to “get the right answer” we want to train children how to express themselves in the written language of mathematics, which is a universal language, and take pleasure in this.

Pennies also lend themselves superbly to subtraction and division exercises, conversion to larger denomination coins and bills, and innumerable other conceptual exercises. Maybe every school should invest $20 in each student for this.