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The tens' place.  The hundreds' place.  The thousands' place.  They each have their place, and when it comes to guiding my kids in their math work, I'm a real stickler about it.

I don't simply mean that my kids have to know which is which, and that they need to line them up when adding.  I mean that they have to understand how the digits filling those places interact with each other.

When adding, for instance, they aren't just adding different columns of digits; they are adding groups of a hundred, groups of ten, and groups of single units.  The worth of each digit should always be in mind as they compute the answer so that the process becomes one in which mathematical truth can be communicated, not merely manipulated.

Keeping a mind on the value of digits during addition or subtraction isn't as tricky as it is during multiplication.  Many of the conventional algorithms simply do not apply, as there is no room in this emphasis for suggestions such as: "Count the total number of zeros that would follow the two digits you're multiplying, and then write the same number of zeros after your answer."  That trick may make the process easier, but it doesn't help a person connect the abstraction of the numbers to the facts in reality.

What is the reality then?  To begin with, when a person multiplies a group of ten by another group of ten, he doesn't end up with a number with two zeros behind it; he has a group of a hundred.  Likewise, a group of ten multiplied hundreds of times will not yield some number with three zeros tacked on the back, but rather some number of thousands.

When language is re-introduced to the process of mathematical calculation, the numbers have a relationship that can be understood and communicated.  When these relationships are kept in mind, errors are more easily avoided and easier to correct.

Just today I asked Aelsa, who is multiplying place values on the other side of the decimal, "Are you trying to tell me that this tenth of a tenth is that number of hundredths, but this tenth of a hundredth is larger than that tenth of a tenth?"

It's helpful to have these relationships memorized, so I created a couple of charts to help to remind us.  The first one covers the multiplication of tens with tens to the multiplication of thousands with thousands.  The second one covers the multiplication of tenths with tenths to the multiplication of hundredths with thousandths, using both decimals and fractions.  You can download a copy for yourself here.


from Genesis 26:

4 And I will make your descendants multiply as the stars of heaven; I will give to your descendants all these lands; and in your seed all the nations of the earth shall be blessed;