Multiplication Chart
The whole table, plus the one picture that turns it from a list to be memorised into a shape you can read: the rectangle every product is counting.
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A multiplication chart printed on paper answers one question: what is 7 times 8. Your phone answers that too, and faster. What a chart can do that a calculator cannot is show you where the answer lives - and once you can see that, the table stops being ninety-odd facts to memorise and becomes a shape with a handful of awkward corners.
Point at any square below.
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Point at any square. The shaded block is what the answer counts - that is why the table is symmetric about its diagonal.
Every answer is a rectangle
Hover 7 and 8 and look at what lights up. It is not just the crosshair through your square - it is the whole block from the top-left corner down to it: seven rows of eight squares, and the answer is how many squares that block contains.
That is what multiplication is. 7*8 is not a fact about the symbols 7 and 8; it is the number of things in a 7-by-8 arrangement. The chart is a picture of every such arrangement at once, filed by its two side lengths.
Two useful things follow immediately.
The table is symmetric, so there is half as much to learn
Turn a 7-by-8 rectangle a quarter turn and it is an 8-by-7 rectangle. Same squares, same count. So the answer in row 7, column 8 must equal the answer in row 8, column 7 - and it does, everywhere.
Run a finger down the diagonal from the top-left: 1, 4, 9, 16, 25 … Everything below that diagonal is a mirror image of everything above it. You are never learning 7*8 and 8*7 as two facts. You are learning one fact and reading it two ways round.
Six facts are hard. The rest are free.
Strike out the rows that cost nothing:
| row | why it is free |
|---|---|
| ×1 | the number itself |
| ×2 | double it |
| ×5 | half of ten times it, so it ends in 5 or 0 |
| ×10 | write a zero on the end |
| ×11 (to 9) | repeat the digit |
| ×9 | one less than the ten times, and the digits always add to 9 |
Now strike out everything below the diagonal, because it is the mirror of something above it. What survives is this: 6*6, 6*7, 6*8, 7*8, 8*8 and 12*12.
Six facts. That is the entire hard part of the times tables, and people spend years reciting the other ninety.
Reading a chart is not learning it
This is the part worth taking seriously. Looking at a completed chart feels like studying and is close to useless, because recognising an answer is a much easier task than producing one. You will recognise 56 as belonging to 7 and 8 long before you can produce it cold.
So switch the chart above into quiz mode. The grid empties, you pick a square, and you have to answer before anything is revealed. The facts you miss are the short list actually worth drilling - and it will be a short list, probably the six above.
Try one
That one is on the hard list for almost everybody, which is why it is here rather than 12*2.
Where the chart stops being enough
The grid is a lookup table for products, and products are only half of what you do with two numbers. The moment a question asks which numbers divide a number - what pairs multiply to give 24, say - you are reading the chart backwards, and that is a different skill with its own name: factors.
The diagonal is worth one more look before you go. Those numbers - 1, 4, 9, 16, 25, 36 - are the ones you get when a rectangle happens to be a square, and they turn up constantly in algebra afterwards.
Common questions
- How do I read a multiplication chart?
- Find your first number along the top row and your second number down the left column. Slide across from one and down from the other; where they meet is the answer. Because multiplication does not care about order, you get the same answer whichever number you start with - which is why the chart is symmetric about its diagonal.
- Why is the multiplication chart symmetric?
- Because 7 rows of 8 and 8 rows of 7 are the same collection of squares, just turned a quarter turn. Any rectangle can be read either way round, so the answer in row 7 column 8 has to match the one in row 8 column 7. That symmetry roughly halves the number of facts there are to learn.
- Which times tables are hardest to learn?
- Once you drop the ones you get free - anything times 1, 2, 5 or 10, and everything you have already learned in the other order - only about six facts are genuinely hard: 6x7, 6x8, 7x8, 7x9, 8x8 and 12x12. Everybody finds roughly the same ones difficult, and drilling those six is far more efficient than reciting the whole grid.
- Should I use a 10x10 or a 12x12 chart?
- Use 12x12. The extra two rows cost almost nothing to learn - the 11 times table is nearly free up to 11x9, and the 12s show up constantly in time, money and measurement. Schools in several countries stop at 10, which leaves a gap you notice later when a question asks for 12 of something.
- What is the fastest way to memorise the times tables?
- Test yourself rather than reread. Looking at a chart feels productive and teaches very little, because recognising an answer is much easier than producing one. Hide the grid, ask yourself a fact, and only then check. Quiz mode below does exactly that, and the facts you miss are the short list actually worth drilling.
Now do it yourself, properly
Reading someone else's working is not the same as being able to do it. The Coddy math course puts you on a board that checks every move you make, so you find out where you actually stand.
Want to run this on your own numbers?
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