### What is the Multiplication Table?

*It is a collection of single-digit number multiplication facts.*

The multiplication facts are equations – as for example m x n = p which states that skip counting by adding m things n .times to gives you a total count of p …….Both m and n are single digit numbers from 0 to 9.

### Why learn the Multiplication Table?** **

*You will need to use it all the time.*

It will be the simplest way for doing arithmetic (add, subtract, multiply, divide) with large numbers. We learn to multiplymuch larger numbers (say 56 x 78 = ? by using simple rules * and* the one-digit multiplication facts we should remember fluently.

Some animals and very primitive people can count as high as: 1, 2, 3, many.

### How large can my numbers get? How small?

Students can learn shortcuts that help them count very large or very small numbers.

Numbers as high as the numbers of stars in a galaxy – or as small as the size of the most elementary particles that make up atoms, molecules and ultimately the substances we can directly see.

And that is awesome!.

You will be able to perform mental calculations with larger numbers quickly and accurately. You will never have time for finger-counting when adding 1023 + 1023 for example — or multiplying 53 by 27 on 10 minute tests – generally losing your count on the wa. Instead, you will be able to use these memorized facts fluently and confidently in simple procedures.

Fluent recall of the Multiplication Table will be one of the most important skills you will ever learn. Remember – EVERYBODY likes to use shortcuts.

### Shouldn’t we just memorize the Multiplication Facts by rote?

*No – you should first understand each Multiplication Fact – only then should you commit them to memory. Memorizing by rote is a short-cut to help you quickly recall each fact.*

We could simply memorize these facts by rote to get some working knowledge.If a multiplication fact is not understood it is not integrated in your store of knowledge. Unless there is understanding – integraation – facts will be easily forgotten.

But if we clearly understand why the facts are true we can better remember and use them.

So we will need to recall what number symbols are, how we count, how we add and multiply, what the equal sign means, and why the commutative law of multiplication holds.

There are 38 such statements of fact that need to be simply memorized once you understand why they are true…

### What are number symbols and what do they mean?

*There are ten number symbols – also called “single digits”:*

We use number symbols to count any like items in a collection such as corn plants in a field, dogs in kennel or dollar bills in a wallet.

Let S be a collection of letters included within brackets as shown: S = {c x o k}. We use number symbols to count how many of each letter there are in S.

- “0” means “zero”count – Given S = {x x x x x x} – S contains 0 (zero) – or no – letter o’s
- “1” means a count of “one” – Given S = {x o o o }, S contains 1 (one) x.
- “2” means a count of “two” – Given S = {x o x o o o o |, S contains 2 (two) x’s.
- “3” means a count of”three” – Given S = x x x o o o}, S contains 3 (three) o’s and 3 o’s
- “4” means a count of “four” – Given S = {x x x o o x}, S contains 4 (four) x’s.
- “5” means a count of “five”- Given S = {x x x x x o o o }, S contains 5 (five) x’s
- “6” means a count of “six” – Given S = {o o o x x x x x x o o}, S contains 6 (six) x’s
- “7” means a count of “seven” – Given S = {o o o o o x x o x o}, S contains 7 (seven) o’s.
- “8” means a count of “eight” – Given S = {c c b v e c c c a z c c c x o}, S contains 8 (eight) c’s.
- “9” means a count of “nine” – Given S = {a a a a a a a a a c c}, S contains 9 (nine) a’s.

### Isn’t there a more compact way to refer to those 38 equations in the Multiplication Table?** **

*Yes – we can using letters to represent single-digit numbers.*

We use a pattern for multiplying two numbers using a letters to represent each number. In general we can refer to any item in the Multiplication Table by writing the question as: a x b = b x a = ? – where we let “a” and “b” be any single-digit numbers we choose from 0 to 9. Once picked we fix them in our discussion.

For example, to multiply two single-digit numbers we can use the letters “a” and “b” and write the multiplication statement as: a x b = c. Each letter stands for one of the single-digit numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

An example will help. If we let (for now) “a” be “7” and “b” be “6”and “c” be the product we have the statement 7 x 6 = 6 x 7 = c before us and we ask what “c” is

Notice that when in using letters to stand for numbers we can write one compact statement rather than write out all 38 questions or statements. This is another useful shortcut in our thinking.

### What does the equal sign mean in these equations?

In — a x b = b x a = ? — t*he equals sign “=” means (is shorthand for) “the same number on the right is the same as as the number on the left”.*

If we let a = 5 and b = 7 then 5 x 7 = 35 = c on the left and 7 x 5 = 35 = c on the right – and that 35 =35 — which is true.

### How do we multiply two single-digit numbers”?

*Recall that multiplication is just a shorthand for addition – or skip counting.*

### What is the commutative rule of multiplication and why is it true?

*It doesn’t matter what order we use in adding. It means that m x n = n x m.*

Notice that we are using the digit symbols 1 through 9 for Ones and place value through Ones, Tens and Hundreds since we need to count that high.

You will need to mentally count by Ones through Nine, skip count by Tens (Ten, Twenty, …, One Hundred Forty) (to 140 since 12 x 12 = 144) and extend the skip count by any needed Ones (for example, “21” equals the same counting number we get by skip counting to twenty and then counting by Ones one more time.).

Multiplication is really skip counting by one factor as many times as given by the other factor. So 3 x 2 means skip counting by 2 Three times as in 2, 4, 6.

The “=” sign means “equals” and says that the number on the left is the SAME number on the right of that sign. Be clear about why we don’t need to multiply by Zero – ANY number multiplied by Zero ALWAYS equals Zero. Be clear about why we don’t need to multiply by One in our table – ANY number multiplied by One ALWAYS equals that same number. Be clear how you can convince yourself and anyone else why each Answer is true.

Silently read the Question to yourself – “Two times Three equals Three times Two equals Six” Do several at a time – get it right for all 55 Items

to yourself Either don’t look at the answer – or re-size your window to hide the “ANSWER” column.

FOR EACH TIMED REPETITION: Note the clocks at the beginning and end of your practice set. You may wish to wait for the time to get to a whole minute + 0 seconds. Remember that minute and get ready to practice the set. After finishing, note the minutes and seconds. Subtract the start time from the finish time to get the repetition time.

Your ultimate goal should be a comfortable 3 minutes elapsed with ALL CORRECT – this represents a fluent, accurate 3 sec/question answer rate – and what you need to do to commit it to memory.

And *NO – this is definitely NOT simply “parrot” or “rote” learning – parrots only mimic nor do they need to balance a checkbook or create new mathematics. *

*You are now ready to commit the multiplication table to memory.*

*You are now ready to commit the multiplication table to memory.*