Multiplication - A Jump Start

WHAT   DO   I   SEE?
         1.   4x4 5x5 6x6 or 7x7 ?

         2.   9 or 2?

         3.   3x7/7x3 or a 1234-Eight?
IF  THE  ANSWER  IS  YES
  First Question:     SUBTRACT 3
  Second and Third Questions: SUBTRACT 1
  NO Questions:     SUBTRACT 2

// standard

You can also press one of the two
"SHIFT" keys to see the next problem.

Click "SHOW ME" to start over as many times as you want.

Click "NEXT" at the top of the page
to go on.

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It may be easier for you to use the number keys across the top of your keyboard instead of using the mouse.

Start over as many times as it takes to get them all right on the first try.

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You can not go to the next level until you get all the answers correct.

You may go back and do
"SHOW ME" or "I'LL TRY"
for practice if you need it.

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If you say each fact OUT LOUD you will learn most of the ones digits in the answers without really trying.

Say them both ways.
Say "Nine times four is Thirty Six" and also say "Four times Nine is Thirty Six" for instance.

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Notice that
the difference between the red number
and the tens digit

is
1
for all the 2 facts.

The tens digits are all ONE.
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Is this a 9 fact?

Is this a 2 fact?

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Get used to asking yourself these questions in this order.

Is it 4x4 5x5 6x6 7x7 ?

Is this a 9 fact?

Is this a 2 facts?

You will soon understand why.

//end eleventhPageA

See the pattern!
The white digits are "Subtract ONE"

The tens digits for facts with 9
1 2 3 4 5 6 7 8

The tens digits for facts with 8
1 2 3 4 4 5 6 7

The tens digits for facts with 7
1 2 2 3 4 4 5 6

//thirteenthPageA

Remember!

First look for problems with 9.

Then look for problems with 2.

Finally look for problems with

8 times a number UNDER 6.

//end of fifteenthPagea

Remember!

If there is no 3 or 8
in the problem
you can click NO!

//end of fifteenthPagea

THINK ABOUT THIS!

Except for facts with 2

All 'Subtract ONE facts must have

7 or 8 or 9

//end of sixteenthPageA

Ask these questions in this order
for each problem?

Is is a 4x4 5x5 6x6 7x7 fact?

Is it a 9 fact?

Is it a 'low' 8 fact, one of the 1234-Eights?

Is it a 'high' 3 fact like 3x7, 3x8 and 3x9?

//end of eighteenthPageA

Good Luck !

Notice that if
ONE
of the numbers is even
then the answer has to be even.

//used as a blank for several pages

Don't rush and stay calm!!

Run through your questions:

One of the 4x4 to 7x7 group?
9?
2?
3 or 8 where one number is above 6 and the other below 6?

Can I swap a little number for a big number?
Have I used two swaps?

The little red numbers are the partners. Multiply them and you get the little blue number. The little blue number and the correct ones digit are the same!

Difference Meter

-1

-2

-3

7x8
8x7

EVEN

ODD

YES

Multiplication - A Jump Start
Copyright 2010 by David W. Billheimer

Dedicated to the staff and students of the Miller Elementary School in Waldoboro, Maine and especially to Mrs. MacCaffray and Mrs. Mayhew and their fourth grade classes of 2009-2010.
Teachers and parents, please click above and read "Notes."

Please email any comments or suggestions to MultiplicationJumpStart(at)gmail.com.
Use @ instead of (at)

Here are a few suggestions:
1. This site is based on some very little known facts about the multiplication tables. If you ignore the 1's. 10's and facts without a tens digit there are only 32 facts to learn. This figure assumes that "turn-around" facts such as 3x7 and 7x3 are really one fact and can be taught at one time. The tens digit in the answer to every one of these 32 facts is 1, 2 or 3 less than the smaller number being multiplied. There are 4 facts that require subtracting 3: 4x4 5x5 6x6 7x7. There are 16 facts that require subtracting 1: all facts in the 9's and 2's, the 8's up to 8x5, and 7's up to 7x3. All of the other 12 require subtracting 2. It is a simple as that. Take a few minutes to try this approach out yourself so that you believe it is true. You probably are wondering, as I am, why we were not shown this in grade school. I have yet to meet a teacher or administrator that knew it.

2. Encourage students to say each fact OUT LOUD as they progress through this program. Although the initial emphasis is on learning tens digits in the answers, experience has proven that students who do say every fact out loud usually learn most of the ones digits without really working on them.

Reading the instructions for each level with the students is also important. Experience has shown that students tend to jump right to the 'action', often missing important tips and concepts.

3. You may want to review what multiplication is all about and to go over the 1's, 10's and single digit facts before children begin. Also make sure they understand what a tens digit is and where it is found in a product.

4. 'Rules' that are given do NOT apply to the above mentioned facts. When we say that all facts with 2 have 1 for a tens digit in the answer, this does naturally not apply to 2x10. Experience has shown that children easily accept this and usally don't even notice it.

5. Tens digits for some of the groups of facts mentioned in suggestion (1) above can actually be learned without even having to subtract 1, 2 or 3. The group 4x4 5x5 6x6 7x7 have tens digits that go 1, 2, 3, 4 in that order. The tens digits in the lower facts in the 8's (if you include 8x2) also go 1, 2, 3, 4 just as the first four tens digits in the 9's do. Make sure children see these short cuts. Of course,they don't have to use them.

6. Many students may already have learned a 'nines trick' by which you can determine the tens digits in the 'nines' by subtracting one from the smaller factor. Build on this knowledge.

6. Facts in the 2's all have the same tens digit, since they are all in the 'teens'. The tens digits in the 3's from 3x1 to 3x9 go 000111222. Some children can benefit from seeing this pattern.

7. Point out to children that they really don't have to memorize which facts are 'subtract two' facts, since they are simply all the facts that are not in the 'subtract three' and 'subtract one' categories. If you feel it will help, encourage children to spot the facts that are not squares and JUST involve 4, 5, 6, and 7 are six of the twelve 'subtract two' facts.

8. If children are having trouble with this approach, PLEASE help them learn NOT to just guess or make 'stabs' at the tens digits. Experience has shown that if they systematically ask themselves these questions, the answer will come easily: Is it 4x4, 5x5, 6x6 or 7x7? Is there a 9? Is there a 2? Is this one of the 8's up to 8x5 or the 7's up to 7x3?

9. The ones digits are introduced after all the tens digits. They are taught using another rarely noticed fact about the multiplication tables. Tens complements can be substituted for numbers. Two numbers are tens complements of one another if they total 10. Eight is two's complement and two is eight's complement, for instance. There are two ways to use tens complements to get the ones digit in an answer.

The first is to substitute their complements for both of the numbers being multiplied. For example, you can multiply 3x2 instead of 7x8 and you will get the correct ones digit.

The second is to substitute its complement for just one of the numbers, multiply the other number times that complement, and then use the complement of the ones digit you get as a product. For example, to multiply 3x8 you can multiply 3x2 instead. You get 6 and the desired ones digit is the complement of that 6.

In either case you must make TWO substitutions.

10. This jump start approach to learning the multiplication facts has been used with success with children from the end of second grade to sixth grade. I provides a child in the upper grades with a way to 'start over' if he or she has bogged down learning the multiplication facts. The fact that it is NOT JUST MORE OF THE SAME OLD THING makes it particularly effective at times.

11. This approach is not intended to replace memorization of the facts. It just seems to get the process going faster and tends to give children a feeling of 'empowerment' that goes a long way. Don't expect overnight results, please. Encourage children to take their time and master facts as they go along, re-doing levels several times. There is a feature built into the program that encourages that: Students can not move on to the next section until they can get all the tens digits correct with no errors. Only then will the "NEXT" button work. For your convenience there is a way to over-ride this block. Press the letter "J" before pressing the Next button. Press "K" to reset the block.

12. To make things easier there are alternative keyboard strokes that can be used instead of clicking the mouse. Either "SHIFT" key will work instead of clicking "NEXT PROBLEM". The number keys at the top of the keyboard, above the QWERTYUIO row, can be used to enter answers instead of clicking.

13. This website is intentionally lacking in graphics. It is entirely contained in one small file which can be viewed with your browser OFF-LINE so that children can use it without your worrying about their using the web. If you have questions on how to do this let me know by email.

14. Email me any comments, questions or suggestions you may have at: multiplicationjumpstart(at)gmail.com. (Substitute @ for the 'at')
This is version 10.05.

Can you subtract 1 from 6 or take away 2 from 3?

Of course you can! It's easy!

Learning the tens digit in all the multiplication fact answers is just as easy, because subtracting
1, 2 or 3 is all you have to do!

Interested?
Click on NEXT> to find out more.

We find out if we have to subtract 1, 2 or 3 by looking at the fact and asking simple questions.

The first question is always "Is this 4x4, 5x5, 6x6 or 7x7?"

4x4=16 5x5=25 6x6=36 7x7=49

If it is one of these four we just
SUBTRACT THREE
from the multiplied numbers to get each tens digit.

These are the ONLY 'Subtract THREE' facts.

Click on the "SHOW ME" button.

See how subtracting 3 from the red number gives you the tens digit in the answer below every time.

To see another fact
click "NEXT PROBLEM."

Click on "I'LL TRY"

See if you can figure out the tens digit.

Subtract 3 from the red number and tell the computer the answer.

Tell it by clicking on the red number on the bar.
Click "NEXT" or hit a "SHIFT" key to see the next problem.

Now you can test yourself to see if you are ready to go to the next level.

"TEST ME" is very much like
"I'LL TRY" only this time you don't get to try an answer again.

If you make a mistake you have to click "TEST ME" again and start over.

To get the tens digit for all the rest of the facts

Just subtact 1 or 2 from the smaller number you are multiplying!!!

We will learn to use questions to find the "Subtract ONE" facts. When we can spot these "Subtract ONE" facts, the rest will be easy because all of the HAVE to be the "Subtract TWO" facts.

(Since you already know the 0's, 1's and 10's facts and 2x2=4 2x3=6 2x4=8 and 3x3=9
we're not going to waste time on any of them. We will just work on
any other facts that have two digits in their answers.)

The first "Subtract ONE"
questions is:"
Is there a 9?

For all problems with a 9 in them, just subtract 1 from the smaller number every time.

The next "Subtract ONE" question is very easy, too:

Is there a 2? The funny thing here is that the smaller number is always the 2
so the tens digit is always a 1.

ALL the facts with 2 are "Subtract ONE" facts

from 2 with 5 up to 2 with 9.

The second "Subtract ONE" question is:

Is there and 8 and a number littler than 6?

8 x 2     8 x 3     8 x 4     8 x 5

2 x 8     3 x 8     4 x 8     5 x 8

Let's look at the facts with 2.

They are easy !

The tens digit for all of them is

ONE

Try all the 9's and these 2's mixed together.

Notice that we are doing both of each pair of 'turn around' facts at the same time.

Why learn 3x9 and then later learn 9x3 when they really are the same thing?

See how you do when we try these same 9's and the 2's, but with the "Subtract THREE" facts mixed in.

Have you noticed that the tens digits
for the "Subtract THREE" facts go

1   2   3   4

There are only four other "Subtract ONE" facts.

7 and 3     8 and 3    8 and 4    8 and 5

It is best to just memorize these four,
but if you need help spotting them, here is an easy 'trick' :
Ask yourself
Is there a 3 or an 8?
If there is a 3 or an 8, start counting with the SMALLER number and
count up to the bigger number.
For 8x5 you would count "five, six, seven, eight".
For 3x7 you would count "three, four, five, six, seven".

Notice that you PASS SIX when you count these.
If you find a six on the way as you count, say to yourself "I found ONE so I'll "Subtract ONE".

Problems with a 3 or an 8 with a 6 between the numbers are "Subtract ONE" problems.

The first half of the 8's are also Subtract ONE facts.

 8 x 2    8 x 3   8 x 4    8 x 5

They are just like all the 9's and 2's.

Just subtract ONE from the smaller number.

8x2=16 8x3=24 8x4=32 8x5=40
Notice that the tens digits go up nicely from 1 to 4, just like the tens digits in the 9's and the tens digits in the 4x4, 5x5, 6x6, 7x7 group.

(Earlier we learned that all facts with a 2 are Subtract ONE facts, so 8x2/2x8 isn't really a new Subtract ONE fact. Only 8x3, 8x4 and 8x5 are new.)

Here are all the facts with a 8.
Most are "SUBTRACT ONE" facts.

All 8 facts below 8x6
and 8x9/9x8

The only ones that aren't are
8x6/6x8 8x7/7x8 8x8

If you see a 'SUBTRACT ONE' fact
click YES or NO.
(You may type 'Y' or 'N' instead)

Now we'll do the same thing except that we will look through ALL 32 facts.

Check first for a 3 or an 8. If you find one THEN check to see if you would PASS SIX when counting from the smaller to the larger number.

The last "Subtract ONE" question is simple:

Is it 7x3 or 3x7?

Subtract ONE from the smaller number 3 to get the tens digit.
7x3=21
There is only 1 fact for this question. You subtract 1 to get the tens digit. And the ones digit is also a 1!
A Neat Trick! 1 Fact, Subtract 1, then Add on a 1.

Let's Review

SUBTRACT ONE FACTS

All facts with 2

All facts with 9

Facts with 8 up to 8x5

The one fact 7x3/3x7

Try all the 'Subtract ONE' facts
again but with the

four

"Subtract THREE" facts

mixed in.

ALL THE REST OF THE FACTS ARE 'SUBTRACT TWO' FACTS !

                                           2   3   4   5   6   7   8   9
                                        --------------------------------
                                   9    -1  -1  -1  -1  -1  -1  -1  -1                
                                   8    -1  -1  -1  -1  -2  -2  -2                                   
                                   7    -1  -1  -2  -2  -2  -3      
                                   6    -1  -2  -2  -2  -3           
                                   5    -1  -2  -2  -3              
                                   4         -2  -3

NO 9's or 2's
NO 8's below 8x6 NO 3x7 / 7x3

Do you think you could spot one of these "Subtract TWO" facts if it was mixed in with the facts you know about?
Suppose you saw 6x8. You would ask yourself the questions:

           Is this one of the 4x4 5x5 6x6 7x7 group ?      NO  
           Is this a fact with 9 or a 2 ?      NO
           Is this a fact with 8 up to 8x5 ?      NO 
           Is it 7x3 or 3x7 ?            NO

If all the answers are 'NO' is has to be a "Subtract TWO" fact! Subtract 2 instead of 1. That's all there is to it!

Here are all the "Subtract ONE" facts with 3 "Subtract TWO" facts mixed together.

If the answers to the
"Subtract ONE" questions
are all NO then you have found
one of the three "Subtract TWO"
facts

Try to figure out what is different about these
three new facts.

I bet you found them easily!

3x4=12 3x5=15 3x6=18

We know that 9x3/3x9, 8x3/3x8 and 7x3/3x7 are Subtract ONE facts.

These are the rest of the facts with 3!

The next few pages will just get you used to subtracting 2.

Remember the numbers in the "Subtract THREE" facts?
4, 5, 6, and 7

Now we'll look at some facts with
JUST these 4 numbers.

Those that are not
"Subtract THREE" facts are all
"Subtract TWO" facts.

What is different about the new "Subtract TWO" facts ??

They had our old friends
4, 5, 6 and 7
times EACH OTHER and not times themselves!

Now we will mix together ALL the facts we've seen so far.

Remember "Subtract TWO" facts have
NO 9's or 2's
NO low 8's NO 3x7 / 7x3

The last three facts are the facts with 8 we haven't seen!

8x6=48 8x7=56 8x8=64

(8x9 is NOT one of these facts. We know already that all facts with 9 are "Subtract ONE" facts.)

I think you will spot them with not trouble.

Here's a little more practice with
just "Subtract TWO"

Decide WHY each one of them is
"Subtract TWO"
and not
"Subtract ONE or THREE"!

Take your time and also think about what the ONES digit is.
If you have been saying the whole facts OUT LOUD the way we asked you too, I bet you know most of the ONES DIGITS by now!

Now more practice with the Difference Meter turned off and without the smaller number being red.

Think about the ONES digits !"

What you have been waiting for!

ALL THIRTY-TWO FACTS

You must
ASK THE QUESTIONS !

If you do, you can come up with
the tens digit in
any multiplication fact
from 2x5 to 9x9 !!!

GOOD LUCK!

C O N G R A T U L A T I O N S !

You made it to the end of the Section on the Tens Digits !

Now we will start working on the Ones Digits.

If you have been saying all the facts OUT LOUD as you were asked to,
you will already know many of these Ones Digits.

First let's see how many of them you do know.
Get a pencil and a piece of paper before you go to the next page. Make two columns on the paper, one for the facts you get right, and one for the facts you need to work on.
Write down each fact in the right column as you go through them.

Good Luck !
You will be surprised how many you know.

Keep track of the ones you know and the ones you need to work on.

Give the ONES a try!

We'll start working on
ONES DIGITS

The first thing we will learn can be a big help.

The only way to get an ODD answer is to multiply TWO ODD NUMBERS.

The odd numbers are:
1 3 5 7 9

Practice until you get good at it!

Facts with a 5

If the other number is even the ones digit is a ZERO.

If the other number is odd the ones digit is a FIVE.

6 + 4 7 + 3 8 + 2 9 + 1

If you're working on ones digits these pairs that add up to 10

A L M O S T S E E M M A G I C A L !

The 2 numbers in each pair can take each other's places!

HERE IS THE 'TRICK'
When you're multiplying you can use
the little number in one of these pairs in place of its big partner.

For Example:
If you are multiplying 8x7 you can multiply their little partners instead.
8's partner is 2 and 7's partner is 3. So you multiply 2x3 instead of 8x7.

If you use a little partner for both big numbers you will always get the right ones digit !!
8x7=56 2x3=6

6 ---- 4 7 ---- 3 8 ---- 2 9 ---- 1
Let's practice figuring out ones digits by using little partners in place of big partners.

If the problem is 6x8, say to yourself "I'll multiply using 6's and 8's little partners instead." Then multiply their partners, 4 and 2. You will get 8 and it will also be the ones digit for 6x8!

Facts with 7
times an EVEN number

Multiply this other EVEN number by 2. Use the ONES DIGIT you get.

7x4 = 28, since 2 times 4 is 8.

7x6 = 42, since 2 times 6 is 12.

7x8 = 56, since 2 times 8 is 16.

Notice that you are expected to
learn the 2's yourself!

You can also use the littler partner for ONLY ONE of the numbers, BUT
YOU STILL HAVE TO USE TWO PARTNERS !!

Say you're doing 8x3. You can
use 8's little partner 2 instead of 8 and not change the 3.

2  x  3  =  6
You have to use the partner of THAT 6 to get the right answer.

6's partner 4 IS the right answer.

Either way, you have to use TWO PARTNERS.

6 ---- 4 7 ---- 3 8 ---- 2 9 ---- 1
Let's practice figuring out ones digits by using a smaller partner in place of JUST ONE number.

Really spend some time on the "SHOW ME" step and make sure you understand what is happening.

To help you the partners have the same color in "I'll Try."

Now let's try the 'trick' on all 32 facts.

Here are some things to remember:

1. Don't bother with the 'trick' if you know the fact.

2. The partners always add up to ten.

3. You always have to use the partners TWICE.

4. You use the partners TWICE if you swap both numbers.

5. You have used the partners only ONCE if you make one

just one swap before you multiply.

You have to swap again. You have to

swap the ones digit in the answer you

get with it's partner.

6 ---- 4 7 ---- 3 8 ---- 2 9 ---- 1
Try the ones digits in all 32 facts.

Remember that you have to use the Partners TWICE.

Don't miss the hints in the "I'll Try" step.

Now try doing BOTH
tens and ones digits.

Do not skip over "Show Me".
It reviews how to do each fact.

Don't just stare at a problem or guess. Start asking yourself the questions about the facts right away.

YOU HAVE ASK THE QUESTIONS FOR THIS METHOD TO WORK WELL.

4x4 5x5 6x6 7x7
9x x9
Low Eights
All 2 less
All 3x
All 8x / x8
All 32 Facts
2 x and x 2
16 Subtact 1 Facts
4 5 6 7 combinations
hi 3's
4x4 5x5 6x6 7x7
7x3
5's
5's
even 6's
High Numbers for 2 complements
high+low for 2 complements