This work was gradually introduced as we continued to work with addition and
subtraction.
    I pointed out that the even numbers could be made with two equal groups, and this
was what multiplication and division were about, equal groups, sets, or parts. It was
made clear that this was a new, but similar, kind of game with different rules.
    Notice that I have switched from using 'parts' to using 'groups' most often.

    Addition and subtraction could work with any two groups or parts, but multiplication
and division were games we played with equal groups or parts. We now needed to
know the size of just one group, and the number of groups in the whole.
    As before I tried to keep the work simple and consistent in the beginning.

    We might start by making up sentences about the even numbers from 6 to 18, using
what we already knew. Blocks were used to demonstrate the equal groups.
    The purpose was to establish the language of multiplication and division, the
meaning of 18 -:- 9 = 2 and 2 x 9 = 18. (-:- will have to do for a division sign)
    The language I used for the above was '18 made into groups of 9 gives us 2
groups', and '2 groups of 9 make 18'. I used this and similar words for some time until
they were used to looking at the -:- and x symbols and thinking of those ideas.

    Much later I introduced the words 'divided by' and 'times'.

NOTE:
    You may wish to reverse the sentences below and the language used because you
disagree with the order, and that is fine as long as you are consistent.

    For me, I taught the children that the sentences were of this form because I liked the
simplicity of the words for the symbols x (groups of) and -:- (made into groups of):

    WHOLE -:- (made into groups of) SIZE OF GROUPS = NUMBER OF GROUPS

    NUMBER OF GROUPS x (groups of) SIZE OF GROUPS = WHOLE

    When this language was becoming familiar we started looking at each whole in
turn, from 2 to 6.

    It soon became clear to all that because this game had to have equal groups we
were not going to have as much to say about certain numbers, and certainly not as
many pairs of numbers as we had for addition and subtraction.

    To promote a clear break from addition and subtraction sentences I started early,
but not immediately, to introduce some new words: factors, and multiples. These
made a real distinction between the pairs of numbers that worked together to make a
whole in addition and subtraction. They were simply parts of a whole.

    A real point had to be made here that we were counting different kinds of things.
2 could tell how many blocks there were, and it could tell how many groups there
were. 1 could count how many blocks there was in each group.

    This was something that had to be constantly stressed.

    Then we did the same for a whole of 3 blocks.
    We recorded these as 2 -:- 1 = 2 and 3 -:- 1 = 3 .
    Next, we put the groups together to make the original wholes and recorded them as
    2 x 1 = 2 and 3 x 1 = 3 .

    At this point we reviewed what equal meant, how it told us that there would not be
a group that was different in number.

    Then I introduced the thought that we could have just one group, and there would
not be a group that was different, so this could still fit our game of multiplication and
division. (Sometimes a child would suggest this before the subject had to be raised.)

    With this in mind, we looked at 1 group of 2 and 1 group of 3, and recorded them.
Now we had 2 -:-2 = 1 , and 3 -:- 3 = 1.

    At this point we had (using 'groups of' for x and 'made into groups of ' for -:- ) :

	2 -:- 1 = 2		3 -:- 1 = 3		or		2 -:- 1 = 2		3 -:- 1 = 3
	2 x 1 = 2		3 x 1 = 3				2 -:- 2 = 1		3 -:- 3 = 1
	2 -:- 2 = 1		3 -:- 3 = 1				2 x 1 = 2		3 x 1 = 3
	1 x 2 = 2		1 x 3 = 3				1 x 2 = 2		1 x 3 = 3

    After this we went back to 1 and recorded it as 1 -:- 1 = 1 and 1 x 1 = 1 .

    Next came 4, and then 5 objects. I won't write out the sentences for them, but we
started to record what we had learned just as we had for addition and subtraction:

1	2	3	4	5
1 | 1	1 | 2	1 | 3	1 | 4	1 | 5
	2 | 1	3 | 1	4 | 1	5 | 1
			2 | 2

    This time we had the numbers underneath the whole as the number in a group on
the left, and the number of groups on the right. It was obvious that these numbers
exchanged places or jobs in the multiplication and division operations.

    It was understood that factors could be the size of the equal groups, or the number
of equal groups, and sometimes both. Factors could change jobs.

    Then, we reviewed the zero rule for addition and subtraction:
    If 0 is one part of a whole, then the whole is the other part.

    It was easy then to discover a similar rule for working with a factor of 1in
multiplication and division:
    If 1 is one factor of a whole, then its related factor is the whole.

    We could see here that our work was getting easier, just as in addition and
subtraction.

    Next we had to discover what happened when we tried to work with 0 in
multiplication and division.
    Could we make 2 into 0 groups? It didn't make sense.
    Could we make 2 groups of 0? Sure, but we would then still have 0.

    After trying this with a few numbers it became obvious that if zero was a factor, the
whole had to be zero, and any number could be the related factor.

    Multiplication and division were not as difficult as they seemed at first.

    We did not have to worry about factors of 0, 1, and the whole itself.

    Out of all the wholes from 0 to 5, only one had something special to remember, 4.

    Then we worked on 6. We checked its behaviour with groups of 1, and 1 group.
They followed the rule for 1.
    We made 6 into groups of 2, and got 3 groups. We made it into groups of 3, and
got 2 groups.
    Working with objects, it was obvious we couldn't use 4 or 5 as factors, just as 3
couldn't be a factor of 4. Many numbers inside a whole could be ignored.

    We could also see that when the related factors were different they still could trade
jobs, or positions in the number sentences.

    We could write what we had to memorize as:

0 - rule	1 - rule	2 - rule	3 - rule	4	5 - rule	6
				2 | 2		2 | 3

    It was time to introduce some new and useful words.
    We reviewed how 1 was known as a unit, and still would be.
    We then looked at 2, 3, 5, and 7.
    They all worked the same way in this game of equal groups.
    The children were told that any numbers like this are called prime numbers.

    Prime numbers have no important factors, just 1 and themselves.

    4 and 6 were not prime numbers. They were called composite numbers.

    Composite numbers have factors besides 1 and themselves.

    Composite numbers were the ones that were going to give us different facts to
remember. Composite numbers could be broken into special equal parts.

    We worked on 8, and then 9 objects. With this we could sum up our findings as: (fig.4)

	4	6	8	9
	2 | 2	2 | 3	2 | 4	3 | 3

    These facts were displayed in the classroom as before.
    With this we could go back to the worksheet frame that was used for addition and
subtraction, on p.12, substitute x for + and -:- for - , and write all the separate
important sentences for 4, 6, 8, and 9 on one sheet with space left over.
    (The picture boxes were used to draw the two groupings (one for 4 and 9) the
factors allowed. For example, with a whole of 6, it was [oo] [oo] [oo] and [ooo] [ooo].)

    Try that with addition and subtraction! This makes multiplication and division seem
so easy, and again, it is important that children think the work is easy. Another way
of saying this is they have confidence in their ability to handle the work to come.

    Now it was time to work with questions. The formation of questions was presented
just as for addition and subtraction; a number was missing from a normal multiplication
or division sentence. Having done similar work before, it was easy to see that 6
questions each could be made up about 4 and 9, and there were 12 each for 6 and 8.

    Expanding this to 10 was so easy since our hands put the important factors right in
front of us. Hold up your two hands and you have 2 groups of 5. Put your hands
together so that just the fingertips touch and you will see 5 groups of 2 .
    Our hands made this a good point to review the ideas taken so far, and to do a lot
of simple multiplication and division questions.

Geometric Shapes:
    As we paused at 10, it was time to add a little interest to the review by noting some
special arrangements of objects placed in equal groups and fitted together so that the
resulting shape had no holes or pieces sticking out.

    Blocks (cubes) were very useful in showing that 2 groups of 2, placed together in
one way, made a shape we call a square. We called 4 a square number.
    6 could form an ordinary rectangle and 9 could form another square. We called 4
the square of 2, and 9 the square of 3.

    8 blocks would form an ordinary rectangle but, since it was made of 2 squares of 2,
if one square of 2 was set on top of the other we could see the three dimensional
shape we call a cube. We called 8 the cube of 2.

    All the prime numbers were found to only form long rectangles, 1 block wide, and
any number could do this.

    The ordinary rectangles could be looked at two ways to show that :
     &nbsp2 x 3 = 6 and 3 x 2 = 6 , 2 x 4 = 8 and 4 x 2 = 8 , 2 x 5 = 10 and 5 x 2 = 10

    With this little bit if interest added we now went on to investigate the numbers to 18.

    11 was prime, but 12 was a most interesting whole. It had two sets of related
factors, 2 and 6, and 3 and 4. You could make 8 sentences about 12, and ask 24
questions, without using 1 or 12.
    12 formed two shapes, both rectangles, and this gave us the chance to talk about
how easy 12 objects are to package up, compared to 10 which only forms one
rectangular shape.
    We talked about how 12 was called a dozen and many things were sold by dozens.
    We noted the relationship between multiplication and division and fractions of
numbers when we talked about half a dozen.

    This gave us a chance to ask about half of 10 (as plain as a hand in front of our
face), 8, 6, and 4. Fractions also talked about equal groups.
    You could have half of 2,4,6,8,10, or 12 without cutting up units, but not the others.
    We were back to even and odd numbers.
    You could have a third of 3, 6, 9, and 12. You could have a quarter of 4,8, and 12.
You could have a fifth of 5 and 10, and a sixth of 6 and 12. Obviously, 12 was the
number that would divide up in the most ways. It had the most important factors.

    Each number was beginning to have a character of its own!

    We talked about factors having multiples , and the reverse, just like children have
parents and parents have children.

    The multiples of 2 were 0,2,4,6,8,10,12... We were back to counting!

    The multiples of 3 were 0,3,6,9,12... This was easy.

    Multiplication and division were related to counting by set amounts.

    Now I asked for predictions. Which numbers to come would be composite? It was
not hard for many to see that this would be all the even numbers, except for 2.

    Would all the odd numbers be prime? Well, 3, 5, 7, and 11 were, but 9 was
composite.

    Going on from 12, the children were asked to make a prediction for 13. Most
thought correctly that it would be prime. 14 was predicted to be composite because it
was even. Most knew by now that numbers ending in even numbers were even.

    14 proved to be an ordinary composite that only had two important related factors,
2 and 7.

    15 also proved to be an ordinary composite with factors of 3 and 5 to pay attention
to, but, like 9, it was special since it was an odd number.

    Now, another break to review. We made counting lists such as (could go to 15):

0	1	2	3	4	5	6	7	8	9	10	11	12
0		2		4		6		8		10		12
0			3			6			9			12
0				4				8				12
0					5					10
0						6						12

    We discussed the patterns in this set and how they related to multiplication and
division. We counted the steps to get to numbers, such as 10, on different lists.

NB: Take care with this, and similar activities, that the children pay attention
to counting the spaces between the numbers. Many naturally want to count what they
see in such a list, when it is the movement from one to another they should count.

    We also used lines such as the first line above by drawing curved lines to show
skip counting by each factor from 2 to 6.

    We set down all the numbers up to the whole, and then used a curved line to join
the related factors.

    If you put in 0, it joins to no other. 1 always joins with the last number, the whole,
and that is all that happens to prime numbers.
    Composite numbers had more lines, but they never went over the halfway mark,
if they even went that far.

    You might at this point look at the expanded fig.4, now fig.5. (could go to 15 now)

	4	6	8	9	10	12	14
	2 | 2	2 | 3	2 | 4	3 | 3	2 | 5	2 | 6	2 | 7
						3 | 4

    Children should try to be able to look at any of these and work out the sentences
that stem from these very basic facts.

    For instance, 10 made from related factors of 2 and 5 produce, starting from 2 and
moving around the triangle of numbers to the right, 2 x 5 = 10. Starting from 5 and
going to the left, 5 x 2 = 10. Starting from 10 and going to the right, 10 -:- 5 = 2 , and to
the left, 10 -:- 2 = 5 .

    Note that 12 has two such triangles.

    I now introduced another short way of recording these facts as multiple and related
factors: MULTIPLE : FACTOR(S)

    For 12 I would write 12: 2 , 3 , 4 , 6 . For 9 it was 9: 3 .

    This led to a chart such as this with patterns to investigate:

    4: 2
    6: 2, 3
    8: 2,   , 4
    9:  , 3
  &nbsp10: 2,  , , 5
  &nbsp12: 2, 3, 4,  , 6
  &nbsp14: 2,  ,  ,  ,  , 7
  &nbsp15:  , 3,  , 5

    Please remember that all the time this is going on the children are doing questions
relating to the facts. These side trips just allow a chance to review and consolidate the
facts in a more interesting way.

    17 was prime, and 18 was like 12 with four factors: 2, 3, 6, 9. 18 had two ordinary
rectangular shapes and could make two cubes of 3.

    At this point we could play another kind of game with the multiples of factors.

    We wrote on the chalkboard or chart paper the following:

  &nbsp2 -> 0 , 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 , 18 , ...

    If we had started two figure addition and subtraction we may have already played
with 'Casting Out Nines'. (See the last part of this booklet for more about this.)
    One of the tricks here is to add the figures in multi-figure numbers until they are
reduced to a single figure, something that happens to give the same result as taking
out all the nines. In this game, nines are like zeros.

    Under the above line of multiples of 2 we could then write:

    0, 2, 4, 6, 8, 1 , 3 , 5 , 7 , 0, ... Guess what the next would be.

    We went on:
  &nbsp3 -> 0 , 3 , 6 , 9 , 12 , 15 , 18 , ...
       0 , 3 , 6 , 0 , &nbsp3 , &nbsp6 , &nbsp0 , ...

  &nbsp4 -> 0 , 4 , 8 , 12 , 16 , ...
       0 , 4 , 8 , &nbsp3 , &nbsp7 , ...

    In each one we were finding a new pattern and this gave us another interesting
way to review the basic facts.
    You will find that each of the single figure factors has a set of multiples that yields a
unique pattern when 'Casting Out Nines' is used on it, with 9 itself having the most
interesting, and useful, pattern (in my opinion).

    From here we went on to 19, which was prime. Then came 20 which had the first
two figure factor, 10.
    Thinking back to addition and subtraction, could this be an important factor to
remember? No, it was obvious, knowing about place value, that 20 was made of 2
groups of 10, and that also meant it had 10 groups of 2. The only factors we really had
to remember were 4 and 5.
    21 gave us factors of 3 and 7.

    From here on we knew we didn't have to remember something about every
number. Any two figure factor, and its partner, could be worked with another way.

    I will not go over every number up to 81. We went on in much the same way as
already described here. The farther we went, the better the children got at predicting
which numbers we would have to pay attention to and what their important single
figure factors would be.

    They were delighted to find that 81 was the end of the multiples that had really
important factors, just as 18, the reverse of 81, was the end of WHOLES with important
PARTS.

    All this time that we were working towards 81 we played the class game described
on p.18 and did the speed tests as on p.15.

    By the time we reached 81, I was able to give the children a single 8.5 x 11 sheet with
all the important multiples and factors, with the square numbers separated from
the ordinary composites. This will be one of the sheets I'll add at the end.

Odd Bits:
    Since the children will have been dealing with two figure addition and understand
working with the tens and ones separately, questions such as 3 x 22 = __ and
44 -:- 2 = __ could be considered for variety.

    Calculations of the area of simple rectangles gives a concrete purpose for
multiplication. It could be as simple as "How many cubes would there be if they were
set out, in a rectangle, 6 cubes on one side and 4 on the other?" You don't really have
to get into square inches or centimeters.
    "How could 24 blocks be formed into a rectangular shape? Is there another way?"
This might be a question for the more able children.

    Questions involving the price of objects provide another real life source of
multiplication and division situations.

    See the next section for ideas about how fractions can be related to multiplication
and division practice.

GSM, Part 1	GSM, Part 2a	GSM, Part 2b	GSM, Part 4	GSM, Part 5
Home Page	Booklets	Terms	CENTER
Start Gordon's Games	Start Not So Simple Math	Start Not Just Math