Somewhere in the early introduction of numbers, children are going to be
introduced to the idea of combining groups of things to make a larger group; and
taking away one group to leave another group. (That is, you can start to learn to add
and subtract before you know all about counting and place value.)

    This little booklet is mainly concerned with these activities in their simplest form,
and as they are recorded with numerals.

    In my opinion, good math or arithmetic instruction in the early grades will prompt
children to say, "This is so easy! I thought it was going to be hard, but its not!"
    In other words, it's the teacher's job to give as many children as possible a good
feeling about their ability to handle numbers.

    There are times when challenges can be set for the more able students, but in the
main, I believe, beginning instruction is best kept as simple and flexible as possible.
    For this reason the examples in this book will be of the simplest form. The aim will
be to start with something small that can later grow to handle more complex questions.

    Wherever possible, rules will be developed and generalizations made that can
eliminate as much of the drudgery of memorization as possible, and make what is
memorized as useful and flexible as it can be.

    I will not deal with word problems, nor will I talk a lot about manipulatives in the
hands of the children. These are for the reader to deal with.

    Where possible, I will use the simplest words, those already familiar to children, to
describe the functions of numbers as used in addition, subtraction, multiplication, and
division questions. This may offend some readers who know the proper terms for each
number in 68 -37 = 31, but , by ignoring these terms I believe I can better tie addition
and subtraction together, and link them to multiplication and division.

    I will also deal with addition and subtraction at the same time, but as related
opposites. The same will be true for multiplication and division.

   What I hope the reader will see is a simple and flexible structure to which
children can attach the facts they must learn.

    It is my aim to make children see that there is little more of real difficulty to learn
about adding and subtracting 18 things than there is for 2 things, and beyond 18 it is
just a matter of learning methods of using what you already know.

    First, its important to have an understanding of 1 as a unit and 0 as a real amount.
With that one can start the game of addition and subtraction.

    At some point in this game you have to teach these words and their symbols:
    MINUS or TAKE AWAY (-), and PLUS or ADDED TO (+), and EQUALS (=) .

    You may feel most comfortable starting with 3 objects. I liked 1in. (2.5 cm.) cubes.
    I would start out by speaking of these 3 as being the WHOLE . In other words, for
the purpose of this game, 3 is all you've got.
    From the 3 you can take away 1. What has been done now is to turn the 3 into two PARTS, a part, 1, that was taken away, and a part, 2, that is left.

    Now reverse this and make a real point of the concept of doing the opposite.
    Take the 2 that was left and bring back the 1 that was taken away so that 3 appears
again. Emphasise the point that the PART that was left and the PART that was taken
away join to make the WHOLE.

    In other words, if you've got 1 thing and a 2 things, it is just the same as having 3.

    If you have taught the words and symbols, then you can simplify the above to be
       3 - 1 = 2, and 2 + 1 = 3.
    Let the children see these objects clearly while you talk about what you're doing.
    (I didn't give children their own objects until I was sure they knew the game.)

    Notice that I have started with subtraction. I think this best defines the WHOLE and
the game you are playing with its PARTS. Subtraction is often misunderstood by
children in terms of the relationship between the WHOLE and the PARTS in it.
    Now its time to take the WHOLE ,3, and from it take the PART, 2, leaving the
PART,1. Recombine them by starting with the PART,1, that was left and bringing back
the PART, 2, that was taken away to make the WHOLE,3, again.
    Take note of how doing the opposite brought us back to the start again.
    In short: 3 - 2 = 1, and 1 + 2 = 3.

    Now compare the sentences to note how the PARTS change places. That is, when
2 is taken away, 1 is left, and when 1 is taken away, 2 is left.
    Inside the WHOLE of 3, the PARTS 1 and 2 are inseparable. They are PARTNERS,
or RELATED PARTS, inside 3, and they can change places. They can be the number
taken away or the number left, the number started with or the number added on, so
long as their PARTNER fills the other position, or does the other job.

This is best seen in simplest form:	3 - 1 = 2		2 + 1 = 3
	3 - 2 = 1		1 + 2 = 3

    This is difficult for some children.

    Make a point of reminding them that this is like a game and the game has to have
two PARTS.
    Make a definite place for the three things, such as on a piece of paper, or inside
string circles.
    When the PART 3 is taken away, nothing is left in that place and one purpose of
zero,0, is to name the amount, nothing.
    It is sensible to say and write, "3 - 3 = 0".

    Now point to the place with the PART 0 in it and bring back the PART 3. You are
now back to the start with the WHOLE 3.
    In short: 3 - 3 = 0, and 0 + 3 = 3.

    Next you can play the game by taking out nothing as a PART, or 0, from the whole.
You are left with a PART 3 of course. Put the PART, 0, back and you have the
WHOLE ,3, again.
    In short: 3 - 0 = 3, and 3 + 0 = 3.

These sentences can again be compared:	3 - 3 = 0		0 + 3 = 3
	3 - 0 = 3		3 + 0 = 3

    We have another set of PARTNERS inside the WHOLE of 3, and they can change
places or jobs.
    As well, we can see that the WHOLE can also be a PART.

    WHOLE, PART, and PARTNERS are all words I would have the children using.

    By now some readers will be saying this is too easy, too obvious.

    It isn't for a number of children at the start. They must know exactly what is going
on, especially when the switch is made to number sentences without objects to see.

    Besides, the purpose of this approach is to build a framework for thinking about any
number.

    Remember, working with numbers is like playing a game.

    You have to know the rules, the language, and the object of the game.

    That is what we are working on.

    List them on the chalkboard or chart paper in an ordered fashion as follows,
leaving space to the left for two more sets:

3	4	5
0 | 3	0 | 4	0 | 5
1 | 2	1 | 3	1 | 4
	2 | 2	2 | 3

    With the whole on the top and the pairs of their related parts below, ask the children
to look for the patterns.
    (You can count down the left and up the right from 0 to the whole).

    It should take little time to develop the sets for a whole of 1, and 2. (fig.1)

1	2	3	4	5
0 | 1	0 | 2	0 | 3	0 | 4	0 | 5
	1 | 1	1 | 2	1 | 3	1 | 4
			2 | 2	2 | 3

    Ask the children if they see something that always happens.

    Hopefully they will see that 0 always works with the whole as a pair of parts, or 0 is
always the partner of the whole. If not, lead them to it.

    Remind them that we can only say this for this game of adding and subtracting.

    Ask if the same will be true if we put 0 into two groups. Will 0 be 0's partner?

    Now to look at this another way:
    Show the children how we can set out the numbers up to 5 in a horizontal line.

0	1	2	3	4	5

    Now join the pairs of parts with a curved line rising up, from 0 to a high point
between 2 and 3, and then down to 5. Do the same for 1 and 4, and 2 and 3.

    You now have a rainbow pattern. Children love to make this, especially with
colors.
    Take a set of 5 things. Have two places to put them. Start with 0 on the left and 5
on the right. Then move 1 to the left to have 1 and 4, another to the left to have 2 and
3, another to the left to have 3 and 2, another to the left to have 4 and 1, and finally
another to the left to have 5 and 0.
    Draw attention to how this mirrors the fig.1 pattern and the rainbow pattern.

    (This might be a good time to introduce odd and even numbers if you haven't
already. Both patterns show plainly how only even numbers can have a pair of
identical parts.)

    By now the children should be ready for a general rule. You might like to discuss
how memory plays a large part in work involving adding and subtracting, how we can
look for things that always happen and call them rules, and how these rules save us
memory work.

    Ask the children if they see something that always happens in the sets of parts and
wholes. If needed, lead them to the zero and its partner, which is always the whole.
Develop this as a rule that seems to be true, but will be checked as bigger numbers
are investigated.

    Go back to fig.1 and erase the pairs of parts that are covered by the rule for zero, or
redraw the set. You may wish to print the rule, eg., "0's partner is always the whole."
Note how many separate sets of pairs of parts are dropped out, and how 1 has only
the rule to remember for it.

    Now may be a good time to go back over what the children need to remember
about how the pairs of parts in a whole are written into number sentences to describe
real situations. I think it is important that this be very clear.

    A number sentence is a quick and easy way to do just that; tell what has happened,
or is to happen, or might happen.

    The movement of objects, such as blocks, while writing number sentences is a way
to demonstrate this. I also had children act out the sentences in groups themselves.

    It was at this time I brought in a worksheet to keep track of the many number
sentences formed by what has been found out so far about 2, 3, 4, and 5.

    I used one letter sized sheet , divided into four equal rectangles. In each section
there was the same format. An approximately 1.5 in. (4 cm) space across the top with
a box in the middle and larger boxes to each side. The middle box was divided in half
by a horizontal line, and the lower part of it divided in half by a vertical line.

    The whole was to be printed in the top of the middle box, and its two parts being
worked with below it in the two smaller boxes. To the left the children drew simple
shapes, such as squares or triangles, to match the number of the part on that side, and
the same for the part on the right on its side.
    Below this was space for four number sentences. Sometimes this could be:

    Either way, the children could observe a pattern involving the pair of parts
and the whole they form.

    We learned a form for each type of sentence:

       WHOLE - PART = PART        and        PART + PART = WHOLE

    This was often placed at the top of worksheets and around the room.

    The children used the worksheet to record the sentences for each of the numbers
investigated to date. For example, for a whole of 3, in parts of 1 and 2:

(picture)	3	(picture)		(picture)	3	(picture)
	1 | 2				1 | 2
	1 + 2 = 3		or		1 + 2 = 3
	3 - 2 = 1				2 + 1 = 3
	2 + 1 = 3				3 - 2 = 1
	3 - 1 = 2				3 - 1 = 2

    The first worksheet (above left) allowed us to observe the opposite action of
addition and subtraction, and for this I always insisted they put the opposites together.
The second worksheet let us see the pairs of parts exchanging roles in the sentences.

    Note that I started these with addition sentences and could have made two more
worksheets by putting subtraction first. These sheets were used often, right up to
wholes of 18. (And, with modification, for multiplication and division.)

    Using these sheets, the even numbers stood out because, for one set of sentences,
two lines were always left blank, and the odd numbers always filled up all the spaces.
At least they did when I convinced those who would insist on writing the same
sentences twice that this wasn't necessary.

    At some time in here you will want to work on addition and subtraction questions, or
incomplete addition and subtraction sentences.

GSM, Part 1	GSM, Part 2b	GSM, Part 3	GSM, Part 4	GSM, Part 5
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