Paying attention to 'whole - part = part' will prove valuable later when the children
apply the same thinking to 2 + __ = 5 or __ - 3 = 2.

    Very early on I showed the children how questions are made by simply taking out
one of the three numbers in a number sentence.

    In fact we worked together to create all the questions one could ask about one set
of two parts in a whole. Starting with 5 made from 2 and 3, the children already knew
they could make four number sentences, each with three numbers.

    This meant each sentence could create three questions:
2 + 3 = 5 becomes 2 + 3 = __ , 2 + __ = 5, __ + 3 = 5.

    All are potential stories with missing numbers (word problems), and all have the
same form, part + part = whole.

    The children also came to see that 4 made from 2 and 2 meant less work, and they
are happy to find ways of doing less.

    At some point here it became obvious that 5 made from 2 and 3 could yield 4
different number sentences, and 12 different questions (or word problems). 4 made
from 2 and 2 could only yield 2 different number sentences, and 6 different questions.
    Knowing that 5 is also made from 1 and 4 makes another 12 different questions
one could ask, for a total of 24 about 5. If you count in those with zero as one part,
there are 36 separate questions one could ask about 5 in two parts.

    Two facts and a rule do all that, if you know how to use them.

    Sometimes I would give a sheet of questions out, all involving a few simple
facts, just to impress the children with how much can be done with so little.
    This makes it all seem easy, and that is important in the beginning.
   

NEXT: When children are comfortable with the basic ideas, it is time to go on to do the
larger numbers up to 10, noting that the amount of memory work is growing, but so are
the savings if it is done wisely. (add to fig.1 and call all this fig.2)

6	7	8	9	10
1 | 5	1 | 6	1 | 7	1 | 8	1 | 9
2 | 4	2 | 5	2 | 6	2 | 7	2 | 8
3 | 3	3 | 4	3 | 5	3 | 6	3 | 7
		4 | 4	4 | 5	4 | 6
				5 | 5

    Time can be spent looking for patterns in all this. You can also note that the five
facts about 10, and the zero rule, will now handle 66 different questions.

    You might enjoy working with odd and even numbers as a way of reviewing. Try
looking at the pairs of parts this time and draw out the rule that even numbers are
made of two like numbers, but odd numbers are made of two unlike numbers.

    You might also review equality and the equal sign by reversing the order of the
usual number sentence and having:

       WHOLE = PART + PART and PART = WHOLE - PART

    What kind of real situations would suit these sentences?

    Before going on past 10 as a WHOLE, I would have done a lot of work to ensure
that the children knew what kind of game we were playing and what the rules were.

    The children will have noted that, going from 1 to 10, the amount of facts to
remember has grown steadily. They would love to know that they won't have to go
beyond 18 with this table of facts (as fig.2), and 18 won't be much more difficult than 2,
and a lot easier that 10.

    I paid particular attention to the facts for 10. There are many of them, but they are
the ones that are going to be most useful in future work.

    After we had worked out the facts for 10, I would start with frequent speed tests of
20 questions each.

    I kept the facts, as in fig.3, on paper above the front chalkboard so the children
could refer to them.

    These were informal tests where they only had to fill in the answer to questions
printed on the paper.
    I sat at the front and as each child finished they brought it to me and I marked down
the number of minutes they finished it in. During lunch hour I marked the tests ( I just
laid them against a score key to make it as quick to mark as possible).
    Children who got a perfect score had their name recorded under the minutes from
1 to 5, according to how long they had taken.
    Children who made mistakes were given back their papers after lunch hour for
corrections to be done.
    All this testing only took about 15 minutes out of the day, but I felt it was very
important, especially as we moved towards 18.

    The tests made the point that these facts had to be memorized if they were ever to
be useful.

    We discussed the kind of multiple figure work that would be coming.

    The children all improved, but it was obvious to all that the fastest were those who
did not have to look up to the front and work out the answer. Memorization helped!

    Eventually I covered the sheets at the front to make the point very clear that they
would not always be there, that they had to be put in their minds.

    I will always remember Betty in Grade 2, who could do 20 mixed questions, up to
amounts of 18, in 30 seconds, and have them all correct. I could never match that!

    Now look at 11 and ask if there is a very obvious pair of parts. It may be helpful to
have a visible set of 10 and 1.

    ( I used egg cartons (dozens) with two sections cut off. This way many objects
could be put in the cartons and it was very obvious when a set of ten had been made.
Children were also reluctant to break a set of ten unless there were no other choice,
and this was helpful later.)

    Tell the children to think of place value if they don't see it right away. Draw out the
set of 10 and 1 as obvious parts. Do the same for 12, and on up to 18. Develop a rule
for one pair of parts in these two figure numbers.
    One pair will always be the ten and the other the number in the one's place.

    Tell the children we have now found something new to make their work easier.

    Next, pay special attention to 12. Remind the children that 0 and 12 are an obvious
pair of parts, as 2 and 10 now are.

    Then have a very obvious way of showing 12 as a set of 10, and 2 ones. Again I
liked my shortened egg carton to show a full set of 10.

    Write the following question for all to see: 12 - 1 = __

    Ask a child to do what the unfinished sentence says and take out 1. I would be very
surprised if the child takes the 1 out of the group of 10. The natural thing to do is to
reach for the 2 and take out 1, leaving 1.
    Make a real point of this. Write down what was really done: 2 - 1 = 1, and note that
the 10 still sits there, untouched, so what remains is that 10, and now only 1 more.

    An adult might see it this way ( This is not for the children as it is too confusing.):
       12 - 1 = (10 + 2) - 1 = 10 + (2 - 1) = 10 + 1 = 11

    Now, all this may seem too simple, but, again it is the process that is important to
get started, not the simple fact, 12 - 1 = 11.

    Do the same for 11 + 1 = __ . Have a child add 1 object to the set of 10 and 1.
The natural thing to do is to add it to the 1, not the 10. Draw attention to the fact that
the real activity was 1 + 1 = 2, and the 10 was untouched again.

    Try the questions the other way around: 12 - 11 = __ , and 1 + 11 = __.

    Note how the real objects still can be kept apart in tens and ones.

    Then start to erase or cross out the pairs of parts that can be handled by the zero
rule, place value, and facts already covered. Note that in all cases one part is a two
figure number and the other is equal to or less than the number in the one's place of
the whole.
    Ask the children what the numbers left have in common. (The pairs of parts are all
one figure numbers.)

    Start a list of pairs of parts under a whole for 11 and 12 as in fig.2 and note that the
list of pairs of parts is shrinking after we pass 10.

    Now of course you will be doing other work all along to reinforce what is being
discovered, and to review and maintain what was learned.
    Keep testing the rules, such as that for odd and even numbers, p.14.

    As you work through 13, note that now two pairs of parts are handled by place
value, and what is already known about 3.
    Keep on gradually, working up to 18 where you will only have 9 and 9 as the pair of
really new parts, just like 2 was made of 1 and 1.

    Try to find a new pair of parts for 19, one that cannot be handled by place value
and facts already discovered. There are none, and the children will be very relieved.

NB: Hopefully, all along you will have been giving them questions such as 14 - 3 = __ ,
2 + 15 = __ , 15 - 0 = __ , and 0 + 12 = __ .
    They need to know that the number of very new facts is shrinking as they move
towards 18 as a whole, but the number of potential questions has continued to grow all
along.

    Add to fig.2 so that in the end you have fig.3 which ends like this:

11	12	13	14	15	16	17	18
2 | 9	3 | 9	4 | 9	5 | 9	6 | 9	7 | 9	8 | 9	9 | 9
3 | 8	4 | 8	5 | 8	6 | 8	7 | 8	8 | 8
4 | 7	5 | 7	6 | 7	7 | 7
5 | 6	6 | 6

    You might want to arrange this differently to show where the other pairs of parts
would fit in, but either way there are lots of patterns to look for. Note how each number
works with all the others as a part, until it works with 9. Then it disappears from the
table.

    In it children challenged two others to be the first to answer
correctly their question about the simple facts, within limits I had set.

    It was important to make the question as difficult as possible since if one
challenged child was beaten by the other, the questioner was up.
    Questions that were too easy resulted in a tie, and the questioner lost the chance to
be up.

    It was simple to lead the children towards asking questions such as: " What number
minus 8 equals 7?"
    I was amazed at how easily these questions were handled, compared to all the
years when I had followed the usual textbook or workbook ideas for introducing and
developing number facts.

    Using the method I have been describing here, children learned to look at a whole
of 15 and its parts of 7 and 8 as something like a triangle with a number at each point.

    They could start with the whole and go either direction to make subtraction
sentences. They could start with either part and head to the other to make addition
sentences.
    (See p.12, middle, for an easy example.)

    To make questions, they simply left out one of the numbers.

    We did have workbooks in Grade 1 and 2, and textbooks in Grade 3, and I did as I
was expected to do and covered the work in them by choosing pages selectively.

    I didn'tuse all the pages, and I did have a lot of worksheets I had made myself.

    A good deal of time was spent on worksheets of the type described on p.12.

    When we reached 18 we made a booklet of these sheets to cover all the really
basic sentences for wholes of 2 to 18, something they could take home.

    As I said on p.1, it has not been my intention to tell you what to do each day.

    I hope you will be able to see some ideas you can use at appropriate times.

    If you are using this as a home schooling guide you will have to supplement it
with other texts and workbooks.

    I have left out many areas of learning that come under the heading of Primary
Arithmetic, such as word problems, telling time, and measurement.

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