Two figure addition and subtraction:
    At this point I will leave the development of multiplication and division to go back
to addition and subtraction. By this time it had been carried into work with two figures.
    I don't intend to say much about this, except that the work you have done with
place value will be important here, as well as a very quick and accurate recall of
number facts .
    I found the children who were unsure at first really needed visual aids, not just
objects to manipulate but little numbers on their paper to remind them what is
happening. I know it is messy, and some think the grouping and regrouping (carrying
and borrowing) should be done in the head, but that is very difficult for some children
at first.
    Children who really weren't clear on the structure of WHOLE - PART = PART
always seemed to have difficulty with subtraction with regrouping (borrowing).
    They would see nothing strange in taking the one's place WHOLE from the one's place
PART if the usual way seemed impossible.
    I found a great deal of value in simply spending a lot of time learning how to
discriminate between the questions that needed grouping and regrouping, and those
that didn't.
    Lastly, especially for those having difficulty, I found it most helpful to spend a lot of
time developing a clear set of steps to take in working out each type of question.

Fractions:
    We did not do a lot of work with fractions in the primary grades, but the work we did
with equal groups suited the study of simple fractions where they were fractions of a
number of things.
    By learning that the bottom number (denominator) named the number of equal
groups in a set of things, and the top number (numerator) counted the number of those
groups the fraction was talking about, it was easy to relate this to division and
multiplication. Eventually the children could find 3/4 of 12 by first dividing 12 by 4
to find there was 3 in a group, and then multiplying that 3 by 3 to get 9.
    This provided another way multiplication and division could be practised.

Fractions and Factors and the Least Common Denominator:
    A knowledge of factors can be used to help when learning how to add and
subtract fractions with different denominators, something older children are faced with.
    a.3/4 + 1/6 = ___ . b.9/10 - 5/12 = ___ . These questions require children to
change the fractions to a common denominator so the numerators can be added or
subtracted. This can be a very trying experience, but it need not be so bad.
    In a., the denominators are 4 and 6. Simple multiplication gets a common
denominator of 24. Trial and error can arrive at the least common denominator of 12.
    But, knowing that the object is to find a number to multiply both parts of the fraction
by to arrive at a common denominator allows us to use our knowledge of factors.
    Factors tell us how a number is built in a basic sense, especially prime factors.

    Prime factors are those that are prime numbers.
    All composite numbers can be represented by a set of prime factors.
    In a. above, the denominator 4 is 2 x 2 , and 6 is 2 x 3. A common denominator will
contain the 4 (2 x 2) and the 6 (2 x 3). That is why multiplying 3/4 x 6/6 , and 5/6 x 4/4
works, but it is not the simplest way. The denominator 24 contains an extra factor
which will later have to be removed to arrive at the simplest form of the answer.
    If, in a., you consider the prime factors, you know that the least common
denominator will contain 2x2 and 2x3. Multiplying these results in 2x2x2x3, but there
is a factor of 2 in there that is not needed. If you see they both have the same prime
factor 2, then you know the least common denominator will drop one of the 2's.
    The LCD (least common denominator) will be 2x2x3 since that will form the
smallest number created by multiplying 4 and multiplying 6. Looking at this you can
see that 4 x 3 ((2 x 2) x 3) and 6 x 2 (2 x (2 x 3) will form the LCD of 12.
    The question becomes 9/12 + 2/12 = __ , something quite manageable.
  &nbspThe answer, 11/12, is already in its simplest form.

    In b. above, the denominator 10 is 2 x 5, and 12 is 2 x 2 x 3.
    10 needs a 2 and 5. 12 needs two 2's and a 3. There is an extra 2 if you simply
multiply and get 2 x 5 x 2 x 2 x 3. We only need two 2's, a 5, and a 3.
    The LCD is 5 x 2 x 2 x 3. In it there is 10, (5 x 2) x 2 x 3 , and 12, 5 x (2 x 2 x3) .
    You can see that 9/10 has to be multiplied by 6/6 , and 5/12 by 5/5.
    The question becomes 54/60 - 25/60 = __ , which looks better than
108/120 - 50/120 = ___ that you would have to do if you just multiplied the
denominator numbers to get the common denominator. Besides, the answer, 29/60 is
already in its simplest form, whereas 58/120 is not.

    Obviously, simplifying fractions is another area where factors can be of use.
    Asked to simplify 36/45, a child can divide top and bottom by 3 and get 12/15, and
by 3 again and get 4/5.
    But a child that can see 36 as 4 x 9, and 45 as 5 x 9 can easily see that dividing by
9 is the quickest way of taking out that factor of 9 and getting to 4/5.
    Using prime factors only, children can learn to cross out common factors from the
numerator and denominator. 36 is 2 x 2 x 3 x 3. 45 is 3 x 3 x 5. Take out the two 3's
from each of them and you are left with the simplest form, 4/5, since 4 is 2 x 2.

    Why all this is a booklet about primary arithmetic?
    New knowledge or skills in Math and Arithmetic are built on top of what has come
before. That is why we speak of basics. But really, every step required to do higher
math is basic to the final outcome.
    I hope you can see that something simple, such as factors and prime numbers,
started early, will have an immediate value, and a later use, just as place value does.
    Think about it. Would you like to hear about factors and prime numbers for the first
time when faced with LCD's, or would you like to have known about them well before?

    2, 3, 2 x 2, 5, 2 x 3, 7, 2 x 2 x 2, 3 x 3, 2 x 5, 11, 2 x 2 x 3, 13, 2 x 7, 3 x 5,
    2 x 2 x 2 x 2, 17, 2 x 3 x 3, 19, 2 x 2 x 5.

    Prime numbers stand out because they are made of one group of themselves.
    Composite numbers have two or more prime factors. Those with three or more
prime factors have smaller composites inside their sets. 12, for example, has 2 x 2,
and 2 x 3.
    Children can spend time picking these out as another way of reviewing
and reinforcing number facts.

111 Day:
    Many schools celebrate or take note of the 100th day at school, but if some effort is
going to be put into it I believe that waiting another 11 school days is worthwhile.
    Doing something for the 111th day allows the observation of the relative size of
100, 10, and 1, and it also places emphasis on place value. For the first time you have
three 1's, all doing different jobs.
    I had thought for years that taking note of the 111th day was of more value and
had done so in my class. In my last teaching year the whole primary school switched
to a 111th day celebration. We carried out similar activities to the 100th day, but
paid more attention to the size of 1 hundred compared to 1 ten, and to 1 one.
    We had many activities in the gym and after that we had a period of time where half
a class would visit another half a class to see what they were doing that was related to
100 and/or place value, and to take part in it.
    My class of Grade 2's had enough 1 inch cubes to form a cube of 1000, a square of
100, a line of 10, and 1. This was a great time to review the meaning of large numbers.
There was a pattern to see: cube,line of cubes, square of cubes (from right to left).
    They also had enough pegs and pegboards to do the same.

    We also helped the visitors to make a simple card I had used for years to mark
the 111th day. A simple sheet of paper folded twice formed the card.
    The front of the card had 111 written in a diagonal line down the page:
       1
             1
                   1
    The children wrote t 1 o , m 1 y, fam 1 ily (or mom, or dad,...) on the front.
    On the inside left the children made 10 rows of 10 x's.
    On the inside right I had already duplicated a diagonal line of 10 large O's.
Inside the O's they printed "I love you", which, if you count the spaces, uses all 10 O's.
    The children colored and decorated the cards as time allowed and filled in their
names on this preprinted section to the lower left of the circles:
       "From ________,
          111 days at school today"

    They liked the idea of taking home 100 kisses (x's), 10 hugs (O's) and 1 "I love you".

    You can think of this as a simple game we can play with ordinary whole numbers.
    Once you learn the game it can be interesting, and useful.

    All numbers in this game become less than 9.

    Any 9 in a number is taken out. Then you only think of the amount that is left.
    The effect of this is to separate all numbers into nine families. There will be those
that have 0 left, 1 left, 2 left, 3 left, ...., 8 left. (The 99 chart in Part 5 may be helpful here.)

    9 becomes 0 in this game, 18 becomes 0, and 0 becomes 0. They are all in the
same family.
    21 becomes 3, and so does 12. 15 becomes 6, and so does 51.
    101 becomes 2. 79 is a 7. 596 is a 2.

    Taking out the nines really involves little or no calculation. You just have to forget
about place value.
    29 is seen as a 2 and a 9. Throw away the 9 and you are left with 2.
    Of course you could subtract 9 three times, or divide by 9, and find the remainder is
2 in either case, but that would take too long for this game.
    929,909 can be seen to be a 2 by just forgetting about all the 9's and the 0, or you
can do it the hard way.

    Most numbers are not so easy, but there is another little trick to use.
    247 is in the family of 4 because I can add the 2 and 7 to make a 9, which I cast out,
leaving 4.
    528 is more difficult. I could start with the 8 and add on the 2 to make 10, which is a
1 in this game. Then I could add the 1 and 5 to make 6.
    Or I could have added all three numbers to get 15, which reduces to 6.

    How can this work? Well, 5 is really 500. 9 is 10 - 1. In 500 there are fifty 10's, but
to find the 9's we have to remember the fifty -1's. Fifty 9's is only 450, leaving 50 to
take 9's out of. There are five 9's in 50, with 5 left over. Altogether, after taking out
fifty-five 9's there remains the same 5 we see in the hundred's place of 528.
    That 5 in 528 really was the remainder after taking out all the 9's. You can take this
farther if you are curious, or doubtful. It really is the same for the 2 and the 8.
    What works is to add the figures as single place numbers, add again if the total isn't
a single place number, and keeping casting out any 9's you can see.

Example: 35928 is to be done. Look only at the 3, 5, 3, and 8. Add any or all of
these together. If the answer is a two place number, add the numbers in those places.
Do this until all are added in and only a one place number remains.
eg. 35928: 3 + 5 = 8, 8 + 2 = 10 and 1+ 0 = 1, 1 + 8 = 9, so 35928 reduces to 0.
or 3 + 5 = 8, 2 + 8 = 10, 8 + 10 = 18, 1 + 8 = 9, so we still see 35928 is in the 0 family.

    You can use this game as a rough check of the sum (answer) of large
addition questions:
       2586 + 3498 = 5214?
    Cast out 9's in the two parts added as if they were one big number, 25863498.
There's an obvious 9, 5 + 4, and 6 + 3 to throw out, leaving 2 + 8 + 8 which reduces to 0.
    Cast out 9's in the whole, 5214. The 5 + 4 goes, leaving 2 + 1, or 3.
    The addition is wrong!
    If both were 0, or 3, the addition would likely be correct.
    The proper sum is 6084. 6 + 8 + 4 = 18, which reduces to 0.

NB: It is possible to make an addition error that would result in a 0 result. That is
why this game is good at finding errors, but not guaranteeing correct answers.

    You can check subtraction by treating it in reverse as a addition question. Again,
cast out all 9's in the two parts and compare the result with that for the whole.

54.7 - 43.8 = 10.9 ? 547 reduces to 7. 438 and 109 reduce to 7. The computation has
a chance of being correct. Notice that decimals have no effect on this checking game.
    You can check large multiplication questions by considering them as repeated
additions.

523 x 314 = 164,222 ? Cast out 9's in each of the factors, 523 and 314.
    523 reduces to 1. 314 reduces to 8. Now multiply the two results, 1 x 8, to get 8.
    Now cast out 9's in the multiple, or product, 164,222, and it reduces to 8.
    The computation could be correct.
    If it were wrong you might use casting out 9's to determine if the error was in your
addition, or your multiplication, saving some time.

    You can check large division questions, but remainders make this tricky.
    Multiply the results from the two parts, and add on the result from any remainder.
    This should reduce to the result for the whole. Try one if you like.

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