Part 6, Multiple figure addition and subtraction:
I believe that children do best at math or arithmetic
when they are confident that this
work is easy for them to do. I tried to start with the simplest
examples and bring them
to understand that most new work really was built on what they already
knew.
We may have been going to start to add and subtract large numbers, but
these bigger
questions were really just a series of little questions they already
knew how to do.
What would be new were the steps that had to be taken in order to do
the bigger
questions.
In my Simple Math booklet, p. 7, I covered only those addition and
subtraction
questions, or incomplete number sentences, which are basic facts.
I spoke of all of these as being formed by TWO PARTS in ONE WHOLE.
I used
these terms because I felt they were already words the children were
comfortable with.
Such number sentences were either
PART + PART = WHOLE or WHOLE - PART = PART
.
Before starting with two figure addition and subtraction questions I
would have had the
children become familiar with two things, using single figure parts.
29b.
One would have been questions having more than two parts, such as
3 + 2 + 8 + 5 = ___ or 17 - 4 - 6 - 2 = ___ .
The purpose of doing this would have been to convince the children that
these could
be done in small steps that they already knew how to do.
What would be new was the need to hold a number, a partial whole, in
their minds as
they went from step to step. Also new would be the need to keep track
of which
numbers had been dealt with.
Such questions allow the introduction of the idea that we can decide
which order we
wish to do the task in, something they might make use of in later
years.
Both examples above are made so that sets of 10 can be created (2 + 8
and -4 - 6) .
Working with sets of 10 provides an opportunity to reinforce the idea
that using place
value can simplify our work, and provides practise for the most
commonly used basic
facts.
That is, seeing 2 + 8 make 10, and 3 + 5 make 8 leads to a simple 10 +
8 = 18.
Seeing that - 4 - 6 are the same as - 10 allows an easy 17 - 10 = 7 and
7 - 2 = 5.
Simply doing the examples in the order given requires more difficult
thinking for most.
The second thing to introduce was questions written in vertical form.
This could have been done while practising the basic facts, and while
working with the
above questions.
In both cases we first worked with real objects so that the children
could
see for themselves that:
1. The larger questions were made of a number of small questions and
steps.
2. The horizontal or vertical arrangement of a task made no difference
to
the answer.
3. The task of addition was to put parts together to make the final
whole.
4. The task of subtraction was to take a part, or parts, from the
whole to leave a final part.
Point 4. above is one area where some children fall down in doing
more complex
multiple figure subtraction. They may try to take a whole from a
part when they
see that the part is larger than the number in the same place in the
whole, as 3 - 5 in
 43
- 25
30.
When we were to begin this multiple figure work, my class sat in a
semicircle on a
carpeted area facing a chalkboard. We had a large sheet of paper, about
16 in. by 24
in., to lay on the carpet, and wax crayons to write on it with. We had
blocks and egg
cartons cut down to use as holders of sets of 10.
I told the children that the most important part of what we were
about to
do was not the answer, but how we were to work to find the answer.
I did not want anyone to say the answer.
The reason for this is that some children’s eyes glaze over when
the answer has been
found. They no longer see the need to pay close attention. In fact,
some of the more
difficult children to teach in the end were those who had learned
elsewhere to do the
simpler types of questions, but not the more complex ones.
We started with a subtraction question written in vertical form on the
chalkboard, and
the paper. (Some children have difficulty working with what
they see if it's not on the
same plane. Some are slow, for example, to copy from the chalkboard
onto their
exercise books at their desks and make frequent errors. Put the same
work on a paper
and set it flat beside them on their desks, or the floor in this case,
and they do better.)
I chose subtraction first to make the point that they were going to
take a
part out of a given whole.
(NB: Some people believe in approaching subtraction through
finding differences
between numbers. This may require there to be two sets of numbers out
in view,
which can be confusing, and requires some more abstract mental
gymnastics.)
The question would have been printed on the chalkboard and the paper.
It would
have been a two figure whole, take away a one figure part, where the
number of
ones in the whole was larger than the number of ones to be taken away:
4 8
-  5
There would be one group of 48 objects separated into 4 tens and 8 ones
on the
carpet.
I would point out the tens and ones and review place value briefly.
Very lightly I would draw a vertical line in a different color between
the 4 and 8, going
down into the answer space, on the chalkboard and on the paper.
I would then ask the children what the question told us to do to the
48, the 4 tens and 8
ones. What I would want to draw out would be that we were to take
away 5 ones.
If someone said, “Take away 5”, I would ask what the 5 were, tens
or ones.
Then I would ask one child to go to the blocks and do what was required
by the
question. He or she would go to the 8 ones and take 5 ones away.
31.
We could see 3 single blocks remained, and I would ask where this
number should be
written in the answer space. This would be the time to stress that this
number can’t be
written just anywhere. It has to be written in the one’s place if it is
counting ones.
How do we know where the one’s place is in the answer space?
The position of the one’s, and any other place is decided by the
first
number written down.
These places extend down in a vertical line. They form a column for
each place.
Some children run into difficulties simply because they don’t keep
the
places clearly separate so this has to be stressed. At first it is
a good idea to use
bigger print and spacing, or light lines, on worksheets to make it easy
to see the
separate places.
Now, going back to the example, I would ask if we were finished
after the 3 was written
down. “How many think, yes? ... How many think, no?...” If any thought
'yes' I would
have someone tell us what we started out to do. Then I would ask if 48
- 5 could be 3.
I would point out that the question told us to take away 5 ones, but 48
has tens and
ones. “Were we asked to take away any tens?” ... “No.”
Then I would lightly print a 0 in the tens place beside the 5. “Is this
how many tens we
were to take away?” ... “Yes.”
There was a second step. We had to answer 4 tens take away 0 tens. The
answer
was 4 tens and 4 was written in the ten's place.
The full answer was 43, and this would be the first time I would wish
the answer to be
spoken as forty-three.
NB: If you haven’t gone through this for the first time with
young children, particularly
those who have difficulty with new learning, you are probably thinking
this is all too
slow and too concerned with obvious little details.
The truth is I have probably forgotten to mention a lot of those little
details.
We older people have done questions like this countless times. What we
do becomes
automatic and effortless. Once automatic it is not easy to see all the
places where
young children can have difficulty.
Certainly there are easier and quicker ways to do 48 - 5, but what
the children
need is to learn a pattern of behavior that will serve to help them
solve
any of the larger and more complicated questions to come.
Later on attention can be paid to methods of doing some questions in
alternate and quicker ways, after they have mastered this pattern of
behavior.
Knowing when to apply other methods requires them to discriminate
between questions, something they probably are not ready for yet.
32.
It hasn’t been my intention to wrap this up into neat little
lessons. There are so many
different classes and situations that teachers have to deal with,
things they have little
control over. It is my hope you can take what you need from what I have
here and
adapt it to your situation.
Going back to the multiple figure work, I would want to do several more
examples with
the children where we were on the floor. After this I would have a
printed exercise for
them to do at their desks, one that practised the very same type of
question, simple
subtraction, where the one figure number taken away was smaller than
the number in
the one's place of the two figure number.
I would stress that although I might have given them ten questions,
there really was
twenty to answer. They had to learn that 48 - 5 is to be done in two
steps. First, 8 - 5,
and second, 4 tens - 0 tens.
I would also ensure that everyone was writing the answers in the
correct
places, that is the 3 directly under the 8 and 5, and the 4
directly under the 4 in 48.
This required some children to shrink the size of the numbers they had
been used to
making.
On the next possible day the class would gather as before with the same
materials.
This time we would do an addition question of the type where a two
figure number is
added to a one figure number.
The whole created in the one’s column will be less than 10.
4 3
+ 5
There would be two sets of objects on the carpet, one with 43 grouped
in tens and
ones, and the other with 5 ones. The question would be on the
chalkboard, and a
large sheet of paper on the floor.
I would ask the children to tell me what we were supposed to do. Likely
someone
would say we were to add the 5 to the 43. I would leave it at that.
Then I would ask one child to do the work. I’m quite sure he or she
would simply put
the 5 in with the 3. (Remember, the 40 blocks were in cut off egg
cartons.)
This would give me the opportunity to ask which two numbers were really
added.
Most children by this time can see that the 4 tens remains untouched.
What we did was simply 3 + 5.
Then we would print the answer, 8, in the one’s column.
“Are we finished?” No, not until we’ve done the second question.
I would ask the children to tell me what was being added to the 4 tens
on the floor.
Obviously, nothing. The second question then was 4 tens plus 0 tens.
The answer, 4, was written in the tens column.
33.
After this we would proceed just as we had before with the subtraction
question, 48 - 3.
We would do more examples together and then practise exercises would be
done at
the desks. Again, I would have watched to see that numbers were being
printed in
their proper places.
Why start with 48 - 5 and 43 + 5 ?
This forces children to start with the one’s place question.
Some children will quite naturally try to start with the place on
the left.
Sometimes this will make no difference.
But, what we are trying to do is to establish good habits that will
help the
children do any multiple figure addition or subtraction question.
Later they can learn to stray from these habits if they have a good
reason.
Next, I chose to go to questions where the number taken away or
added on had
no ones.
4 8
-2 0
This time I would ask what is to be taken away from the 8 ones, and
from the 4 tens.
We would then do 8-0, and write the answer, 8, in the one’s column.
I would explain that we started with the one’s place question because we
needed to develop a good habit, one that would help us later.
Right now the way we worked was more important than seeing or getting
the answer.
Then I would ask a child to go to the 4 tens and 8 ones of blocks
and do what the ten’s
place part of the question asked us to do. From the 4 tens the child
would take 2 tens.
The 2 tens left behind were recorded with a 2 in the ten’s column.
More of these would be done before practising them independently.
Again, I would
pay close attention to answers to see the numbers were recorded in the
proper places
A similar type of addition question would be worked with next.
2 3
+2 0
This one would be dealt with just like the others.
The next step was to put non zero numbers in both ten’s and one’s
places of the
number being taken away or added on. In subtraction, the number of ones
in the
whole, the number being taken from, was more than the one’s being taken
away. In
addition the number of ones or tens being added formed a whole less
than 10.
34.
36 -24 = ___ , and 52 + 33 = ___ , written in vertical form, would be
examples to do.
We would have proceeded with these, separately, just as before.
We would use real
objects, grouped in tens and ones, start with the one’s place question,
and record the
answers. Then we would practise similar examples.
By this time I would start to say, “What is the complete answer?” when
we finished.
I would start to remind the children that although we might be doing 6
- 4 = 2 ones, and
3 tens - 2 tens = 1 ten, what we were really doing, in steps, was 36 -
24 = 12 (twelve).
It was time now to start practising some discrimination between types
of questions.
First we had some discussion about mixed examples of addition and
subtraction
questions on the chalkboard.
What I wanted to review was the practise of checking the operation
sign, the + or -,
before starting a question.
Some of the addition questions were of the type where there were zero
one’s in the
part being added to, as 70 + 23 = ___ , in vertical form.
Some subtraction questions might have had the zero in the answer space
of the one’s
place, as 43 - 23 = ___ , in vertical form.
Zero may mean nothing but it is surprising how much trouble it can
cause
some children. There are times when it is important to put 0 in an
answer, and times when we customarily leave it out.
I also needed to remind the children that although the operation
sign was
only printed once, it applied to the little questions in both places. I
have
had children make errors by subtracting in the one’s column and adding
in the ten’s.
We spent a few days doing practice exercises with mixed addition and
subtraction
questions of the type we’d studied. When the children were fairly
comfortable doing
these we moved towards the more difficult questions.
We started with those addition questions where the one’s place
answer will be
less than ten, but the ten’s place answer will be more than nine.
6 3
+6 4
We worked with real objects grouped in tens and ones.
When a child pulled the 3 ones and 4 ones, and 6 tens and 6 tens
together I asked
what the rule was for our place value number system. 1->|<-10
The 7 ones were fine, but there were 12 tens. What did that mean?
We piled 10 of the ten’s groups together and saw that we had 1 hundred,
and 2 tens.
What would we write as an answer under the ten’s column? 2.
Where would we write the 1 hundred? To the left of the ten’s place!
That was where the hundred’s place was.
35.