We proceeded as before, but again, in checking practice work I had
to look carefully to
see that the answers to each step were written in their proper columns.
Now that we knew that sometimes the answer in one addition step can
move into the
next place to the left, we could start to try the most difficult
addition.
But, before going on it was time for a great confidence booster. The
whole class would
do a 12 place addition question together on the chalkboard. I wrote
down:
2 3 6, 4 8 3, 5 9 1, 6 1 8
+3 4 1, 4 1 5, 4 0 5, 1 0 1
Note that I have taken care not to let any of the parts in one place
total over nine.
Then we did the same for subtraction. I may have put up another two and
allowed a
pair of children to try them at the chalkboard. They felt good about
this and it was a
reminder that doing one big question can just mean doing a lot of
little ones.
After this it was time to go back to the blocks, as first described on
p. 25, to do
questions where the parts in the one’s place form a whole greater
than 9
when a one figure number was being added to a two figure number.
4 7
+ 5
As before, we did this in steps, starting with the blocks. When a child
pulled the 7 ones
and 5 ones together we knew we had 12 blocks. I wrote 12 down away from
the
question. Obviously it was a two place number. Could we write it in the
one’s place?
No. We could by now see the 4 tens waiting to come down into the
answer, and 412
would not be a reasonable answer.
The 12 tens needed to be grouped. Our number system required it.
I wrote down the old 1->|<-10 to emphasize this.
When grouped, we had 1 ten and 2 ones.
What could we write in the one’s column? 2. There couldn’t be any more
ones.
Where should the 1 ten of blocks go? They belonged with the other tens.
Where would we write it? We could write it at the bottom of the ten’s
column, but there
were more tens to be considered, and they had to go there too.
(Here I differ from some teachers.) I taught the children to print the
1 as a tiny number
above the ten’s column. My feeling is that this crutch can be discarded
later, but right
now it a very important visual aid. It is especially helpful when it
comes to correcting
errors.
Now we had to add 1 ten, 4 tens, and 0 tens. We recorded 5 in the ten’s
column.
The full answer was 52.
36.
After doing some more examples I showed the children how we could
picture these
questions on scrap paper.
We practised putting enough x’s down to equal each part of a question.
The x’s were grouped by circling 10’s, and these picture numbers were
arranged with
one part above the other, tens above tens and ones above ones.
We circled 10 ones and drew an arrow from that set to the top of the
tens.
Then we drew a horizontal line under it all and wrote the number
answers at the
bottom.
Each child was given some scrap paper, and a set of questions of the
type above.
They were to draw pictures of the parts being added together if they
had difficulty.
I spent a lot of time going around to help, and to check to see the
procedures we had
practised together were being used correctly. A few children might have
had to go
back to the blocks.
When the children were comfortable with this type of question we moved
on to those
addition questions where both parts have two figures, but only
those in the one’s
place form a whole larger than 9.
5 6
+2 8
Note: It is important to see that, after grouping, the ten’s place
is not more than 9.
The lesson for this type of question went as before. The
children were given scrap
paper to picture the parts they were to add while doing practise
questions at their
desks.
Next to be learned was the similar question where both places will
form a whole
larger than 9, sometimes only after the one’s are grouped to form a new
ten.
4 5
+5 6
After going through the same teaching routine with this type, I
told the children that
they may hear others say they were “carrying” when describing
the act of forming a
set of ten and moving it to the place to the left as one more there.
At this point some of the more advanced children might have enjoyed
trying questions
with 3 or more figures. If this didn’t seem advisable, I demonstrated
on the chalkboard
how any very large addition question (They would give me one.) could be
done using
just what we had learned so far.
This was a good time to review how doing large questions can be nothing
more than
doing a lot of little questions, carefully, and in the correct order.
37.
When the children were comfortable with all the addition questions
requiring grouping,
it was time to have them learn to discriminate between those
simple ones that we
did first with two little adding questions, and the more complex ones
we had just
worked on that required grouping as well.
We put up one question of each type on the chalkboard and discussed how
we could
tell which one would need grouping, or carrying, and which would be
simple addition.
We could see that if the one’s place parts added up to more than 9 we
had a question
with an extra step. If the one’s place part added up to less than 10 we
just had two
little addition questions.
Of course, this could apply to the ten’s place as well, but at this
stage with only two
figure numbers this was not really important.
The children then did a number of questions at their desks that were of
both types.
A few children would put a 1 above the ten’s place when they had not
needed to
create a ten. Some would forget to use 1->|<-10 when it
was needed.
When they were comfortable with this work we reviewed the simple
subtraction
questions we had done before and added some of them to the mixed review
questions
so they had two basic types of addition, and one of subtraction.
After this we got out the usual things and went back together to the
carpet. First we
studied questions like 10 + 4 = 14 to see where the 10 and 4 went in
the answer, 14.
Next we put down a whole set of blocks grouped in tens and ones,
with a
small number in the one’s place, a number less than the ones to be taken
away. Then I wrote the following on the paper and chalkboard:
5 2
- 4
I asked the class what the question asked us to do with the 52 blocks
on the floor.
They usually responded that we had to take away 4 ones. It was
emphasised that
what we could see was the whole, and we were to take part of it away.
Then I asked one child to come forward and do just that. Often a child
would move
towards the blocks, take a long look at the single blocks, and back
away. They were
puzzled. But there obviously was enough blocks to take 4 from.
Eventually someone would suggest that we take the 4 from one of the
tens.
I would respond that if we did, it wouldn’t be a ten anymore.
If it wasn’t a group of ten, what would it be? Ones.
Where should these ones be? With the other ones.
Then I took one egg carton ten and dumped the blocks in with the 2.
How many ones did we have then? 12.
Could we take 4 ones out of 12 ones? Yes.
How many ones would be left? 8.
Now we went to the printed figures.
38.
We had to have a way of showing what we had done to give us enough ones
so we
could take away the 4 ones.
Well, the first thing we had done was to go to the ten’s place and take
out one of the
ten’s. I then drew a diagonal line through the 5 ten’s and wrote a
little 4 above it.
The second thing we had done was to spill the set of tens in with the
ones.
How do you show that a single figure number is now ten more?
You can put a 1 in the ten’s place to the left of that figure.
I then printed a tiny 1 to the left of the 2. I told the children this
was something we were
allowed to do to remind ourselves what would have to happen with real
things.
Now they could see that the little questions were 12 - 4 in the one’s
place, and
4 - 0 in the ten’s place.
We then did another example much like this one in the same manner.
The third example had the whole as a number of ten’s, but no ones. This
was a little
disturbing for some, but they soon saw that if we proceeded as before
we could do the
question.
There was just one extra step, regrouping.
The children then did some of this type of subtraction at their desks,
a two figure
number minus a one figure number, where the one figure was larger
than the
number of ones in the two figure number.
On the next day we reviewed what we had done and practised some more.
Then we went back to the blocks on the carpet to try questions with two
figures in the
whole, and the part to be taken away, and with the one’s place part
being larger
than the one’s place whole, and the ten’s place part at least two less
than the ten’s place whole.
7 3
- 5 9
We went on just as in the last example, except we had to do 6 - 5 in
the ten’s place
instead of 6 - 0.
Again, while the children were doing practise questions it was
important to go around
to see that they were following the correct procedure, and printing the
numbers in their
places.
Some children had to have some real objects to work with to be
comfortable that what
they were doing was correct.
At this point I could mention that some people would say,"borrowing",
when they
talked about the regrouping that had to be done in these questions.
39.
On the next day, while reviewing the above and using blocks as before,
I would put in
a question where the ten’s place part was only 1 less than the
ten’s place
whole.
8 3
- 7 5
This type of question bothered some children. They could see the
question in the ten’s
column became 7 - 7, but should they write the 0? Should the answer be
08?
We looked at our number chart and saw 08 in the first row. I asked what
the zero told
them. They would respond that it meant there were zero tens.
I asked how many tens were there when we just printed 8. They would
answer that
there was zero.
Then we considered if the usual way of writing a number of ones
required a 0 in the
ten’s place. Obviously it didn’t. If this was true we didn’t need to
print anything under
the 7 - 7. Putting a 0 didn’t make the answer wrong. It just wasn’t
needed, so it
needn’t be put in.
Again, we practised these questions, and then again with some two
figure minus one
figure questions, all of which required regrouping.
After this it was time to learn to discriminate between subtraction
questions that required regrouping and the simple ones that didn’t.
This is a very important step. Regrouping is a difficult concept for
some
to deal with. They get so wrapped up in doing it they will regroup when
they don’t need to.
This was also a great time to review the need to make a habit of
starting with
the one’s place. If a question required regrouping, one could get
into a real mess
by starting with the ten’s place.
Back on the carpet, I put out a single set of blocks grouped in tens
and ones, say 45.
Then I asked what we would do if a question asked them to take 6 ones
away, 7 ones,
8 ones, and 9 ones. In each case we would have to regroup.
But what if we were to take away 5 ones, 4 ones, 3 ones, 2 ones, 1
ones, and 0 ones.
In each case we would have a simple subtraction question.
Then the number was changed so an 8 was in the one’s place (any number
of tens).
The children found that only one amount, 9, to be taken away would
cause
regrouping. We could see that regrouping was needed when the amount
of
ones to be taken away was more than the number of ones in the whole.
We could then start some mixed subtraction practice, questions
requiring regrouping
and questions only requiring simple subtraction.
40.
Gradually I began to add some simple addition questions as well, ones
that didn’t
require grouping.
When they were quite comfortable with these it was time to bring in all
the types.
Together we made a classification table on the chalkboard. At the top,
to the left of a
vertical line we put "Addition", and on the right, "Subtraction". Then
a horizontal line
was added in the middle, giving us four spaces. ( I didn't print the
following!)
Top left: Addition where the parts in any place's column do not add
up to
more than 9.
Here we simply add and record the whole at the bottom of this column.
Bottom left: Addition where the parts in any place's column add up to
more than 9.
Here we have to group the whole in that place's column into 10’s
and add that number of 10’s to the top of the place's column to the
left.
Then we record the remaining ones of that place's column at the bottom
of their column.
Top right: Subtraction where the part to be taken away in any place's
column is less than the number in the whole above.
Here we simply subtract and record the part left at the bottom of this
column.
Bottom right: Subtraction where the part to be taken away in any
place's
column is more than the number in the whole above.
Here we must take one from the place to the left in the whole, cross it
out, and put a number that is one less. Next we must take that one to
the
place to the right where it becomes ten. After that we must put a
little 1
to the left of the number in that place to show it is ten more. Then we
can
take the part away from it and record the part that is left at the
bottom of
this column.
To these we added examples of different ( not all) situations within
each type, which I
shall put in horizontal form to save space. (See the example after
p.42. )
Top left: 45 + 3 = __, 45 + 20 = ___ , 45 + 23 = ___
Bottom left: 42 + 73 = ___ , 45 + 7 = ___ , 45 + 27 = ___ , 45 + 57 =
___
Top right: 45 - 3 = ___ , 45 - 20 = ___ , 45 - 23 = ___
Bottom right: 45 - 8 = ___ , 40 - 8 = ___ , 45 - 28 = ___ , 45 - 38 =
___
These were all the types of questions we had gone over together. Now we
could do
practice work that was fully mixed. This is always a difficult task for
some children.
41.
Again, you may wonder why I have gone into so much detail with this
work. What you
must remember is that for young children, doing this for the first
time, there is a lot of
little details. It’s all new. I'm sure I haven’t gone into all the
problems some will face.
Some of the main points I hope you have gathered are:
1. Children must have a good understanding of place value. They must
be able to see 48 as a single quantity called forty-eight, and as two
separate amounts, 4 tens and 8 ones.
2. Children must understand that the way to do large questions is to
break them down into little questions.
3. Children must have a good knowledge of the basic addition and
subtraction facts.
4. Children must be confident enough with place value to see that they
can change 15 ones into 1 ten and 5 ones, and they can change 4 tens
into 3 tens and 10 ones for a special purpose.
5. Children must be able to discriminate between the questions that
require no changes before proceeding, and those that do. That is, they
must be able to see which ones require the extra grouping (carrying) or
regrouping (borrowing) steps.
Once again I’ll remind you that I haven’t tried to lay out lessons to
follow.
Some of the factors and pressures that can affect individual lessons
are: children in a
class, teachers (yourself and others), principals, schools, timetables,
time allocations,
curriculums, training, classrooms, texts, workbooks, supplies, climate,
weather,
painters, lawn mowers, announcements, fire drills, lost teeth, .....
What I set out to do was to give a sense of the many little things that
a child can find to
be new, and often puzzling; things an adult handles automatically. My
purpose was to
make you aware of these, and to tell a little about how I handled them.
I did not describe my lessons in great detail. Much more went on than
what I have
spoken about here. But much of what I did not say was related to my own
particular
class, my own interests and knowledge, and other factors at that
moment.
There were also many other times of the day when something would come
up that
could be related to what we had been learning in math, or arithmetic.
I hope I've at least made you think about the topics in this booklet.
Please contact me if you have any comments or suggestions.
My addresses are on p.1.
42.
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