Part 2,
Reading,writing, and understanding the value of larger numbers:
These topics remain important to us throughout our lives. It is amazing
how easy it is
for adults to be deceived when comparing or judging very large numbers.
For
example, when government officials and politicians speak in terms of
millions and
billions being spent out of taxes, or carried as debt, do we really
know how much
money this is?
A country like Canada with a relatively small population, roughly 30
million people,
may have a debt of roughly 600 billion dollars. What does this really
mean?
Look at it this way: 30,000,000 people owe $600,000,000,000.
If we take the same number of zeros off each we keep the same ratio.
This becomes: 3 people owe $60,000, or each person, no matter what age,
owes
$20,000, and that is on top of any personal debt they may have.
I’m afraid not all people of voting age are capable of dealing with
what millions, and
billions, and trillions really mean.
I hope you will read the following carefully to find those places that
seem natural for
adults, but can really be confusing for children. Perhaps you may find
you have not
fully understood the value(numerical quantity) of a number yourself.
In my class we had enough blocks (wooden 1in. (2.5 cm) cubes) to
show an amount
just above 1000. This was a great advantage. It allowed us to see the
basic value of
each of the first four places in our number system, and to see an
emerging pattern.
Our place value number system, the system we use to represent
amounts, is
based on groups of 10. If we see this many x’s, xxxxxxxxxxxx, we
write 12.
What we are doing is grouping them in 10’s, (xxxxxxxxxx) xx, and
writing in a place
value system that there is one set of 10 x’s and two single x’s.
For young children, there is something awkward about this system that
adults may not
think about.
In many languages , but not all, we read from left to right, and young
children have to
be trained to move their eyes accordingly.
As we read large numbers we have to read from left to right.
As we write numbers we have to write from left to right.
But, when we judge what a particular number is to be called because
of
the place it is in, we have to make a judgment based on the number of
places it lies to the left of the one's place, which lies on the right.
In other words the system requires us to be totally aware of a right to
left sequence
before reading from left to right.
7.
The four place number my class formed with all their blocks was
1111, or 1,111.
This can be read as “eleven hundred and eleven” because many people
read and
write four figure numbers this way. This is something people may do,
but it strays from
the actual system.
Even the name ‘eleven’ does not follow the pattern through which most
numbers are
named. All the numbers from 11 to 19 are the same. They have two
places, with the
place on the right something other than 0. All the other two place
numbers without
a 0 on the right have a combination name. 41 is “fortyone”. 69 is
”sixtynine”. Those with
a 0 on the right have a single name. 50 is “fifty”.
People who write 1,111 tend to stick to the system. They would read
this as “one
thousand, one hundred and eleven”.
This is further confused since some people also skip out the “and” in
either way of
reading the number. This isn’t a big problem to us, but it could be for
children.
My advice on this point is to stick to a system while the children are
young and let them
learn to stray from it later.
The Systems:
I hope you can see that writing, reading, and understanding numbers
is not so simple.
Writing: This is the most consistent system. We first need enough
unique numerals to
name each single amount that can only be named separately. By numerals
I mean
shapes we form, and their names we use, to represent a particular
amount. (2 and two)
Our place value number system is based on groups that could be matched
onetoone with this: (xxxxxxxxxx). We call that amount, “ten”. All
around the world
other languages name it differently, but the amount remains the same.
We say we have a base ten place value number system.
We need then, a 0,1,2,3,4,5,6,7,8, and 9 as numerals to write with.
They represent the amounts that could stand alone when our system is
applied.
Beyond an amount of 9, we use those numerals in a special way.
You may be aware of other place value number systems. One system
requires two
numerals, 0 and 1. It is a base two system, using groups equivalent to
(xx).
Using this system you can write any amount with 0 and 1 that we can
write using 0 to
9, but it takes a lot more space. However, it is ideally suited to
computers that can
represent the 0 by an electrical current of low voltage, and 1 by a
slightly higher voltage.
Another system that has had limited use was based on groups of 12, and
another in
use is based on groups of 16. Both systems require additional single
figure numerals.
8.
In our base ten system, 10 represents the amount that is 1 more
than 9.
Children have to learn this is not a new shape.
It is a 1 and a 0 used in a very special way.
The 1 and 0 in 10 each represent what they did on their own. They also
represent
something new based on their placement in relation to each other. That
is, 01 is far
different from 10.
In 10, the 1, being one place to the left of the 0, now represents 1
group of ten, or a
group equivalent to (xxxxxxxxxx).
0 has two very important functions.
The 0 in 10, being on the right of the 1, represents an empty group,
AND
serves to push the 1 into what we call the ten’s place.
In 01, the 0 really has no use, but we sometimes put it there. It does
no harm.
My Simple Math booklet describes a number line made from adding machine
paper that ran across the
front and along most of one side. We moved a marker along it to
indicate the number of days at school.
This was numbered from 0 to 203 and allowed the children to always have
a sense of where a number was
in relation to others. It was not a crucial part of understanding the
writing of numbers and place value.
In my Simple Math booklet I also described another activity we
did every morning in
my class. We had a board with three hooks, side by side, with numeral
cards from 0 to
9 hanging on each hook, 0 being at the front to start. The set on the
right was the one
we began using on the first day of school to record the number of
school days.
I kept the 0 in the middle set and left set facing the children as we
went through the
cards on the right, putting yesterday’s numeral to the back of its pack
to expose the
new one. On the first day we had 001.
On the tenth day the middle zero was sent to the back of the pack and 1
was exposed.
Of course, at the same time the 0 on the right appeared again. We had
010.
On the 40th day of school my class would have seen 040 on our display.
The 0 on the
left stayed where it was until the 100th day.
Children need to be reminded often that 0 serves two important
functions in our place
value number system. 0 represents an empty set, but it also serves
to fill a
place. There is a big difference between 1,000,000 and 10 , and
that difference is
made with 0’s that force the 1 a special number of places to the left.
At this point I should remind you that young children may not have a
well developed
sense of right and left. Even up to age 8 and beyond, children will mix
up ‘no’ and ‘on’.
It can happen to adults. Just recently I received a lengthy fax message
that wasn’t
intended for me, 22 pages of confidential information. I looked at the
phone number of
the party it was meant for and tried to phone them. But, I made the
same mistake as
the fax sender. My phone/fax number starts 531, and the fax should have
gone to 513.
9.
I saw this, but when I typed it into my computer phone, I typed 531.
Luckily, there was
only an answering machine to receive my message about an errant fax,
and a
correction when I realised my mistake.
This was the same mistake the original sender had made, a reversal of
numerals.
My wife and I arrived home from a trip recently to find an elderly,
respected criminal
lawyer, and politician, had left a message on our answering machine
containing
confidential, and not flattering, information for another lawyer about
one of his criminal
clients. He had mixed up some numbers when he dialled, but was
confident enough
in his own abilities to leave a message. It could have been a very
serious mistake.
Now if adults can make this kind or error, certainly we can expect
children to.
Our left to right reading and writing is not something we are born to
do.
Other people arrange their symbols in other ways and get along just
fine.
Children have to develop this left to right movement we require to read
and write.
To some degree this requires them to be physically aware of their own
left and right
sides, and that of others, and of other objects.
We spent time in my classes practising just this left and right
orientation and I will
elaborate on these activities in another booklet.
In my Simple Math booklet ,p. 5, and my Games booklet, p.9, I’ve
described activities where children have
to decide the larger or smaller of 40 and 04. This is not easy for
some. It requires a confident sense of left
and right that not all have, but practise pays off for them.
Now, going back to my class counting the days at school, when we
got to the ninth day
at school, we had to start thinking about place value.
On a ledge above this display I had put one block for each day. These
were plastic
blocks that snapped together, but they were left loose at this time.
We could see that there were no more numerals in the card set we had
been
changing. We had 009. Hanging behind the 9 was a 0.
The next day I introduced the idea that we could put a set of things
together and call
them 1 group. We would have a special rule for naming the number of
days from now
on. Every time we got a set like (xxxxxxxxxx) we would call it 1 group
of ten.
This meant our set of plastic blocks now were snapped together in a
line.
That 0 in the next place to the left would go behind to expose a new 1,
a 1 that was
special and didn’t mean just what it had on the first day of school.
The 9 on the right
went behind to let us see the 0 again.
The blocks showed us what had happened. All the blocks had left the
one’s place and
were now being counted in a new way in the ten’s place.
10.
Thereafter we referred to the numerals we put up as going into the
one’s or ten’s
place. This time the one’s place had nothing in it, as shown by the 0.
We had 010. The 0 on the right, filling the one's place, gave the 1 its
special value.
On the chalkboard I left this reminder: 1><10 . (Ten
could be shown as 1 in the
ten’s place, but that group of 1 ten could be broken up to make ten
separate ones.)
The following day was great for demonstrating the effect of a place
giving a numeral
value. We had 011. Each 1 was a 1, but that on the left was special. It
was 1 group of
ten. We had 1 group of ten, and a simple 1. The blocks above
demonstrated this. We
had two 1’s of different value, yet they were both 1.
The day after was the introduction to the idea that a 1, because of its
place, could be
worth more than a 2. The blocks helped us see this. We had 012.
This idea was reinforced through to 019, but some children were still
uneasy about that 9
being worth less than the 1. It’s difficult for some children to break
away from their
early learning about what is big and what is small, and come to see
that big and small
can depend upon the placement of a numeral. The blocks above 019 helped.
(This difficulty persisted for some time, with some having trouble
understanding that
21 was bigger than 19, or 68 was smaller than 71. They could see it if
given real
objects, but when looking at the two written numbers they weren’t sure.)
When we got to the twentieth day we reviewed the notion of (xxxxxxxxxx)
being called
1 group. We now had enough blocks to demonstrate this again. We had 020.
This went on until we had 099. Everyone knew what number came next, but
not how it
all came to be.
On the hundredth day we found we had 10 sets of 10 blocks. We reviewed
the rule
that any set of 10 was grouped together to become a new set of 1.
Luckily these
blocks could be placed so that all the sets of ten joined. They formed
a square.
(I added some tape to the back of them just to make sure they stayed
together.)
Obviously there were no separate blocks, so the 9 in the one’s place
was put to the
back of its set so 0 would show. There were no separate sets of ten
blocks, so the
same thing happened there. We went to the third place to the left, put
the 0 to the
back, and we had 100.
This place that had waited so long to be used was the hundred’s place.
Most years all the primary grades celebrated this day, but I also
celebrated the 111th
day in my class. I called it Place Value Day. On that day we had 1 in
each place, and
with the blocks it was easy to see that each 1 counted something
different, and that the
value of the 1 on the left was far more than the 1 on the right.
11.
One of the activities for this day was to use other wooden 1 in.
(2.5 cm) blocks we had
to form sets of 1000, 100, 10, and 1. (Luckily, we had a lot of blocks.)
We also did the same with pegs in pegboards, with room for 100 in each.
We stacked
10 full pegboards to show 1000. We set one full one of 100 to the
right, one with a
single row of 10 to the right of that, and one with 1 peg in the last
place on the right.
At this point I’ll add an activity also described on p. 30 of my Simple
Math booklet.
My class made a simple card to mark the 111th day. A sheet of paper
folded twice
formed the card.
The front of the card had 111 written in large numerals, and in
a diagonal line
down the page, starting at the top left.
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".
But, going back to the 1,111 wooden blocks, we kept these on the floor
for many days
where we could gather together to discuss them.
Remember that the blocks used were 1 in. (2.5 cm.) cubes. I believe
these were a very
good size for what we were to do next. We did have 1 cm. cubes, but
these were too
small to be handled easily, and too small to provide a lasting image in
children’s
minds of the rate at which the value of 1 increases as it moves to
places to the left.
I think it’s important for everyone to have some image that allows
them to
judge the relative value of numbers.
These blocks provided a way to build a simple picture in the mind of
what happens as
we go to the left from the one’s place, a picture that remains
consistent no matter how
large the number is, and one that is consistent with our method of
writing large
numbers. (It also operates to the right of the one’s place.)
We had 1,111 blocks. Some would write 1111 , but that’s
not my preference.
The 1 on the far right was represented by a single block, a
single cube.
(That cube was important to the picture we were developing.)
The 1 in the 10’s place to the left of it was a line of 10
cubes.
The 1 in the 100’s place to the left again was a square of
100 cubes.
The 1 in the 1,000’s place to the left again was a CUBE of
1000 cubes.
12.
With a little imagination it was easy to see that a pattern had
started.
To show 1 in the 10,000’s place we would need a line of 10 cubes
of 1,000.
To show 1 in the 100,000’s place we would need a square of 100
cubes of 1,000.
To show 1 in the 1,000,000’s place we would need a cube of 1000
cubes of 1,000.
A pattern had formed that hinted at the value of the commas
in the numbers:
square line cube, square line cube, square line cube, square line
cube ...
We discussed the amount of space each of these large numbers of blocks
would take
up. (I’ve added approximate measurements. With the children, we just
used our eyes,
comparisons with other objects, the room, the school, and the
playground.)
The thousand cube was 10 in. (25 cm) on a side, so the 10,000 line,
having 10 such
cubes, would be 100 in. long (8 ft. 4 in. or 2.5 meters), 10 in. wide
and 10 in. high.
The 100,000 square would cover the floor for 100 in. on each side and
be 10 in. high.
The 1,000,000 cube would be the 100,000 square stacked 10 high and
would be 8 ft.
4 in. (2.5 meters) on each side. In other words it would be close to
the ceiling.
Older children might like to work out the weight of such a cube. We
wondered if the
floor would support its weight. Obviously a million is a lot of blocks,
and we did learn
the word ‘million’.
We went on to think about 10,000,000. That would be a line of 10 of the
cubes holding
a million blocks, a line 83 ft. 4 in. (25 meters) long, 8 ft. 4 in.
wide, and the same high.
This would have to be made outside.
100,000,000 blocks would form a square of cubes of a million blocks
each, 83 ft. 4 in.
on each side, and 8 ft. 4 in. deep.
1,000,000,000, or a billion, blocks would form a cube, 83 ft. 4 in. (25
meters) on each
side. Obviously a billion is far, far bigger than a million; a 1000
times bigger.
10 billion would form a line of billion cubes 833 ft. 4 in. long, 83
ft. 4 in. high and wide.
100 billion would be 10 such lines side by side and form a square of a
billion cubes, 833
ft. 4 in. on each side, and 83 ft. 4 in. deep.
(My country’s debt would be a stack of about 6 such squares, if each 1
in. cube
represented one of our dollars.)
A trillion, or 1,000,000,000,000 , would be a cube formed by a stack of
10 such
squares, and would be 833 ft. 4 in. on each side.
There is a system at work here. Every time a 0 is added to the right of
the 1, or any
number, it becomes worth ten times more.
13.
A ten is ten ones.
A hundred is a hundred ones, or ten tens.
A thousand is a thousand ones, or ten hundreds.
A million is a thousand thousands.
A billion is a thousand millions.
A trillion is a thousand billions.
When children begin to multiply they can see this as:
10 = 10 x 1
100 = 10 x 10 x 1
1,000 = 10 x 10 x 10 x 1
10,000 = 10 x 10 x 10 x 10 x 1
100,000 = 10 x 10 x 10 x 10 x 10 x 1
1,000,000 = 10 x 10 x 10 x 10 x 10 x 10 x 1
1,000,000,000 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 1
1,000,000,000,000 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x
10 x 10 x 1
Probably I’ve gone on with this longer than you would wish to, but I’ll
end this with a
note that once you get to decimals, to the right of the one's place,
you will find there is
a mirrored symmetry centred on 1.
To the left of 1, each numeral represents a number of 10’s. To the
right of 1, each
numeral represents a number of 1/10 ths. Instead of each place becoming
worth 10
times more for each move to the left, the places to the right are worth
10 times less.
Before leaving this section on writing numbers, and understanding their
value, you
should note that I have used commas to divide the large numbers into
groups of 3
places. This is not always done, but it is an aid to reading, and
understanding.
Remember, children have learned to scan to the right. Understanding and
reading a
number requires one to judge a number of places from the right back to
the left.
Commas help.
Writing large numbers spoken by others, or thought of, is aided by the
knowledge that
a comma goes after each new name beyond ‘hundred’. There is a pattern.
No matter how large the number, it has a hundred's, ten's, and one's
place, followed
by a name such as trillion, billion, million, thousand. Only with the
smallest three
places do we not add a name. It is understood to be ‘ones’.
Reading Numbers:
There is a system to reading numbers, but it isn't so simple and
consistent as that for
writing numbers.
First, the numerals 0 to 11 have unique names: zero, one, two, three,
four, five, six,
seven, eight, nine, ten, eleven.
14.
Twelve has a hint in of two in it as it starts the same, as does
twice. Thirteen also
starts like three, and like third. Fourteen has four in it. Fifteen
hints at five through fifth.
Sixteen, seventeen, eighteen and nineteen all have their number of ones
in their
name. From thirteen on the ‘teen’ hints at the ten in the number.
Learning to read these numbers is no easy task for young children, even
if reading
them only implies knowing what to say when looking at the numerals, as
17.
Then we get to twenty and the task becomes much simpler as a real
system starts to
operate.
We now have to learn twenty, thirty, forty, fifty, sixty, seventy,
eighty, and ninety.
All of these are related to what we’ve learned before.
We also have to learn that after each of these names we can use all the
names from
one to nine before we go to the next new name.
(A counting chart with a first row from 0 to 9, a second row of 10 to
19, and so on,
is a great place to note how all the first row gets used over and over
again.
Children like to look for patterns, and its easy to see the columns of
0, of 1, of 2, etc.,
beside the changing numerals in the ten’s place. It’s also easy to look
down these
columns and see the numerals from 1 to 9 appearing beside each 0, 1, 2,
...
I liked to start my first row with 01,02,03, ... . This way the
numerals 0 to 9 in the ten’s
place appeared before each 0 to 9 in the one’s place.)
Then we get to the hundred's place. Children are usually delighted to
learn how far
they could possibly count once they have mastered everything to one
hundred.
There are no new names until we get to one thousand.
There’s only a flexible procedure. We may say “one hundred and one” ,
which I prefer, or
“one hundred one” for the next number. We just repeat everything from
one on to
ninetynine. After that we say “two hundred” and begin all over again.
Eventually we get to one thousand.
Here it can get confusing.
We can continue on in the same way saying, “One thousand, ....”, and
add all the
names we have used up to one thousand. Then we can say “two thousand,
....” and
so on using all the numbers before ‘thousand’ that we used up to one
thousand, and
all the numbers after ‘and’ that we used up to one thousand.
But when the number has four figures, some people choose to break from
this system.
They would read the two largest places as a two figure number and put
‘hundred’ after
it. Then they would add the last two figures.
4176 would be read as, “fortyone hundred and seventysix”.
15.
To be consistent with the way in which most large numbers are read
this would be,
“four thousand, one hundred and seventysix”, and would be written as
4,176.
This only happens when there are four figures, but it can be confusing
for young
children.
After conquering all of the thousands, we get to the million mark. That
is handled just
the same as the thousands numbers, as would be the billions, and so on.
Comparing numbers:
As I mentioned before, some children have difficulty judging the
relative size of
numbers. Which is bigger, 35 or 29? Which is smaller, 28 or 42?
Some will have difficulty deciding between 42 and 24, or 56 and 65.
These children need to really focus on the value each numeral has in
its place.
They may have difficulty seeing the numerals as separate entities, each
with its own
value. They may have trouble deciding which numeral is on the left, or
the right.
Take 35 and 29. Children have to learn that the place that decides
their relative value
is the place that is worth the most, the ten’s place. The one’s place
has no bearing on
the question, “Which is bigger (smaller)?” They have to come to see
that it's a
question of which number will be worth more (less), one with 3 tens or
one with 2 tens.
Counting real objects can make this point. I liked to use dozen type
egg cartons with
two spaces cut off. Any small objects could be counted this way and
grouped clearly
into sets of 10 with the ones remaining ungrouped.
The number line we had stretched along two walls of our classroom also
helped.
The larger number was farther on, the smaller was farther back.
Our number chart helped in the same way. The larger number was farther
down, the
smaller was farther up.
As I also mentioned before, we had activities where we practised
judging the larger or
smaller number, or where we intentionally formed a larger or smaller
number with two
given numerals.
I’ve spent a lot of time talking about something that doesn’t get a lot
of space in math
texts. I did so because, in my experience, children can have real
problems working
with larger numbers simply because they have not completely come to
terms with the
meaning of place value. They don’t have the ability to look at a two
figure number and
see it as one amount in one sense, and as two separate amounts in
another.
That is they can’t comfortably switch between 43 and 4 tens, 3 ones.
These children have difficulty with the more complex two figure adding
and subtracting
questions, and they certainly will have difficulty with multiple figure
multiplication and
division, particularly division by two or more figures.
Time spent on understanding numbers is time well spent.
16.
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