Part 3, More than, Less than, and Differences:
This seems to be a subject that goes in and out of favor. There were
times during my
teaching years when we did teach > (more than) and < (less than)
at some point
during the primary grades. There were times when it was not required.
The concept of more than and less than is an important part of
understanding number.
It can provide practice of place value knowledge, practice with
addition and
subtraction number sentences, and an introduction to a special
subtraction situation.
I’m sure most teachers of the youngest children have come across a
little story, or
made up one, to teach the difference between the symbols > and <
.
I like the greedy crocodile’s mouth myself. This crocodile always opens
his mouth
towards the bigger number. Then there's the big bird that does the same
thing.
On p. 8 of my Games booklet you’ll find an activity that requires young
children to
practise the ideas of ‘more than’ and ‘less than’. In the section
before this I’ve already
mentioned other activities and games requiring the same place value
judgments.
Before starting to teach this work, I think you should make children
aware that what is
being done falls under the general category of comparing two of
anything.
That is, you can say: Bill is taller than Tom. My coat is lighter than
yours. This apple is
sweeter than that one.
Of course you can say these things the other way around: Tom is shorter
than Bill.
Your coat is heavier than mine. That apple is not as sweet as this one.
Note that , with the exception of one case, all the comparison words
end with ‘er ’.
It is important that children see you can go both ways.
It is also important they see that the comparison is only about one
particular quality.
The quality we wish to judge here is number size, or value.
This work really can start with the youngest children. 5 > 3 is not
a hard concept.
It may take awhile before they are comfortable changing this to 3 <
5.
Later, children can be asked to write a comparison statement about
larger numbers,
say 59 and 61. This is a set where some children have difficulty. They
see the 9 and
think the number with 9 must be bigger.
It also requires them to make the shift from the fifties to the sixties.
I didn’t make a point of this in the previous section, but children do
have the most
problem judging relative size where one is close to the next amount of
tens, and one is
just over it. It is also the place where there can be the most
difficulty in learning to
count.
17.
Asking them to use > and < with 59 and 61 brings out this problem
so it can be dealt
with.
With objects to count at first, a number line and/or 100 chart to use
later, this is not a
hard skill for most children to master.
Differences:
At some point one can say to children, “ Yes, 5 is more than 3, but how
much
more?” “3 is less than 5, but by how much?"
One has to think of these questions in terms of real things. There are
two simple but
different situations where you might ask these questions.
Sally has 5 cookies in her lunch. Jane has 3. Who has more cookies? How
many
more?
ooooo
ooo
Put these separate sets side by side and you can see the difference.
Jim and Sam walk home together. Jim walks 5 blocks, and Sam walks 7.
Who walks
farther? How many blocks farther?
_ _ _ _ _ _ _
This time the count is along the same line, as it would be on a number
line.
(_ _ _ _ _) _ _
There are a number of strategies for working these out when they aren’t
obvious.
One can count up from the smaller amount to the larger one, or the
reverse.
In the cookie example, one can take a group of the larger one equal to
the smaller one
and then count to see how many there are beyond that amount.
One can also go to the smaller set and count on enough imaginary
cookies to make
the larger set.
In the number line example, one can count on from the smaller distance
to the larger,
or the reverse.
At some point children can learn to subtract the smaller from the
larger to find the
difference. This is a more abstract activity.
If children are familiar with number sentences they can reason out that
the smaller
amount plus the difference equals the larger amount, as 3 + __ = 5.
Or the larger amount minus the difference equals the smaller amount, as
7 - __ = 5.
18.
That is if you take away the difference from the larger, or add on the
difference to the
smaller, both would be equal.
5 > 3 , by 2 , or 7 > 5 , by 2, and 3 < 5 , by 2, or 5 < 7
, by 2 were the type of
statements I liked children to be able to make as a final answer.
With larger numbers, 61 > 49 , by 12 , since 61 - 49 = 12 might have
been
more appropriate.
At some point you will like to bring in comparisons with three or more
numbers. Here
the comparison words change. With 3, 7, 9, the largest is 9. 3
is the smallest.
This can provide good review: 3 < 7 , 3 < 9 , 7 < 9 , 9 > 7
, 9 > 3 , 7 > 3.
This might be shortened to: 3 < 7 < 9 , or 9 > 7 > 3.
Another set of questions can be more open: 3 > _____ , or 5 < __
, __ , __ , or
5 > ______ < 3.
Part 4, Measurement:
In this section I’d like to look at those places where we
have to group things according
to a rule. Mainly this is when we measure.
I’d like you to see that this is an area that needs careful attention.
We’ve already dealt with this under place value. Here we have a fixed
rule:1->|<-10.
If we get a set of 10, we group it and count it as 1 in the place to
the left.
Sometimes we have a need to go the other way, as in questions where the
1 on the
left may be changed to 10 on the right to allow subtraction to take
place.
This 1->|<-10 rule is a very common one in many
countries.
It is the basis of the metric system, and decimal based monetary
systems.
This simple rule is not always held to as we count certain types of
things, and this can
create problems for children. At the same time, dealing with other
rules can reinforce
the concept of grouping in order to count in larger sets under a
different name.
Usually children’s texts will not ask them to deal with conversions,
but some may be
part of their lives.
Take eggs, for example. We normally group them in sets of 12 and name
those sets
as a dozen.
12 is a number that has advantages over 10 for packaging. It can be
efficiently
arranged in two ways, although most often we see eggs in cartons of 2
rows of 6.
It can also be evenly divided in more ways than 10.
19.
Commerce has seen the use of dozens in packaging many other things. For
large
amounts it is common to find a dozen dozens, as in a shipment of
pencils.
Children now love to hear the name given to 12 sets of 12, a gross.
Units of time can be a problem.
To go from seconds to minutes we use a ratio of 60 to 1.
To go from minutes to hours we use the same ratio, 60 to 1.
To go from hours to days, we use 24 to 1.
But most of our clocks only show hours up to 12, so children have to
learn there is a
morning set of 12 which starts at midnight, and an afternoon set of 12
which starts at
noon.
To go from days to months children have to learn this is flexible, but 30
to 1 is often
used.
To go from days to years the ratio is usually 365 to 1, but
every fourth year this
becomes 366 to 1.
To go from months to years the ratio is fixed, 12 to 1.
After this last, it seems we finally go back to a ratio of 10 to 1.
Years become a
decade, decades become a century, and centuries become a millennium.
(A long time ago at least one nation toyed with changing to completely
metric time.)
When it comes to measuring distance or length, weight, and volume, we
in Canada
have a unique problem.
Many, many years ago our government decided it knew that what would be
best for us
was to switch to the metric system, despite the fact that our largest
trading partner was
still using, and continues to use, measures based on the English
Imperial systems.
Now we have to learn to operate in both systems, the old and the new.
For length we deal with inches to feet, a ratio of 12 to 1,
inches to yards at 36 to 1,
and feet to yards at 3 to 1. These are the measurements that
are people sized.
Then we have feet to miles at 5280 to 1, or yards to miles at 1760
to 1.
In the metric system we have millimetres to centimetres at 10 to 1,
centimetres to
decimetres at 10 to 1, and decimetres to metres at 10 to 1. (Decimetres
are seldom
used.)
Then we go from metres to kilometres at 1000 to 1.
Still we see children’s rulers measuring 30 cm. long since this is
about equal to 12 in.,
a handy length to use. In fact many rulers show both inches and
centimetres and
children have to learn to use the proper scale.
Classrooms will be equipped with metre sticks, and sometimes yard
sticks, for longer
distances. 3 rulers may make the same distance as 1 yardstick, but they
won’t equal
one metre stick.
20.
Ask children their height, or adults, and they will probably answer in
feet and inches.
Of course these units of measurement are also linked to measurements of
area, and
volume. Such forms of measurement each have their own unique names and
ratios to
deal with as well, but young children seldom have to deal with them.
The measurement of weight also uses two systems. Officially ours is the
metric one,
with grams and kilograms. The reality is that when grocery shopping one
finds both in
use. Many people feel the gram is too small and the kilogram is too
large. Meat
especially is often labelled as so much per pound, and so much per
kilogram.
Besides that, a lot of food items come from places where ounces and
pounds are
used.
Ask Canadian children or adults what they weigh and their answer will
likely be in
pounds. I know mine would be. Bathroom scales don’t wear out that
quickly.
In any event we have to be prepared to switch between ounces and pounds
with a
ratio of 16 to 1.
Grams and kilograms, as the name indicates, have a ratio of 1000 to
1.
Young children won’t have much to do with the larger units of both
system, which is
just as well as they can be confusing.
Liquid volume is another common type of measurement where
attention needs to be
paid, and it is even more confusing. (The flight of a large jet full of
people almost
ended tragically because of an error in converting the measurement of
fuel on board.)
In Canada, we used to, and to some extent still do, use the Imperial
system.
We measured liquid volume with ounces, cups, pints, quarts, and gallons.
Ounces to cups had a ratio of 8 to 1. Cups still are commonly
used in cooking.
Ounces to pints had a ratio of 20 to 1.
Pints to quarts had a ratio of 2 to 1.
Cups to quarts had a ratio of 5 to 1.
Ounces to quarts had a ratio of 40 to 1.
Quarts to gallons had a ratio of 4 to 1.
In the U.S.A. the same words are used, but they don’t always mean the
same.
Ounces to cups stay the same.
Cups to pints became 2 to 1, which meant a U.S. pint is 16
ounces, instead of 20.
21.
Pints to quarts stayed 2 to 1, so a U.S. quart became 32
ounces, instead of 40.
Quarts to gallons stayed the same, so a U.S. gallon had 128 ounces,
instead of 160.
Shopping for liquids in Canada means one can find containers measured
with
the metric millilitres and litres, with a ratio of 1000 to 1.
One can also find containers holding an Imperial pint or quart, or a
U.S. pint or quart.
Large quantities, such as pails of paint, may come in Imperial or U.S.
gallons, or litres.
Ever since this government’s decision to switch
measuring systems there has been a lot of money
expended for charts, pamphlets,gadgets, and even calculators to help
citizens make the switch between
the metric system and the Imperial system.
Despite all these we still have puzzles to work out. In the U.S. for
example, cars are rated as achieving so
many miles per gallon of fuel. In Canada, our gallons are different,
our cars measure distance in kilometres,
and our gas stations sell fuel in litres. Canadian cars are rated as
using so many litres per 100 kilometres.
If you want some work, try comparing the reported mileage of cars from
each of the two countries.
Following recipes can also be a challenge, as when the recipe calls for
so many ounces of chocolate chips,
and the stores only have bags measured in grams.
Besides just becoming familiar with the units we use to measure,
you may ask children
to convert some of the simpler measures from large units to small
units, or the reverse.
If you were working with cups and quarts for example, you could ask how
many quart
jars would be filled by 12 cups of water. Depending on which quart, you
would be
asking how many times you could get 5 cups out of 12, or 4 cups out of
12, and how
many would be left over.
You could ask how many cups you could get from 2 quarts.
For the youngest children these examples would require working with
real objects.
I have not seen a text that spends a lot of time with converting units
of measure.
Still, working with measurement provides practise opportunities for
children who have
begun multiplication and division, and for those working on multiple
figure addition
and subtraction, which will be the subject of Part 6.
3 metres, 5 centimetres 3 m. , 18 cm.
+2 metres, 8 centimetres - 1 m. ,
9 cm.
These questions require one to work in columns, keeping the like units
together, just
as required by the more abstract work in Part 6.
One last topic in this area is money. Money in a lot of countries is
related to decimal
fractions. I don't intend to say more than this: there is an
opportunity to practice
changing from one money form to another, and to relate the patterns of
our place value
system to the right of the one's place where we get a mirror image of
the left side.
22.