Transcript 2. Quantities Versus Numbers

Taking the Fear
out of Math
next
What’s My Noun?
Quantities
vs.
Numbers
Adjective/Noun
© Math As A Second Language
All Rights Reserved
next
In our previous lesson, we
emphasized the importance of this fact…
When we add two quantities the
correct result is the sum of the
adjectives (numbers) only if the
nouns (the units) are the same.
Even though addition tables tell us that
3 + 2 = 5, the fact is that 3 dimes plus
2 nickels is neither 5 dimes nor 5 nickels.
© Math As A Second Language
All Rights Reserved
next
To find the amount of money,
we converted both quantities to
a common unit and then added.
3 dimes + 2 nickels =
30 cents + 10 cents = 40 cents
© Math As A Second Language
All Rights Reserved
next
Note… There may be more than
one common unit.
For example, the value of 2 nickels is
1 dime. Hence, we may restate the problem
in the form…
3 dimes + 2 nickels =
3 dimes + 1 dime =
note
1
4 dimes1
In our adjective/noun theme we do not distinguish between singular and plural.
The fact is that while “dime” and “dimes” are different nouns they represent the
same unit.
© Math As A Second Language
All Rights Reserved
next
We could also have used nickels as
the common unit, in which case we could
have replaced 3 dimes by 6 nickels
to obtain…
3 dimes + 2 nickels =
6 nickels + 2 nickels =
8 nickels
And even though 40, 4, and 8 are different
adjectives (numbers), 40 cents, 4 dimes
and 8 nickels all describe the same
quantity.
© Math As A Second Language
All Rights Reserved
next
A Preview of
Coming Attractions
At first glance, our emphasis on the
adjective/noun theme might seem like
little more than just a novelty, but as we will
see throughout the study of
arithmetic, this theme can greatly improve
students’ ability to internalize
all of arithmetic.
© Math As A Second Language
All Rights Reserved
next
We will discuss this in greater
detail as the course
progresses, but for now let’s
focus on just one aspect of
how the adjective/noun theme
simplifies arithmetic algorithms
that often befuddle students.
© Math As A Second Language
All Rights Reserved
next
Note… On the Terms Numerator
and Denominator
Too often the “numerator” is introduced as
a synonym for “top” and “denominator” as
a synonym for “bottom”.
top = numerator = adjective
bottom denominator
noun
This obscures the fact that the numerator
is the adjective and the denominator is the
noun and leaves many students confused
when they are asked to add fractions.
© Math As A Second Language
All Rights Reserved
next
When adding two fractions, students
feel it is more natural to add the two
numerators to obtain the numerator of the
sum and to add the two denominators to
obtain the denominator of the sum.
For example, they would prefer that adding
1/ + 1/ would mean to do the following…
2
2
1 + 1
2
2
=
1+1
2+2
=
2
4
…which is a result that they will most
likely recognize as being incorrect”.
© Math As A Second Language
All Rights Reserved
next
However, once the proper definitions
are given for numerator and denominator,
these students will quickly realize that this
is not the correct way to add fractions.
Namely, when they are called upon
to compute a sum such as
6 nickels + 2 nickels, they would add the
two adjectives (6 + 2) but then keep the
common denomination (nickels).
Even though it is true that a nickel and a
nickel is a dime, in no way would they
have felt that the answer was 8 dimes.
© Math As A Second Language
All Rights Reserved
next
Guess My Noun
Guess My Noun is a fun way to
reinforce the notion of the
adjective/noun theme and how
3 + 2 = 40 can be a true statement.
For example, if there are certain facts you
want the students to know (such as the fact
that 7 days = 1 week) you might ask them to
supply the nouns for…
7 ______ = 1 ______.
© Math As A Second Language
All Rights Reserved
next
If we want to emphasize that there are
12 months in a year, fill in the blanks for the
missing nouns in…
months = 1 ______.
year
12 ________
Notice that there can be more than
one correct answer.
12 inches = 1 foot
12 eggs = 1 dozen eggs
© Math As A Second Language
All Rights Reserved
next
You might wonder why 12 was chosen
rather than 10 for the number of inches
in a foot. Such a question can lead to
the “discovery” of whole number
fractional parts.
For example, since “teen” means plus ten,
one might naturally assume that the first
teen should come after ten. That is, the
number we call eleven should have been
called “oneteen”.
So why does the first teen come after
twelve not ten?
© Math As A Second Language
All Rights Reserved
next
As surprising as it might seem, the
concept of ten was not considered to be
important until the advent of place value.
Until that time, people preferred to avoid
the need for using fractions whenever
possible. Therefore, since 12 had more
proper divisors than 10, it meant that by
having a foot consist of 12 inches, more
fractional parts of a foot would be a
whole number than if there had been
10 inches in a foot.
© Math As A Second Language
All Rights Reserved
next
Aside from the “folk lore”
values of these examples, it
might be reassuring to
students for them to know
that hundreds of years ago
people were already learning
how to invent nouns that
would minimize the need for
using fractions.
© Math As A Second Language
All Rights Reserved
next
Depending on the grade level you can ask
more difficult questions by having the
students add different quantities such as…
2 _____
feet
+
12 inches
______
= 1 ______
yard
You may prefer additional examples, and
you should feel free to create problems of
your own choosing. Students also may
want to create their own problems to
challenge their classmates.
© Math As A Second Language
All Rights Reserved
next
Examples such as 3 + 2 = 40 are too
sophisticated for children in grades K – 2.
Instead, colored rods of different lengths
may help to reinforce the adjective/noun
theme at the lower grade levels.
© Math As A Second Language
All Rights Reserved
next
From a different perspective,
you could see that…
1 blue rod = 12 red rods
1 blue rod = 6 green rods
© Math As A Second Language
All Rights Reserved
next
1 blue rod = 4 yellow rods
1 blue rod = 3 white rods
1 blue rod = 2 purple rods
© Math As A Second Language
All Rights Reserved
next
Therefore…
1
= 2
1
= 3
1
= 4
1
= 6
© Math As A Second Language
All Rights Reserved
next
In summary…
12
6
6
4
4
3
3
2
1
2
1
4
1
© Math As A Second Language
3
2
1
1
All Rights Reserved
3
2
1
1
2
1
1
2
1
1
1
next
You could give your students examples in
adding quantities such as…
1 white rod + 1 green rod = 1 purple rod
In later grades, the above equality could
become a visual model for showing that
1/ + 1/ = 1/ (that is, 1 third + 1 sixth = 1 half).
3
6
2
© Math As A Second Language
All Rights Reserved
next
Another example might be…
5 red rods + 1 white rod = 3 yellow rods
(that is, 5 twelfths + 1 third = 3 fourths)
© Math As A Second Language
All Rights Reserved
next
Guess My Noun
If time is limited, assign them as
homework under the heading of such
phrases as “Fun With Math” and make sure
that students know that it is just for fun and
that they will not be graded. Rather they
should be encouraged to work on the
problems and share their results with the
class (as time permits).
© Math As A Second Language
All Rights Reserved
next
Guess My Noun
We know that you are under pressure to
cover a certain amount of prescribed
content, and that as a result you may feel
that there is no time for such “games” in
your class. However, our approach to
helping students to better internalize
mathematics hinges on their thorough
grasp of adding quantities using the
adjective/noun theme, and the previous
examples are fun ways in which to learn.
© Math As A Second Language
All Rights Reserved