#1 t quantities vs. numbers adjective noun

What’s My Noun?What’s My Noun?

Adjective/Noun

nextTaking the Fear

out of Math

© Math As A Second Language All Rights Reserved

Quantities vs.

Numbers

Quantities vs.

Numbers

In our previous lesson, we emphasized the importance of this fact…

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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.

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When we add two quantities the correct result is the sum of the

adjectives (numbers) only if the nouns (the units) are the same.

nextnext To find the amount of money,we converted both quantities to a common unit and then added.

3 dimes + 2 nickels =

30 cents + 40 cents

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10 cents =

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For example, the value of 2 nickels is 1 dime. Hence, we may restate the problem

in the form…


3 dimes + 1 dime =

4 dimes1

Note… There may be more than one common unit.

note

1 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.

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And even though 40, 4, and 8 are different adjectives (numbers), 40 cents, 4 dimes

and 8 nickels all describe the same quantity.


6 nickels + 2 nickels =

8 nickels

We could also have used nickels as the common unit, in which case we could have replaced 3 dimes by 6 nickels to obtain…


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.

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We will discuss this in greaterdetail 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.

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Too often the “numerator” is introduced as a synonym for “top” and “denominator” as

a synonym for “bottom”.

Note… On the Terms Numerator and Denominator


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.

topbottom

numeratordenominator

adjectivenoun

= =

nextnext 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 1/2 + 1/2 would mean to do the following…


24

12

12

+ = 1 + 12 + 2

=

…which is a result that they will most likely recognize as being incorrect”.

nextnext 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.


Guess My Nounnext

Guess My Noun is a fun way to reinforce the notion of the

adjective/noun theme and how3 + 2 = 40 can be a true statement.

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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 ______.


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If we want to emphasize that there are 12 months in a year, fill in the blanks for the missing nouns in…

12 ________ = 1 ______.

Notice that there can be more than one correct answer.

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12 inches = 1 foot

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months year

12 eggs = 1 dozen eggs


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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.

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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”.

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So why does the first teen come after twelve not ten?

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As surprising as it might seem, the concept of ten was not considered to be important until the advent of place value.

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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.


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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.


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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.

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Depending on the grade level you can ask more difficult questions by having the

students add different quantities such as…

2 _____ + 12 ______ = 1 ______feet inches yard

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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.

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From a different perspective,you could see that…

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1 blue rod = 12 red rods

1 blue rod = 6 green rods

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1 blue rod = 4 yellow rods

1 blue rod = 3 white rods

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1 blue rod = 2 purple rods


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Therefore…

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2=1

3=1

4=1

6=

1

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12

12

34

6 64 4

3 3 32 2 2 2 2

1 1 1 1 1 1 1 1 1 1 1

In summary…

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You could give your students examples in adding quantities such as…

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1 white rod 1 purple rod+ 1 green rod =

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In later grades, the above equality could become a visual model for showing that

1/3 + 1/6 = 1/2 (that is, 1 third + 1 sixth = 1 half).

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Another example might be…

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5 red rods 3 yellow rods+ 1 white rod =

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(that is, 5 twelfths + 1 third = 3 fourths)

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If time is limited, assign them ashomework 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).

Guess My Noun


Guess My Nounnext

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 tohelping 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.


#1 t quantities vs. numbers adjective noun

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