#1 t introduction to adjective noun

Introduction to the

Adjective/Noun Theme.

Introduction to the

Adjective/Noun Theme.

© 2012 Math As A Second Language All Rights Reserved

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Taking the Fearout of Math

The basis of “Math as a Second Language” is that most students see numbers as

quantities.

If you ask students to tell you what the number 3 is, they

might hold up 3 fingers.

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In other words, we have seen 3 fingers, 3 apples, 3 tally marks, etc. but never

“threeness” by itself.

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A quantity is a phrase consisting of an adjective and a noun.

The adjective is a number, and the noun is the unit.

Definition

3 fingers is a quantity in which the adjective is 3 and the noun (unit) is fingers.

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In a similar way, 3 inches is a quantity in which the adjective is 3, and

the noun (unit) is inches.

Definition

As quantities, 3 fingers is not the same as 3 inches. However, as adjectives, the “3”

in “3 fingers” means the same thing as the “3” in “3 inches”.

Key Point


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Definition

Hence, at least in English grammar, it is rather vague for someone to say

“This is a blue”.

The above concept transcends mathematics. Although a blue pencil

doesn’t look like a blue sweater, the adjective “blue” means

the same thing in each case.


nextnextWith the above concept in mind, our

innovative approach to teaching basicmathematics, which we call

“Mathematics as a Second Language”, is to introduce numbers in the same way

that people from all walks of life use them; namely as adjectives that modify nouns.

Our technique is to show that by using this concept, all of basic arithmetic can be done by just knowing the addition and multiplication tables from 0 through 9.


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The greatest obstacle to this approach is the tendency for presenting numbers to students only in the form of adjectives. That is, we often will talk about 3 without reference to what noun 3 is modifying.1


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1 In our opinion it is amazing how much clearer the various computations in both arithmetic and algebra become when students are allowed to visualize the adjectives

as modifying nouns of their own choosing.

Since the noun is usually omitted, we have to understand a few things about quantities.

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When we write such apparently simple statements as 1 = 1, we are assuming

that the 1 on one side of the equal sign is modifying the same noun as the 1 on

the other side of the equal sign.

First…

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1 inch ≠ 1 mile2,

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2 To negate a relationship, it is a common mathematical procedure to put a “slash mark” through the symbol that expresses the relationship. Thus, to negate a

statement such as b = c, we would write b ≠ c, which we read as “b is not equal to c” or “b is unequal to c”.

even though as an adjective the 1 that modifies “inch”

means the same thing as the 1 that modifies “mile”.

Secondly…

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On the other hand, as adjectives 12 ≠ 1, but it is true that 12 inches = 1 foot.

There are other interesting things that occur when we study the arithmetic of quantities

that we will mention briefly here but explore in greater detail as the course unfolds.

Thirdly…

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When we write that 3 + 2 = 5, we are assuming that 3, 2, and 5 are modifying

the same noun.


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3 Of course if the nouns are present, it is possible that 3 + 2 = 5 even if the nouns aren’t all the same. For example, 3 dimes + 2 nickels = 5 coins. However, if we are

thinking in terms of the amount of money, 5 coins doesn’t mean the same things as 3 dimes and 2 nickels. On the other hand, if we are thinking in terms of the number of

coins it does make sense to replace “dimes” and “nickels” by “coins” and write3 coins + 2 coins = 5 coins.

3 dimes + 2 nickels = 40 cents, but as adjectives it is false that 3 + 2 = 40.3

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Why it is Important!

For example, young students might be overwhelmed by an addition problem

such as 3,000,000,000 + 2,000,000,000 because of the number of digits.

The fact that 3 + 2 = 5 whenever 3, 2, and 5 modify the same noun is extremely important because it can be used to

explain many things in a simple manner.


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Based on how we add quantities, one does not have to know what a gloog is to know that…

3 gloogs + 2 gloogs = 5 gloogs.

However, this problem is simply theplace value version of 3 billion + 2 billion for which the answer is 5 billion because

the 3, 2, and 5 are each modifying “billion”.


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In demonstrating that…3 dimes + 2 nickels = 40 cents,

we changed dimes and nickels to a common denomination (cents).

Something similar to this occurs in a beginning algebra course when students are asked to simplify 3x + 2x. We do not

have to know what number x represents in order to know that 3 of them plus 2 more of

them is 5 of them.


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For example, to add 3/7 and 2/5, think of the problem as being written in the

form 3 sevenths + 2 fifths.

The same thing happens when we add fractions.


We cannot add the 3 and the 2 because they are modifying different units

(sevenths and fifths).

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On a report card if you got 3 A’s and 2 B’s you do not say that

you got 5 AB’s. You simply say that you got 3 A’s and 2 B’s.4


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4 Schools have solved the problem of adding A’s and B’s by going to a 4.0 grade point scale. An A is worth 4 points and a B is worth 3 points. Without going into how

the computation is formed, the student with 3 A’s and 2 B’s gets a GPA (grade point average) of 3.6.

Report Card

ASocial Studies

AScience

AMathematics

BLanguage

BReading

4321Grading Period

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3 feet × 2 pounds = 6 foot pounds 5

The statement 3 × 2 = 6 is always true, but what the 6 modifies depends on what

the 3 and the 2 are modifying.


3 kilowatts × 2 hours = 6 kilowatt hours

3 hundred × 2 thousand = 6 hundred thousand

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5 When we multiply 2 quantities, we multiply the two adjectives (numbers) to obtain the adjective part of the product, and we multiply the 2 nouns (which we do my

writing them side by side) to obtain the noun part of the product.

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Why it is Important!

…students mechanically multiply the 3 by the 2 to obtain 6 and then annex the total

number of 0’s to obtain 600,000.

In doing multiplication problems of the form…

300 × 2,000


However, as seen above, our adjective/noun theme gives us the answer

in an easy to understand format.

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In terms of our adjective/noun theme, the reason is that the numerators are the adjectives and the denominators

are the nouns.6

In multiplying two fractions, we multiply the two numerators to obtain the numerator of the product, and we multiply

the two denominators to obtainthe denominator of the product.


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6 The rule for multiplying two fractions might seem “self evident”. However, the “rule” doesn’t work when we add two fractions. Namely, we can only add the numerators (i.e., the adjectives) if they modify the same noun (i.e., denominator).

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However, using our above “rule”, when we multiply 3x by 2y, we multiply 3 by 2 to

obtain 6 and we multiply x and y (which we may view as the nouns) by

writing them side by side.

In algebra, if we are given a problem such as 3x + 2y, students often want to add the 3 and 2, not recognizing that the 3 ismodifying x and the 2 is modifying y.


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In other words… 3x + 2y ≠ 5xy, but 3x × 2y = 6xy.

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It follows rather simply that…3 tens + 2 tens = 5 tens.

A rather nice way to have students see the difference between adding and

multiplying is to have them compare how we add 3 tens and 2 tens with how we

multiply 3 tens by 2 tens.


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However, 3 tens × 2 tens ≠ 6 tens.

Rather, 3 tens × 2 tens = 6 “ten tens”.

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And since ten tens is equal to a hundred we see that…

3 tens × 2 tens = 6 hundreds.7

According to our rule, (multiply the adjectives and multiply the nouns)…

3 tens × 2 tens = 6 “ten tens”.


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7 Don’t confuse 3 tens × 2 tens with 3 × 2 tens. If we take 2 tens, 3 times (that is 3 × 2 tens) the answer is 6 tens.

However, 3 tens × 2 tens = 30 × 20 = 600 = 6 hundred.

#1 t introduction to adjective noun

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