© 2006 Herbert I. Gross

byHerbert I. Gross & Richard A. Medeiros

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The Game of AlgebraPrelude to Signed

Numbers

The Game of AlgebraPrelude to Signed

Numbers

Lesson 3

In a preceding course, “Math as a Second Language”, we emphasized that most of us visualize numbers as

adjectives rather than as nouns.

This prelude to signed numbers reviews this concept.  Understanding

this presentation will make the subsequent study of signed

numbers more meaningful and easier to visualize.

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Adjective

Noun

0123456789Adjective

Noun© 2006 Herbert I. Gross

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Numbers can be viewed

either as nouns or adjectives.

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0 1 2 3

In this case, 2 is a noun that names the point P.

P

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0 1 2 3

In this case, 2 is an adjective that modifies (measures) the distance

between points Q and P.

2PQ

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Most of us see numbers as adjectives. That is, we’ve

seen:3 people

3 apples

3 tally marks

1 2 3

1 2 3

1 2 3© 2006 Herbert I. Gross

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But never “threeness”

by itself.© 2006 Herbert I. Gross

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Let’s explore this Adjective / Noun

theme.

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True or False.

1 = 1

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True or False.

1 = 1

True or False.

1inch = 1mile

False

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An amount such as 1 mile is called a quantity. A quantity such

as 1 mile consists of 2 parts.

1. The adjective (in this case the number 1).

2. The noun (in this case “mile” which is referred to as the “unit”).

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When the nouns (units) are not present, and

we write 1 = 1, we are assuming both 1’s

modify the same noun.

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First Fundamental PrincipleFirst Fundamental Principle

Language of MathWhen we write a = b

we assume that a and b modify the same noun

(units are the same).© 2006 Herbert I. Gross nextnext

True or False.

3+2 40

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True or False.

3 dimes+ 2 nickels

40 cents

True

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If the nouns do not appear, and we write

3 + 2 = 5, we are assuming

3, 2, and 5 modify the same unit (noun).

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Second Fundamental Principle

Second Fundamental Principle

Language of MathWhen we write a + b = c we are assuming that a, b, and c modify the

same noun (unit).© 2006 Herbert I. Gross

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3 + 2 = 5

3 apples + 2 apples = ?

5 apples

when the adjectives modify the same noun.

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1 + 2 = 3

1 cookie + 2 cookies = ?

3 cookies

when the adjectives modify the same noun.

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4 gloogs + 2 gloogs = 6 gloogs

For example, we do not have to know what “gloog” means

to be able to say …

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when the adjectives modify the same noun.

4 + 2 = 6

4 + 2 = 6

6x

4x + 2x = ?when the adjectives modify the same noun.

xxxx xxIn a similar way with respect to algebra, we do not need to

know what number x represents to know that 4 of them plus 2 more of them equals

6 of them.© 2006 Herbert I. Gross nextnextnextnext

True or False.3 tens × 2 tens = 6 tens

False

× = 600 30 20600 = 6 hundred

Not 6 tens© 2006 Herbert I. Gross

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True or False.3 tens × 2 tens = 6 “ten tens”

True

6 “ten tens”× =6 “ten tens”“ten tens” = hundred

6 “ten tens” = 6 hundred© 2006 Herbert I. Gross

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When we multiply two quantities, we separately multiply the

numbers (adjectives) to get the adjective part of the product, and

we separately multiply the two units (nouns) to get the noun part of the product. When we multiply two nouns we simply write them

side-by-side.© 2006 Herbert I. Gross

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Examples

1. 3kw × 2 hrs = 6kw hrs

2. 4ft × 2 ft = 8ft ft = 8 ft²

3. 5ft × 2 lbs = 10ft lbs

(measuring electricity)

(measuring area)

(measuring work)

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Third Fundamental PrincipleThird Fundamental Principle

Language of MathIf a and b are adjectives and x and y are nouns,

then (ax) × (by) = (ab) × (xy).

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Example

3 hundred x 2 thousand =

6 hundred thousand =×

6 hundred thousand

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Compare with the following traditional recipe.

300 × 2,000 =

6 00

1) Multiply the non zero digits.

,000

2) Annex the total number of zeros.

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SummarySummaryMost of us see numbers concretely in

the form of quantities.

A quantity is a phrase consisting of a number (the adjective) and the unit

(the noun).

For example, we don’t talk about a weight being 3. Rather we say 3 ounces, 3 grams, 3 tons, etc.

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In this context, our course will be based on the following three

principles.© 2006 Herbert I. Gross

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First PrincipleFirst Principle

When we say two numbers (adjectives) are equal, we assume they are modifying

the same unit (noun).

For example, 3 ounces is not equal to 3 pounds because an ounce does not equal a pound, even though 3 means

the same thing in each case.© 2006 Herbert I. Gross

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Second PrincipleSecond Principle

When we say a + b = c, we will assume that a, b, and c modify the same unit

(noun).

For example, we don’t write 1 + 2 = 379 even though 1 year +

2 weeks = 379 days. (Except in a leap year.)© 2006 Herbert I. Gross

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Third PrincipleThird Principle

When we multiply 2 quantities we separately multiply the adjectives, and we

separately multiply the units (nouns).

For example:

3 hundred × 2 million = 6 hundred million (Notice how much simpler this might seem to a beginning student than if we had written 300 × 2,000,000 = 600,000,000).

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