© 2006 herbert i. gross by herbert i. gross & richard a. medeiros next the game of algebra...
TRANSCRIPT
© 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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