The Game of Algebra Prelude to Signed Numbers Lesson 3 by Herbert I. Gross & Richard A. Medeiros © 2006 Herbert I. Gross next

numbers more meaningful and 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. next © 2006 Herbert I. Gross

1 2 3 5 7 4 6 8 9 Adjective Adjective Noun Noun next 9 Adjective Adjective Noun Noun next © 2006 Herbert I. Gross

Numbers can be viewed either as nouns or adjectives. next © 2006 Herbert I. Gross

In this case, 2 is a noun that names the point P. 1 2 3 In this case, 2 is a noun that names the point P. next next next © 2006 Herbert I. Gross

2 Q P 1 2 3 In this case, 2 is an adjective that modifies (measures) the distance between points Q and P. next next next © 2006 Herbert I. Gross

as adjectives. That is, we’ve seen: Most of us see numbers as adjectives. That is, we’ve seen: 3 people 1 2 3 3 apples 1 2 3 3 tally marks 1 2 3 next next next next © 2006 Herbert I. Gross

But never “threeness” by itself. next © 2006 Herbert I. Gross

Let’s explore this Adjective / Noun theme. next © 2006 Herbert I. Gross

True or False. 1 = 1 next © 2006 Herbert I. Gross

True or False. 1 = 1 True or False. 1inch = 1mile 1 = 1 True or False. False 1inch = 1mile next next next © 2006 Herbert I. Gross

1. The adjective (in this case the number 1). 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”). next next next © 2006 Herbert I. Gross

When the nouns (units) are not present, and we write 1 = 1, we are assuming both 1’s modify the same noun. next © 2006 Herbert I. Gross

First Fundamental Principle Language of Math When we write a = b we assume that a and b modify the same noun (units are the same). next next © 2006 Herbert I. Gross

True or False. 3 +2 40 next © 2006 Herbert I. Gross

True or False. 3 dimes + 2 nickels 40 cents next next © 2006 Herbert I. Gross

If the nouns do not appear, and we write 3 + 2 = 5, we are assuming 3, 2, and 5 modify the same unit (noun). next © 2006 Herbert I. Gross

When we write a + b = c we are assuming that a, b, and c modify the Second Fundamental Principle Language of Math When we write a + b = c we are assuming that a, b, and c modify the same noun (unit). next next © 2006 Herbert I. Gross

3 + 2 = 5 3 apples + 2 apples = ? 5 apples when the adjectives modify the same noun. 3 apples + 2 apples = ? 5 apples next next © 2006 Herbert I. Gross

1 + 2 = 3 1 cookie + 2 cookies = ? 3 cookies when the adjectives modify the same noun. 1 cookie + 2 cookies = ? 3 cookies next next © 2006 Herbert I. Gross

4 + 2 = 6 4 gloogs + 2 gloogs = 6 gloogs when the adjectives modify the same noun. For example, we do not have to know what “gloog” means to be able to say … 4 gloogs + 2 gloogs = 6 gloogs next next © 2006 Herbert I. Gross

4 + 2 = 6 when the adjectives modify the same noun. 4x + 2x = ? x x x x x x In 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. 6x next next next next © 2006 Herbert I. Gross

True or False. False × 30 20 = 600 3 tens × 2 tens = 6 tens 600 = 6 hundred Not 6 tens next next next next next © 2006 Herbert I. Gross

True or False. True = 6 × “ten tens” 6 “ten tens” “ten tens” = hundred 3 tens × 2 tens = 6 “ten tens” 6 × “ten tens” = 6 “ten tens” “ten tens” = hundred 6 “ten tens” = 6 hundred next next next next next next next © 2006 Herbert I. Gross

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. next © 2006 Herbert I. Gross

Examples 1. 3kw × 2 hrs = 6kw hrs 2. 4ft × 2 ft = 8ft ft = 8 ft² (measuring electricity) 2. 4ft × 2 ft = 8ft ft = 8 ft² (measuring area) 3. 5ft × 2 lbs = 10ft lbs (measuring work) next next next © 2006 Herbert I. Gross

Language of Math If a and b are adjectives and x and y are nouns, then Third Fundamental Principle Language of Math If a and b are adjectives and x and y are nouns, then (ax) × (by) = (ab) × (xy). next next © 2006 Herbert I. Gross

Example 3 hundred x 2 thousand = 6 × hundred thousand = next next next next next © 2006 Herbert I. Gross

Compare with the following traditional recipe. 300 × 2,000 = 6 , 1) Multiply the non zero digits. 2) Annex the total number of zeros. next next next next next © 2006 Herbert I. Gross

Summary Most 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. next next next next © 2006 Herbert I. Gross

In this context, our course will be based on the following three principles. next © 2006 Herbert I. Gross

When we say two numbers (adjectives) are equal, we assume they are modifying the same unit (noun). First Principle 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. next next next © 2006 Herbert I. Gross

When we say a + b = c, we will assume that a, b, and c modify the same unit (noun). Second Principle For example, we don’t write 1 + 2 = 379 even though 1 year + 2 weeks = 379 days. (Except in a leap year.) next next next © 2006 Herbert I. Gross

When we multiply 2 quantities we separately multiply the adjectives, and we separately multiply the units (nouns). Third Principle 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). next next next © 2006 Herbert I. Gross