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

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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

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Adjective Noun 0 1 2 3 4 5 6 7 8 9 Adjective Noun © 2006 Herbert I. Gross next

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

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0123 In this case, 2 is a noun that names the point P. P © 2006 Herbert I. Gross next

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0123 In this case, 2 is an adjective that modifies (measures) the distance between points Q and P. 2 PQ © 2006 Herbert I. Gross next

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Most of us see numbers as adjectives. That is, we’ve seen: 3 people 3 apples 3 tally marks 123 123 123 © 2006 Herbert I. Gross next

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

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

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

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

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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”). © 2006 Herbert I. Gross next

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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. © 2006 Herbert I. Gross next

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First Fundamental Principle 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). © 2006 Herbert I. Gross next

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

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

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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). © 2006 Herbert I. Gross next

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Second Fundamental Principle 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). © 2006 Herbert I. Gross next

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

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

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

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4 + 2 = 6 6x6x 4x + 2x = ? when the adjectives modify the same noun. xxxxxx 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. © 2006 Herbert I. Gross next

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

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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 next

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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 next

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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) © 2006 Herbert I. Gross next

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

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

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Compare with the following traditional recipe. 300 × 2,000 = 600 1) Multiply the non zero digits.,000 2) Annex the total number of zeros. © 2006 Herbert I. Gross next

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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. © 2006 Herbert I. Gross next

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

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First 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 next

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Second 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 next

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Third 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). © 2006 Herbert I. Gross next

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