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Key Stone Problem… Key Stone Problem… next Set 3 © 2007 Herbert I. Gross

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You will soon be assigned five problems to test whether you have internalized the material in Lesson 3 of our algebra course. The Keystone Illustration below is a prototype of the problems you'll be doing. Work out the problem on your own. Afterwards, study the detailed solutions we've provided. In particular, notice that several different ways are presented that could be used to solve the problem. Instructions for the Keystone Problem next © 2007 Herbert I. Gross

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As a teacher/trainer, it is important for you to understand and be able to respond in different ways to the different ways individual students learn. The more ways you are ready to explain a problem, the better the chances are that the students will come to understand. next © 2007 Herbert I. Gross

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next Problem #1 What signed number is named by… ? Keystone Illustration for Lesson 3 next Answer: - 26 © 2007 Herbert I. Gross

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Solution for Problem 1: The rule is that when we add two numbers that have different signs, we subtract the lesser magnitude (in this case, 67) from the greater magnitude (in this case, 93) and keep the sign of the number that had the greater magnitude (in this case - 93). In other words the answer is - (93 – 67) or next © 2007 Herbert I. Gross

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In adding a positive number and a negative number we are never subtracting a greater magnitude from a lesser magnitude. In other words the magnitude of both and = 26. However, in the former case, the answer is - 26 because the negative number has the greater magnitude. In the latter case, the answer is + 26 because the positive number has the greater magnitude. next Note 1 © 2007 Herbert I. Gross

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next It's easy to confuse adding two numbers that have opposite signs with subtraction. Namely to find the sum of a positive and a negative number we subtract the smaller magnitude (the adjective) from the greater magnitude (the adjective) and keep the sign of the number (the noun) which has the greater magnitude. Caution © 2007 Herbert I. Gross

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next That is, we are not subtracting one signed number from another. Rather when we add a positive and a negative number, we subtract the adjective parts of the numbers. How we subtract signed numbers will be discussed in our next lesson. Caution © 2007 Herbert I. Gross

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Some Ways to Visualize the Process Using the Profit and Loss Model In this model, positive represents a profit transaction and negative represents a loss transaction. The sum of two numbers represents the net result of the two transactions. In this case, we have a loss of $93 followed by a profit of $67 (or equivalently, a profit of $67 followed by a loss of $93). The net result is a loss of $26. next © 2007 Herbert I. Gross

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Using the Temperature Model In this model, positive represents an increase in the temperature and negative represents a decrease in the temperature. In this case, the temperature decreases by 93 degrees and then increases by 67 degrees. The net change is a decrease in temperature of 26 degrees. next © 2007 Herbert I. Gross

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Some Background The profit and loss model is a nice way to see that the same rules that apply to adding whole numbers also apply to adding signed numbers. next Justifying the Rule Mathematically. If each signed number represents a transaction the net result does not depend on the order in which we list the transactions, nor does it depend on the way we group the transactions. For Example next © 2007 Herbert I. Gross

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If the three transactions are a $6 loss, a $12 profit and a $2 loss the net result is a $4 profit regardless of the order in which we read or group the three transactions. next $6 loss/negative + $12 profit/positive + $2 loss/negative $2 loss/negative + $12 profit/positive + $6 loss/negative $12 profit/positive + $2 loss/negative + $6 loss/negative $2 loss/negative + $6 loss/negative + $12 profit/positive $4 profit/positive © 2007 Herbert I. Gross

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The key to the rules for adding signed numbers, in addition to the rules we already use, is the definition that the sum of a number and its opposite is 0. This too can be seen rather easily from our physical models. next © 2007 Herbert I. Gross

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next The net result of a $3 loss and a $3 profit is that we “broke even". That is, the net result was a change of “$0”. In terms of a change in temperature, if the temperature increases by 3 degrees one hour and then decreases by 3 degrees the next hour, the net result is that there was no change in temperature. For Example next © 2007 Herbert I. Gross

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Let's now apply these ideas to the problem to show why the answer is next If the problem had been the answer would have been 0. Knowing this we rewrite - 93 as or , whereupon becomes… © 2007 Herbert I. Gross ( ) next

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( ) next We may then regroup the numbers in the above expression into the equivalent form ( ) and since = 0, we may rewrite the above as and since = - 26, we may rewrite the above as © 2007 Herbert I. Gross

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The justification for saying that - 93 = rests on the fact that the signs are nouns. Hence since 67 and 26 modify the same noun means that we simply add them and keep the common noun. In more symbolic language… 67 negative + 26 negative = ( ) negative next Note 1 © 2007 Herbert I. Gross

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The profit and loss model gives us a visual picture of the above. Namely to “neutralize” the $67 profit we may think of the $93 loss as two separate transactions, the first being a loss of $67 and the second being a loss of $26. Pictorially… ProfitLoss $67 $93 $0 $26 next Net $67

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Summary Adding Signed Numbers next Case 1: If the two numbers have the same sign, we add the two magnitudes and keep the common sign. © 2007 Herbert I. Gross Case 2: If the two numbers have opposite signs, we subtract the lesser magnitude from the greater and keep the sign of the number that had the greater magnitude. next

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