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The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard.

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Presentation on theme: "The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard."— Presentation transcript:

1 The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 4

2 Signed Numbers: Subtracting Signed Numbers Signed Numbers: Subtracting Signed Numbers +n+n -b-b © 2007 Herbert I. Gross next

3 To read 5 – 3 = 2 as “5 take away 3 is 2” is not a problem. © 2007 Herbert I. Gross next After all, if there are 5 candies in a dish; the most you can “take away” is 5 candies, while the least number of candies you can “take away” is none. On the other hand, to read 5 – - 3 as “5 take away negative 3” would be a harder pill to swallow.

4 © 2007 Herbert I. Gross next To avoid the “take away” dilemma we agree to read “5 – 3” as “the number which we must add to 3 to obtain 5 as the sum”. This makes it much easier to give a “sensible” meaning to + 5 – - 3. Namely, + 5 – - 3 is the number we have to add to - 3 in order to obtain + 5 as the sum”. This definition does not contradict our “take away” interpretation of 5 – 3. Namely, if we take away 3 from 5, what’s left is the number (2) we have to add to 3 to obtain 5 as the sum.

5 Recall, for example, that before the advent of electronic cash registers: if you bought something for $6.85 and paid for it with a $10-bill, the clerk would give you change by adding-on to $6.85 the amount necessary to make $10. © 2007 Herbert I. Gross next Note

6 That is, he would say “$6.85” and then start giving you money by counting something like “and a nickel makes $6.90 and a dime makes $7 and three $1’s make $10”. © 2007 Herbert I. Gross next Note $6.85 +$.05 +$.10 +$1.00 + $1.00 $6.85 +++ $2.00 $3.00 $3.10 $3.15 = $10.00 next

7 When we view subtraction in this way, we are finding the “gap” between the two numbers. © 2007 Herbert I. Gross next As an aside that will become quite important in subtracting signed numbers: it is important to keep in mind that, as the years go on, the difference in their ages will always be 2 years. That is, the gap remains constant. next For example: Suppose Mary is 5 years old and John is 3 years old. To find the difference in their ages (that is, the gap between the two ages), we subtract John’s age from Mary’s age.

8 An Application to Whole-Number Subtraction © 2007 Herbert I. Gross next Some people have trouble with the regrouping (“borrowing ”) concept when given a problem such as 1,000 – 278. One way to deal with this more easily is to subtract 1 from each number; thus, keeping the gap the same. next 1000 – 278 999 722722 – 1 The gap between the numbers stays the same. – 277 next

9 To help visualize the gap concept, suppose one museum item is 1,000 years old and another is 278 years old. Then, the gap between their ages is given by 1,000 – 278. © 2007 Herbert I. Gross next Notice that this gap is the same as it was the year before, when one item was 999 years old and the other was 277 years old. In other words; 1,000 – 278 = 999 – 277. Even without borrowing, we easily see that 999 – 277 = 722. Hence; 1,000 – 278 is also equal to 722. next

10 This same application also helps simplify the problem of subtracting a negative number from a given number. Suppose, for example, that we are given the following problem… + 8 – - 5 = ? © 2007 Herbert I. Gross next This really asks us to find the gap between - 5 and + 8; let’s do this on the number-line. 0 +1+1 +2+2 +3+3 +4+4 +5+5 +6+6 +7+7 +8+8 +9+9 + 10 + 11 + 12 + 13 + 14 -1 -2-2 -3-3 -4-4 -5-5 -6-6 -7-7

11 © 2007 Herbert I. Gross next So imagine one person standing at the point - 5 and a second person standing at the point + 8 0 +1+1 +2+2 +3+3 +4+4 +5+5 +6+6 +7+7 +8+8 +9+9 + 10 + 11 + 12 + 13 + 14 -1 -2-2 -3-3 -4-4 -5-5 -6-6 -7-7

12 © 2007 Herbert I. Gross next If each person moves 5 units to the right (moves + 5), the person at - 5 will get to 0, while the other has gone from + 8 to + 13. 0 +1+1 +2+2 +3+3 +4+4 +5+5 +6+6 +7+7 +8+8 +9+9 + 10 + 11 + 12 + 13 + 14 -1 -2-2 -3-3 -4-4 -5-5 -6-6 -7-7

13 © 2007 Herbert I. Gross next The gap between them (which was between - 5 and + 8) remains the same, and is therefore equal to the gap between 0 and + 13. 0 +1+1 +2+2 +3+3 +4+4 +5+5 +6+6 +7+7 +8+8 +9+9 + 10 + 11 + 12 + 13 + 14 -1 -2-2 -3-3 -4-4 -5-5 -6-6 -7-7 13 next In terms of subtraction… + 8 – - 5 = + 8 + + 5 = + 13

14 The fact that the answer is + 13 tells us two things… © 2007 Herbert I. Gross next Note (1) The length of the gap between - 5 and + 8 is 13, and (2) to get from - 5 to + 8, we have to move in the positive direction. In other words… the directed distance in going from - 5 to + 8 is given by + 8 – - 5 next

15 If instead we measured the distance in the negative direction… © 2007 Herbert I. Gross next Note The person standing at + 8 moves 8 units to the left to get to 0 and, keeping the gap the same, the person at - 5 also moves 8 units to the left, thus arriving at the point - 13. The gap is still 13, but the direction is now from right to left. next

16 More visually in terms of a number line, + 8 – - 5 is the directed distance in going from - 5 to +8… © 2007 Herbert I. Gross next Note 0 +1+1 +2+2 +3+3 +4+4 +5+5 +6+6 +7+7 +8+8 +9+9 + 10 + 11 + 12 + 13 + 14 -1 -2-2 -3-3 -4-4 -5-5 -6-6 -7-7 0 +1+1 +2+2 +3+3 +4+4 +5+5 +6+6 +7+7 +8+8 +9+9 + 10 + 11 + 12 + 13 + 14 -1 -2-2 -3-3 -4-4 -5-5 -6-6 -7-7 …while - 5 – + 8 is the directed distance in going from + 8 to - 5. + 13 - 13

17 More generally in terms of a number line a – b is the directed distance in going from b to a. © 2007 Herbert I. Gross next Note a – b b a

18 Similarly, b – a is the directed distance in going from a to b. © 2007 Herbert I. Gross next Note b a b – a

19 Therefore for any two numbers a and b, a – b and b – a have the same magnitude, but in different directions. In summary: while a + b = b + a, a – b and b – a are opposites. So keep in mind the order is very important when we subtract. © 2007 Herbert I. Gross next Note

20 Our definition of subtraction also gives us a nice connection between arithmetic and geometry. © 2007 Herbert I. Gross next Note Namely: if the signed numbers a and b represent two points on the number line, a – b represents the directed distance from b to a while b – a represents the directed distance from a to b. next

21 In terms of going from the concrete to the abstract, it’s not difficult to count the “steps” we have to take in getting from, say, - 2 to + 3. © 2007 Herbert I. Gross next Special Note 0 +1+1 +2+2 +3+3 +4+4 +5+5 +6+6 +7+7 +8+8 +9+9 + 10 + 11 + 12 + 13 + 14 -1 -2-2 -3-3 -4-4 -5-5 -6-6 -7-7 You can see that you have to count 2 steps to the right to get to 0, and then 3 more to the right to get to + 3. next

22 However, if one started at the point - 123 and moved to the point + 567, it would be quite tiresome to count the number of spaces between these two numbers. © 2007 Herbert I. Gross next Special Note

23 Fortunately, the answer is given by + 567 – - 123 (= + 567 + + 123 = + 790) which tells us (without our having to draw a picture) that we have to go 790 units in the positive direction. © 2007 Herbert I. Gross next Special Note In particular… we have to first move 123 units to the right to get to 0 and then, another 567 units to the right to get to 790. next

24 As a practical example: to find the change in elevation when we go from 123 feet below sea level to 567 feet above sea level, we have increased our elevation by 790 feet. In more mathematical terms… + 567 – - 123 = + 567 + + 123 = + 790 © 2007 Herbert I. Gross next A Practical Example sea level 123 feet 567 feet 790 feet next

25 With respect to our sea-level example: one might argue that it should be clear from the context that if we reversed the order of the numbers (to - 123 – + 567, instead of + 567 – - 123), we would realize that we had subtracted the numbers in the wrong order when the answer turned out to be - 790 instead of + 790. © 2007 Herbert I. Gross next What a difference a sign makes! (Because to get from below sea level to above sea level, our elevation has to increase, not decrease.) next

26 However: if a wrong sea-level answer was merely an intermediate step in a longer problem, the error could become both huge and undetectable. © 2007 Herbert I. Gross next What a difference a sign makes! For example, the numbers, 1,590 and 10, do not differ by just a sign. However, 800 + 790 = 1,590 while 800 + - 790 = 10. In other words, it makes quite a difference whether we are adding + 790 to + 800 or adding - 790 to + 800. next

27 All of the previous discussions were actually based on the number line model. So in order to “unhitch” this lesson from any particular model, we can use the concept of the opposite or additive inverse. © 2007 Herbert I. Gross next To this end, notice that the easiest addition problem is when at least one of the numbers is zero. Hence, by adding + 5 to - 5, we obtain 0 as the sum; and if we then add + 8 to 0, we obtain + 8 as the sum. All in all, we added + 13 (i.e. + 5 + + 8) to - 5 to obtain + 8 as the sum.

28 Stated in a more mathematical format, the process for solving the subtraction problem + 8 – - 5 can be summarized by the following sequence of steps… © 2007 Herbert I. Gross next Start with -5-5= Add + 5 - 5 + + 5=0 Add + 8( - 5 + + 5) + + 8=0 + + 8= +8+8 Use the associative property. - 5 + ( + 5 + + 8)= +8+8 Therefore - 5 + + 13= +8+8 next From the last line, we see that + 13 is the number that we have to add to - 5 to obtain + 8 as the sum. Rewritten in subtraction form, + 8 – - 5 = + 13.

29 The above discussion of subtraction is usually abbreviated as the add the opposite rule. © 2007 Herbert I. Gross next Note The add the opposite rule says: subtracting a signed number is equivalent to adding its opposite. Note that this is analogous to the invert and multiply rule for division; namely, to divide by a fraction, we multiply by its reciprocal. next

30 In showing that subtracting - 5 from + 8 is equivalent to adding + 5 to +8, we have successfully justified the validity of the add the opposite rule. © 2007 Herbert I. Gross next Note next + 8 – - 5 = + 13 Check: - 5 + + 13 = + 8 next + 8 – - 5 =

31 The add the opposite rule also works effectively with a physical model, such as the profit and loss model. For example, let’s see what the next transaction must be if we want to turn a $5 loss into an $8 profit. In table form, the situation is… © 2007 Herbert I. Gross next Business TransactionProfitLoss First Transaction5 ??? Net8 next The Question Asked: - 5 + ? = + 8

32 To consider how to answer this question… ( - 5 + ? = + 8)… © 2007 Herbert I. Gross next Note Then, once we do break even: we will still have to make an additional $8 profit in order to end with the net profit of $8. So, let’s visualize the process of converting a $5 loss into an $8 profit by the following sequence of steps. next We first see that a $5 profit would be needed just to “break even”.

33 We want to convert the $5 loss into a “break even” situation. We do this by making another transaction, this one yielding a profit of $5. © 2007 Herbert I. Gross next Business TransactionProfitLoss First Transaction5 Second Transaction5 Net00 next Step 1 - 5 + + 5 = 0 next

34 We then follow the break-even point (no profit or loss) by a transaction that yields an $8 profit. © 2007 Herbert I. Gross next Step 2 Next Transaction8 Net00 Business TransactionProfitLoss First Transaction5 Second Transaction5 next

35 © 2007 Herbert I. Gross next Step 3 Net00 Business TransactionProfitLoss First Transaction5 Second Transaction5 next Next Transaction8 The previous Table can be condensed into the form…

36 All of the profits are then totaled ($5 + $8 = $13), and then the $5 loss is subtracted from the $13 total profit; thus showing that the net profit is $8. next Step 4 next Next Transaction8 Business TransactionProfitLoss First Transaction5 Second Transaction5 Net8 Total Profit13 next From the above table we see that + 8 – - 5 = + 13

37 Notice the difference between + 8 + - 5 and + 8 – - 5. © 2007 Herbert I. Gross next Reading Comprehension is Important! In terms of the profit and loss model, in the expression + 8 + - 5, + 8 is used to indicate one of the transactions. However, in the case of + 8 – - 5, + 8 represents the net profit. next

38 © 2007 Herbert I. Gross next Reading Comprehension is Important! In words… + 8 + - 5 asks, “What is the net result of an $8 profit, followed by a $5 loss?” …while + 8 – - 5 asks, “What transaction do we need in order to convert a $5 loss into a net $8 profit?” next

39 The Chip Model can be used to show how the “take away” idea can be applied to the subtraction of a negative number. © 2007 Herbert I. Gross next Note PPPPPPPP next For example, we may view + 8 – - 5 in terms of positive and negative chips. Let’s start with a collection of 8 positive chips...

40 In this form, we do not have any negative chips to take away. © 2007 Herbert I. Gross next PPPPPNNNNN However, by the add the opposite rule: if we add on 5 negative chips and 5 more positive chips the net result is 0. PPPPPPPP

41 If we now add our 5 negative chips and 5 positive chips (whose total value is 0) to the 8 positive chips that we started with, we have not changed the total value that © 2007 Herbert I. Gross next PPPPPPPP PPPPPNNNNN represents...

42 In this equivalent form, we can now “take away” the 5 negative chips to obtain... © 2007 Herbert I. Gross next PPPPPPPPPPPPPNNNNN Thus verifying that + 8 – - 5 = + 13

43 In explaining subtraction in several different ways, we may have lost sight of the forest because of the trees. In any event, the bottom line is that: if a and b are any two numbers, a – b represents the gap (directed distance) in going from b to a. © 2007 Herbert I. Gross next Summary

44 In Terms of the Number Line © 2007 Herbert I. Gross next b a a – b b – a b a next Directed Distance Directed Distance

45 Using any model that helps you visualize the result, the answer to the subtraction problem a – b is the same as the answer to the addition problem a + - b. © 2007 Herbert I. Gross next In other words, the add the opposite rule allows us to rewrite any subtraction problem as an equivalent addition problem (and, we already know how to add two signed numbers).

46 © 2007 Herbert I. Gross next There is only one number that can be added to b in order to obtain a as the sum. By definition; that number is denoted by a – b. That is… (a – b) + b = a On the other hand, the add the opposite rule says that… a – b = a + - b Note on Subtraction

47 © 2007 Herbert I. Gross next That is, (a + - b) + b should equal a. The following sequence shows that this is indeed the case… (a + - b) + b = a + - b + b =() 0 0 = next a a


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