1. Key Stone Problem… Key Stone Problem… next Set 4 © 2007 Herbert I. Gross

2. You will soon be assigned five problems to test whether you have internalized the material in Lesson 4 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

3. 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

4. next Problem #1a What signed number is named by… 20 – - 7 – - 8 ? Keystone Illustration for Lesson 4 next Answer: + 35 © 2007 Herbert I. Gross

5. Solution for Problem 1a: Prelude: As written, the expression is ambiguous. That is, it makes a difference whether we read it as (20 – - 7) – - 8 or as 20 – ( - 7 – - 8). Recall that in Lesson 2 we agreed to use the convention PEMDAS whenever there was an ambiguity caused by the lack of grouping symbols. By this agreement when only addition and/or subtraction appear in the same expression we perform the operations in order from left to right. next © 2007 Herbert I. Gross

6. Solution for Problem 1a: Hence we read Problem 1(a) as if it were written… (20 – - 7) – - 8. next © 2007 Herbert I. Gross (20 – - 7) – - 8 To solve the expression, we work within the parentheses first… next

7. Solution for Problem 1a: next © 2007 Herbert I. Gross 20 – - 7 By the “add the opposite” rule, we may rewrite… next + + 7 Recalling that 20 means the same thing as + 20 we may rewrite the expression. + = + 27

8. Solution for Problem 1a: next © 2007 Herbert I. Gross (20 – - 7) – - 8 If we now replace 20 – - 7 by + 27 we obtain… next + + 8 Which by the “add the opposite” rule is equivalent to… + 35. + 27 = + 35

9. The “add the opposite” rule is derived from the definition of subtraction being the inverse of addition. That is, 20 – - 7 means the number we must add to - 7 to obtain 20 as the sum. So we first have to add + 7 to - 7 to obtain 0 as the sum, and we then add 20 to 0 to obtain 20 as the sum… next © 2007 Herbert I. Gross Note That is… - 7 + + 7 + + 20 = next ( ) 0 20

10. next Problem #1b What signed number is named by… 20 – ( - 7 – - 8) ? Keystone Illustration for Lesson 4 next Answer: + 19 © 2007 Herbert I. Gross

11. Solution for Problem 1b: To find the signed number that is represented by… + 20 – ( - 7 – - 8). next © 2007 Herbert I. Gross 20 – ( - 7 – - 8) We start within the parentheses first… next

12. Solution for Problem 1b: next © 2007 Herbert I. Gross - 7 – - 8 By the “add the opposite” rule, we may rewrite… + + 8 And if we now replace the expression - 7 – - 8 by + 1. = + 20 – (– - 8) +1 +1 Thus, evaluating the expression we get... = 19 next

13. Part (b) looks exactly like part (a) when the grouping symbols are omitted. The grouping symbols determine whether the term - 8 is increasing or decreasing the value of the expression. More specifically, when we write…(20 – - 7) – - 8 or equivalently (20 – - 7) + 8. We see that the 8 increases the value of the expression by 8. next © 2007 Herbert I. Gross Note

14. On the other hand when we write 20 – ( - 7 – - 8) or equivalently 20 – ( - 7 + 8). We see that the 8 increases the amount we are subtracting by 8 thus decreasing the value of the expression by 8. next © 2007 Herbert I. Gross Note next The difference between subtracting 8 and adding 8 results in a total difference of 16 and this is precisely the difference between the answer to part (a) and the answer to part (b). 35 – 19 = 16

15. If we prefer to perform the subtraction without using the “add the opposite” rule, recall that we are looking for what has to be added to - 7 – - 8 to obtain 20 as the sum. next © 2007 Herbert I. Gross Note next If we add - 7 to + 7 the sum is 0, and to “undo" subtracting - 8 we add - 8. In summary ( - 7 – - 8) + ( + 7 + - 8) = 0 Therefore… Hence ( + 7 + - 8 + 20) or 19 is the number we must add to - 7 – - 8 to obtain 20 as the sum. ( - 7 – - 8) + ( + 7 + - 8) + 20) = 20

16. next Problem #1c How are… 20 – ( - 7 + - b) and 27 + b related? Keystone Illustration for Lesson 4 next Answer: They are equivalent. © 2007 Herbert I. Gross

17. Solution for Problem 1c: 20 – ( - 7 + - b) means the number we must add to ( - 7 + - b) to obtain 20 as the sum. To this end we may mimic the steps we used in our last note above. More specifically, if we add + 7 to - 7, the sum is 0, and if we add b to - b, the sum is 0. next © 2007 Herbert I. Gross next -

18. Solution for Problem 1c: In other words… ( - 7 + - b) + ( + 7 + + b) = 0 next © 2007 Herbert I. Gross next Therefore… ( - 7 + - b) + ( + 7 + + b) And … ( + 7 + + b + + 20) = b + 27 + 20) = 20 (b + + 7

19. The significance of part (c) is that it indicates the power of paraphrasing. More specifically, given any value for b it is much simpler to add 27 to b than to change the sign of b, add it to - 7 and then subtract this result from 20. next © 2007 Herbert I. Gross Note

20. For example, if we start with 15 and take its opposite ( - 15) then add – 7 ( - 22) and then subtract this from 20 (20 – - 22), we obtain 42 as the answer. However the same answer could have been obtained simply by starting with 15 and adding 27. next © 2007 Herbert I. Gross Note Clearly, the fewer steps that are involved in a computation, the less the chances are for making a computational error. next