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CHAPTER 3
The Four Fundamental Operations of Arithmetic
Copyright © Cengage Learning. All rights reserved.

2. Understanding Subtraction
SECTION 3.2
Understanding Subtraction
Copyright © Cengage Learning. All rights reserved.

3. What Do You Think?
Why do most current elementary texts discourage the term “borrow”?
What other contexts for subtraction are there besides “take away”?

4. Understanding Subtraction
Five subtraction problems: 1. Joe had 7 marbles. He lost 2 in a game. How many does he have left?
2. Billy has 23 marbles and Yaka has 32. How many more does Yaka have?
3. At the beginning of the week, the hospital had 32 ounces of insulin. During the week, 14 ounces of insulin were used. How much insulin did the hospital have at the end of the week?

5. Understanding Subtraction
4. Alicia was 45 inches tall at the beginning of the first grade and 53 inches at the end of the third grade. How many inches did she grow during this time?
5. The Jones farm has 35 pigs and 18 chickens. How many more pigs are there than chickens?

6. Contexts for Subtraction

7. Contexts for Subtraction
One way in which the problems are different is connected to the two contexts we discussed for addition.
Problems 1, 2, and 5 are discrete problems. Problems 3 and 4 are measurement (continuous) problems.
These problems differ in another way, and this difference is connected to the idea of parts and wholes that we find in all four operations. In “take-away” problems, a part is taken away from a whole.

8. Contexts for Subtraction
In “comparison” problems, two amounts are being compared. In some cases, the comparison is between two subsets of a large set (two parts of a whole); in other cases, the comparison is between two sets (wholes). Problems 1 and 3 are take-away problems, whereas the other three problems fit the comparison and missing-addend models.

9. Contexts for Subtraction
Figure 3.6 shows the subtraction problem 7 – 2 in the two contexts.
comparison and
missing-addend
take away
Figure 3.6

10. A Pictorial Model for Subtraction

11. A Pictorial Model for Subtraction
One way to express the commonality of all five problems just discussed is that in all cases, we have a large amount and two smaller amounts whose sum is equal to the larger amount.
As we know the pictorial model for addition: If we invert that diagram, we have a pictorial representation of a general model for subtraction (Figure 3.7).
Figure 3.7

12. A Pictorial Model for Subtraction
It is important to note that whole and part refer to the numerical values rather than the contexts. For example, in Problem 2, we are comparing two wholes and saying that one whole contains 9 more marbles than the other. We define subtraction formally in the following manner: c – b = a iff a + b = c That is, the difference between two numbers c and b is a if and only if c is the sum of a and b.

13. A Pictorial Model for Subtraction
In mathematical language, c is called the minuend, b the subtrahend, and a the difference. This model also highlights the connections between addition and subtraction problems (Figure 3.8): = = – 3 = – 5 = 3
Figure 3.8

14. Representing Subtraction with Number Lines

15. Representing Subtraction with Number Lines
Many measurement situations, such as time and distance, naturally lend themselves to a number line representation. Let’s use an example to explore this model. Joanne had 8 feet of rope and used 3 feet to stake a tent. How much rope does she have left? Represent this problem with a number line. As with addition, we can use a number line to solve the problem in either of two ways (Figure 3.9).
Figure 3.9

16. Representing Subtraction with Number Lines
As with the addition number lines, the left diagram more closely resembles the physical representation of the problem.

17. Properties of Subtraction

18. Properties of Subtraction
Some students think that subtraction has an identity property, because if we take away zero from a number, its value does not change; that is, a – 0 = a. This is true; however, if we reverse the order, the result is not true—that is, 0 – a  a. Therefore, we generally say that the operation of subtraction does not have an identity property. After examining a few cases, you can see that the operation of subtraction does not possess the commutative property or the associative property.

19. Properties of Subtraction
The commutative property is not immediately understood by children. I recall one first-grader arguing that 3 – 5 was 0 because “you can’t have less than nothing” and another arguing that it was 2 because “you just turn them around.” Finally, the operation of subtraction is not closed for the set of natural numbers because the difference of two numbers can be a negative number.

20. Investigation A – Mental Subtraction
Do the following computations in your head. Briefly note the strategies you used, and try to give names to them.
1. 65 – – – 125
– – 648
Discussion:
As with addition, there are a variety of subtraction strategies. Do they make sense to you? If not, ask yourself why.

21. Investigation A – Discussion
cont’d
Is it because you don’t understand how the strategy works? Nearly all of the strategies work better with certain kinds of numbers than with others. In Problem 1, one alternative is to add up—that is, to ask how we get from 28 to 65.
One student’s actual strategy looks like this: = 58
= 65
= 37, which is the answer.

22. Investigation A – Discussion
cont’d
The add-up strategy is nicely illustrated with a number line.
Another variation of adding up looks like this:
= 30
= 65
= 37, which is the answer.

23. Investigation A – Discussion
cont’d
This is illustrated on the number line.
In Problem 2, we can use compensation, which does not work exactly the same way with subtraction as it does with addition.
This is how compensation looks with Problem 2: 62 – 39 Add 1 to both numbers. We now have 63 – 40, which is easily solved: 23.

24. Investigation A – Discussion
cont’d
Some students understand the compensation strategy better by using a number line.
On a number line, 62 – 39 can be interpreted as the distance between the two numbers.
If we increase both numbers by 1, we have not changed the distance between the numbers.

25. Investigation A – Discussion
cont’d
In Problem 3, we can add up:
184 – 125
= 185
185 – 1 = 184, so the answer is 59.
In Problem 4, we can adapt the compensation strategy:
132 – 36
136 – 36 = 100.

26. Investigation A – Discussion
cont’d
Because we increased 132 by 4, we need to give back the 4, so the answer is 100 – 4 = 96.
We can also solve Problem 4 by using compatible numbers and adding up:
132 – 36
= 100
= 132, so the answer is = 96.
In Problem 5, we can use compatible numbers:1000 – 648.

27. Investigation A – Discussion
cont’d
Realizing that = 100, we can add up:
= 700
= 1000, so the answer is
= 352.
Problem 5 can also be solved by adding up:
1000 – 648
= 948
= 1000, so the answer is

28. Children’s Strategies for Subtraction

29. Children’s Strategies for Subtraction
Below are three very different methods that children have invented for subtracting larger numbers. As you explore algorithms that children and adults have invented, you will come to appreciate both the complexity of subtraction (more children struggle with subtraction than addition) and how this process is made easier by connecting addition to subtraction and by understanding how place value works. The problem is 64 – 27.

30. Children’s Strategies for Subtraction
Break the second number apart Essentially, this approach breaks apart the 27 and subtracts in pieces.
64 – 7 = 57
57 – 20 = 37 This method is similar to adding up, where we begin with one number, break the second number into parts (expanded form), and add one place at a time. Here, we begin with the minuend and then subtract one place at a time.

31. Children’s Strategies for Subtraction
As with addition, different representations illustrate and illuminate the problem in different ways. On the number line, we move 7 units, which brings us to 57, and then we move 20 units, which brings us to 37.

32. Children’s Strategies for Subtraction
We can also represent the process numerically – 27
= 64 – (20 + 7)
= 64 – (7 + 20)
= 64 – 7 – 20
= 57 – 20
= 37

33. Children’s Strategies for Subtraction
This method works like this: Subtract each place, beginning at the left. Next, combine the two differences: 40 – 3 = – 20 = – 7 = – – 3 = 37

34. Children’s Strategies for Subtraction
Adding up This method is very common, especially if the context of the problem is not take away but rather comparison or missing addend = = The answer is = 37. When asked to think out loud, the strategy sounds like this: “ = = 64, so the answer is = 37.”

35. Children’s Strategies for Subtraction
Note that a common variation is to overshoot the number and then come back. For example,“ = 67. Now I need to take away 3 to get to 64, so the answer is 40 – 3 = 37.” A number line nicely illustrates this strategy.

36. Investigation B – Children’s Strategies for Subtraction with Large Numbers
What if the numbers were bigger, for example 832 – 367? Look back on the approaches described above.
Discussion:
“Break the second number apart” strategy extends but is now more cumbersome: Begin with the minuend and subtract each place.

37. Investigation B – Discussion
cont’d
Using negative numbers extends quite well. In fact, many of my students actually find this strategy easier than the algorithm they grew up with.
500 – 30 = 470; 470 – 5 = 465
Adding up also becomes a bit cumbersome, but interestingly, some students feel quite comfortable with it.

38. Investigation B – Discussion
cont’d
This is what they write:
832
= = = 832.
The answer is = 465.
Some children add up from left to right:
= 767; = 827; = 832.
The answer is = 465.

39. Understanding Subtraction in Base Ten

40. Understanding Subtraction in Base Ten
Let us now examine some standard and nonstandard algorithms for subtraction. Let us examine the following problem: 300 – 148.

41. Justifying the Standard Algorithm

42. Justifying the Standard Algorithm
A key to understanding this algorithm is to understand the equivalence of 3 hundreds and 2 hundreds, 9 tens, and 10 ones.
Although they look different, and the numerals are different (300 versus 2910), the amounts are equal.
Another key to understanding this algorithm is to know why we needed to do the regrouping.

43. Justifying the Standard Algorithm
This is easier to see at the physical level; without these regroupings, we cannot literally “take away” 1 flat, 4 longs, and 8 units.

44. Investigation C – An Alternative Algorithm
People who grew up in many European countries learned a very different subtraction algorithm. Below are two examples of its use and what a person using this algorithm would say.
For the first problem, “You can’t take 8 from 4, so you put a 1 next to the 4, cross out the 6, and put a 7. Now subtract: 14 – 8 = 6, 8 – 7 = 1, 9 – 3 = 6.”

45. Investigation C – An Alternative Algorithm
cont’d
For the second problem, “You can’t take 8 from 3, so you put a 1 next to the 3, cross out the 5, and put a 6.
Now, you can’t take 6 from 2, so you put a 1 next to the 2, cross out the 1, and put a 2. Now subtract:13 – 8 = 5, 12 – 6 = 6, 6 – 2 = 4.”

46. Investigation C – Discussion
The justification for this algorithm is very different from the justification for the previous algorithm, in which the “borrowing” didn’t change the value of the minuend, but only its representation.
Here, however, the value of both the minuend and the subtrahend do change: The value of the minuend and the subtrahend are now 994 and 378.

47. Investigation C – Discussion
cont’d
Because the value of both minuend and subtrahend have increased by 10, the difference (distance) between them remains the same.

48. Justifying the Standard Algorithm
The discussion of these two algorithms is a nice place to introduce a framework for examining algorithms that was presented by Hyman Bass in the February 2003 issue of Teaching Children Mathematics. Bass proposes that we examine five qualities of an algorithm: • Accuracy (or reliability). The algorithm should always produce a correct answer. • Generality. The algorithm applies to all instances of the problem, or class.

49. Justifying the Standard Algorithm
• Efficiency (or complexity). This refers to whether the cost (the time, effort, difficulty, or resources) of executing the algorithm is reasonably low compared to the input side of the problem. • Ease of accurate use (vs. proneness to error). The algorithm can be used reasonably easily and does not lead to a high frequency of error in use.
• Transparency (vs. opacity). What the steps of the algorithm mean mathematically, and why they advance toward the problem solution, is clearly visible.

50. Investigation D – Children’s Mistakes in Subtraction
There is a very powerful video in Integrating Mathematics and Pedagogy to Illustrate Children’s Reasoning that shows a second-grade child not understanding why 70 – 53 is not equal to 23. How do you think she got 23? What misunderstanding(s) of place value and/or subtraction might be causing her problems?

51. Investigation D – Discussion
In the video, you hear her say that “0 take away 3 is 3,” which is a common misconception among young children. Interestingly, even when the interviewer had the child do the problems with base ten blocks and a hundred chart, even though she got the correct answer using both of those representations, she trusted the answer she got using pencil-and-paper more!

52. Investigation E – Rough and Best Estimates with Subtraction
In each of the problems below, first obtain a rough estimate (5 to 10 seconds). Then obtain the best estimate you can. A. In 1995, Acstead State College had an enrollment of , whereas Milburn College had an enrollment of How many more students did Acstead have than Milburn?

53. Investigation E – Rough and Best Estimates with Subtraction
cont’d
B. The attendance at the Yankees game was 73,468, whereas the attendance at the Mets game was 46, How much greater was the attendance at the Yankees game?

54. Investigation E – Discussion
A. You can get a quick estimate by rounding: – 3500 = To get a more accurate estimate, you could use adding up: = 4275, then back off 40 to get The estimate is 800 – 40 = B. Leading digit: 7 – 4 = 3, giving an estimate of about ,000.
Rounding: 70,000 – 50,000 = 20,000.
Rounding and adding up: 47, makes 50, Adding up to 73,000 gives 23,000. Our estimate is ,000 = 26,000.

55. Investigation F – Number Sense with Subtraction
A. Fill in the blanks to make this subtraction problem. Discussion: At first glance, some students panic. “There is no way I can do this problem!” However, if we stop for a moment and remember the relationship between addition and subtraction, we can see that we can quickly determine the missing number by simply adding !

56. Investigation F – Number Sense with Subtraction
cont’d
B. Take no more than 5 to 10 seconds to determine whether this answer is reasonable. That is, make your determination, using number sense, without doing any pencil and paper work.
Discussion: For most people the simplest way to proceed here is to add 3125 and 2847 and see if the answer is close to A quick estimation of the leading two digits shows = 6000, so the answer is not true.

57. Carrying and Borrowing

58. Carrying and Borrowing
We have noted that we often need to regroup when adding and subtracting. However, using the traditional algorithms we use different words—we use carry for addition and borrow for subtraction. The following problem illustrates this point.

59. Carrying and Borrowing
In the addition problem, because > 9, we put the 4 in the ones place and “carry” the 1 to the tens place. In the subtraction problem, in order to subtract in each place, we need to “borrow” a 1 from the tens place and move it to the ones place whose value is now 14.