CHAPTER 10 – EXPANDING OUR NUMBER SYSTEM Reconceptualizing Mathematics Part 1: Reasoning About Numbers and Quantities Judith Sowder, Larry Sowder, Susan Nickerson © 2010 by W. H. Freeman and Company. All rights reserved. 1

Positive and negative numbers are often called “signed numbers” because of the + or – signs that are used. The + sign is normally omitted. 10-2

DISCUSSION 10-3

10-4

The way we arrange these chips together can ultimately represent numbers as well… 10-5 And these are the same too…

10-6

10-7

Although -6 can be read as “negative six,” “the opposite of six,” or “the additive inverse of six,” -a should be read as “the additive inverse of a” or “the opposite of a,” but not “negative a.” This is because if a itself happens to be negative, then -a is actually positive. The context usually clarifies the situation. For instance is subtraction, -5 is the use of “negative,” and -x would refer to an additive inverse. 10-8

Signed numbers, including all the integers, positive and negative fractions, and the positive and negative repeating or terminating decimals, comprise the rational number system. Positive or negative numbers that have non-terminating and non-repeating decimals are irrational numbers. The rational and irrational numbers together comprise the real number system. 10-9

DISCUSSION 10-10

10-11

Just as there are several ways to think about signed numbers, there are several ways to think about adding and subtracting them. Combining the colored chips that we discussed in the last section is one way to do this: 10-13

EXAMPLE 10-14

EXAMPLE We can use the “take away” model of subtraction to model how to subtract using colored chips: 10-15

ACTIVITY 10-16

ACTIVITY Now, turning our attention to the number line, how could one picture a sort of “hopping on the number line” for each of the following? a) 6 + 5b) 3 + 4c) d) e) f) g) h) 11 - (-4) 10-17

10-18 Example:

ACTIVITY 10-19

10-20 Discussion Is |a – b| = |b – a| always? Is |a| + |b| = |a + b| always? Is |a – b| = |a|  |b| always?

10-21

ACTIVITY Calculate each below referring to the rules given for addition and subtraction. For each subtraction problem, first rewrite as an addition problem:

10-23

ACTIVITY 10-24

The properties of whole numbers and rational numbers continue to remain true when negatives and irrational numbers come into play. For example, = or (3 + -7) + -5 = 3 + ( ) 10-25

10-26 Example:

EXAMPLE 10-27

10-28

10-30 You may have heard the rhyme: “Minus times minus is plus, the reason for this we need not discuss.” The reasoning for assigning the sign of the answer when multiplying signed numbers has often been thought of as just using a rule. But here we want to consider this multiplication much more deeply.

10-31

So from the discussion on the previous slide, we have found that in general: (positive)  (negative) = (negative) But we know that the multiplication of integers must be commutative. For example, -2  4 = 4  -2. In general then, using the statement at the top, we derive: (negative)  (positive) = (negative) 10-32

ACTIVITY 10-33

10-34 Discussion

DISCUSSION 10-35

10-36

10-37

Considering the division of signed numbers becomes a matter of applying what we’ve already learned regarding multiplication. Consider Example:

10-39

10-40

We could also use the colored chips to demonstrate how we handle signed integers in multiplication

continued….

10-43

From the previous two sections we’ve summarized that five properties were true for addition, five for multiplication, and one additional property, which connected multiplication and addition. When all 11 of these properties hold true on a set of numbers, then mathematicians call this set a field

DISCUSSION 10-45

Some number systems do not have infinitely many numbers. One such number system is called clock arithmetic. Suppose you have a five-hour clock as shown below: When one adds in this arithmetic, it is like going around the clock. So would start at 2, then go clockwise four spaces such that the result is 1.

ACTIVITY 10-47

ACTIVITY 10-48

By the sixteenth century negative numbers began to appear in algebraic expressions but were treated as fictitious numbers, and were referred to as “false.” In 1796, Frend, a Cambridge mathematician, produced an algebra text that completely avoided negative numbers. He said that those who consider multiplying a negative by a negative find their supporters “...amongst those who love to take things upon trust and hate the labor of serious thought.” 10-50

Mathematicians now view the use of negative numbers as obvious and a necessity. Students too show resistance to negative numbers. But because we use them in regard to temperature, or for debits in finance, we accept their existence often in terms of usefulness in these realms. However, when we begin operating on these numbers, it quickly becomes difficult to justify all the operations

With signed numbers, especially in terms of multiplication and division, children are not always able to build their knowledge with intuition. In terms of educating kids regarding some of these more challenging concepts, it is important to remember that although some of the “rules” involved are inventions, they are inventions at work

10-53