
Reasoning about Negative Numbers Using Known Relationships
Purpose: Students will demonstrate why the "rules" for
operations with negative numbers are what they are. Students will
discuss the type of reasoning with algebra.
Background: This lesson was used in a seventh grade
prealgebra class as the third way to demonstrate why negatives work
the way they do. This lesson should be coordinated with: 
Reformatted: January 2010 

Introduction: To establish the value of negatives it is probably easiest to start
by showing a small minus a large (eg: 34=1). Once "rules" for
subtraction are established everything else follows quite naturally
from the very nature of arithmetic. This sequence assumes some other
method has been used to generate the "rules" for either addition or
subtraction. From that first set of "rules" all other "rules" will be
determined.
[a] Subtraction <> Addition
What we know about one tells us what we know about the other. We
may use the "rules" for subtraction to infer the "rules" for addition
or visa versa. Assuming subtraction in known, what is implied below
about addition?
4  3 = 1

<>

1 + 3 = 4

Motivating Example

(4)  3 = a

<>

a + 3 = 4

a =

(2)  5 = b

<>

b + 5 = 2

b =

(2)  (5) = c

<>

c +(5) = 2

c =

[b] Addition > Multiplication
Once the "rules" for addition are known then we may infer most of
the "rules" for multiplication.
3 * 4 = a

3+3+3+3 = a

a =

2 * 3= b

(2) + (2) + (2) = b

b =

2 * 5

(5) + (5) = c

c =

Patterns will be needed to show a negative times a negative.
{Some may say word problems may be used for negative times negative,
but most word problems not using physics <eg: negative current *
negative voltage = delivered (positive) power> are poorly
structured, confusing, and frequently wrong about the concept.} 
Related pages at this site


[c] Multiplication <> Division
Once the "rules" for multiplication have been established the
"rules" for division follow quite naturally. Knowing the one means
knowing the other.
3 * 5 = a

a / 5 = 3

a / 3 = 5

a =

(3) * 5 = b

b / (3) = 5

b / 5 = 3

b =

3 * (5) = c

c / (5) = 3

c / 3 = 5

c =

(3)*(5) = d

d / (5) = 3

d / (3) = 5

d =

Summation: If you know nothing about the
"rules" for negatives, but follow this and the preceding three
lessons, you will discover on your own the "rules" for negatives
without needing anyone to tell them to you. What's more, the "rules"
you find in any of the three sections will match the "rules" you find
in the other two sections. 
