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 pre-algebra 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: 3-4=-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

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

• integers motivated by temperature
• intergers motivated by physical examples
• integers motivated by patterns
• multiplying integers

[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.