Reasoning About Negative Numbers Using Patterns

Purpose: How do we know the "rules" for negatives? Did somebody just make them up? What can we do when we have forgotten the "rules" for negatives? All these questions can be answered by using patterns to search for what we want to know. In all of these cases the "rules" for negatives may be discovered from the pattern.

Background: These lesson were used in an urban seventh grade pre-algebra class. At the end of each pattern students successfully told me what the "rules" were for each operation with negatives, with out me telling them.

Last Updated: January 2010

  What numbers will complete these patterns?

Addition:

3 + 2 = 5

Once the first column is understood, this one follows:

3 + 1 = 4

3 + 0 = 3

2 + (-3) = -1

3 + (-1) =

1 + (-3) = -2

3 + (-2) =

0 + (-3) = -3

3 + (-3) =

-1 + (-3) =

3 + (-4) =

-2 + (-3) =

 

 Subtraction: complete the pattern in both directions

3 - 5 =

Once the first column is understood, this one follows:

3 - 4 =

3 - 3 = 0

3 - (-3) = 6

3 - 2 = 1

2 - (-3) = 5

3 - 1 = 2

1 - (-3) = 4

3 - 0 = 3

0 - (-3) = 3

3 - (-1) =

-1 - (-3) =

3 - (-2) =

-2 - (-3) =

3 - (-3) =

Multiplication:

3*5 = 15

Once the first column is understood, this one follows:

2*5 = 10

(-2)* 2 =

1*5 = 5

(-2)* 1 =

0*5 = 0

(-2)* 0 =

(-1)*5 =

(-2)* (-1) =

(-2)*5 =

(-2)*(-2) =

Division:

Follow the same pattern idea as in multiplication. The same pattern will follow.

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Comments:

In each of these cases we start a pattern using operations on positive numbers that we know how to solve. In each case there is only one consistent way to complete the pattern, and that way happens to demonstrate the correct answers for problems with negatives.

Ask students, "what numbers will complete these patterns?" Once the patterns are completed, ask, "For these patterns to hold, what must the 'rules' for this operation must be?"

  
 

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