Intermath | Workshop Support

Write-up

 


 


Title Determine a rule that relates arithmetic operations on four consecutive odd and even numbers.

Problem Statement
Determine a rule that relates arithmetic operations on four consecutive odd and even numbers. Choose four consecutive odd counting numbers. Take the product of the middle two numbers and subtract the product of the first number and the last number. Try a few samples and formulate a rule. Explain why the rule works.


Problem setup

The problem is to determine the pattern that 4 consecutive odd numbers follows, then subtract the product of the outer two numbers with the product of the inner two numbers.

 

Plans to Solve/Investigate the Problem

My initial plan was to try a few sequences on paper and see what the difference of the products would be to determine if there was a pattern. After that try to determine a procedure to follow to use with excel.

 

Investigation/Exploration of the Problem

I first tried the obvious, 3, 5, 7 and 9.

(5x7) (3x9) = 8

then, 23, 25, 27, 29

(25x29) - (23x29) = 8

 

A pattern for odd consecutive numbers is: (2n +1), (2n +3), (2n +5), (2n+7)

Using this for my four consecutive odd numbers and subtracting the products:

(2n +3) (2n +5) - (2n +1) (2n+7)

4n2 + 10n + 6n + 15 - 4n2 - 14n - 2n - 7 = 8, the magic number!

Click here to see an excel worksheet for the above problem

 

Extensions of the Problem

If I used 4 consecutive even numbers, the outcome was an eight.

A pattern for 4 even consecutive numbers is: (2n +2), (2n +4), (2n +6) (2n+8)

(2n +4) (2n+6) (2n +2) (2n +8)

4n2 + 12n + 8n + 24 - 4n2 - 16n - 4n - 16 = 8

Click here to see an excel worksheet for the above problem.

Author & Contact
Insert name and contact information.
Insert Email

Link(s) to resources, references, lesson plans, and/or other materials
Link 1
Link 2