A Chance To Argue

Name: A Chance to Argue In this unit we have been studying writing quadratic expressions in factored and expanded form. Since this class of 8th graders loves to argue, I wanted to give you an opportunity to argue in the context of the algebra we are learning. You will need to use this knowledge to prove or disprove the following relationships. Write your proofs on a separate piece of paper. I.

The sum of any two consecutive triangular numbers is a square number. An example of a triangular number is 10. Observe: * * * * * * * * * *

10 = 1+ 2 + 3 + 4

The ancient Greek mathematicians studied triangular numbers intensively. They discovered many interesting relations involving this number family. This is an example of a relationship that they discovered. A geometric "picture" proof of this statement is very easy to do, of course. And you can verify it by testing with a few numerical examples. Demonstrating that several numerical examples work will not be enough to prove the relationship is true. Since this is an Algebra class, you must prove its truth by way of algebra. You can, however, demonstrate that a given relationship is not true by citing one numerical example that does not work. Even though a numerical example will not serve as a proof, it is a very good place to start your thinking about the problem.

II. The product of 8 and any triangular number, increased by 1, is a square number.

III. The product of any four consecutive integers, increased by one, is always a square number.

IV. Given eight consecutive numbers, it is always possible to arrange their squares in two sets of four with the same sum.

V. If you double the sum of two squares, the result will always equal the sum of two squares.