Breaking Up a Square

Materials: Handouts, grid paper, Geoboard

Goals (you need to fill in the blanks):

¨ Follow a pattern.

¨ Collect data from a sequence and make a conjecture for the formula of the nth term of the sequence.

¨ Make a connection between visual and analytic representations of mathematical concepts.

¨ Use the formula for triangular numbers in other contexts.

¨ Represent with a diagram.

¨ Prove analytically.

¨ Use in problem solving.

Step 1:

¨ Complete the following table. What conclusions can you make from your results?

n = The largest number in the pattern	Adding the integers from 1 up to to n and back down to 1	The sum
1	1	1
2	1 + 2 + 1	4
3	1 + 2 + 3+ 2 + 1
4	1 + 2 + 3 + 4 + 3 + 2 + 1
5
6
7
8
9

¨ Complete Activity 4.14: Number Ideas: Proofs Without Words from Mathematics for Elementary Teachers via Problem Solving: Student Activity Manual.

¨ Complete Problem 2: Laying Blocks in a Patio from Mathematics for Elementary Teachers via Problem Solving: Student Activity Manual.

¨ Do problem 8b on page 203.

¨ Be able to make the connections between the equation and a drawing.

¨ What is a triangular number? What is the formula for finding the nth triangular number? What is the formula for finding the (n – 1)st triangular number? What connection do triangular numbers have to do with the equation .

¨ Using the formula for triangular numbers, prove the equation works no matter what n is?