### Geoboards - Squared Numbers, Roots, and Pythagoras

Recently I dug into another cupboard at the school and pulled out a pile of Geoboards and rubber bands that have probably sat there for ages. There were 2 stacks small yellow ones or larger clear ones with grid lines drawn on the boards. I opted for the larger; they were labeled overhead manipulative kits as an example of how dated they were.

I knew I wanted to work on finding area of squares to look at squared numbers and square roots to use as a progression to the Pythagorean Theorem. I did not expect my 8th graders to buy into it as much as they did.

Day 1: Started with making as many squares as possible that were aligned to the pegs, so they were all vertical and horizontal with no skewed squares. They counted the area until a few were able to make a connection to length times width. We also looked at perimeter distinguishing the outside.This took nearly the entire class period after the initial warm up.

Day 2: We began looking at triangles. Very few of my 8th graders had the area of a triangle memorized or if they did they couldn't apply it to the right triangles we were making on the Geoboard without being prompted (which I didn't do). When I realized this we started just by estimating by counting whole squares then began talking about shapes that were easy to count the area of which we agreed were squares and rectangles. We then composed rectangles/squares around our right triangles to then count a total area and assumed that the triangle represented half. We also spent some time talking about diagonals/hypotenuses originally it was assumed that across a 1 by 1 peg the hypotenuse would originally be 1. Using the Geoboard, which was 10 by 10, we discussed a race would you rather start from the corner or would you rather start on the edge. Students then agreed that traveling the hypotenuse would be longer, but it took some convincing to see this with a 1 by 1 square as they look more similar in distance on the smaller scale.

There will be several more days exploring this concept as students will bridge their way to calculating approximately how long the diagonal should be eventually transitioning to seeing a proof of the Pythagorean theorem. The closest attempt to using Geoboards past early elementary school within the concept of squared numbers, square roots, and the pythagorean theorem was this written article: http://www.watsonmath.com/wp-content/uploads/2011/03/EKW-Pythagoras-Article.pdf

I knew I wanted to work on finding area of squares to look at squared numbers and square roots to use as a progression to the Pythagorean Theorem. I did not expect my 8th graders to buy into it as much as they did.

Day 1: Started with making as many squares as possible that were aligned to the pegs, so they were all vertical and horizontal with no skewed squares. They counted the area until a few were able to make a connection to length times width. We also looked at perimeter distinguishing the outside.This took nearly the entire class period after the initial warm up.

Day 2: We began looking at triangles. Very few of my 8th graders had the area of a triangle memorized or if they did they couldn't apply it to the right triangles we were making on the Geoboard without being prompted (which I didn't do). When I realized this we started just by estimating by counting whole squares then began talking about shapes that were easy to count the area of which we agreed were squares and rectangles. We then composed rectangles/squares around our right triangles to then count a total area and assumed that the triangle represented half. We also spent some time talking about diagonals/hypotenuses originally it was assumed that across a 1 by 1 peg the hypotenuse would originally be 1. Using the Geoboard, which was 10 by 10, we discussed a race would you rather start from the corner or would you rather start on the edge. Students then agreed that traveling the hypotenuse would be longer, but it took some convincing to see this with a 1 by 1 square as they look more similar in distance on the smaller scale.

There will be several more days exploring this concept as students will bridge their way to calculating approximately how long the diagonal should be eventually transitioning to seeing a proof of the Pythagorean theorem. The closest attempt to using Geoboards past early elementary school within the concept of squared numbers, square roots, and the pythagorean theorem was this written article: http://www.watsonmath.com/wp-content/uploads/2011/03/EKW-Pythagoras-Article.pdf

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