My fourth grade students are deep in the study of geometry. Our unit is about complete and I am just reviewing major concepts and misconceptions with some of the kiddos. As I was attempting to reteach/reinforce/deepen a select group of students’ understanding of angles as turns, I was reminded of how effective using the back of the geoboard was for this.

When I originally taught this lesson, I was searching for a manipulative that would help students see how angles open, or turn as they become bigger. I was coming up short. A quick tweet tweet to #MTBOS and Kristin Gray (@MindsMinds) suggested using the clock to show this. That got me thinking… I liked the idea of using a circle and mobile hands, but I didn’t want to attach the clock face numbers to it. I was worried about some of my most striving students getting confused by attaching other numbers that were not angle measurements. As I was brainstorming how I could use a brad and attach layers of laminated strips of paper together at a vertex so that students could open them and close them, creating bigger and smaller angles, it dawned on me…the back of the geoboard! That weird circular arrangement of pegs sure does look a lot like a clock.

Now, if you are like me, you have never ventured to the back side of the geoboard. I had noticed the circular arrangement of pegs before, but had never found a use for it. We’ve made lots of angles using the square side, however. And the geoboard is always an engaging tool for students. So how could I use it to help students think about angles as turns? What could I ask students to help them make some discoveries on their own?

When I was planning this lesson I was at home, and without a geoboard of my own to play with. So I just jotted down some questions to start a conversation with students and decided I would just go from it. I was going to wing it (*gasp*)! I was able to find a great resource to use to help me demo on our Promethean board what students were doing on their individual geoboards (http://www.mathlearningcenter.org/web-apps/geoboard/).

I should let you know that, up until this point in the unit, I had been really annoyed by how disjointed this unit felt from “regular” math. There was very little work with operations or properties and I was feeling like this unit was just going to set my kids back. I wanted to get to the good stuff! Fractions! Multiplication and division! Geometry was just slowing us down.

Lesson time. I gathered students, and passed out geoboards. I asked them to flip it over and take a look at that circular side. “What do you notice?”

“It’s a circle”

“It looks like a clock”

I asked, “Could we use this side to make angles?”

“Umm, can we have some rubber bands so we can try?” I passed out bands. At this point they started experimenting. They very quickly started making times. Someone made 3:00, another made 6:00. They were excited by this discovery and kept sharing out the different times they were able to create. I stopped them and asked, “Are you creating angles?” Here there was a little “aha!” moment as they recognized that 3:00 is a 90 degree angle, and 6:00 shows a 180 degree straight angle. They even recognized that 9:00 would be a 270 degree angle. At this point I asked them to place one rubber band so that it was “pointing to the 12” and then put another one on top of it. Then I instructed them to take the top rubber band and turn it to show 90 degrees, then turn it more to be 180 degrees and turn it again to be 270 degrees, then turn it again to be back to 12:00. At this students exclaimed, “Hey! 360 degrees and 0 are at the same place!” Wow, I hadn’t anticipated this, but it felt big. “Ok, explain that to a neighbor.” The class erupted in talk and as I listened in, I realized that the idea of 360 degrees being a whole circle and a whole turn around one point was starting to sink in. Many kids were realizing that in a circle, the starting and finishing point are the same. Very cool.

I wanted to keep discovering! “If 3:00 shows 90 degrees, could we figure out the measure of each peg? What is the measure of 1:00 and 2:00?”

Students again got to work, excited to figure this out. It didn’t take long for students to reason that if you make the 1:00, 2:00 and 3:00, you have created a right angle and then split it into three parts. Then they reasoned that each part was equal and so must be 30 degrees each. On our geoboards at school, there is a peg in each corner of the circular side of the geoboard, outside of the circle of pegs. Students created an angle here and decided that it showed a 45 degree angle because it was splitting the right angle in half. I hadn’t even noticed that peg before! And I had not even wanted to venture into the realm of decomposing angles into smaller angles, but that’s where they were headed.

Because I wanted to emphasize the turn of an angle, we put two bands on 12:00 again and “opened” the angle by 30 degrees all the way around to 360. We counted by 30 degrees as we went. I passed out some scrap paper and asked students to make the 1:00, 2:00 and 3:00 angles again. I wanted them to write a number sentence that would represent their thinking. I got a lot of variation here:

30 + 30 + 30 = 90

60 + 30 = 90

3 x 30 = 90

The multiplication representation stuck with me, remember, I had been itching to get to multiplication talk and there it was. It appeared all on its own 🙂

We created other angles and split them and wrote equations to match their thinking. It was fun. Finally, I decided to let them create their own angles and split them however they wanted. One student did this and I replicated it on the board so that we could talk about it:

Whoa! Do you see it? There are four 60 degree angles here. This student saw that there were two groups of two 60 degree angles. I definitely didn’t see that coming, but I sure was thrilled to see it! There it was, the connection to multiplication and the properties of operations that I had been missing all along. It took a student to show it to me. Wow, did that get us talking!

I was so pleased by the discoveries that students made during this investigation. And I was even more pleased that inquiry in math was again validated. I love setting my students free to explore, but sometimes “winging it” doesn’t make me feel like the hyper prepared, forward looking teacher that I am suppose to be. This lesson was a great reminder that I need to always allow room for discovery in math. I have another inquiry lesson prepared for next week. I can’t wait to see what they discover about types of triangles and their angle measures.