Multiple Representations of Patterns

I have all my classes doing a Visual Pattern from Fawn as a warm up once a week. With my Algebra 1 students I decided to extend the assignment to create posters showing how the patterns relate to the table, words, graph, and general formula.

I did a similar assignment last year for exponential growth and decay when we got to that section, but I figured I should start with linear patterns. When we do the project again for exponential patterns hopefully the connections will be even stronger.

The process:

The warm up was a simple linear pattern from Visual Patterns.

Pattern #11 from Fawn Nguyen’s VisualPatterns.org

I always have them fill in a table and attempt to come up with a rule. After they did this, I asked one student to share the results with the whole class using the projector. While she was explaining I had another student plot the table points on a graph on the board.

We had a quick discussion on where the pattern “up 2” showed up in the four representations (drawn pattern, table, algebraic expression, graph) and had presenting students mark the 2s on the examples. Then I asked where the plus one in the rule came from. The student at the front said it was the one red star in every pattern. The one drawing said: “Oh, if I extend this back, its right here!” (pointing to the y-intercept). I asked them to find it in the table. (It wasn’t there….So add it! Where would to show up?)

I then handed each student a new pattern and asked them to make a table, graph, rule, and color code the connections between the representations. When they were confident, I had them make post sized versions to hang around the room.

A few examples:

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Things I noticed: Some students drew in a Stage 0. I asked why, and for the most part these students thought of stage 0 as taking away the pattern part of stage 1. I asked the others why they didn’t draw a stage 0. The response was generally, “Is there a Stage 0? I see the start value here in 1…” I want to dig deeper into this difference.

Note On Slope: We haven’t talked slope yet. Students saw the pattern as change in number of squares or dots exclusively since n was going by 1s, and I didn’t push them on it yet although I did start those discussions one on one with the students as they worked. My plan is to give them Step 1, 3, and 6 of a pattern and ask them to come up with a rule. As well as to introduce slope/rate of change with a similar task (and non-integer changes) and have the discussion on rates of changes of 1.5 vs 3/2; and where those show up in the different representations. Then have them go back and update the current posters and with creating new examples with differing x changes as well. I’m not sure if I should have done this during this task, waited to do a connection of representations until we had already talked about this, or if this do it now and revisit is best.  The students have definitely benefited from looking at the different representations, and there ability to come up with pattern rules has been improved but part of me is still worrying I should have done the Step 1,3,6 thing at the same time…maybe giving that to my students who needed more of a challenge for their original poster and then had a class discussion about how they were the same/different.

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Measuring Angles (Plus a Little Dancing)

Geometry A is working through a unit on definitions and introduction to proof, with a good dose of reviewing key ideas that may or may not have been learned in middle school math. Our topic today was recognizing: vertical angles, linear pairs, corresponding, alternate interior, alternate exterior, and same-side interior and how to use that information to start building proof.

I put up a set of lines and a transversaIMG_0065l and had student call out angles that fit the above terms. Then I had them all choose a set of taped lines that were spread around the room. We went through each location again, trying to find as may pairs that fit said relationship. After students were comfortable with each, I hooked up “Dance, Dance, Transversal.” I have no idea who I first saw the idea for “Dance Dance Transversal” from or I’d give credit, but there are lots out in the inter-webs of math land, so if you want more, Google returns many hits. We played a few rounds to warm up. Basically, I had a PowerPoint set up to cycle through the terms (I limited it to vertical, AIA, AEA, and corresponding) and students had to jump to the correct location. We had music going and in later rounds, judges. We crowned the class champion and I let them know they’d have to dance on the unit exam.

Then, each student was tasked with getting a protractor and measuring the eight angles formed by the transversal. I had purposefully used both parallel and non-parallel set ups (About 2 parallel for every non-parallel). After measuring, they walked around and had to write down at least two notices IMG_0058and two wonders. (Examples “I noticed all the vertical angles were the same.” “I won
der why some gro
ups have only two different angle measurements and others 4.”   “I noticed an error, or what I think must be an error in set B.”   “I’m wondering if my tape was messed up, because it doesn’t look the same as most of them.”)

We were running short on time, so we shared out a few of the Noticings and Wonderings as a class and we’ll pick back up there tomorrow. The plan is to agree upon, prove, and then practice with the key ideas.

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