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Chapter 5 – Option 3 (Action)

02/08/2019 10:05 PM | Anonymous

Anytime you plan to do a thinking-focused task with students, it is important to try it yourself in the role of a learner to experience the thinking process and predict the mathematical approaches students might try.  Go to www.visualpatterns.org and pick one of the patterns, or create your own.  Try to find several different ways to generate the rule for the repetition in your pattern.  Pay attention to your process and materials and tell us about the experience.


Comments

  • 02/10/2019 12:05 PM | Cindy Noftle
    I picked two patterns randomly #11 and #16.. I found that I immediately went to algebra to solve the problem. I struggled to get myself to try different ways to write the rule. I tried but found myself guessing at what I thought my students would do. It did not feel authentic. When I introduced finding a pattern in class last fall, my students solved it mainly by algebra but they are in algebra 2. I saw that they didn’t really think for themselves during the individual think time. They wanted to chat right away. I do think this is because once they are afraid to be wrong. Having practice with individual think time will improve their self confidence. I see that I need to go through this a few more times myself. I hope to be able model it better.
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    • 02/10/2019 8:28 PM | Karen Campe
      Cindy,
      I agree that we as teachers /experienced math students gravitate to algebra solutions. What I'm wondering is, can you see more than one way to express the growth of the pattern? I find it harder to generate more than one way after I've seen that way. And of course, one student doesn't need to come up with multiple ways, but rather the multiple ways the class generates can be shared to the whole group.

      For example, in pattern #11 you mentioned, did you think of it as 2 vertical + 1 more horizontal, then 3 + 2, then 4 + 3? Or did you see 1 corner +1 +1 for the "legs", then 1 corner +2 +2 etc.? Or some other way?

      Each of these ways of noticing the repetition yields an algebra rule, and then students can check for equivalent expressions.

      Karen
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  • 02/11/2019 2:54 PM | Sarah Giaquinta
    I have a lot of trouble "thinking like a student", so I went to the website and picked a few patterns to try. The first one I picked was actually the same as the one discussed in the first part of chapter 5, #15. When I started to work through it, I thought of a diamond around each square, counting 4 circles per square. Then I had to adjust for double counting, so I subtracted (n-1) from my total. I always see shapes first, which I think drew me to the diamond that jumped out at me for the first iteration of the pattern. But I felt like students wouldn't go a route where they had to account for the double counting. So then I tried my best to think like a student. My next approach was actually the same as what Larena did in chapter 5, and my third approach was Briana's thinking, so I guess I'm getting better at thinking like a kid!

    The second pattern I pulled apart was #9- snowflakes. In my first attempt, I started counting "towers 3 high", and thought of the leftover "L" on the right as 3 I needed to add to my total. The second method I came up with was thinking of a large rectangle on left that has a height of 3 and the base increases by 1 at each step. I still had that leftover "L" with 3 snowflakes to add to the end. I tried thinking of another way to count the "L" for how a student may see it otherwise, but I was really stuck on that shape! Like I said before, I see shapes first and I couldn't get it out of my head! I'm curious to see if other people did this one a different way.

    One of the things that I loved about this routine was the stems for the discussion (p. 106): "Every time I..." and "I always...". These really help lead students to the process and not the product, where the process is the goal. They are so answer driven, that it is extremely helpful to have a guide to get them to move in a different direction. By finishing these stems, they have to slow down and think about what they are doing. This can be hard! I think another thing to help them think about the process is the ability to use manipulatives to complete the problems. When I was doing them, I just drew and used paper, but I couldn't help but think about how helpful it would be to actually have something to move around. In the second book example, the teacher has them use unit cubes to actually build the designs and see their repeated movements. This is so helpful for kids to FEEL their repetition. I was thinking that for the #15 pattern, if kids were using unit cubes for the squares and bingo chips for the circles, they could keep track of their grouping and movements. When I had to account for double counting, this would be a great way to have students actually see that there are two bingo chips in between each unit cube.

    On a side note, this routine reminds me of when I have my geometry students come up with a process for finding the sum of the angles in a polygon. They start with a triangle, then we add a side each time and they have to break it up into triangles. By the end, they figure out that there are two less triangles than sides in the polygon, and each triangle has 180 degrees. I find that they find the pattern but have trouble making it into a formula. I'm curious what would happen if they were used to this routine before tackling that problem!
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    • 02/15/2019 11:04 AM | Todd Butterworth
      Sometimes I don't even put the answer to the question at the end if I'm looking at the process and at patterns. I don't let them simplify either (it drives some of them crazy). If I write the answer or I let them simplify, then they're focusing on that and not looking for patterns (this is true in any class I teach).

      What I'm realizing about teaching is that I need to focus on patterns more. I need to connect content together more. In elementary school, they're always looking for patterns, it's a natural thing. We then start teaching them as if what we're teaching are a lot of distinct ideas, but that's silly. Almost everything we teach can be connected and built from other things, so let's embrace that and help their understanding by only using what they know to do what they don't know (I've had more success doing that in precalc (trig proofs) and calc (everything)). I think I just need to be more flexible in my thinking, something to work on :)
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  • 02/24/2019 5:15 PM | Katie Chuchul
    I chose this activity because when in high school and college, I always dreaded questions that were focused on determining a rule for a pattern. I felt as though I never had strategies to use to apply my thinking to a rule that would work for any # in the pattern and was often left confused and frustrated.

    So, I figured I'd put my student hat back on and give it a try after learning about approaches and routines to use with students.

    I worked on Pattern #32 (purple rectangles) and thought of the pattern this way...

    Rectangle 1: 1 x 2 array = 2 boxes
    Rectangle 2: (1+1) x (2+1) = 6 boxes
    Rectangle 3: (1+2) x (2+2) = 12 boxes
    The next in the pattern = Rectangle 4: (1+3) x (2+3) = 20 boxes

    My starting rectangle has l=1 and w=2, so we can figure out any rectangle (m) in the sequence using this expression.

    Rectangle m: (1+m-1) x (2+m-1)

    As I worked through this problem, I visually drew out some of the rectangles to test my thinking. I also thought a lot about my starting rectangle and how what I was naming each rectangle. By naming the first one Rectangle 1, I noticed that Rectangle 2 had +1 length and +1 width so its number in the sequence (i.e. 2) was one greater than what I was adding to each dimension. That helped me apply my thinking to Rectangle m (or any rectangle in the pattern) as m-1 for both length and width.
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  • 02/25/2019 2:55 PM | Laura Larson
    Some of my seventh-grade math classes have been writing linear equations from tables, word problems, etc. for a while now, so I thought it would be nice to try some of the visual patterns with them, with a couple of non-linear growth patterns in the mix.

    I gave them a choice of at least two from patterns
    #2, #4, #9, #15, #17, #18, #21, and #101

    While I did not use the Recognizing Repetition routine for this (the lesson straddled February vacation week -- it didn't seem wise!), I tried using a list of "Ask Yourself" questions to help focus their thinking:

    * Is the rate of change constant (linear)?
    * Would it help to make a data table?
    * Could you use color to identify different “chunks” within the figure?
    * Is there a “core” element that is always in each figure? (start value?)
    * Can you find “figure ZERO”?
    * How do you see each new figure growing? What piece(s) get(s) added on?
    * Do you see the figure number itself within each figure? That is, do you see groups of 2 in Figure #2, groups of 3 in Figure #3, etc.?

    It was fun to work through the patterns with a colleague first, especially since we got our expressions in completely different ways. I like activities like this to reinforce equivalent expressions. For instance, I see the number of circles in pattern #15 as:

    2n
    (top and bottom row of circles)

    plus

    n - 1
    (interior circles)

    plus

    2
    (on each end)

    Which works out to 2n + n - 1 + 2. If I were to make a table of the number of circles in each figure, I come out with the simplified form: 3n + 1. But I don't SEE it that way first, necessarily, even though the expressions are both equivalent and both represent the number of circles in figure #n.

    I would love to expand on this so that students can see the value in noticing what part of a relationships is constant and what part is changing. I could see color-coding playing a part here, but I would also love to try having students draw or construct the figures, so that the repetition was more apparent in their repeated drawing or building motions.

    Teaching about writing equivalent expressions, in particular combining like terms and use of the distributive property, is notoriously dry stuff, so these types of activities appeal to me for their natural "puzzle" traits.
    --Laura Larson
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  • 03/04/2019 9:57 AM | Michele Hanly
    I had not been to this site before but will definitely note it for the future. I chose the pattern with the dogs in it. My first instinct told me to simply start predicting what number would come next in the sequence. Then as I wrote out the numbers realized there was a constant rate of change and connected that to a linear equation. I then made a chart of the x and y coordinates and graphed the points. My final step was to write the equation from seeing the line. I solved the problem by putting in x=43.

    Some students might not need to graph the points to be able to write the equation. Others may just add up the entire sequence.

    This type of exercise could be used for many different levels of math. I think the students would love to see all of the interesting items and could find the work challenging and yet fun!
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