Yesterday, in my post Fawn Nguyen’s New “Math Talks” Site Enhances her “Visual Patterns” Site, I discussed how Fawn and I saw Pattern # 115 in her Visual Patterns site differently. I also sent Fawn a pattern that fit my first erroneous way of interpreting Pattern # 115. Shed added the pattern I sent to the site. It is Pattern # 116. Under Pattern 116, were instructions to go to the link to get the equations. The link, however, was to yesterday’s Watsonmath post, which only included the equation to Pattern # 115, not # 116. I’m creating this post so that if Fawn wants to change the link to #116, it will help students to see how two different people interpreted how the shape was constructed.
Below I will explain how I envision Pattern # 116 and how I guessed that Fawn might envision the same pattern, based on the different ways she and I interpreted Pattern # 115.
The fact that both formulas simplify to y = 8x – 1 does not mean that the simplified formula provides information about the shape. The most efficient way to show the 10th pattern, which would have 8(10)-1 = 79 circles, would be to draw a large rectangular array of circles that has a vertical dimension of 8 and a horizontal dimension of 10, with one circle taken out somewhere. The total number of circles is the same, but the pattern looks nothing like the Patterns 1, 2 and 3.
Using my formula y = 3(2x – 1) + 2 (x + 1), I can easily create the 10th pattern that has the same general “shape” as the original 3, but would have 79 circles. I know that there would be a large rectangle that has a vertical dimension of 3 circles and a horizontal dimension of 2(10) – 1 = 19 circles. There would be 10 + 1 = 11 towers that are 2 circles tall that are evenly spaced along the top of the large rectangle. The outer 2 towers would be placed alongside the top row of the large rectangle. The remaining 9 would be placed on top of the large rectangle and there will be a space that is one circle diameter wide between each tower.
Although these pattern problems aren’t “real world” problems, they provide students with a way to practice problem solving. They elicit student use of the following CCSSM Practice Standards:
1. Make sense of problem and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics. (Although they are not real world situations, an abstract model is being developed.)
5. Use appropriate tools strategically. (The tool of the x/y table helps students to keep track of their thinking.)
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
If you missed my previous post, make sure to check it out to see how Fawn uses Patterns in her classroom on a regular basis.