Experience First

After going over our Unit 6 test, we started Unit 7 off with our first visual pattern task. We want students to start to see relationships between the figure number and the number of small squares which later translates into the term number and the term value. In this intro activity, we keep academic vocabulary pretty light, knowing that students will build on this activity tomorrow. We use words like rule or pattern instead of explicit formula and we encourage students to consider how they see the shape growing. This is a critical question for helping students “see” the equation. For example, if students consider that the two “wing” squares are the same in every figure, then we know that the formula will have a “+2” at the end. Any part of the figure related to the figure number itself will depend on n.

Arrays like the one presented here give students a chance to visualize quadratic sequences. While these kinds of explicit formulas can be difficult to find analytically, seeing rows and columns and connecting this with the area of the shape (counting unit squares!) is a huge help.

The question about the zero term is an interesting one. Most students found that the number of small squares would be zero, but they had different ideas about where the two squares would be located. Finding the zero term will be important moving forward when writing explicit rules for arithmetic sequences or for finding the constant term of an explicit formula for a quadratic sequence.

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Formalize Later

While light in notation and academic vocabulary, this intro activity provides loads of opportunities to connect concepts across grades. Students may struggle to articulate a constant second difference. Many students said “it goes up by the same amount”, to which you can ask if the pattern is linear. Students quickly corrected themselves to say that the difference goes up by the same amount every time. This concept of a constant second difference is discovered in Algebra 2, explored further in Precalculus, and deepened in Calculus with derivatives! Once students realize that the equation will be quadratic, you can have them write the general equation for a quadratic as ax^2+bx+c and find the values of a, b, and c. We started off by finding the value of c (y-intercept, how many small squares in Figure 0), then we found the value of a by comparing the constant second difference of our parent function y=x^2 to the constant second difference of this pattern. Since both have a constant second difference of 2, we can deduce that a=1. We then plugged in a point to find the value of b.

Another key idea that surfaces here is the idea of equivalence. While most students at this point know that n(n+2)+2 can be expanded to make n^2+2n+2, we then asked students to look for n^2+2n+2 in their picture. Students found a square of width n, 2 columns of length n, and 2 squares as the “wings”.

As time permits, consider projecting additional patterns from visualpatterns.org. We loved this one in particular, as students found multiple ways to “see” the pattern and come up with equations. The site challenges students to find the number of circles in the 43rd figure.