Experience First

We originally came across this visual pattern on Jo Boaler’s YouCubed site and knew this context was rich with sequences and series. After yesterday’s intro activity, students were much more comfortable finding explicit rules from a visual pattern, though today’s rule proved more challenging since there was no rectangular array.

Students were quick to notice that the figure number was added each time and thus were able to come up with a recursive formula. Students also knew the patterns would be quadratic because of the constant second difference and a few found the values of a, b, and c in the general equation for a parabola. When monitoring groups, ask students if they see any squares or rectangles in the picture, tying back to yesterday’s activity. Students may be able to relate each figure to a triangle, to half a rectangle, or to a little more than half of n^2. In fact, each staircase is half the square plus n additional triangles (half squares) on the main diagonal. Since each triangle is half the square, students can reason that the explicit formula is ½*n^2 +n/2.

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Formalize Later

In this lesson we reinforce the new notation for term numbers and term values and also add sigma notation. Students tend to easily grasp the lower and upper limits of notation but struggle with the use of the index i in conjunction with the variable n. When finding sums, make sure students know how to evaluate these by hand and on a calculator.

Check Your Understanding question 2b proved to be slightly more challenging for students. Remind students that rules for a(n) can themselves be fractions, allowing students to think about the rule for the numerators and the rule for the denominators separately.