Visual Patterns and Quadratic Sequences
• Look for structure in number sequences arising from visual patterns. • Model the patterns with quadratic functions given by recursive descriptions, expressions, or equations.
In the previous section students saw a quadratic function where the change over successive intervals of a fixed length grows linearly. It is not until calculus that students will be able to describe precisely the growth law for quadratic functions (namely that their derivatives are linear). However, they can see a discrete version of this law by restricting to quadratic functions whose domains are the whole numbers, that is, quadratic sequences. This allows students to focus on some basic features of quadratic functions without getting bogged down in analysis of real world data and contexts. In this and subsequent sections these sequences often arise from sequences of visual patterns. Each visual pattern lends itself to a geometric interpretation for how it grows, in such a way that the number of additional squares (or other objects) in each successive pattern grows linearly. It is then natural to model the number with a quadratic sequence given what students learned in the previous section. Students conjecture the number of squares at a given stage, create a table, and express the number of squares algebraically. Since the prompts are visual patterns, there are multiple points of entry and students have a context with which to check their conjectures. A note on equations, expressions, and functions: students might initially write an expression, for example $n^2$, to represent the number of squares in the pattern. This is a natural thing to do. At some later point, in order to help them see that a sequence is a function, it might be helpful to introduce another variable $S$ for the number of squares in the pattern and write an equation $S=n^2$, or to use function notation $f(n)=n^2$. All are acceptable, but it is important to use terminology correctly and not refer to expressions as equations or vice versa.
WHAT: Students study Visual Pattern #16 at VisualPatterns.org in which the number of squares in the $n$th pattern is $n^2$. This is the simplest integer sequence that gives rise to a quadratic function. Visual Pattern \#16 shows this sequence in a triangular pattern, providing opportunities for students to discover that square numbers are sums of consecutive odd numbers. Students could be given the form provided by visualpatterns.org to help scaffold their thinking. Students should look for structure in the sequence of visual patterns, make predictions about visual patterns later in the sequence, and begin writing explicit expressions, equations and recursive formulas A-SSE.A.1, A-CED.A.2, F-IF.A.3, F-BF.A.1a, MP.7, MP.8.
WHY: This first sequence, $1, 4, 9, 16, \cdots,$ is easily recognizable, so students have an opportunity to explore the structure in the visual pattern and think about what it tells them about the growth law for the function.
WHAT: Students study Visual Pattern #3 at VisualPatterns.org. This visual pattern arranges the triangular numbers in a staircase formation. Students could be given the form provided by visualpatterns.org to help scaffold their thinking.
WHY: The purpose of this task is to build on the discoveries in the previous activity and identify commonalities and differences. The pattern bears a superficial resemblance to the previous one, but the function that arises from it turns out to be different. The staircase arrangement allows for the possibility that students can discover the formula for the sequence by fitting two copies of the pattern together to make a rectangle. As with the previous task, this task is designed to elicit tables, expressions, recursive formulas, to describe a growing pattern A-SSE.A.1, A-CED.A.2, F-IF.A.3, F-BF.A.1a. The discussion should focus on the growth in the various representations, developing the idea that quadratic functions show linear average rates of change over successive intervals of the same size MP.7, MP.8.