We need to see which function has an integer solution for each number of given tiles. Pattern A has 404 tiles in step 20, Pattern B has 195 tiles in step 14, and Pattern C has 432 tiles in step 12.For example, step $n$ of Pattern A has a square of $n \times n$ tiles with 4 tiles tacked on, so $g(n)=n^2+4$ makes sense. Another method would be to see how each pattern grows visually. This tells you that the first step of the pattern defined by $f$ has 3 tiles, so $f$ must go with Pattern C. You could evaluate each function for $n=1$. ![]() $f(n)$ defines Pattern C, $g(n)$ defines Pattern A, and $h(n)$ defines Pattern B.In part (c), students have an opportunity to express regularity in repeated reasoning (MP.8) by applying these same undoing operations to an equation to isolate a variable of interest in terms of other variables. The purpose of part (b) is to motivate solving a quadratic equation where the variable of interest, $x$, can be isolated by undoing operations. Part (a) could be scaffolded with more specific instructions for example, students could be asked to evaluate $g(3)$, find that $g(3) = 13$, and conclude that this means the pattern defined by g(n) must have 13 tiles in step 3. For example, they could be asked to "draw the next step" for one or more patterns. If not, students might benefit from additional questions to familiarize themselves with the idea. Ideally, students will have had some experience working with visual patterns prior to this task. ![]() ( Here is a good primer on what that conversation might look like.) For example, students might evaluate each function at $n=1$, or they might analyze the way the patterns grow visually. For example, each step of Pattern A consists of $n^2$ tiles with four tiles tacked on, and each step of Pattern B consists of $n^2$ tiles with one tile removed. Part (a) has many possible points of entry. The other tasks in the sequence are Quadratic Sequence 2 and Quadratic Sequence 3. ![]() That the domain of these quadratic functions is the set of positive integers provides an interesting wrinkle which students might not be used to thinking about in the setting of quadratic functions. The functions roughly increase in complexity through the three tasks, with the intent that the techniques learned in each will be used and expanded in the subsequent tasks. However, seeing structure is emphasized in the standards because of how it connects and helps in understanding many foundational concepts, and these tasks develop the ability to see structure when working with quadratic expressions and equations. Students are asked to analyze the functions in the context of the tile sequence, a process which involves manipulating the quadratic expressions into different forms (identifying square roots at first, then completing the square in the third task.) With solving quadratics, there can be an impulse to put everything in standard form and just use the quadratic formula. Quadratus is Latin for square.This task belongs to a series of three tasks that presents students with a sequence of tile figures with the property that the $n$-th figure in the sequence has $f(n)$ tiles, for some quadratic function $f$. ![]() In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms.
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