![]() 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 equations in this particular task highlight algebraic moves that are useful when students learn to solve quadratic equations by completing the square. The purpose of this task is to motivate solving a quadratic equation where the variable of interest, $n$ can be isolated by "undoing" operations. Or students could be asked to discuss how they "see" the pattern growing ( here is a good primer on what that might look like.) Or, they could be asked to evaluate d(1) and explain its meaning in terms of the number of tiles in Pattern D. 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. 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. The other tasks in the sequence are Quadratic Sequence 1 and Quadratic Sequence 3. 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 questions about the sequence of figures, 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. In general, if we start looking for this second difference at the $n^$, then the first few terms are summarised below.This task belongs to a series of three tasks that has students process a sequence of tile figures with the property that the $n$-th figure in the sequence has $f(n)$ tiles, for some quadratic function $f$. Then the second-level differences are $(4-2),(6-4),\ldots$ and happen to always be $2$. The second difference being referred to is the difference between adjacent differences. ![]() The difference between the first two terms comes from writing down the first and second terms and taking their difference: $(a*2^2+b*2+c)-(a*1^2+b*1+c) =a*3+b $ The first-term formula comes from substituting in $n=1$, since $n$ is the variable being used to denote which term we're looking at. "Quadratic" basically means $an^2+bn+c $ (historically related to things like "a square has four sides" and "quad is the Latin root for 'four'"), so that formula could be treated as true by the definition of "quadratic sequence". For the formula, I think you may have the idea backwards.
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