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Throwing Horseshoes


Alignments to Content Standards: A-SSE.A.1

Task

The height (in feet) of a thrown horseshoe $t$ seconds into flight can be described by the expression $$ 1\frac{3}{16} + 18t - 16t^2.$$ The expressions (a)–(d) below are equivalent. Which of them most clearly reveals the maximum height of the horseshoe's path? Explain your reasoning.
 

  1. $\displaystyle 1\tfrac{3}{16} + 18t - 16t^2$
  2. $\displaystyle -16\left(t-\frac{19}{16}\right)\left(t + \frac{1}{16}\right)$
  3. $\displaystyle \frac{1}{16}(19-16t)(16t+1)$
  4. $\displaystyle -16\left(t-\frac{9}{16}\right)^2 + \frac{100}{16}$.

 

IM Commentary

Variations on this problem might ask which expression is useful for finding the starting height of the horseshoe, or the time when it lands.

The task illustrates A-SSE.1a because it requires students to identify expressions as sums or products and interpret each summand or factor.

Solution

(d)

This expresses the height as the sum of a negative number, $-16$, times a squared expression, $\left(t-\frac{9}{16}\right)^2$, plus a positive number, $\frac{100}{16}$. Since the squared expression is always either positive or zero, the value of the entire expression for the height is always less than or equal to $\frac{100}{16}$, and is only equal to zero if the squared expression is zero, when $t = \frac{9}{16}$. So the maximum value is $\frac{100}{16}$.

Dev Sinha says:

about 4 years

Anyone have thoughts about how this task could be reworked so that it could more confidently be used for a (perhaps high-stakes) summative assessment?

Cam says:

about 4 years

We've struggled with questions like this a lot. Have you seen https://www.illustrativemathematics.org/illustrations/1344?

Noah Heller says:

over 5 years

What if a student picked (a) because it is easiest to transform into standard form, where they may then want to use the equation t= -b/2a and plug in. It's also the easiest expression to take the derivative of.

I realize they would have missed the point of the task, but would they be wrong?

Kristin says:

over 5 years

That is a debate we have been having ourselves. Can a student be wrong for saying one form of the equation is "most useful" for some purpose? I think the answer is "no" as long as the student has a compelling argument. For this reason, I wouldn't personally put this kind of problem on a high-stakes assessment, but I do think it is a great opportunity to generate a classroom discussion.

But as I said, this is an ongoing debate, and I'd like to hear what others have to say.

Heather_Brown says:

over 5 years

What about using the language of the standards and say "which of the following expressions reveals the maximum height of the horseshoe's path?" It seems that d immediately "reveals" it and the others require additional computation.

Cam says:

over 5 years

Great idea . We had a similar task with a similar wording difficulty, and found that that verb "reveals" pretty compellingly saved the day. I've taken a stab at new language -- feedback welcome.

Heather_Brown says:

over 5 years

Love it!