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Right now I’m teaching precalculus in high school and I want to propose a project to my students about polynomial functions. They already know enough about quadratic functions and we study variation models recently. We start working with polynomials this week and I really want to do a 2 weeks project for them with this topic but I’m falling short in ideas.

I already have a bacteria growth project prepared for them in exponentials functions but I want also to make something that meaningful to them in this topic too.

Any ideas?

Thank you in advance.

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  • $\begingroup$ Any suggestions? $\endgroup$ – Grouper Nov 4 '17 at 17:23
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Given a circle, cut it with 1 straight line. You obtain 2 pieces.

Now cut it with 2 straight lines. If you pick the second line correctly, you obtain 4 pieces.

What is the largest number of pieces you can obtain with $n$ lines?

The answer is a cubic (OEIS A000125, MathWorld:Cake Number), although you might not manage to prove that to the students' satisfaction. It can be used to explore identifying probable polynomials using iterated finite differences, and extrapolation of polynomials using finite differences. I remember that aspect of it making for the most interesting project we did in secondary school maths.

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You can identify polyhedra with polynomials. Let $a_0$ be the number of vertices, $a_1$ the number of edges and $a_2$ the number of faces, then the polynomial for a 3D polyhedron is $p_3(x)=-x^3+a_2x^2-a_1x+a_0$, for a 2D polyhedron (an n-gon) it is $p_2(x)=x^2-a_1x+a_0=x^2-nx+n$, a 1D polyhedron (a line) $p_1(x)=-x+a_0=-x+2$

If you want to get a prism with an n-gon base, multiply the base with the line segment: $$p_\text{prism}(x)=(x^2-nx+n)(-x+2)$$

If you want a cube, take the third power of a line: $$p_\text{cube}(x)=(-x+2)^3$$

You can also explore higher dimensional polyhedra (polytopes) and read off the number of $k$-dimentional cells from its polynomial. Say, how many edges does a fourdimensional cube have? No problem, just compute $(-x+2)^4$ and read off the first power coefficient. Is there some way to compute a pyramid with a given basis like it worked for a prism? Explore examples!

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Perhaps curve fitting of experimental data with more and more terms in a polynomial?

first term: average, constant

second term: slope, line

third term: curvature, quadratic

fourth term: rate of change of curvature, cubic

(and maybe up to 4th order)

Can show that (depending on the data) that most of the explanation can come from very simple models. Also gives a little intuition about value of a constant (average) versus line (increasing/decreasing quantity), etc.

This might be more than what they need or are ready for, but it is an idea. Definitely don't do something to be cool that is too much for them. (I wonder if any projects are needed. But if you do some, make sure they work for the students.)

P.s. You don't need any fancy stats modeling programs, you can do it in excel.

http://www.statisticshowto.com/excel-multiple-regression/

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  • $\begingroup$ You are right, I don´t want to do something that is too hard on them but I like your idea. Thx. $\endgroup$ – Grouper Nov 6 '17 at 1:02
  • $\begingroup$ So I am really interested in this idea and want to maybe do some data fitting exercises without making it too complex. In searching based on this thread I found this which may work. Creating a Polynomial function to fit a table $\endgroup$ – Jeffery Thompson Nov 7 '17 at 0:33
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I think doing something with polygonal numbers might be fun and instructive. You could ask them to write down the sequence of triangular numbers $1, 3, 6, 10, 15, \dots$ and have them see that the first difference increaseas by one and the second difference is constant. This suggests that the sequence is given by a quadratic. They have three points and so the could solve for the coefficients of the quadratic. The square numbers are easy. The pentagonal numbers can be handled in the same way as the triangular numbers. The might even by able to figure out the formula for the $k$-gonal numbers.

They could then maybe explore a famous theorem due to Fermat that says every positive integer $n$ can be written as the sum of $k$ $k$-gonal numbers (if we allow 0 to be a $k$-gonal number for all $k$).

Among other things, it will reinforce the following:

  • figuring out patters
  • solving systems of equations
  • multiplication as area
  • there is a unique degree $n$ polynomial through $n+1$ points (as long as the points have different $x$-coordinates
  • math has these things called theorems that can be pretty cool
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    $\begingroup$ Building on this, what about characterizing an $n^{th}$ degree polynomial as one whose sequence of successive differences becomes constant after $n$ steps? Students do this a lot by intuition, but they usually must be led to the general observation. That is, the successive differences of a linear function are constant, the differences of the differences of a quadratic are constant, the diff's of the diff's of the diff's of a cubic are constant, etc. Since this is a precalculus course, this would be a nice precursor to the derivative concept. $\endgroup$ – Nick C Nov 10 '17 at 21:31
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This may not be easy to make clear without computation, but the intersection of two ellipses leads to a 4th-degree polynomial.


          enter image description here
          (Image from Elliet Noma.)
See Dave Eberly's "Intersection of Ellipses" (PDF download) for the (formidable) details.

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Since a precalculus is, by its nature, designed to prepare students for calculus, I recommend gently introducing students to the concepts of calculus using the tools they already have. In this case, you can use quadratic functions to introduce students to the concept of a tangent line and a limit, without really using that terminology.

See the first project ("How fast does the pumpkin fly?") in the link below.

https://www.dropbox.com/s/60qdz1npspwu7o0/Precalculus%20Projects%20-%20Brendan%20W%20Sullivan.pdf?dl=0

To celebrate autumn, I bought a a small trebuchet with which to launch pumpkins and investigate their flight paths. I am able to control the initial launch height and velocity of the pumpkin, but I need your help to determine where it will hit the ground and how fast it will be traveling at that moment.

I put my trebuchet atop a hill so that the pumpkin is launched from exactly 80 feet above ground level. The launch imparts a vertical velocity of 64 feet per second and a horizontal velocity of 40 feet per second. Because I know the strength of gravity on Earth, the height h of the pumpkin, t seconds after launch, is given by this function: $h(t) = -16t^2 + 64t + 80$.

  1. Use factoring or the quadratic formula to determine when the pumpkin returns to ground level. For the purposes of the rest of the project, let's say your answer is $T$ seconds after the launch.
  2. Use the given horizontal velocity to identify where the pumpkin will hit the ground. (You may assume there are no effects on the flight path like friction or wind, so the horizontal velocity is constant.)
  3. Your next goal is to find the speed of the pumpkin as it hits the ground. Suppose $d > 0$ is a small positive value, and consider the height of the pumpkin at time $t = T - d$ and at time $t = T$. Explain why the ratio $\frac{h(T)-h(T-d)}{d}$ represents the average velocity of the pumpkin over the time interval $T-d\leq t\leq T$. (Possible hint: Think about the units of measurement of the top and bottom of that fraction.)
  4. Use a calculator to estimate that ratio for the values $d = 1, d = 0.1, d = 0.01$, and $d = 0.001$. Make a prediction for what you think the vertical velocity of the pumpkin is as it hits the ground.
  5. Now, instead of using specifi c numbers, use the given function $h(t)$ to simplify that expression, $\frac{h(T)-h(T-d)}{d}$, as much as possible so that it no longer involves division. Then, substitute $d = 0$ to fi nd the instantaneous vertical velocity of the pumpkin at that moment. Compare to your prediction.
  6. In fact, the overall speed of the pumpkin as it hits the ground is $S = \sqrt{V^2+H^2}$, where $V$ is its vertical velocity and $H$ is its horizontal velocity. Find $S$.
  7. Find the equation of the line that passes through $(T, 0)$ and has slope $V$. Graph this line together with the function $h(t)$ itself on the domain $0\leq t\leq T$ to verify that you found the tangent line to the graph of $h$ by fi nding that instantaneous velocity.

Extra Credit Follow a procedure similar to the one used above to find the instantaneous vertical velocity at the moment where $t = 2$. Graph $h$ and the tangent line together. Explain why your result makes sense physically.

(Note: $t=2$ is where the maximum height is reached so velocity is 0.)

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