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I have to choose a topic to give a presentation. The topic should be high-school level or at most Linear Algebra 1 and Calculus 1. Conics and polygons in the Euclidean geometry are some fine topics but have already been worked out by the professor so they are not available. I am required to bring up a question about the topic and answer it, using Mathematica or Geogebra as a visual tool, and then show mathematical theory related to the problem. My question is, what would be a nice topic fitting this requirements?

The presentation is between 30 minutes and an hour long.

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    $\begingroup$ Has the professor actually showed how to rotate coordinates and/or complete the square to put tilted or shifted conic sections into a more standard form ? There is a lot to conic sections, perhaps something is still left there to do ? Also, from the perspective of physics, quadratic forms are interesting. See en.wikipedia.org/wiki/…, the significance of eigenvectors of the intertia tensor and the formula for rotational KE is the conic section math, but it looks really different at first glance. That wiki has too much, but it might get u started. $\endgroup$ – James S. Cook May 9 at 13:53
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    $\begingroup$ Maybe this idea, which I cited in a recent answer here (thus, why it's on my mind). $\endgroup$ – Dave L Renfro May 9 at 17:58
  • $\begingroup$ Welcome to the site! "What is a cool high-school level topic" is too broad, but I think your real question can be fixed to be more specifically answerable. Can you go a bit more into detail about the purpose of the presentation? For example, is the purpose to inspire the students? Is the purpose to show that you can make clear presentations, like a job interview? Is the purpose to take up their time because you are a substitute for the day? You might also have multiple of these purposes in mind. $\endgroup$ – Chris Cunningham May 9 at 19:24
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    $\begingroup$ Surely there is more to Euclidean geometry than has been worked out by the professor! How about connecting the midpoints of the sides of a quadrilateral (link)? The result - including in the case where the quadrilateral is concave - may be a bit surprising, and can likely be proved in full along with some auxiliary results. It is definitely "high school level" and could be explored in Geogebra. $\endgroup$ – Benjamin Dickman May 10 at 2:13
  • $\begingroup$ My guess is that you are in a math for high school teachers course, and the teacher is having everyone do presentations. Is this correct? $\endgroup$ – Sue VanHattum May 14 at 18:01
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If it doesn't have to be related to Calculus and/or Linear Algebra, why not give the story of the seven bridges of Königsberg and show how Euler developed graph theory and solved that problem at the same time? No prior mathematics required :)

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    $\begingroup$ How would you use either Mathematica or GeoGebra to demonstrate this? I think this answer could be improved by suggesting how to use the required applications. $\endgroup$ – Nick C May 17 at 22:23
  • $\begingroup$ I agree. I’ve seen examples of both but I can’t answer here how to help implement such examples because I’m not really familiar with either software packages. $\endgroup$ – cheyne May 18 at 13:27
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I've used Geogebra to show the Taylor polynomial of a function. Adding terms one by one shows how the series converge to the function and how approximation improves.

Here is my Taylor polynomial of exponential https://ggbm.at/tszpraxe . However, I think sine function could be more effective.

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    $\begingroup$ Here's one that uses the sine function, including a slider to show the successive Taylor polynomials and their equations. You can also change the original function to something other than f(x) = sin(x). $\endgroup$ – Nick C May 22 at 22:53
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Related to Joseph O'Rourke's cycloid idea, I've used the following to discuss the motion of a train wheel, part of which is always moving backwards when the train is moving forwards (which now has an explanation on MSE. The length of the code is due to the ornamentation of the demonstration. The color-coded horizontal components of the velocity (which are constant at a given level above or below the rail) are plotted in the background; they are also plotted below for some of the actual positions of the point on the wheel. There are some spokes and points on the wheel to help visualize the rolling motion when the t slider is moved. The distance R of the traced point from the center is adjustable, too.

Manipulate[
 With[{
   cyc0 =   (* full cycloid path, computed once for all t *)
     Table[{θ}~Join~cycloid[R, θ], {θ, 0, 3 π, π/18}],
   bg = {   (* background: vertical grid, horizontal arrows *)
     {Thin, LightGray,
      Line[
       Table[{{t0, -2.2}, {t0, 2.2}}, {t0, π/3, 3 π, π/3}]],
      Gray,
      Line[{{{0, -2.2}, {0, 2.2}}, {{2 π, -2.2}, {2 π, 2.2}},
        {{-2.1, 0}, {3 π + 2.1, 0}}}]},
     {Arrowheads[Small],
      Table[{Hue[(y + 1)/5, 0.7, 1],
        Arrow[
         Partition[
          Table[{t0, y}, {t0, 
            If[y > -1, 0 - 5 π, 3 π + π], 
            If[y < -1, 0 - 5 π, 3 π + π], (y + 1) π/6}],
          2, 1]]
        }, {y, {-2, -15/10, -5/10, 0, 5/10, 1, 15/10, 2}}]},
     Thickness[Medium], Line[{{{-2.1, -1}, {3 π + 2.1, -1}}}]}},
  Dynamic@Show[
    With[{   (* cycloid path 0 ≤ θ ≤ t *)
      cyc = 
       Append[Cases[cyc0, x_?(#[[1]] <= t &) :> Rest[x]], cycloid[R, t]]},
     Graphics[{
       bg,
       Line[{{{-2.1, -1}, {3 π + 2.1, -1}}}],  (* rail *)
       With[{L = Max[R, 1], l = Min[R, 1]},    (* spokes *)
        {Lighter@Gray, 
         Line[Table[{circle[{t, 0}, L, θ + t], 
            circle[{t, 0}, -L, θ + t]}, {θ, 0, 2 π/3, π/3}]]}],
       {   (* rest of wheel *)
        Thick, Circle[{t, 0}], 
        Thickness[Medium], Circle[{ t, 0}, R], PointSize[Medium], 
        Point[Table[
          circle[{t, 0}, 1, θ + t], {θ, 0, 5 π/3, π/3}]],
        Line[{{ t, 0}, cycloid[R, t]}],
        Disk[{t, 0}, 1, {-t + π, -t + 5 π/6}], 
        Disk[{t, 0}, 1, {-t, -t - π/6}]},
       {   (* Velocity vector plots below plot *)
        Arrowheads[Small],
        Table[{Hue[(R Sin[-π/2 - t0] + 1)/5, 0.7, 1],
          Arrow[With[{y0 = 
                 If[#[[1, 1]] > #[[2, 1]], -2.4, 
                  If[t0 > 2 π, -2.8, -2.6]]},
            {{#[[1, 1]], y0}, {#[[2, 1]], y0}}]
           ] &[{cycloid[R, t0 - π/12], cycloid[R, t0 + π/12]}]
          }, {t0, 0, t, π/6}]},
       {        (* cycloid *)
        {Thickness[0.008], Gray, Line[cyc]}, (* frames the cycloid *)
        {       (* leading point of the trace of the cycloid *)
         Hue[(R Sin[-π/2 - t] + 1)/5, 1, 1], EdgeForm[Black], 
         Disk[cycloid[R, t], 0.075]},
        Thick,  (* the cycloid & mesh points *)
        Line[cyc, VertexColors -> (Hue[(#[[2]] + 1)/5] & /@ cyc)], 
        Black, Point[cycloid[R, #] & /@ Range[0, t, π/6]], Gray, 
        Point[cycloid[R, #] & /@ Range[π/12, t, π/6]]}
       }]
     ],
    ImageSize -> 600, 
    PlotRange -> {{-2.1 , 3 π + 2.1}, {-2.9, 2.2}}
    ]
  ],
 {t, 0., 3. π}, {{R, 2}, 0., 2.},
 Initialization :> (
   cycloid[rad_, θ_] := {θ, 0} + 
     rad {Cos[-π/2 - θ], Sin[-π/2 - θ]};
   circle[ctr_, rad_, θ_] := 
    ctr + rad {Cos[-π/2 - θ], Sin[-π/2 - θ]};)
 ]

Mathematica graphics

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Having linear algebra as an option opens up a whole lot.

Related to @cheyne's answer, you could introduce the adjacency matrix of a graph and how it and its matrix powers can be used to count paths. The rules for matrix multiplication are essentially the same as those for path composition.

Another thing would be to write the recurrence relation for the Fibonacci numbers as a two by two matrix. Then the asymptotic formula can be expressed in terms of eigenvalues.

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How Archimedes figured out the value for pi is a pretty cool construction. The lower bound is found through inscribed hexagon, then 12-gon then ..., there is lots of algebra with the Pythagorean theorem, along with constructing angle bisectors (which is also the perpendicular bisector of the chord, and you could go into proving that).

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The geometry of the cycloid could make a nice demonstration. You could derive (for a unit-radius circle): \begin{eqnarray} x(t) & = & t - \sin t \\ y(t) & = & 1 - \cos t \\ t & \in & [0,2 \pi] \end{eqnarray} Then you could derive the length of the curve: $L(t) = 4 - 4 \cos t/2$, which then shows that $L=8$ for $t=2\pi$, i.e., the length of one arch is four diameters of the circle.


         
          Image from Wikipedia.
Either Mathematica or Geogebra could be used to make the animation above.

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