Monday, 20 June 2016

Developing Derivatives Graphically

In this activity students develop the visual connection between a quadratic function and its derivative. Each student is given a quadratic function and an x value. They are to determine the point and the instantaneous rate of change of their function at that x value. Once they do this they plot their point and then also plot their rate of change (as a point at that x value) on a piece of chart paper. Since everyone is given a different x value, a graph of both the original function and its derivative should be constructed. This would be a good activity to introduce the concept of the derivative shortly after students know how to find the instantaneous rate of change by using a limit.
Once students are done the graphs on the charts, they can then move to extend their thinking with this Desmos activity where they visually estimate the slope of the tangent to eventually build the derivative (based on a original sketch by @eluberoff). As they slide the sliders to estimate each tangent, once they get them all close to where they should be, the derivative appears (see animated gif below)They do this for several simple quadratic functions and (hopefully) see the beginnings of how the power rule works as they move towards simple cubic functions.

  • MCV4U-A2.2 - generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function, f(x), for various values of x (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative, f ’(x) or dy/dx , and make connections between the graphs of f(x) and f ’(x) or y and dy/dx
  • A3.1 - verify the power rule for functions of the form f(x) = xn , where n is a natural number
  • Grid chart paper (one per function - write a different function on the top of each piece). You may wish to draw the grid on each piece before hand. Because the scale can get disproportionally large on the y vs x axis you may want to spread the scale out on the x so that it's not so crowded. Note that if you choose to us a disproportionate scale on the chart paper then the kids must use a similar scale in their mini graph papers ( I've done this activity a few times and only now, as we created this post, did we figure out that this is why the slopes of the tangents didn't quite match the graphs physically). 
  • Copies of each mini graph for each student via the handout (see image to the right for a sample). There are four functions (f(x) = 1/4x2, f(x) = 1/2x2, f(x) = x2, and f(x) = 2x2) and potentially nine x values for each function (-4, -3, ....3, 4). You may not want to give out the -4 or 4 for the f(x) = 2x2 function as it requires a big y axis scale. Depending on how big your class is, you may not want to hand out all of the mini graph sheets. For example, if you had 25 students you might want to use 3 of the functions. Cut these out ahead of time. You may wish to have more than you need ready so that if someone finished quickly, they can be given another one. 
  • Scotch tape for students to stick their graphs on the chart paper.
  • Markers. Preferably different colours (one colour for the function points, derivative points, function line, derivative line and tangent lines).
  • Laptops, Chromebooks, iPads if you will be extending this to the Desmos version
  • Place chart paper on the walls and distribute one mini graph to each student. Make sure there is enough of each function so that the shape will be visible
  • Each student then does the following:
  1. Determine the point on that function at that x value: A(     ,      )
  2. Determine the slope of the tangent at that x value for your function: Slope of tangent = _____
  3. On your mini graph, draw a line with the same slope as the slope you just calculated and going through the middle point on the mini graph
  4. Plot the point from #1 on the large graph. Stick your mini graph on the large graph paper so the point on the mini graph paper is on top of the point you just plotted (be careful with your orientation)
  5. Plot a second point that has your x value and its y value is the slope of your tangent: B(   ,   )
  • As students plot their points by sticking their mini graphs on the chart paper, you may wish to use the markers to draw over their points and lines to make them more visible. Note: If a student makes a mistake, do not correct it right away (or make it permanent with a marker). Instead ask students if they think anything is out of place either at the time or during the consolidation
  • Once all the graphs are done consolidate the ideas
  • Once consolidated then move to the Desmos activity. If you have not done a Desmos activity before you might like to watch some of these tutorials first: Navigating, starting an activity, and teacher dashboard.
  • Note that it is not necessary for students to get all the way through the activity. As long as they get through the quadratics. The cubical are the extension. The whole idea behind this is they discover the power rule.
Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

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