what is path of midpoint of a falling ladder?

Saturday 24 November 2018

Visualizing ... in your head

Something which I think that is often not emphasized or valued enough when teaching mathematics is encouraging students to visualize a graph, diagram, relationship between objects, motion of an object, a locus of points, etc.  There are several topics for which mentally visualization can help a student's understanding or ability to solve a problem.  These include: transformations of graphs, finding domain & range of a funcion, inverse functions, the unit circle and values of trigonometric functions, vectors in 2d and 3D (especially planes in 3D), definite integration to find areas & volumes, modelling linear motion (kinematics) - and there are many more not listed here.  I often find that even students who have been very successful in mathematics are quite weak at visualizing graphs, diagrams (2D & 3D), motion and paths of points 'in their head'.  I think this is mostly due to the fact that they have not been given enough opportunities to practice visualizing, or required to do so without the aid of a calculator, diagrams in a book or some kind of computer software.  So, I strive to ask questions in class - and on assignments (for example, sketch the graph of a function given a graph of its derivative) - where students can gain practice in visualizing in their head.

Here is one simple example. 

Imagine a ladder is leaning against a wall.  As the foot (bottom) of the ladder moves away from the wall, what shape of the path that the midpoint of the ladder moves along?

If students are having difficulty in visualizing this path, you can give them the following sequence (A to F) of six images that show the ladder at different stages as the foot of the ladder moves away from the wall.  (These images could also be used with the althernative version of the question below).  download Word file: falling ladder images

The question could also be asked in such a way that would demand a more 'mathematical' answer rather than just a description of the path.

A line segment of length one unit is such that one of its endpoints moves along the x-axis and its other endpoint moves along the y-axis.  Imagine the y-axis endpoint is at (0,1) and the x-axis endpoint is at (0,0).  The x-axis endpoint then moves along the x-axis to (1,0).  What is the equation for the path of the midpoint of the line segment?


The Geogebra applet below can be used to convince students of the correct answer.  Thiis is best done with trace turned on for the midpoint.  With the trace turned off, you can also use the applet to present the question.  Even when seeing the movement of the line segment some students will still struggle to correctly describe the path of the midpoint.


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