Looking over my blog and realizing that I neglected to reflect on our lesson on manipulatives, I thought I would use this post to discuss one particular application about which I have been thinking a great deal. I had my first real day in a school last week and I sat in on two math classes, one grade 9 and the other grade 11. In the grade 11 class (a fairly large one), the topic of the day was the Sine Law and the "ambiguous case". For those who may be a bit rusty, the ambiguous case refers to the situation where more than one triangle can be constructed from the information given (2 sides and an angle). Mathematically, the ambiguity can be seen from the fact that sin(x) is not one-to-one, where x ranges from 0 to 180 degrees. Some of the students in the class had difficulty grappling with the idea of there being more than one solution. Although I was mostly watching, bits of conversation I heard made me think that the deeper issue related to the qualities of sin(x) as a function. Although its not a term they would use, the students seemed to be assuming that sin(x) was bijective. I was trying to think of ways to discuss the Sine Law and the different cases (no solution, one solution, two solutions) as simply as possible. The diagram as I've often seen it drawn shows the fixed side and angle, with the two possible fixed-length sides superimposed.
I included the red line in order to emphasize that the triangle outlined by the green lines is an isosceles triangle. The difficulty with the above diagram is that although there are only two triangles of interest, there are six triangles that appear. This leads to confusion with respect to where we should be looking, and what each line represents. One diagram I prefer, though I can't find many examples of it online, is to build the three cases from the intersection of the two black lines show above with a circle.
Shown above are the same two black lines as before. Three different segments were chosen for the second fixed-length side. Imagine that these were rods rotating on an axis fixed at the upper vertex of the triangle, I traced out the circle that the free end of each rod would draw. The intersection of the horizontal black line with any given circle marks the position the rotating segment would have to make in order to complete the triangle. The blue circle never intercepts and so has no solution. The red circle touches the line tangentially at one point and so has one solution. The green circle has two intercepts and so has two solutions (the ambiguous case). I could have more thorough and drawn a fourth, larger circle that also has only one solution.
I like this diagram because it contains a lot of information yet presents it in what I feel is the simplest and most natural way. The fixed properties remain fixed and it is clear that the only variable is the orientation of the second side. Practically, this diagram could be made in a few ways. I would be drawn from scratch by a student using a compass and a rule. What's nice about this approach is that as the circles get larger and larger, the nature of the solutions changes. This would lead to a natural path of investigation through which the student can be guided into deriving expressions that define each of the four cases. More tangibly, the black lines could be drawn on a piece of construction paper with a marker. The length of the second side could be chosen and a compass could be used to find the orientation(s) of that side. Alternatively, the side could be made out of strip of construction paper and rotated by hand until a solution is found. In whatever approach is used, it would to a good demonstration of the Sine Law and the ambiguous case, as well as connection between calculation and measurement.
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