1) I reminded you that I spend a lot of time preparing Saturday Math Club materials and lessons, and that I expect you to take the assignments seriously. This means not waiting until Thursday night to start thinking about the assignments, and it means calling me on the telephone if you are having difficulty with an assignment. You agreed.
2) We spent most of our time discussing Situations 5, 6, and 7 from the assignment given for January 10. For Situations 5 and 6, we discussed the idea of "dependence" in a diagram, and the idea that how a diagram behaves in Sketchpad when you moved a point in it is completely dependent on the construction that made the diagram.
We constructed "dependency diagrams" to show how one part of a diagram is immediately dependent upon others (and therefore how a part of a diagram is remotely dependent on other things in the diagram).
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Situation 5 (from previous assignment): Start with 
, with midpoint C, and 
, with midpoint E.
Construct line k
perpendicular to 
at C.
Construct line n
perpendicular to 
at E. Construct point F, the point of intersection of lines k and n.
Construct a circle with F as its center and which passes through point D (first select point F, then shift-select point D, then choose Circle by Center + Point in the Construct menu).
Dependency diagram for Situation 5.
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Situation 6 (from previous assignment): Use the circle tool to make a circle centered at A
and passing through B. Put two more points on the circle. Construct 
and 
. Construct perpendicular bisectors to 
and 
.
Move: Point A.
Do this: Describe everything that is wrong with this "explanation" of why the circle moves as it does when you move point A.
When you move point A, the lines still have to go through
A, so they have to move to follow A. Also, when the lines move, 
and 
have to move so that the lines are still perpendicular to
them and they have to move so that the lines still go through the segment's
midpoints. Also, the circle has to go through points B, C, and D, so wherever
points B, C, and D move, the circle has to follow them.
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What we
ended up deciding was that everything that is wrong with these explanations can
be summarized by one statement: This person has all the dependencies
backward.
The dependency diagram we constructed is at the right. Lines k and n do not depend on point A, as the
explanation above would have us believe. Also, the line segments BC and CD do
not depend on lines k and n
as the explanation above claims. Rather, lines k and n depend on segments BC and CD. The
last sentence is partially correct. The circle depends on point B, but it does
not depend on points C and D. Rather, points C and D depend on the circle.
Questions for 1/17/92:
In the diagram of Situation 6 given above, lines k and n appear to pass through point A.
(1) If the lines do in fact pass through point A (and not just so close that we are fooled into thinking they do), is this because having the lines pass through point A is part of the diagram's construction? Or is it the case that the lines "just happen" to pass though point A as a consequence of the construction? Explain.
(2) If you answered "they pass through point A as a consequence of the construction," then answer this: Will the lines always pass through point A no matter how we move points A, B, C, or D? Explain. Relate your explanation to the dependency diagram for this construction.
Situation
7: Construct a circle that passes through all three vertices of ÆABC, so that no matter how you move any vertex of the triangle, the circle continues to pass through all three vertices.
Our Answer: Construct the perpendicular bisectors to each side. Construct G, the perpendicular bisector's point of intersection. Then construct a circle centered at G and passing through any one of the triangle's vertices.
Question: Why doesn't it matter which of the triangle's vertices you use as the "circumference" point when you construct the circle centered at G? (For instance, some people might think that you can only use G and C to make the circle.)
In
class we came to the conclusion that the reason it doesn't matter which of
points A, B, or C we pick as the point on the circle, because each of them is
equidistant from point G (this "fact" itself needed justification,
but we assumed it was clear from before why they are equidistant from point G).
Since they are equidistant from point G, and will remain equidistant from point
G no matter how the diagram is altered (that is, altered in ways allowed by the
diagram's construction). Since points A, B, and C will always be equidistant
from point G, they will always all be on one circle centered at G, so it
doesn't matter which of them you use as a point through which the circle will
pass.