- What could have been done in class to better prepare you for the PSAT 10?
- What could you do to better prepare yourself for the PSAT 10?
Most of the feedback centered around the heavy algebra focus on the PSAT. There were some concepts that they had never seen. I mentioned that, next year in Algebra II, they would likely see much of this content. They also mentioned how there were many problems where values weren't given but they were asked for a solution. They also mentioned vocabulary that they didn't know or hadn't remembered. Finally, the mentioned the complexity of wording for some questions; in general, they found these confusing and didn't how to proceed.
The main suggestion was to review algebra pieces throughout the year. This would help keep concepts fresher.
They didn't have many suggestions on what they could do for themselves to prepare. I asked how many had actually worked through the practice test they were given. Only a handful of students had taken this step to prepare for the test. Only one student made use of the Khan Academy SAT preparation material.
With that, we moved on to circles. I decided to develop the concept of circle and use this to drive the need for finding distances, which I'll use to move to a formula for circles.
I asked the class what a circle was. The response tended toward, "It's a rounded figure with no sharp edges." I asked the class if the wall clock was a circle; they responded, "Yes." I asked about a circular disk magnet on the board and, again, the response was, "Yes."
I drew a rough circle on the board and asked whether this was a circle or just a representation of a circle. The class said it was actually a representation of a circle. I then went back to the clock and magnetic disk and asked the same question. They agreed that these were also just representations of a circle.
I briefly discussed the origin of geometry and how ancient Greeks separated the physical manifestation of a concept from the concept itself. I asked students how many had heard of the Greek philosopher Plato. I was pleased that almost one third of the class had.
I briefly explained Plato's concept of ideal form and then handed out a brief explanation that was suitable for high school students:
Plato's Theory of Forms
Written by Michael Vlach (http://www.theologicalstudies.org/)
Plato is one of the
most important philosophers in history. At the heart of his philosophy is his
“theory of forms” or “theory of ideas.” In fact, his views on knowledge,
ethics, psychology, the political state, and art are all tied to this theory.
According to Plato,
reality consists of two realms. First, there is the physical world, the world
that we can observe with our five senses. And second, there is a world made of
eternal perfect “forms” or “ideas.”
What are “forms”?
Plato says they are perfect templates that exist somewhere in another dimension
(He does not tell us where). These forms are the ultimate reference points for
all objects we observe in the physical world. They are more real than the
physical objects you see in the world.
For example, a chair
in your house is an inferior copy of a perfect chair that exists somewhere in
another dimension. A horse you see in a stable is really an imperfect representation
of some ideal horse that exists somewhere. In both cases, the chair in your
house and the horse in the stable are just imperfect representations of the
perfect chair and horse that exist somewhere else.
According to Plato,
whenever you evaluate one thing as “better” than another, you assume that there
is an absolute good from which two objects can be compared. For example, how do
you know a horse with four legs is better than a horse with three legs? Answer:
You intuitively know that “horseness” involves having four legs.
Not all of Plato’s
contemporaries agreed with Plato. One of his critics said, “I see particular
horses, but not horseness.” To which Plato replied sharply, “That is because
you have eyes but no intelligence.”
I gave students time to read the article and then we discussed it. Students thought this made sense and could understand that we held a perfect circle in our minds while representing this perfect circle through objects in the real world.
I then revisited how could we define this perfect circle. Students understood the challenge but were a bit perplexed about how to proceed. I had a xy-coordinate grid with a circle centered at (0,0) projected on the board. I pointed out there was a center, which was a fixed given point. I asked what they could say about the points lying on the circle. They readily recognized that these were all the same distance from the center.
I drew an arrow to the center and wrote "Fixed, given point that we call the center." I then drew an arrow to the circle and wrote "Set of points that are all the same distance from a fixed, given point."
The class seemed comfortable with this definition. I asked if any other figure could fit this definition, i.e. if I gave them a fixed point, could they think of any other figure that could result if the set of points were all the same distance from the center? They agreed that we would end up with a circle.
I picked the point a point on the project circle, point at (-3, 4). I asked the class what the distance was from this point to the center of the circle. At this point they struggled a bit because they didn't remember how to calculate distances. I reminded them about trying to use the Pythagorean theorem by identifying a right triangle to use. With this hint, students started determining the needed triangle side lengths and determining the desired length was 5 units.
I picked several other points around the circle and asked how far these points were from the center. At first, some students wanted to start calculating a new distance. Soon, most realized that each of these points was still 5 units away from the center. I emphasized that this value represented the radius but that the radius wasn't just one segment, it was defined between every point on the circle and the circle's center.
I wanted to practice calculating distances and midpoints, since we hadn't done these in a while. I provided a series of problems I found online. Students grabbed whiteboard grids and markers, then got to work. I let them work through two problems at a time and then we discussed their results.
This was perfect review. The problems that asked for a midpoint, I related to having the endpoints of a diameter and trying to determine the coordinates of the circle's center.
I asked the class to review midpoint and distance formulas on Khan Academy, if they wanted additional practice.
We are set to use this work to try to derive the equation of a circle next class.
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