Today we continued to work on circle equations. We went through the assigned problems and the only problem that posed difficulty was one in which the circle equation was given as
(x2 + 2x + 1) + (y2 + 4y + 4) = 121. The issue came with factoring the expression.
I reviewed the area model for multiplication, which seemed to jog some students' memories. We worked through the factoring of the first term and more students seemed to remember. I left the factoring of the second expression to them. Once factored, the class had no problem identifying the circle's center and radius.
I used the last set of problems as an assessment. I asked students to work on them by themselves initially to see how well they understood the material. Afterward, if they needed to discuss problems with their group, they could.
The hang-up on this set of problems again came with algebraic manipulation of equations. I had to remind students that equations would need to be re-written if they did not look like the equation of a circle: (x - h)2 + (y - k)2 = r2. Dividing through by common factors, adding or subtracting constants to both sides of an equation and realizing that 2 (x - h)2 / 2 just equals (x - h)2 seemed to baffle most students.
I left the remainder of problems they didn't finish as homework. We'll go through the results next class. The intended progression will force students to practice more algebraic skills. It will be interesting to see how the remainder of the semester plays out.
Friday, April 29, 2016
Wednesday, April 27, 2016
More practice with equations of circles
Today started with going through the responses of the homework problems. The only issue appeared on the problem where 3 tangent lines defined the boundary of the circle. I drew out the representation and asked the class what they knew about the center of the circle in relation to the lines x = 8 and
x = 14? The class understood the center would lay half way between these two lines. From this, I asked what the radius had to be? Students saw that the radius was 3 units. Given the tangent line y = 3, students realized the circle center had to be above this line.
We then worked on problems from the Great Minds unit lesson on circles. The first set of problems went quickly. Students appeared to be comfortable with writing equations or giving the center and radius when given an equation. Few students were making the common mistakes that I saw last class.
We'll continue practicing with circle equations on Friday and start to move toward reversing the process, i.e. re-writing expressions to standard equation form.
x = 14? The class understood the center would lay half way between these two lines. From this, I asked what the radius had to be? Students saw that the radius was 3 units. Given the tangent line y = 3, students realized the circle center had to be above this line.
We then worked on problems from the Great Minds unit lesson on circles. The first set of problems went quickly. Students appeared to be comfortable with writing equations or giving the center and radius when given an equation. Few students were making the common mistakes that I saw last class.
We'll continue practicing with circle equations on Friday and start to move toward reversing the process, i.e. re-writing expressions to standard equation form.
Tuesday, April 26, 2016
Practicing with equations of circles
My intent for the next couple of classes is to allow students to get comfortable working with equations of circles. To this end, I had students work on problems 1-4 and 9-14 of the practice sheet (pages 6 and 7 from Michelle Bousquet's Equation of a Circle lesson plan).
Before working through these problems, I displayed the answers to the four examples we worked through last class. The main question focused on example one; how do you find the equation of a circle when given just the end points of a diameter. One student presented their approach, which was to graph the points out and work from there. We revisited how to find mid-points and worked through the example again.
There were a few common issues that arose on the assigned problems; this was not unexpected. First, when given a value for the square of the radius, many students initially treated this as the radius. I had students reference the general equation of a circle that they had recorded yesterday. For the few that were still unclear, I reminded them that the equation says that we have the square of the radius. This seemed to be enough of a nudge to get them going.
Some students wanted to pull the center coordinates directly from the equation, so (x - 2)2 + (y + 3)2 would have a center of (-2, 3). I worked with these students on relating how to translate the new center back to (0, 0). So, if the center were at (-2, 3), would subtracting 2 from -2 and adding 3 to 3 move the center back to (0, 0). Most students seemed to understand this idea, but a few continued to struggle through a couple more problems.
Problem 11 presented the same situation as example 1 from yesterday. I would point out these were the same and that we had already discussed how to work through the problem.
I had to explain what tangent meant in the sense of problems 12 and 13. Students were trying to make sense of the opposite side/ adjacent side definition that was used in right triangle trig. After explaining the idea of a line tangent to a circle, I would tell them that the line was acting, in essence, as a boundary for the circle. Students seemed comfortable with this and appeared to be able to proceed ahead successfully.
The final question came up on problem 4 where the square of the radius is 14. Students didn't feel comfortable with the radius being √14, but understood that this was what the radius must be.
Questions 5-8 require students to complete the square, which we have not covered yet. We'll get there, but I still want the class to become even more comfortable with what they are doing with the equations of circles. It will be easier for them to reverse the process if they know what form they are trying to achieve.
Before working through these problems, I displayed the answers to the four examples we worked through last class. The main question focused on example one; how do you find the equation of a circle when given just the end points of a diameter. One student presented their approach, which was to graph the points out and work from there. We revisited how to find mid-points and worked through the example again.
There were a few common issues that arose on the assigned problems; this was not unexpected. First, when given a value for the square of the radius, many students initially treated this as the radius. I had students reference the general equation of a circle that they had recorded yesterday. For the few that were still unclear, I reminded them that the equation says that we have the square of the radius. This seemed to be enough of a nudge to get them going.
Some students wanted to pull the center coordinates directly from the equation, so (x - 2)2 + (y + 3)2 would have a center of (-2, 3). I worked with these students on relating how to translate the new center back to (0, 0). So, if the center were at (-2, 3), would subtracting 2 from -2 and adding 3 to 3 move the center back to (0, 0). Most students seemed to understand this idea, but a few continued to struggle through a couple more problems.
Problem 11 presented the same situation as example 1 from yesterday. I would point out these were the same and that we had already discussed how to work through the problem.
I had to explain what tangent meant in the sense of problems 12 and 13. Students were trying to make sense of the opposite side/ adjacent side definition that was used in right triangle trig. After explaining the idea of a line tangent to a circle, I would tell them that the line was acting, in essence, as a boundary for the circle. Students seemed comfortable with this and appeared to be able to proceed ahead successfully.
The final question came up on problem 4 where the square of the radius is 14. Students didn't feel comfortable with the radius being √14, but understood that this was what the radius must be.
Questions 5-8 require students to complete the square, which we have not covered yet. We'll get there, but I still want the class to become even more comfortable with what they are doing with the equations of circles. It will be easier for them to reverse the process if they know what form they are trying to achieve.
Monday, April 25, 2016
The general equation of a circle
Today's class started with looking at how to express the distance between a point (x, y) on a circle with radius 5 and the center located at (0 , 0). Because I had left expressions written out with the center point still being shown, many got to the equation (x - 0)2 + (y - 0)2 = 52. Some students wanted to swap out the zeroes but realized this wouldn't work.
I asked the class why this expression described points on the circle. It took some thought but, finally, students referenced the triangles being formed and the distance formula. I asked the class what would happen if we dilated the circle and now had a radius of 1? While some students were still unclear about this, most said we would only have to replace the 5 with a 1 in the equation, resulting in
(x - 0)2 + (y - 0)2 = 12. I next asked what would the equation be if the dilation resulted in a radius of 10? At this point, students quickly stated the equation would be (x - 0)2 + (y - 0)2 = 102.
Next, I asked what would happen if the center were translated three units to the right. While some students wanted to replace the x and y values with the new center of (3, 0), many realized the equation would be (x - 3)2 + (y - 0)2 = 52. I repeated this with translating the center to the point
(-2, 0). Students wrote out (x - -2)2 + (y - 0)2 = 52. I asked what happens when you subtract a negative value? Students hesitated but one student said it would result in adding the value. Other students agreed, so I wrote out (x + 2)2 + (y - 0)2 = 52.
I pointed out that as we move the center in the positive direction we subtract the value and as we move in the negative direction we add the value. A colleague had mentioned that she thinks of the addition and subtraction process from the perspective of what does it take to move the center back to (0, 0)? I mentioned this to the class and many students related to this idea.
At this point, students started asking what would happen if the center were translated along the y-axis? Others pointed out that it should then change the value of the y-coordinate of the center being used. I asked what the equation would be if the center were translated to the point (4, 2)? Most students wrote out (x - 4)2 + (y - 2)2 = 52. There were still some students that wanted to replace the x and y values with 4 and 2.
At this point, I passed out a practice sheet (pages 4 and 5 from Michelle Bousquet's Equation of a Circle lesson plan). Students started by writing out the general equation of the circle. A few students wanted to replace h and k with numbers, but most were able to write the general equation:
(x - h)2 + (y - k)2 = r2.
The other issue that came up in these problems was the first example. Students tried using the endpoints as the center or couldn't remember how to find the midpoint of a line segment. The other thing that some students failed to realize was they needed to calculate the length of the diameter in order to find the circle's radius.
Most students completed the four example problems by the end of class. We'll go through these next class and complete the remainder of problems on the practice sheet . The plan is to focus on working through these and another set of practice problems before moving on to determining a circle's properties from an expanded expression.
I asked the class why this expression described points on the circle. It took some thought but, finally, students referenced the triangles being formed and the distance formula. I asked the class what would happen if we dilated the circle and now had a radius of 1? While some students were still unclear about this, most said we would only have to replace the 5 with a 1 in the equation, resulting in
(x - 0)2 + (y - 0)2 = 12. I next asked what would the equation be if the dilation resulted in a radius of 10? At this point, students quickly stated the equation would be (x - 0)2 + (y - 0)2 = 102.
Next, I asked what would happen if the center were translated three units to the right. While some students wanted to replace the x and y values with the new center of (3, 0), many realized the equation would be (x - 3)2 + (y - 0)2 = 52. I repeated this with translating the center to the point
(-2, 0). Students wrote out (x - -2)2 + (y - 0)2 = 52. I asked what happens when you subtract a negative value? Students hesitated but one student said it would result in adding the value. Other students agreed, so I wrote out (x + 2)2 + (y - 0)2 = 52.
I pointed out that as we move the center in the positive direction we subtract the value and as we move in the negative direction we add the value. A colleague had mentioned that she thinks of the addition and subtraction process from the perspective of what does it take to move the center back to (0, 0)? I mentioned this to the class and many students related to this idea.
At this point, students started asking what would happen if the center were translated along the y-axis? Others pointed out that it should then change the value of the y-coordinate of the center being used. I asked what the equation would be if the center were translated to the point (4, 2)? Most students wrote out (x - 4)2 + (y - 2)2 = 52. There were still some students that wanted to replace the x and y values with 4 and 2.
At this point, I passed out a practice sheet (pages 4 and 5 from Michelle Bousquet's Equation of a Circle lesson plan). Students started by writing out the general equation of the circle. A few students wanted to replace h and k with numbers, but most were able to write the general equation:
(x - h)2 + (y - k)2 = r2.
The other issue that came up in these problems was the first example. Students tried using the endpoints as the center or couldn't remember how to find the midpoint of a line segment. The other thing that some students failed to realize was they needed to calculate the length of the diameter in order to find the circle's radius.
Most students completed the four example problems by the end of class. We'll go through these next class and complete the remainder of problems on the practice sheet . The plan is to focus on working through these and another set of practice problems before moving on to determining a circle's properties from an expanded expression.
Friday, April 22, 2016
Connecting the distance formula to the equation of a circle
Today was focused on establishing a connection between calculating distances and the equation of a circle. I am hoping to build the foundation that enables students to derive the equation of a circle.
There are two different online resources that I could use to work on this. I elected to track along with material found on the Great Minds site. This site does require registration, but the teacher and student resources are free in pdf format. I am using chapter 17 of the circle unit as a basis for my lessons. The other resource would be the circle material from the Mathematics Vision Project. A colleague has elected to follow this tract and found teacher resource material for these lessons.
To start things off, I presented to different distance calculation problems. These tied in directly with the practice we had last class. Students seemed comfortable calculating out the required distances and had no issues or questions.
Next, I gave students a whiteboard grid, a compass, and tissue paper. I told students to draw a point near the center of the tissue paper; this was their given fixed point. I then had students measure off five units on the whiteboard grid using the compass. Next, they drew a circle of radius five units centered on their point.
Surprisingly, this took a lot longer than I expected. Students had trouble getting the measurement correct, or they tore their tissue paper as they attempted to draw their circle, or the tissue paper slid as they were drawing. Finally, we had our circles.
I asked students to place their circle such that the center was at the origin. I was projecting up an example to make clear what should be done. Next, I labeled the points (0, 5) and (5, 0) on the graph. Although these points didn't actually form triangles, we could think of one side length as a length of zero and one with a length of five.
I wrote out the equations (0 - 0)2 + (5 - 0)2 = 52 and (5 - 0)2 + (0 - 0)2 = 52 to represent the process we actually use to calculate the distance between these points. I then asked students to use this as a model and to identify other points on the circle that aligned with grid intersections on the graph. I used a radius of 5 specifically because the points with a combination of 3 and 4 (either positive and negative or in reverse order) would meet this criteria.
Students were able to find the points but struggled on the process idea. Many were confused because they basically were asking, "Wouldn't the result just be a distance of 5?" I explained to them we wanted to focus on the process they used versus the result that came out.
This idea bothered students. Working with the class this year, I have seen that students tend not to think about what they are doing and just do something. Something as simplistic as finding the distance between two points is difficult for students to break down as to what they are actually doing.
Many students came to the conclusion that they were counting grid lines to find the lengths. I finally had to return the opening problem of finding the distance between the points (7, 15) and (3, 9). When asked how they found the side lengths of their right triangles, they said they counted grids. I changed the points to (70, 150) and (30, 90) and asked if they would still count grid lines? They all responded that they wouldn't. I pushed on what would they do?
Finally, students started to realize they would take the difference between the x-coordinate values and then the difference between the y-coordinate values. I wrote out (70 - 30)2 + (150 - 90)2 to represent what they had just said.
At this point the light bulbs seemed to turn on. The class was able to go back and write out equations like (-3 - 0)2 + (4 - 0)2 = 52 and (3 - 0)2 + (4 - 0)2 = 52. Then questions started coming. When would you do something like (-3 - 4)2 + (4 - 3)2 or (7 - 3)2 + (15 - 9)2? Other students responded that those would represent the distance between the points (-3, 4) and (4, 3) or the distance between (7, 15) and (3, 9). Another student asked why all the results were equaling 5? Another student responded that the circle had a radius of 5 and all points on the circle are the same distance from the center.
It took time to get to this point but students were getting the point. I drew a point on the circle and labeled the point (x, y). For homework, I asked students to use what we had just worked on to write out how they would calculate the distance from this point to the center of the circle located at (0,0).
Next class we'll focus on solidifying the idea that the equation of a circle centered at (0, 0) is given by x2 + y2 = r2. From there, we'll move to looking at what happens when a circle is not centered at the origin.
There are two different online resources that I could use to work on this. I elected to track along with material found on the Great Minds site. This site does require registration, but the teacher and student resources are free in pdf format. I am using chapter 17 of the circle unit as a basis for my lessons. The other resource would be the circle material from the Mathematics Vision Project. A colleague has elected to follow this tract and found teacher resource material for these lessons.
To start things off, I presented to different distance calculation problems. These tied in directly with the practice we had last class. Students seemed comfortable calculating out the required distances and had no issues or questions.
Next, I gave students a whiteboard grid, a compass, and tissue paper. I told students to draw a point near the center of the tissue paper; this was their given fixed point. I then had students measure off five units on the whiteboard grid using the compass. Next, they drew a circle of radius five units centered on their point.
Surprisingly, this took a lot longer than I expected. Students had trouble getting the measurement correct, or they tore their tissue paper as they attempted to draw their circle, or the tissue paper slid as they were drawing. Finally, we had our circles.
I asked students to place their circle such that the center was at the origin. I was projecting up an example to make clear what should be done. Next, I labeled the points (0, 5) and (5, 0) on the graph. Although these points didn't actually form triangles, we could think of one side length as a length of zero and one with a length of five.
I wrote out the equations (0 - 0)2 + (5 - 0)2 = 52 and (5 - 0)2 + (0 - 0)2 = 52 to represent the process we actually use to calculate the distance between these points. I then asked students to use this as a model and to identify other points on the circle that aligned with grid intersections on the graph. I used a radius of 5 specifically because the points with a combination of 3 and 4 (either positive and negative or in reverse order) would meet this criteria.
Students were able to find the points but struggled on the process idea. Many were confused because they basically were asking, "Wouldn't the result just be a distance of 5?" I explained to them we wanted to focus on the process they used versus the result that came out.
This idea bothered students. Working with the class this year, I have seen that students tend not to think about what they are doing and just do something. Something as simplistic as finding the distance between two points is difficult for students to break down as to what they are actually doing.
Many students came to the conclusion that they were counting grid lines to find the lengths. I finally had to return the opening problem of finding the distance between the points (7, 15) and (3, 9). When asked how they found the side lengths of their right triangles, they said they counted grids. I changed the points to (70, 150) and (30, 90) and asked if they would still count grid lines? They all responded that they wouldn't. I pushed on what would they do?
Finally, students started to realize they would take the difference between the x-coordinate values and then the difference between the y-coordinate values. I wrote out (70 - 30)2 + (150 - 90)2 to represent what they had just said.
At this point the light bulbs seemed to turn on. The class was able to go back and write out equations like (-3 - 0)2 + (4 - 0)2 = 52 and (3 - 0)2 + (4 - 0)2 = 52. Then questions started coming. When would you do something like (-3 - 4)2 + (4 - 3)2 or (7 - 3)2 + (15 - 9)2? Other students responded that those would represent the distance between the points (-3, 4) and (4, 3) or the distance between (7, 15) and (3, 9). Another student asked why all the results were equaling 5? Another student responded that the circle had a radius of 5 and all points on the circle are the same distance from the center.
It took time to get to this point but students were getting the point. I drew a point on the circle and labeled the point (x, y). For homework, I asked students to use what we had just worked on to write out how they would calculate the distance from this point to the center of the circle located at (0,0).
Next class we'll focus on solidifying the idea that the equation of a circle centered at (0, 0) is given by x2 + y2 = r2. From there, we'll move to looking at what happens when a circle is not centered at the origin.
Wednesday, April 20, 2016
Introducing circles and PSAT 10 debriefing
Today started with a debriefing on yesterday's PSAT 10. I wrote 2 questions on the board:
Plato's Theory of Forms
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.
- 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.
Monday, April 18, 2016
Prepping for PSAT
This week is a little weird from a scheduling standpoint. Today is Monday and is a normal class schedule. Tomorrow is the state-mandated PSAT for all 10th grade students. Since we just completed the car project on Friday, it didn't seem to make sense to start our next unit today. I have a number of colleagues who have focused on prepping students since last week. I chose to wrap up the surface area volume unit and catch up with the other teachers. (Inquiry and letting students make sense of the mathematics always seems to take longer than I plan. You just can't anticipate how long it will take for students to absorb the material.
After reviewing what students should do on test day (time to show up, what to bring, where to find class assignments, and such), I pulled up some sample practice problems for the class to tackle. The first set of PSAT Math Practice was 10 questions, mostly focused on algebra. Students worked through these and then we checked answers. Overall the questions went well, although some students struggled with using expression relationships that involved a variable.
The other question that was an issue dealt with finding the average of 1/3 and 1/6. Students always seem to freak out when they see fractions. They also seem to lack any sense of magnitude and of operational proficiency with fractions. As a result, many students chose 1/9 as their answer, since they simply added the numerators and the denominators together to form 2/9. Well, the average of this value is 1/9.
I drew out a pie graph and colored in 1/3 of the graph. I then drew 1/6 of the circle, non-adjacent to the 1/3 slice, and colored it in. I then asked the class if it made sense that adding these two segments together would result in a smaller slice of the pie? The agreed that it didn't. I reviewed the idea of common denominators and students quickly realized the answer should be 1/4.
The last piece we discussed had to do with the meaning of the term, "product." It was surprising to see that students did not know the mathematical meaning of this term. As a result, they added values together rather than multiplying them. In addition, they ignored the trailing decimal values, and rounded heavily to get to one of the given answers. It's a bit disheartening to see the lack of thinking and reasoning that takes place to arbitrarily truncate or round values just to get to an answer that doesn't match any of the given answers. And then to take this a step further and to select the answer with the closest value to the calculated result.
We wrapped up with look at some of the College Board's PSAT 10 practice problems and resources. These problems were a bit tougher than the first ten problems, although all were accessible to the class' ability level. Students did get stuck on solving a quadratic equation. This was a no calculator test and the problem involved a linear term coefficient of 14 and a constant of -51, not something most of my class would want to tackle by using the quadratic formula, especially without a calculator. It turned out (what a surprise) that the equation could be factored.
With that, we were done for the day. I encouraged students to look through the practice exam they had previously received. I think it's helpful for students to become familiar with the structure of the test and questions, even if they don't answer the practice questions. The more comfortable they are with what they are going to face, the less chance they have to panic when confronted with the test.
I won't be posting again until Wednesday, unless something really interesting and/or unusual happens at the PSAT.
We'll dive into the world of circles after the PSAT.
After reviewing what students should do on test day (time to show up, what to bring, where to find class assignments, and such), I pulled up some sample practice problems for the class to tackle. The first set of PSAT Math Practice was 10 questions, mostly focused on algebra. Students worked through these and then we checked answers. Overall the questions went well, although some students struggled with using expression relationships that involved a variable.
The other question that was an issue dealt with finding the average of 1/3 and 1/6. Students always seem to freak out when they see fractions. They also seem to lack any sense of magnitude and of operational proficiency with fractions. As a result, many students chose 1/9 as their answer, since they simply added the numerators and the denominators together to form 2/9. Well, the average of this value is 1/9.
I drew out a pie graph and colored in 1/3 of the graph. I then drew 1/6 of the circle, non-adjacent to the 1/3 slice, and colored it in. I then asked the class if it made sense that adding these two segments together would result in a smaller slice of the pie? The agreed that it didn't. I reviewed the idea of common denominators and students quickly realized the answer should be 1/4.
The last piece we discussed had to do with the meaning of the term, "product." It was surprising to see that students did not know the mathematical meaning of this term. As a result, they added values together rather than multiplying them. In addition, they ignored the trailing decimal values, and rounded heavily to get to one of the given answers. It's a bit disheartening to see the lack of thinking and reasoning that takes place to arbitrarily truncate or round values just to get to an answer that doesn't match any of the given answers. And then to take this a step further and to select the answer with the closest value to the calculated result.
We wrapped up with look at some of the College Board's PSAT 10 practice problems and resources. These problems were a bit tougher than the first ten problems, although all were accessible to the class' ability level. Students did get stuck on solving a quadratic equation. This was a no calculator test and the problem involved a linear term coefficient of 14 and a constant of -51, not something most of my class would want to tackle by using the quadratic formula, especially without a calculator. It turned out (what a surprise) that the equation could be factored.
With that, we were done for the day. I encouraged students to look through the practice exam they had previously received. I think it's helpful for students to become familiar with the structure of the test and questions, even if they don't answer the practice questions. The more comfortable they are with what they are going to face, the less chance they have to panic when confronted with the test.
I won't be posting again until Wednesday, unless something really interesting and/or unusual happens at the PSAT.
We'll dive into the world of circles after the PSAT.
Saturday, April 16, 2016
Car Project - wrap-up
The final day of the car project focused on students building their car models. As it turned out, a couple of groups realized, as they built their design, that they could do better in terms of using less material and having greater volume. Others were still fine tuning their write-ups. I also had to have a few students put more comparison in their write-ups as they had focused on their final design without justifying their decision by explaining other designs they had considered.
I was pleased to see that many of the final designs contained curved surfaces. I know that students had primarily started with rectangular prisms. Seeing the curved surfaces was a clear indication that they had discovered that they could create more volume using less surface area through curved surfaces.
Tuesday is the PSAT test in Colorado. I have asked the class to take the practice PSAT this weekend. We'll go over questions and materials in on Monday.
The day after the PSAT test we'll begin our next unit, Circles. We'll be focusing on how a circle is defined, the equation of circles derived through the distance formula, completing the square, and transformations of circles: translations and dilations.
I was pleased to see that many of the final designs contained curved surfaces. I know that students had primarily started with rectangular prisms. Seeing the curved surfaces was a clear indication that they had discovered that they could create more volume using less surface area through curved surfaces.
Tuesday is the PSAT test in Colorado. I have asked the class to take the practice PSAT this weekend. We'll go over questions and materials in on Monday.
The day after the PSAT test we'll begin our next unit, Circles. We'll be focusing on how a circle is defined, the equation of circles derived through the distance formula, completing the square, and transformations of circles: translations and dilations.
Wednesday, April 13, 2016
Car Project - Day 3
Work continued on the car project. Groups finalized designs and began building their scale models. Students have two nights to write-up their report on which design they selected and why. We'll finish building the models next class and have students present their designs.
Tuesday, April 12, 2016
Car Project - Day 2
Today was the second full day of working on the car project. Since I saw most students working with rectangular-shaped cars last class, I decided to challenge them to push beyond this traditional shape. I asked students how they knew that a rectangular-shaped car would work better than, say, a hexagonal-shaped car? With that, I let students work.
As I walked around, I could see that students were pushing themselves to look at different shapes. Some played around with pyramids, others with spheres and cylinders. Issues brought to the forefront included finding the area of a trapezoid and the surface area of a sphere.
As the day progressed, I could see that the design for many groups was evolving to elongated cylinders with spherical end-caps. This was good to see as these shapes produce the best volume-surface area ratios.
Next class, I am hoping to wrap up the work on the designs and have students build their scale models. Each student will write-up their findings and the final report will be due on Friday. The plan would be for students to turn in their write-up and then have each group present their final design.
As I walked around, I could see that students were pushing themselves to look at different shapes. Some played around with pyramids, others with spheres and cylinders. Issues brought to the forefront included finding the area of a trapezoid and the surface area of a sphere.
As the day progressed, I could see that the design for many groups was evolving to elongated cylinders with spherical end-caps. This was good to see as these shapes produce the best volume-surface area ratios.
Next class, I am hoping to wrap up the work on the designs and have students build their scale models. Each student will write-up their findings and the final report will be due on Friday. The plan would be for students to turn in their write-up and then have each group present their final design.
Monday, April 11, 2016
Car Project - Day 1
Today was a productive day on the car project. I recapped requirements and told the class that they should be comparing at least three different car configurations/shapes. Several students had questions about their approach. I encouraged them to consider what the results of their designs were showing them so that they could use that information to make decisions as to the direction they should move, for example, if one design provides larger volume then how can they push that design to yield even more volume.
Several designs used curved surfaces. In essence, these designs were using a sector of a cylinder. I used this as an opportunity to address this with the entire class. I was able to relate the sector directly to a sector in a circle. The total degrees for a circle as we sweep around the center is 360. If we measure the angle of the sector created, we can determine the percentage of the circle used. This then related to the portion of the cylinder used. This works well to determine how much surface area is being used.
For volume of the sector, students would need also need to determine the volume of the triangular prism created with vertices at the circle center, and the end points of the cylinder arc that is being used.
To find the center of the circle, I had students use a compass to determine the center of the circle being used. They used a protractor to determine the angle measurement of the arc.
As I checked on student work, I saw that students were trying to move beyond just rectangular designs. I was pleased to see students considering the surface area and volume impact of wheel wells and other design nuances they had drawn.
I will continue to push students on their designs next class, especially trying to get them to explore pentagonal or hexagonal designs. I expect students to start honing in on a design choice by the end of next class.
Several designs used curved surfaces. In essence, these designs were using a sector of a cylinder. I used this as an opportunity to address this with the entire class. I was able to relate the sector directly to a sector in a circle. The total degrees for a circle as we sweep around the center is 360. If we measure the angle of the sector created, we can determine the percentage of the circle used. This then related to the portion of the cylinder used. This works well to determine how much surface area is being used.
For volume of the sector, students would need also need to determine the volume of the triangular prism created with vertices at the circle center, and the end points of the cylinder arc that is being used.
To find the center of the circle, I had students use a compass to determine the center of the circle being used. They used a protractor to determine the angle measurement of the arc.
As I checked on student work, I saw that students were trying to move beyond just rectangular designs. I was pleased to see students considering the surface area and volume impact of wheel wells and other design nuances they had drawn.
I will continue to push students on their designs next class, especially trying to get them to explore pentagonal or hexagonal designs. I expect students to start honing in on a design choice by the end of next class.
Friday, April 8, 2016
Car Project performance assessment
Today's class was spent on wrapping up work with the practice problems and moving to work on a performance-based assessment.
Students said they found the practice problems on volume difficult. I asked questions to get a better feel of what they struggled with. The first was just a conceptual understanding of volume. They recognize volume and how to calculate it for rectangular prisms. They don't translate the idea of volume to other shapes, even though I have tried to emphasize the idea that volume is area being stacked on top of each other. Thus, the volume of cylinders posed a road block.
After explaining the idea of volume as base area times height, I had students work the two volume of cylinder problems that I gave them. Things went better and they could at least apply the idea of calculating the base area (B) and then multiplying by height (h) to find volume (V), i.e. how much base area is being stacked (in essence).
The next problems that students got stuck on were ones in which they had to reverse the process. For example, you are given the volume of a cylinder (150π cm3) and the height of the cylinder (6 cm), what is the radius of the cylinder? I referenced students back to the idea of V = B x h. Students were able to find the base area of 25π cm2. I then asked what the radius of a circle would be if the area of the circle was 25π cm2? This was something the class could do and they found the radius was 5 cm. I asked them to reflect on this problem. What was in this problem that was so difficult? How could they overcome their fears and uncertainty to forge ahead and attempt something?
The next problem was similar in that it provided the volume of a cube as 2744 cm3. Students were asked to determine the side length of the cube. I asked students what they knew about cubes. The cube has 6 faces and the side lengths are all the same. For cubes, we can think of volume as
V = l x w x h. But in this situation, the side lengths are all the same, so V = l3 or, more specifically,
2744 cm3 = l3.
I asked students if they remembered how to find cube roots. None did. I then told them they were all capable of determining a number, when multiplied to itself three times would equal 2744. I then left them to their own devices. I walked around to check how they were doing. Many gravitated to using a calculator and guessing, with 10 being a popular first guess. The next guess tended to be 20 and students readily recognized the value they needed was a number between 10 and 20. After a couple of minutes, everyone in the class had found the cube side length was 14.
I used this as an opportunity to recognize that they didn't need to know a formula or algorithm to find a cube root, they just needed to think about the problem and have a way they could come to a reasonable answer. I think we have, too often, taught students to believe that math is about learning formulas and applying formulas rather than reasoning and problem solving. As a result, students find math confusing and boring. Who wouldn't if all they were walking away with were a bunch of memorized formulas that didn't make any sense?
I continued to use October Sky as a bridge for Mindset discussions and to provide students an opportunity to view high school students from a past age and to see what they were able to accomplish through effort and perseverance. We got to the point where the four teenagers had their first successful rocket launch. The discussion afterward was rich and provided many insights. We talked about characters and their mindsets. We discussed things that stood out or touched them. One student commented on how being successful took a lot of hard work. I am so glad I decided to bring this movie into the class room.
At this point, we have covered the core concepts in surface area and volume. I didn't cover ideas like apothems and surface area of cones because they are not something that the general student really needs and they just promote the idea that math is about memorizing meaningless stuff. It was time to unveil their performance-based assessment for the unit.
The "Car Project" is something that I thought up in my first or second year of teaching geometry. It requires students to consider surface area, volume, and the interplay of the two. It requires proportional thinking and ties back to dilations as they must use a scale factor to translate between their model design and reality.
Below are the instructions and rubric for the Car Project. I presented these to the class, let them absorb what the task was about and then ask questions. I received a lot of good, clarifying questions and was pleased to see they were actually thinking about what they needed to do and how they would need to approach it.
I limited groups to two or three people. A student can work on their own if they choose. For this project, students can form their own groups. This avoids personality conflicts that may arise with random groupings or groupings that I may put together.
Students took the last 15 minutes of class forming their groups and discussing initial designs. I made clear that groups needed to consider multiple designs as they had to convince the reader that their final design was indeed best. I also encourage students to think outside the box as to the shapes they would use. There is a tendency for students to stick with rectangular shapes.
I am giving students three class periods to work through their designs and to construct a model. They will then have two nights to complete their write-up of the project, which will be due at the start of the 4th class. I'll use this period to have students present their design and why they believe it is the best design.
From past experience, I will have to push students to consider other designs and other shapes, to challenge their thinking on convincing me their design choice is best, and to pose questions that force them to consider additional options. I'm looking forward to seeing this year's designs.
The next problem was similar in that it provided the volume of a cube as 2744 cm3. Students were asked to determine the side length of the cube. I asked students what they knew about cubes. The cube has 6 faces and the side lengths are all the same. For cubes, we can think of volume as
V = l x w x h. But in this situation, the side lengths are all the same, so V = l3 or, more specifically,
2744 cm3 = l3.
I asked students if they remembered how to find cube roots. None did. I then told them they were all capable of determining a number, when multiplied to itself three times would equal 2744. I then left them to their own devices. I walked around to check how they were doing. Many gravitated to using a calculator and guessing, with 10 being a popular first guess. The next guess tended to be 20 and students readily recognized the value they needed was a number between 10 and 20. After a couple of minutes, everyone in the class had found the cube side length was 14.
I used this as an opportunity to recognize that they didn't need to know a formula or algorithm to find a cube root, they just needed to think about the problem and have a way they could come to a reasonable answer. I think we have, too often, taught students to believe that math is about learning formulas and applying formulas rather than reasoning and problem solving. As a result, students find math confusing and boring. Who wouldn't if all they were walking away with were a bunch of memorized formulas that didn't make any sense?
I continued to use October Sky as a bridge for Mindset discussions and to provide students an opportunity to view high school students from a past age and to see what they were able to accomplish through effort and perseverance. We got to the point where the four teenagers had their first successful rocket launch. The discussion afterward was rich and provided many insights. We talked about characters and their mindsets. We discussed things that stood out or touched them. One student commented on how being successful took a lot of hard work. I am so glad I decided to bring this movie into the class room.
At this point, we have covered the core concepts in surface area and volume. I didn't cover ideas like apothems and surface area of cones because they are not something that the general student really needs and they just promote the idea that math is about memorizing meaningless stuff. It was time to unveil their performance-based assessment for the unit.
The "Car Project" is something that I thought up in my first or second year of teaching geometry. It requires students to consider surface area, volume, and the interplay of the two. It requires proportional thinking and ties back to dilations as they must use a scale factor to translate between their model design and reality.
Below are the instructions and rubric for the Car Project. I presented these to the class, let them absorb what the task was about and then ask questions. I received a lot of good, clarifying questions and was pleased to see they were actually thinking about what they needed to do and how they would need to approach it.
I limited groups to two or three people. A student can work on their own if they choose. For this project, students can form their own groups. This avoids personality conflicts that may arise with random groupings or groupings that I may put together.
Students took the last 15 minutes of class forming their groups and discussing initial designs. I made clear that groups needed to consider multiple designs as they had to convince the reader that their final design was indeed best. I also encourage students to think outside the box as to the shapes they would use. There is a tendency for students to stick with rectangular shapes.
I am giving students three class periods to work through their designs and to construct a model. They will then have two nights to complete their write-up of the project, which will be due at the start of the 4th class. I'll use this period to have students present their design and why they believe it is the best design.
From past experience, I will have to push students to consider other designs and other shapes, to challenge their thinking on convincing me their design choice is best, and to pose questions that force them to consider additional options. I'm looking forward to seeing this year's designs.
Wednesday, April 6, 2016
Struggling through surface area of cylinders
The problem set I gave my students included surface areas of rectangular prisms; triangular, square, and hexagonal pyramids; and cylinders. The going was slow finding the surface are for all of these. Part of the issue was that I asked for the lateral surface area and the total surface area.
For pyramids, the lateral surface area was easy to describe, since it was all the sides other than the base upon which the pyramid stood. This then caused some minor confusion for cylinders, especially, as students put all areas other than the bottom into lateral surface area. I had to explain that we don't include the top in the lateral area either.
With this understanding, students worked through most of the pyramid and rectangular prisms. The hexagonal prism still posed problems. As I drew diagonals and divided the figure into triangles, students could see that they could find the area of the hexagonal base by finding the area of each triangle. We revisited the interior angle sums for polygons and used this to determine the interior angle size of a regular hexagon. Some students saw that the diagonal would bisect the angle so each triangle had half of this angle measurement (θ). With some prodding, they could see that the height (h) of the triangle bisected the hexagon side (s). Working with the base angle, students then had to revisit their trigonometric ratios to find that tan(θ) = h / (s/2).
For cylinders, students just had trouble making connections. I try to use an analogy of wrapping paper around the cylinder. It's a easy, hands-on way for students to visualize the lateral surface area of the cylinder. This helped, but students were not comfortable with the whole idea. This presented the biggest struggle of the day.
A couple of questions dealt with reasoning and problem solving. For example, given the surface area of a cube, what is the cube's side length? One question was, for a given cylinder, would doubling the height or doubling the radius result in more surface area? Most of the students felt doubling the height would produce more surface area. For this, and as part of our brain break, I took students outside and had them stand in a tight circle. I had students kneel or squat down and said this represented the original cylinder. Standing up is equivalent to doubling the height. Next, I paced off the diameter of the circle we formed. I told students that we would step back to simulate doubling the radius. Looking at the much larger circle, I asked which cylinder had the larger surface area. Students could see doubling the radius results in a bigger surface area.
I had another set of problems that concerned volume that I gave to students for homework. We'll discuss these problems and work through a few additional surface area and volume problems next class.
For pyramids, the lateral surface area was easy to describe, since it was all the sides other than the base upon which the pyramid stood. This then caused some minor confusion for cylinders, especially, as students put all areas other than the bottom into lateral surface area. I had to explain that we don't include the top in the lateral area either.
With this understanding, students worked through most of the pyramid and rectangular prisms. The hexagonal prism still posed problems. As I drew diagonals and divided the figure into triangles, students could see that they could find the area of the hexagonal base by finding the area of each triangle. We revisited the interior angle sums for polygons and used this to determine the interior angle size of a regular hexagon. Some students saw that the diagonal would bisect the angle so each triangle had half of this angle measurement (θ). With some prodding, they could see that the height (h) of the triangle bisected the hexagon side (s). Working with the base angle, students then had to revisit their trigonometric ratios to find that tan(θ) = h / (s/2).
For cylinders, students just had trouble making connections. I try to use an analogy of wrapping paper around the cylinder. It's a easy, hands-on way for students to visualize the lateral surface area of the cylinder. This helped, but students were not comfortable with the whole idea. This presented the biggest struggle of the day.
A couple of questions dealt with reasoning and problem solving. For example, given the surface area of a cube, what is the cube's side length? One question was, for a given cylinder, would doubling the height or doubling the radius result in more surface area? Most of the students felt doubling the height would produce more surface area. For this, and as part of our brain break, I took students outside and had them stand in a tight circle. I had students kneel or squat down and said this represented the original cylinder. Standing up is equivalent to doubling the height. Next, I paced off the diameter of the circle we formed. I told students that we would step back to simulate doubling the radius. Looking at the much larger circle, I asked which cylinder had the larger surface area. Students could see doubling the radius results in a bigger surface area.
I had another set of problems that concerned volume that I gave to students for homework. We'll discuss these problems and work through a few additional surface area and volume problems next class.
Monday, April 4, 2016
Working through surface area and volume problems
Here in Colorado. we use PARCC for our state mandated assessment. A colleague had put together a table of PARCC approved formulas.
I made copies of this table and told students to paste or tape these into their notes. To be sure the notation was clear, I explained and illustrated the meaning of b versus the meaning of B in the formulas. Students seemed to understand that B was the area of the base in a 3-dimensional object while b was, essentially, the width in a 2 dimensional object. I asked students to read through the formulas in case there were any other questions about the notation. One student checked to make sure that d referenced the diameter of a circle.
With that, I asked students to work through the surface area and volume problems they were given last class. These problems weren't really worked on as the students were confused and the substitute teacher was not helpful, although they said he was a nice guy.
I walked around the room and checked with students as to how they were doing and to hear about their thinking. There were some common points of misunderstanding, such as a 3-d representation of a square pyramid not looking square even though the labeling said it was a square pyramid.
There were also issues with finding the base area of a regular hexagonal pyramid. For this, I showed how the base could be divided into triangles by connecting the polygon's center point to each vertex. The triangle sides were angle bisectors of the interior angles. I reminded students that we had looked at the sum of interior angles for polygons. Using this information, the base length, and the fact that the triangle height was a perpendicular bisector of the hexagon's side, to establish a right triangle in which we knew one side length, needed a second side length, and knew all the angles measurements of the right triangle. This leads us right back to using trigonometric ratios to find the height of the triangle. This process works for any regular polygon, the only things that change are the size of the interior angles, the number of triangles that are formed, and whatever the given side length is.
I continued to walk around and check on student progress. While there continued to be a bit of uncertainty and lack of confidence in reading diagrams, for the most part, the class seemed to grasp the general ideas of lateral surface area, total surface area, and finding base areas.
I asked students to complete their work on these problems before next class. We'll look at results and move onto another set of practice problems. I may introduce some additional variations from the problems, for example, finding the height of the pyramids they worked with and then using that information to calculate volumes.
I made copies of this table and told students to paste or tape these into their notes. To be sure the notation was clear, I explained and illustrated the meaning of b versus the meaning of B in the formulas. Students seemed to understand that B was the area of the base in a 3-dimensional object while b was, essentially, the width in a 2 dimensional object. I asked students to read through the formulas in case there were any other questions about the notation. One student checked to make sure that d referenced the diameter of a circle.
With that, I asked students to work through the surface area and volume problems they were given last class. These problems weren't really worked on as the students were confused and the substitute teacher was not helpful, although they said he was a nice guy.
I walked around the room and checked with students as to how they were doing and to hear about their thinking. There were some common points of misunderstanding, such as a 3-d representation of a square pyramid not looking square even though the labeling said it was a square pyramid.
There were also issues with finding the base area of a regular hexagonal pyramid. For this, I showed how the base could be divided into triangles by connecting the polygon's center point to each vertex. The triangle sides were angle bisectors of the interior angles. I reminded students that we had looked at the sum of interior angles for polygons. Using this information, the base length, and the fact that the triangle height was a perpendicular bisector of the hexagon's side, to establish a right triangle in which we knew one side length, needed a second side length, and knew all the angles measurements of the right triangle. This leads us right back to using trigonometric ratios to find the height of the triangle. This process works for any regular polygon, the only things that change are the size of the interior angles, the number of triangles that are formed, and whatever the given side length is.
I continued to walk around and check on student progress. While there continued to be a bit of uncertainty and lack of confidence in reading diagrams, for the most part, the class seemed to grasp the general ideas of lateral surface area, total surface area, and finding base areas.
I asked students to complete their work on these problems before next class. We'll look at results and move onto another set of practice problems. I may introduce some additional variations from the problems, for example, finding the height of the pyramids they worked with and then using that information to calculate volumes.
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