From circles to ellipses and functional transformations

The post is a wrap-up to the geometry class. I've run out of time and won't be able to have the class finish this, but I wanted to outline how the course would proceed to the end.

Students have just learned how to find the area of a circle's segment. This is exactly what is needed to find the area of overlap between two circles. The next step would be to give students some simple situations where the sector angle measurement and radii are known.

The next step would be to introduce the idea that all they know are the centers and radii of two circles. After considering this situation for a bit, the hope would be that they recognize they need to find the points of intersection of the two circles.

Because of the messiness of finding the points of intersection for two circles in the general case, I would focus on the situation where the centers fall on either the same vertical or horizontal line. These situations allow students to write out equations in standard form and then substitute expressions to obtain a quadratic equation in one variable. From here, using the quadratic formula or some other technique, they can find the two values for that variable. Back substituting into the original equations would provide the two values of the other variable.

When developing this approach, I discussed this idea with a colleague. He teaches pre-calc and I wanted to see if there were things they were doing in that class that I could bridge. Basically there wasn't anything to connect to directly. He did agree that the general case would be messy and that keeping the situations simpler would probably be best.

Knowing the points of intersection of the two circles now provides additional information to use. In this case, the radius is given and the base of the triangle has been calculated. These two pieces of information, along with an appropriate trigonometric ratio, can be used to find the central angle. This leads students to be able to calculate the segment area.

This investigation and work provides students many connections between their circle work, the ideas of area, and trigonometric ratios. I feel this is a good way to begin a review process without actually stopping everything to review for a final.

The last unit was to cover functional transformations. The idea was to do some algebra review as students start thinking about algebra 2 for the following year. My feeling was that this should be developed from the circle work rather than moving into linear or other functional forms. A colleague really wanted to move toward connections with ellipses.

With all the circle work, the idea of translations and how this affects the equations of circles was a central part of the circle unit. This was an introduction to functional transformations, but in a much more natural way.

The next step was going to be "normalizing" the equation of a circle. This would be accomplished by dividing all terms in the standard form of a circle equation by the square of the radius:

(x - h)² + (y - k)² = r² goes to (x - h)² / r² + (y - k)² / r² = 1

The "normalizing" is drawn from the concept in statistics of dividing through by a standardized value. I wanted a term to distinguish a standard form of the equation from this new form.

The next step would be to re-write the normalized equation as
((x - h) / r)²  + ((y - k) / r)²  = 1

A final modification in notation and we'll be ready to have some fun. Re-write the equation with subscripts for the radii as
((x - h) / r₁)²  + ((y - k) / r₂)²  = 1

The investigation would allow students to change any of the values for h, k, r₁ or r₂. What deformations of the circle can be made? How would a dilation that makes the figure bigger look? How would a dilation that makes the figure smaller look? What happens as one of the r values gets very large or very small.

This could be expanded into taking two ellipses and asking how you could tell if they overlap. You could look at approximating the overlap of the ellipses. You could ask students to re-write these in the form y = or to expand the equation out and ask what completing a square, as they did with circles, would look like. What is similar and what is different? How can they determine if the equation results in a circle or an ellipse?

That would be the concluding unit for geometry. If any one has the class time to go down this path, I'd love to hear how it worked for you. If you do want to try this, but are unclear about some of what I wrote, please feel free to contact me.

Next year I am back to teaching discrete math and statistics class only. I'll be blogging about changes to my discrete math content as we move from a single semester course to a full-year course.

I'll also be highlighting changes to my probability and statistics content as I add additional subjects into the course. I found myself short on content and having students doing too many projects during the second semester.

Have a great end to the school year!

Finding the area of a segment.

Today is the last day of instruction as we only have one more class before the final exam.

We started by checking results from the homework problems. Students seemed to do well with these. I dove into the idea of finding the area of a segment within a circle as this involves review material that students need for the final.

I revisited the overlapping circle scenario and told students to focus on the sectors that overlap. In essence, although the circles are overlapping, what we really need to be concerned with are the sectors that overlap. I colored in the segments that are formed and asked students to consider what could be done to find the area of the segments.

As I walked around, many groups were thinking about the triangle that gets formed. I was pleased that they were considering how to make use of the triangle. One group worked out the idea of finding the area of the sector and then subtracting the area of the triangle. (see image below)

I asked what the area of the triangle would be if the central angle formed by the sector was 50^o and the radius was 6 units?

As students looked at this situation, I asked the class what they needed to know about the triangle to calculate the area. Most of the students recognized they needed the base and the height. The base of the triangle is the chord forming the segment. The height is the perpendicular bisector from the circle's center to the chord.

With these pieces in place, students started to realize they would need to use trigonometric ratios to find the missing lengths. This took some time but students started to see that the triangle's height was found using the cos(25^o) and the base could be found using sin(25^o).

I provided two more problems for additional practice, using 60^o and 40^o with radii of 3 and 5, respectively.

With that, we used the remainder of the class to take the circle test. I had students work 20 minutes on their own. At this point, I had students circle the problems they had completed on their own. I then gave the class an additional 20 minutes to work in their groups. They were not to give each other answers. What I asked them to do was to ask each other questions. For example, in the equation of a circle, if they couldn't remember whether the constant value represented the radius or the square of the radius, they could ask their partners that question for clarification. I was hoping to focus their collaboration and growing their learning.

Next class will be looking at more review problems and then we'll take the final two days after that.

I'll be outlining where I was taking the rest of the circle unit and how I was going to tie it into function transformations in my next post.

Working with sectors

Today was focused on having students become more comfortable working with sectors. As usual, this took a lot longer than I had hoped. The good news was that most of the class seemed to be getting more comfortable working with sectors.

We started by looking at the homework problem: if a sector has an area of 2π and the circle has a radius of 3, what is the arc measurement of the circle? The issue students faced was how to find the portion of the circle covered by the sector and how to convert this portion to the corresponding degrees.

Fundamentally, students were not recognizing that the problem 9π x ___ = 2π, was the same as solving for x when you have the equation ax = b. Students knew they should divide to find x = a/b. For some reason, having values with π in the expression baffled them.

Once they realized these were the same problem, they found the sector covered 2/9 of the circle. The next struggle was to translate the 2/9 of the circle into an equivalent number of degrees. The first suggestion was to divide 360^o by 2/9. I asked students to do this and to see if this made sense. Students quickly saw that the result didn't make sense. When students multiplied 360^o by 2/9 they got an answer of 80^o, which did make sense.

I then gave the class two problems to work on, one in which the radius and arc measurement were given and they needed to find the area of the sector, and one in which the radius and the area of the sector were given and they needed to find the arc measurement.

These problems went slowly with different struggles occurring for different students. Simplifying expressions such as 5π / 25π to 5 / 25π were not uncommon. There is also a tendency to take two given values and either multiply or divide the values. When asked what the calculation represents, the typical answer is, "I don't know." I had to continue to reinforce the idea that a calculation should represent the physical reality.

After working through these two problems, we had some time left. I wanted students to get more comfortable with the work they were doing, so I provided two additional problems, changing the radii, area sizes, and arc measurement values. I provided easily divisible values, for example a sector with area 3π and radius 6 or a circle with radius 2 and a sector with arc measurement of 45^o.

Many students finished one or both of the questions in the last few minutes of class. We'll go through the answers and then focus on the overlap of the two sectors and dissecting those into component parts.

We're supposed to have an assessment on the circle unit tomorrow. I think I am going to make this a take-home assessment in order to allow time to work on overlapping.

There are only two more classes before the final exam. I won't get through transitioning from circles to functional transformations. I'll outline that once I finish my work with circles.

Overlapping circles and area of sectors

Today was a bit rough. With this being the last full week of school, it was showing in student effort. I started by asking students to brainstorm ideas of how they could find the area of overlap between two circles. I was being much more active in cycling through the class in an attempt to keep them focused and on task. This seemed to help, but the brainstorming was not producing much.

First, several groups came up with the idea of creating an equation. Good thought, but what would you include in your equation and what would the equation attempt to model. Second, students would suggest different calculations. Okay, what do the calculations you suggest actually represent in the problem situation.

It is discouraging to see that a large portion of the class continue to just slap two numbers together in the hope it may lead to something without thinking about what their equation or calculation actually represents. Several groups looked at triangles they could form. I like the idea. Invariably, the triangles they created were not actually helpful in modeling the situation.

I used the triangle idea to identify the sectors that overlapped. I wanted to focus on finding the area of sectors and this seemed like a good way to acknowledge their ideas while putting the discussion on a better course.

The sector piece moved along slowly. First, students couldn't remember how to calculate the area of a circle. I reminded them that we had dealt with circle area before and that they should know this. Next came the issue of central angles and the corresponding arc measurement. Again, students claimed they had never seen this.

I briefly touched on their intuition as to how central angles corresponded to arc measurement. I also tried to address the percentage of the circle covered by the sector. Eventually, we were able to do things, with some help, like what is the area of a sector if the circle radius is 3 units and the central angle formed by the sector is 55^o. We could also look at what the area of the sector is when the circle radius is 3 units and the arc measurement of the sector is 120^o.

I left as homework the problem of reversing the process. Given a sector has an area of 2π and the circle has a radius of 3 units, what is the central angle formed by the sector?

We'll practice this a bit more and move into forming triangles to subtract off the non-overlapping piece of the sector. This should bring us back into using some trigonometry.

When do circles overlap?

The last problem on the Circle Challenges sheet deals with overlapping circles. Several students had questions regarding this problem. This fit in nicely, as I intended to start the class looking at when circles overlap.

I displayed a graph with two circles drawn on it. One circle was centered at (4.3) with a radius of 5 units. The second circle was centered at (-6, 4) and had a radius of 2. I asked the class to consider what they knew about these two circles and how they could determine, using just the centers and radii, if the two circles overlapped.

This took quite a while. Students were not overly focused and had to be redirected several times to focus on the question. I eventually placed at circle centered at (-2, -8) with a radius of 10 on the graph. I asked how they knew the new circle overlapped the circle with radius 5?

After much thought, one group said you could see if the circles had common points. I responded that, yes, finding if the circles had points of intersection would work. I still wondered if there were other ways to do this.

Finally, somebody brought up the idea that you could compare the distance between the circles' centers and the size of the radii. We clarified what was meant and people agreed that this could determine if the circles did not overlap. I wrote:

r1 + r2 < distance between centers --> circles do not overlap

I then asked what would determine if the circles were externally tangent or intersected. At this point students talked about the distance between centers equaling the sum of the radii or being less than the sum of the radii. Our list was now:

r1 + r2 < distance between centers --> circles do not overlap

r1 + r2 = distance between centers --> circles are externally tangent

r1 + r2 > distance between centers --> circles overlap

The question I posed next dealt with how much overlap was occurring. Specifically, when are two circles internally tangent and when is one circle completely enclosed within another circle.

This took some time, but did proceed ahead a bit better than the first investigation. Several groups were coming up with good ideas but were struggling to express these mathematically. After a while, I let one group describe their idea. I demonstrated their thinking using the graphic on the board.

After a bit more time, a group expressed the idea of comparing the radius of the larger circle with how close the centers were with each other. We discussed at what point the circles become internally tangent. I used this to expand to what conditions would need to exist to indicate that the smaller circle was completely contained within the larger circle.

Our list expanded to:

r1 + r2 < distance between centers --> circles do not overlap

r1 + r2 = distance between centers --> circles are externally tangent

r1 + r2 > distance between centers --> circles overlap
r1 - r2 = distance between centers --> circles are internally tangent

r1 - r2 > distance between centers --> smaller circle completely contained within larger circle

This seemed to make sense to the class. A few people were still pondering the results but most felt comfortable with the relationships displayed.

At this point we were at the end of the period. I pointed out that we'll be interested in how much area is contained in the overlap. We'll dive into this idea next class.

Additional practice with circle equations

Today started with me passing back the self assessment problems I had collected last class. I discussed the two major issues that I saw while going through the work: switching signs for the circle's center and not adding additional value when completing the square.

With that, I had students work on the last two problems on the self-assessment sheet, since no one had got that far. The first problem dealt with showing why two circles with given equations were externally tangent. The second dealt with the idea of picking a random point inside a circle and determining how large a radius it could have while still being completely enclosed inside the circle.

Students had questions about the wording of the first problem. Specifically, they were having trouble understanding what the problem meant by externally tangent. A drawing helped clarify this point. A second issue arose from how to determine that the circles were externally tangent. It took a while for many students to realize they needed to compare the distance between the two centers to the sum of the radii of the two circles. Some students still did not understand this point.

I wanted students to consider what they would need to use or to do to help determine the size of an internal circle centered at a random point A. Many students recognized that the radius for a circle centered at A would have to become smaller as A moved toward any point lying on the circle. They also recognized that the radius would become larger as point A moved closer to the center of the original circle.

At this point, I wanted students to practice more with circle equation problems. I gave students the seven Circle Challenge problems from the MVP curriculum. Students worked on these problems in class. I left the completion of the problems as homework, which I'll collect at the start of next class.

We'll start looking at the area of overlapping circles next class. This will enable students to build on the ideas of externally tangent circles and circles lying within circles. We'll also be able to look at solving quadratic equations, use some trigonometry, central angles, area of sectors, and review area of triangles.

Fear of fractions and assessing recognizing circle equations

Today we started by looking at problems that were troublesome for students. The questions students had issues with were ones that involved fractions. One problem, in particular, caused students to shut down.

Students were asked if the following equation was the equation of a circle, defined a point, or resulted in the empty set:

2x² + 2y² -5x + y + 13/4 = 0.

Students tend to immediately shut down when they see fractions. In working through this problem, it is evident that most students do not have a solid conceptual understanding of fractions. It appears that, over the years, students memorize rules to deal with working with fractions but don't understand why the rules work or when they should be applied. This lack of fundamental understanding then leads to an inability to work with fractions.

We worked through this problem as a class. I continually asked the class what they could do to convert the equation into a form which looked more familiar or that they felt more comfortable working with.

The first suggestion was to subtract 13/4 from both sides of the equation. We now had
2x² + 2y² -5x + y = -13/4.

I asked the class how they felt about working with squared terms that had a coefficient in front of them. Was this something that looked familiar? Students suggested dividing everything by 2. We now had
x² + y² -5x/2 + y/2 = -13/8.

This last step posed some issues as students did not understand that dividing by two meant that we went from fourths to eighths. I tried to help their understanding by drawing a box and splitting it into two halves. I then asked what would happen if we divided the half by 2. I then drew a dotted line through the middle of each half. Students could see that a half divided in two results in a fourth.

Students did not like the look of this equation. Again, they found all the fractions troubling. We proceeded ahead. Students suggested rearranging the terms to get x terms together and y terms together. We now had
x² - 5x/2 + y² + y/2 = -13/8.

At this point, students knew they could use the area model but were still struggling with the fractions. Splitting 5/2 in half confused them. They could still not see that one half of 5/2 was 5/4. Nor could they see that one half of 1/2 was 1/4. We worked through these using the area model.

I had to remind students that when multiplying two fractions together, they had to multiply the numerators together and then the denominators together. As a result of completing the square, we had added an extra 25/16 and 1/16 together.

We now had
(x - 5/4)² + (y + 1/4)²  = -13/8 + 26/16.

Students knew to find a common denominator for the fractions before adding or subtracting. At this point the equation resulted in
(x - 5/4)² + (y + 1/4)²  = 0.

This result most students recognized as a point located at (5/4, -1/4).

After this, I had students work on a set of problems to assess how they were doing. While most students showed some knowledge of finding circle equations, there were still a few students that were getting the positive and negative signs wrong for the coordinates. There were also quite a few students that were not recognizing the value that was being added when completing a square. They were basically ignoring this fact.

Next class, we'll spend some more time working on circle equation problems. I'll be focused on helping those students that still show the misunderstandings that I described above. I am still hopeful of being to work on the area of overlapping circles next class as well.

Continued work with recognizing circle equations

Today we continued work with recognizing circle equations. I checked with how the three assigned homework problems went. About half the class actually worked on the problems. For these students, most had success, with a few still struggling with the algebraic manipulations that are needed.

I walked through the first example problem and shared thinking of how to separate and break down the problem into parts. After this, I let students work through the next set of problems.

For these problems, students had to add a constant in order to complete the square. While there was some struggle at first, most students started to see how they would need to split the values in the area model we've been using. Most initial confusion came from dealing with negative signs.

As students worked through these problems, some still didn't recognize that the two factors had to be exactly the same, since we are squaring the expression that is created. Working with these students helped them to understand that squaring the value meant that both factors had to be the same value.

The next issue arose with the added constant to complete the square. Students found the value they needed to add but didn't see that they were adding value to one side of the equation. I had these students multiply out their squared terms and compare them to the original equation. They could now see that there was added value. Again, algebra skills were a bit lacking as some students now wanted to subtract off the added value.

I referenced the original equation and broke out their expression to match the original expression plus the added value. This helped students to see they needed to add value to the other side of the equation.

At this point, most students were rolling along on the practice problems. As I walked around, I addressed individual questions, but most were looking for reassurance they were proceeding correctly, which they were.

I left the remainder of the problems for homework. We'll try to wrap up these problems with a discussion and then use the final set of exercises as an assessment of how well they are understanding the material.

The plan is to begin looking at areas of overlapping circles, which will pull in central angles, areas of sectors, trigonometry, and coordinate planes, after this.

Getting back into recognizing circle equations

Today was a bit of re-orientation to recognizing circle equations. I wanted students to try to re-engage on their own as much as possible. With that in mind, I had students focus on two problems that were previously assigned as homework but with which they had little success.

The two equations were:

1. 9 = 2y - y² - 6x - x²
2. 16 + x² + y² - 8x - 6y = 0

For the first problem, students were stumped about how to get started. I know that this equation looks different because of the negative in front of the squared terms. As I walked around, I talked with students about trying to re-format the equation into something that they would feel more comfortable working with. While students liked the idea, they were still stumped. I told them that I am more used to working with equations where the squared terms were positive and I would use that as my starting point. This helped some students, but not all. For these students I had to talk about ways to manipulate the equation so that equality was maintained, either by adding and subtracting terms to both sides of the equation or by multiplying both sides of the equation by the same value, such as -1.

Students ended up with a variation of the equation -9 = -2y + y² + 6x + x², which can be written as -9 = y² -2y + x² + 6x. For some students, this was enough to get them going. For others, we re-visited the area model.

Using the x² + 6x expression, we set up the area model as follows:

x2

The two areas adjacent to the x² area need to evenly split the 6x term. Students saw this led to:

x2	3x
3x

Students saw that they needed to square the expression (x + 3). But by doing so, they had the following:

x2	3x
3x	9

(x + 3)² produced the expression x² + 6x but also added an extra 9. This got the students rolling and they were able to successfully complete the square for the y terms and find the additional value being added in the equation.

Students worked on the second equation and most were able to complete this problem successfully.

I left 3 similar problems, although somewhat easier, as homework so that they can build confidence and practice the process.

We'll hopefully complete working through problems in the next class or two, at which point we'll look at finding the area of overlap of intersecting circles. Time permitting, we'll then explore connections between circle equations and ellipses.

Approaches to students finding/re-writing circle equations not in standard form

Today we had our final MAP test of the year. I asked students to continue their work with circle equations when they finished the test. We'll continue to work with finding/re-writing circle equations when not given in standard form.

The approach to take on this was discussed during our geometry team meeting this morning. As previously described in my last post, I introduced the task by giving non-standard forms of the equation and allowing students to play around to see if they could re-write the equation. We worked on a couple of simpler examples and used a graphic representation of an area model as support.

A colleague is taking a different approach. She started with using circle equations in standard form and having students multiply the expressions out, combining like terms, and placing variables on one side of the equals sign and constants on the other. She then had students undo the process on these same equations.

From here, see used a visual support, such as writing x² + 6x + ___ = 40 + ___. This was to reinforce the idea that once a perfect square was identified, students needed to add the same value to both sides of the equation.

As I discussed on my last post, my students are not understanding when two expressions or an expression and a constant have the same value. While the above approach does help students focus on the aspect of maintaining equality, my sense is that it does nothing for helping students understand the equivalence of value being expressed.

It will be interesting to see how my class progresses. I anticipate that it may take them a bit longer to work through the process but that they will have a stronger sense of the mathematical properties at play.

Recognizing circle equations - introducing completing the square

As expected, students did not have much success with finding the circle equations for the three unfinished problems:

1. y² + 4x - 20 - 2y = -x²
2. 9 = 2y - y² - 6x - x²
3. 16 + x² + y² - 8x - 6y = 0

One student started to build toward the correct equation with the first listed problem. He ended up with (x + 2)² + (y - 1)²  = 20. He failed to realize that he had added an additional quantity in his expression. Even still, he was on the right track for finding squared expressions.

With that, I used the Great Minds' material on recognizing equations of circles to push students along. I had students work on the first page. These practiced multiplying out squared expressions and factoring trinomials. The issue came with the final problem of completing the square when
x² + 6x = 40.

Even though they had an area model as a guide, students were confused. What the confusion boiled down to was that students do not recognize that x² + 6x and 40 are equivalent values. This means they do not recognize that one value can be substituted for another in expression.

So, what was happening was students saw that they would have the squared expression (x + 3)². They would then set (x + 3)² = 40 or 31 or just stare blankly because they didn't know how multiplying out the expression could result in x² + 6x = 40. As I walked around and talked with students, I realized they were not seeing the equivalence of value. Multiplying out (x + 3)² results in the expression
x² + 6x + 9. Students could not see that the first part of this expression (the x² + 6x)  had a value of 40. If they had, they would have recognized that x² + 6x + 9 is the same as 40 + 9 = 49, so that
(x + 3)² = 49.

I worked with students on this idea and they were slowly grasping the concept. We worked through problem 1 above, since it was nearly complete anyway. I wanted to provide a second example of how this works. I left the last two problems as homework. I'll see how they do on this, but the intent is to spend another day or two working through problems of this nature.

Finding equations of transformed circles

Today, we started by going through the self-assessment exercises that were assigned last class. Some students had questions about the last problem. I went through the answers with the class and there were few questions. Even for the last problem, the questions were not major hindrances to them answering the questions.

I asked the class to indicate with one to five fingers, how well they did on these problems. A one would indicate they were totally lost to a five indicating they got all the problems correct. Almost the entire class held up three or four fingers. I had a couple of students who held up two fingers and none that held up one finger. So, the class seemed on firm footing with the exception of two students who continue to struggle a bit.

With that, I wanted to focus on transforming circles. I started with a circle of radius 5 centered at the origin. I asked the class a series of questions about different transformations and what would happen to the equation of the circle.

For example, we translate the circle using T(x + 2, y -3). Students were able to readily re-write the equation after translations. Next, I asked what the equation would be if the circle was reflected over the x-axis. This gave students some pause, but they soon realized they just needed to find the image of the center after the reflection. Reflecting over the y-axis then proved no problem. Reflection over the line y = x proved a bit more challenging, but students managed to determine the new center and the resulting equation.

Finally, I asked students what would happen if there was a dilation with a scale factor of 3. This baffled the class for a bit because the squaring of the radius was confusing them. After a while, some students started to realize that the radius changes and that the result of multiplying the radius and scale factor is what ends up being squared in the circle equation. To check things, I then asked what happens if the scale factor is 1/5?

At this point, I had students re-visit the four problems we had skipped on the first practice sheet of problems. These require completing the square. I asked students to tackle these. I don't expect them to get the correct answers, but by attempting the problems, they will have a better appreciation of what we will cover next class.