The inaugural full year discrete math class is rapidly closing in on the half-way mark. The content is working out that for the first semester we'll get through the counting and graph theory units. The counting unit test became our mid-term and the graph theory test will be our final exam.
I have a colleague who is teaching discrete math for the first time this year. It is also her first time teaching an inquiry-based course. She has been a tremendous help in making suggestions and helping to clarify meaning of teaching suggestions for different lessons.
It has also been helpful having three sections of students taking the class. I have been able to clean up typos, re-word confusing question prompts, and make corrections in printed solutions. In total, there are now over 250 pages of textbook, teacher notes, and solutions.
As for the classes, we are now looking at conjectures and proofs related to Euler and Hamilton paths and circuits. As in past years, students struggle with communicating precisely what they see that leads them to believe something is either true or false.
I had attended an AP Computer Science Principles training this summer that focused on cooperative learning strategies. Seeing some students trying to work through their explanations while others sat watching, I decided to mix things up a bit.
Students had been asked to prove or provide counterexamples for four different statements. I told students that every group was going to put their proof for the first claim on the board. I divided my whiteboards into sections and assigned each group to a different section. After students wrote their proofs on the board the fun began.
First, I told the class that when I read a proof, not only theirs but any proof, I look for clear supporting evidence of any claims made. Basically, I ask myself if this is a proof or if the statements are basically saying, "Trust me, it works." With that preface, I asked students to consider the first proof. I asked the class is this a proof or is the group saying, "Trust me." The class concurred it was a trust me statement.
I pointed out that what was on the board was a good start. However, a proof cannot stop there. There needs to be statements of why the statements are being made. A proof needs to go into the "because" realm to explain the features that support the claim.
One proof actually went beyond a simple "trust me" it works statement. Students recognized that the proof was more than simply a trust me statement but that it still wasn't totally convincing.
I asked students to consider what they just did and to attempt a proof of the second claim. I prepared the board for a second round of proof writing while the students worked in their groups.
The second time through there was a marked improvement in what they wrote for proofs.We went through these proofs and students could see the improvement. While there was only one trust me statement, students also recognized that none of the proofs were totally convincing. We discussed how to take the proofs further.
Besides helping students to better differentiate trust me statements versus a proof, I had more engagement from all students in the proof writing process. Different students were required to go up to the board and write down their thinking. I had other students check what was being written, watching to make sure the proof wasn't simply a trust me statement. This checking led students into discussion about what should be said and how to state it clearly.
I actually had students who normally don't participate be the ones from their groups to go up and start writing their proof on the board. During the second go around, I hadn't asked students to start writing their proofs on the boards. Groups just got up and started writing when they were done at their tables. Other groups would see the activity at the board and start working to try and finish quicker.
I felt like this was a big step in the proof writing journey that we are taking during the school year. I recognize that this won't necessarily make it easier for students to get their thoughts down on paper, but I think they will recognize when they are tending toward a "trust me" proof. My hope is this recognition will spur them to push further in getting additional support and trying to be more specific in their explanations.
Thursday, November 17, 2016
Thursday, September 8, 2016
Discrete Math - Establishing What Constitutes a Proof
As in past years, Discrete Math started with the counting unit. The class first looked at patterns and Pascal's triangle, moved on to polygonal numbers (specifically triangular and square numbers), and then looked at using finite differences to extract polynomial formulas. This work includes extensive work with recursive and closed formulas for triangular numbers.
At this point, I introduced a lesson on writing proofs based on an article in NCTM's Mathematics Teacher magazine. The article "Empowering Students' Proof Learning through Communal Engagement" appeared in the volume 109, number 8, April 2016 issue. The lesson followed the suggestions in this article using triangular number formulas.
As suggested in the article, you want to use a formula or concept that students are familiar with. The work with triangular numbers was a perfect choice given the different work students had performed on triangular numbers.
First, we briefly discussed the idea of proof. Proof had previously been discussed in the context of providing a solid mathematical foundation. Students were told that a formula like the triangular number formula appears to work, but how do you know the formula won't break down at some point? Proving the formula always works instills confidence that you can always rely on the formula. It is also not enough to just show a few examples because you don't know if the next example might be the one that doesn't work.
With that, groups were tasked with writing out in words and illustrations why the triangular number formula (recursive or closed) always works. Students were told to think and discuss how they would do this before writing anything on the poster.
This was a difficult task for students. The challenge of actually getting their thinking on paper was much harder than students initially thought. As students worked on their posters I walked around and posed questions or made suggestions to help students clarify their meaning. Specifically, I reminded students that no one would be left behind to explain their poster. The poster had to be self-explanatory.
I initially set a time limit of 15 minutes for the poster task. This was overly optimistic. As time was gradually extended, the total time came closer to 35 minutes. Once posters were completed, I told students to have a notebook to record their thoughts on each of the other posters. Students walked around making notes about which posters proved the formula and what was it about the posters that helped convince them about the truth of the formula. Circling through the posters took approximately 10 minutes.
Students returned to their groups and then discussed their notes. The purpose of the discussions was to identify characteristics of posters that helped prove the formula. I specifically instructed students to focus on the characteristics, not which poster was best. Each group created a list of what they thought should be included in a proof.
At this point we shared out and constructed a class list. Any items added to the list were agreed to by the class. Students were allowed to question or express concerns about any suggested additions. After all the student suggestions were out, I reviewed the list and asked the class permission to add one last item, mathematical terms are used properly. The class agreed this would be okay to add.
I went through this process with two classes. While the individual posters were okay in terms of proving the triangular number formula, students in both classes were able to extract key characteristics. Both classes came out with almost identical ideas. I revised and combined ideas to shorten and clarify the lists. Looking at the result, I was amazed that such a good list was able to be created from the posters I saw.
The result of this lesson was the following list:
Students stayed engaged in this task for approximately 50 minutes. They worked hard on trying to effectively communicate their thinking, on evaluating other posters, and on considering key characteristics that should go into proof.
As the article suggests, this is an effective way to help students build an understanding of what should go into a proof. All of the ideas and thinking came from the students. They weren't told, here's what needs to go into a proof. The students determined from their work and assessment of others' work, what should go into a proof. There is class buy-in as to how future proofs will be evaluated and, the generated list provides a solid foundation on which to further build the concept of proof.
I was pleased about the results and excited about the prospects of building on the work that the classes did today. I am glad I was able to implement the suggestions made in the article and the value that I was able to bring into the classroom as a result of this work.
I did mention to a colleague what I had done in class. She is teaching geometry and is just getting ready to introduce geometric proofs after working through some algebraic proofs. She was interested in what I did and how it went. She is now going to make use of this idea to help her students establish some criteria for proof that can then be transferred to geometric proofs. It's exciting to think of the possibilities for using these ideas in different classes throughout the school.
At this point, I introduced a lesson on writing proofs based on an article in NCTM's Mathematics Teacher magazine. The article "Empowering Students' Proof Learning through Communal Engagement" appeared in the volume 109, number 8, April 2016 issue. The lesson followed the suggestions in this article using triangular number formulas.
As suggested in the article, you want to use a formula or concept that students are familiar with. The work with triangular numbers was a perfect choice given the different work students had performed on triangular numbers.
First, we briefly discussed the idea of proof. Proof had previously been discussed in the context of providing a solid mathematical foundation. Students were told that a formula like the triangular number formula appears to work, but how do you know the formula won't break down at some point? Proving the formula always works instills confidence that you can always rely on the formula. It is also not enough to just show a few examples because you don't know if the next example might be the one that doesn't work.
With that, groups were tasked with writing out in words and illustrations why the triangular number formula (recursive or closed) always works. Students were told to think and discuss how they would do this before writing anything on the poster.
This was a difficult task for students. The challenge of actually getting their thinking on paper was much harder than students initially thought. As students worked on their posters I walked around and posed questions or made suggestions to help students clarify their meaning. Specifically, I reminded students that no one would be left behind to explain their poster. The poster had to be self-explanatory.
I initially set a time limit of 15 minutes for the poster task. This was overly optimistic. As time was gradually extended, the total time came closer to 35 minutes. Once posters were completed, I told students to have a notebook to record their thoughts on each of the other posters. Students walked around making notes about which posters proved the formula and what was it about the posters that helped convince them about the truth of the formula. Circling through the posters took approximately 10 minutes.
Students returned to their groups and then discussed their notes. The purpose of the discussions was to identify characteristics of posters that helped prove the formula. I specifically instructed students to focus on the characteristics, not which poster was best. Each group created a list of what they thought should be included in a proof.
At this point we shared out and constructed a class list. Any items added to the list were agreed to by the class. Students were allowed to question or express concerns about any suggested additions. After all the student suggestions were out, I reviewed the list and asked the class permission to add one last item, mathematical terms are used properly. The class agreed this would be okay to add.
I went through this process with two classes. While the individual posters were okay in terms of proving the triangular number formula, students in both classes were able to extract key characteristics. Both classes came out with almost identical ideas. I revised and combined ideas to shorten and clarify the lists. Looking at the result, I was amazed that such a good list was able to be created from the posters I saw.
The result of this lesson was the following list:
- Clear statement of formula or concept being proved.
- Clear explanation why the formula or concept is true and works all the time using specific details and steps in a logical order.
- Clear explanation of what symbols, variables, and labels represent.
- Clear use the formula or concept through examples.
- Supportive illustrations, diagrams, graphs, and tables.
- Correct use of mathematical terms.
Students stayed engaged in this task for approximately 50 minutes. They worked hard on trying to effectively communicate their thinking, on evaluating other posters, and on considering key characteristics that should go into proof.
As the article suggests, this is an effective way to help students build an understanding of what should go into a proof. All of the ideas and thinking came from the students. They weren't told, here's what needs to go into a proof. The students determined from their work and assessment of others' work, what should go into a proof. There is class buy-in as to how future proofs will be evaluated and, the generated list provides a solid foundation on which to further build the concept of proof.
I was pleased about the results and excited about the prospects of building on the work that the classes did today. I am glad I was able to implement the suggestions made in the article and the value that I was able to bring into the classroom as a result of this work.
I did mention to a colleague what I had done in class. She is teaching geometry and is just getting ready to introduce geometric proofs after working through some algebraic proofs. She was interested in what I did and how it went. She is now going to make use of this idea to help her students establish some criteria for proof that can then be transferred to geometric proofs. It's exciting to think of the possibilities for using these ideas in different classes throughout the school.
Saturday, August 20, 2016
Discrete Math: An Inquiry Based Approach for High School textbook available for trial
This summer I decided to organize my discrete math problems and notes into a textbook. The first two units, which cover combinatorics and graph theory, of Discrete Math: An Inquiry Based Approach for High School is now available for viewing or download in pdf form. All I ask, if you elect to use the textbook, is to provide feedback. Just email me updates on how the problem sets are working for you and your students. Please keep in mind this is the first draft and will be subject to updates and revisions.
Also, I have developed an accompanying book containing extensive teaching notes, a slide set for each section of the textbook, assessments, and answer keys for the textbook problems and the assessments. I will release these for use to a very limited number of teachers. In return, I need feedback on the accuracy and usefulness of the materials being provided. Email me your request, along with the number of sections of discrete math you are teaching and how many students total you have in your classes.
I am excited that I have a co-worker teaching discrete math along with me this year. We have three sections; I am teaching two of the sections and my co-worker is teaching the third section.
I hope your school year is off (or gets off) to a great start. I will be posting updates throughout the year on how things are progressing and if there are any major changes from posts I have previously made regarding the discrete math course.
All the best.
Also, I have developed an accompanying book containing extensive teaching notes, a slide set for each section of the textbook, assessments, and answer keys for the textbook problems and the assessments. I will release these for use to a very limited number of teachers. In return, I need feedback on the accuracy and usefulness of the materials being provided. Email me your request, along with the number of sections of discrete math you are teaching and how many students total you have in your classes.
I am excited that I have a co-worker teaching discrete math along with me this year. We have three sections; I am teaching two of the sections and my co-worker is teaching the third section.
I hope your school year is off (or gets off) to a great start. I will be posting updates throughout the year on how things are progressing and if there are any major changes from posts I have previously made regarding the discrete math course.
All the best.
Thursday, May 19, 2016
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!
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
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!
Wednesday, May 18, 2016
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 50o 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(25o) and the base could be found using sin(25o).
I provided two more problems for additional practice, using 60o and 40o 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.
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 50o 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(25o) and the base could be found using sin(25o).
I provided two more problems for additional practice, using 60o and 40o 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.
Tuesday, May 17, 2016
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 360o 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 360o by 2/9 they got an answer of 80o, 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 45o.
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.
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 360o 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 360o by 2/9 they got an answer of 80o, 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 45o.
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.
Monday, May 16, 2016
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 55o. 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 120o.
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.
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 55o. 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 120o.
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.
Friday, May 13, 2016
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
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
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 --> 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.
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.
Wednesday, May 11, 2016
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.
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.
Tuesday, May 10, 2016
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.
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.
Monday, May 9, 2016
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.
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.
Friday, May 6, 2016
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:
The two areas adjacent to the x2 area need to evenly split the 6x term. Students saw this led to:
Students saw that they needed to square the expression (x + 3). But by doing so, they had the following:
(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.
The two equations were:
- 9 = 2y - y2 - 6x - x2
- 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:
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.
Wednesday, May 4, 2016
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.
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.
Tuesday, May 3, 2016
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:
- y2 + 4x - 20 - 2y = -x2
- 9 = 2y - y2 - 6x - x2
- 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.
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.
Monday, May 2, 2016
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.
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.
Friday, April 29, 2016
Assessing work on equations of circles
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.
(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.
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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