I am pleased to announce that Discrete Math: An Inquiry Based Approach is now done. The textbook is designed for a full-year course in high school (post algebra 2) or a single semester introductory course for college. There are six units covering Combinatorics, Graph Theory, Elementary Number Theory, Cryptography, Set Theory and Boolean Algebra, and Logic and Proof.
This material was developed from material I have used in discrete math classes taught over the past seven years. Last year, I had a colleague who taught the material for the first time and provided valuable feedback on corrections and typos.
In conjunction with the book, I have also created corresponding slides, answer keys, assessments, and extensive teacher notes. I am looking for a handful of teachers to try the material out and provide any feedback on the course materials.
Contact me through my website if you are interested.
I hope your school year gets off to a good start. Have a great school year.
Saturday, August 5, 2017
Thursday, November 17, 2016
Discrete Math update - engaging students in proof writing
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.
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, 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.
Subscribe to:
Posts (Atom)
