Today we worked through a series of problems using basic probability properties. Problems included looking at a series of coin tosses, rolling two 6-sided die, rolling two 4-sided die, and working with information presented in tabular format.
The biggest hurdle seemed to be in understanding what the question was asking exactly. We discussed some of the issues in deciphering scenarios and I am hopeful that additional exposure will help.
We also discussed the phrase "at least" when used in probability. I told them that many times this results in calculating the complement event and subtracting from 1. I know that I'll need to reinforce this idea numerous times throughout the semester.
View the class summary to read a student's perspective of the class.
Friday, January 25, 2013
Thursday, January 24, 2013
Discrete Math - Day 8
This class started with a discussion of the second part of the pizza problem. Students were stuck on trying to use 5! as part of the solution, typically multiplying this by 6 to account for the sauce-crust combinations. I asked them what the 5! represented in the context of the problem. Most responded that it represented all the different ways that the five toppings could be created. I then asked if placing the five toppings on top of the pizza in different orders created a different pizza. Students realized that this did not change the pizza. I reinforced that 5! would represent all the possible arrangements of the 5 toppings but once they were placed on the pizza it was still, in essence, the same pizza.
I had students revisit the first part of the problem in which they calculated the number of 1-topping pizzas. I tried to get students thinking about how many 2-topping, 3-topping, 4-topping, and 5-topping pizzas could be created. At this juncture many students started creating lists to count these pizzas.
A couple of students caught on quickly to what they needed to count and found a pattern of 5, 10, 10, 5, 1 for the number of topping pizzas that could be created. They summed these and multiplied by 6 to account for the sauce-crust combinations.
Other groups started to get a better idea of what they were doing and I had these two students visit other groups to discuss what they had done.
At this point one group called me over to show me a connection to the pattern that they saw. They had included pizzas without any toppings and saw that the number of pizzas for each topping was
1, 5, 10, 10, 5, 1
which happens to be the 6th row of Pascal's triangle. They were very excited to make this discovery.
Students were ready to discuss the problem and the emphasis was that there were multiple ideas at play in the problem, which is more typical of what happens in counting problems. I showed how the number of 3-topping pizzas is connected to 5C3 since were are selecting a group of the items from the 5 available.
At this point I had the one group share their discovery of the connection to Pascal's triangle. I wondered whether it was a coincidence or if this would occur for 6 toppings instead of the 5 we worked with. Since pepperoni wasn't listed we added this and students confirmed that the counts matched the next row of Pascal's triangle.
I had told student's that Pascal's triangle shows up in unusual places and here it was showing up in the pizza problem. I don't go into the binomial expansion in depth but this would be one place where an investigation of binomial expansions could be inserted. I did lay out binomials and after writing the first few terms out I asked students to focus on the coefficients of the binomial expansions. They were amazed to see Pascal's triangle showing up. I simply stated that the coefficients in a binomial expansion can be characterized using nCr which is why there is a connection between Pascal's triangle and the pizza problem results.
Invariably students ask why they were never shown this in Algebra as it would have made life so much easier. I have taught Algebra 1 classes and introduced some of these ideas. My feeling is that if students are able to make connections and sense of the mathematics then it is appropriate to introduce the ideas.
We wrapped up this piece with students recording their thoughts on working with permutations, combinations, and general ideas about counting.
We then looked at problems involving the pigeon-hole principle. The sock problem is fairly easy for students to grasp and they quickly understand why five socks need to be drawn to form a pair when there are four colors available. It's easy for students to consider what the worst case scenario is when drawing socks. This makes sense to them.
The gum ball problem tricks students a bit because the inclination is that with three children and six colored gumballs they need to draw 18 to guarantee that all three children have the same color gum ball. After some more thought and discussion students come to the realization that they only need to draw 13 gumballs in order to guarantee that three match.
I cover the general idea of the pigeon-hole principle and ask students to consider of the sock and gumball problems what represents the coups and what represents the pigeons in each problem. I like to do this so that they restructure their thinking slightly to allow for more flexible solutions.
I then presented three scenarios and asked them to explain why the statements were true. The first two are fairly easy for students to explain. The third scenario provides a bit more of a challenge. Only a couple of students were able to reason through the third scenario. I asked students to think some more about this problem.
We'll work on a couple of more problems next class just to be sure that these ideas make sense for students.
Visit the class summary to read a student's perspective of the class and to view the lesson slides.
I had students revisit the first part of the problem in which they calculated the number of 1-topping pizzas. I tried to get students thinking about how many 2-topping, 3-topping, 4-topping, and 5-topping pizzas could be created. At this juncture many students started creating lists to count these pizzas.
A couple of students caught on quickly to what they needed to count and found a pattern of 5, 10, 10, 5, 1 for the number of topping pizzas that could be created. They summed these and multiplied by 6 to account for the sauce-crust combinations.
Other groups started to get a better idea of what they were doing and I had these two students visit other groups to discuss what they had done.
At this point one group called me over to show me a connection to the pattern that they saw. They had included pizzas without any toppings and saw that the number of pizzas for each topping was
1, 5, 10, 10, 5, 1
which happens to be the 6th row of Pascal's triangle. They were very excited to make this discovery.
Students were ready to discuss the problem and the emphasis was that there were multiple ideas at play in the problem, which is more typical of what happens in counting problems. I showed how the number of 3-topping pizzas is connected to 5C3 since were are selecting a group of the items from the 5 available.
At this point I had the one group share their discovery of the connection to Pascal's triangle. I wondered whether it was a coincidence or if this would occur for 6 toppings instead of the 5 we worked with. Since pepperoni wasn't listed we added this and students confirmed that the counts matched the next row of Pascal's triangle.
I had told student's that Pascal's triangle shows up in unusual places and here it was showing up in the pizza problem. I don't go into the binomial expansion in depth but this would be one place where an investigation of binomial expansions could be inserted. I did lay out binomials and after writing the first few terms out I asked students to focus on the coefficients of the binomial expansions. They were amazed to see Pascal's triangle showing up. I simply stated that the coefficients in a binomial expansion can be characterized using nCr which is why there is a connection between Pascal's triangle and the pizza problem results.
Invariably students ask why they were never shown this in Algebra as it would have made life so much easier. I have taught Algebra 1 classes and introduced some of these ideas. My feeling is that if students are able to make connections and sense of the mathematics then it is appropriate to introduce the ideas.
We wrapped up this piece with students recording their thoughts on working with permutations, combinations, and general ideas about counting.
We then looked at problems involving the pigeon-hole principle. The sock problem is fairly easy for students to grasp and they quickly understand why five socks need to be drawn to form a pair when there are four colors available. It's easy for students to consider what the worst case scenario is when drawing socks. This makes sense to them.
The gum ball problem tricks students a bit because the inclination is that with three children and six colored gumballs they need to draw 18 to guarantee that all three children have the same color gum ball. After some more thought and discussion students come to the realization that they only need to draw 13 gumballs in order to guarantee that three match.
I cover the general idea of the pigeon-hole principle and ask students to consider of the sock and gumball problems what represents the coups and what represents the pigeons in each problem. I like to do this so that they restructure their thinking slightly to allow for more flexible solutions.
I then presented three scenarios and asked them to explain why the statements were true. The first two are fairly easy for students to explain. The third scenario provides a bit more of a challenge. Only a couple of students were able to reason through the third scenario. I asked students to think some more about this problem.
We'll work on a couple of more problems next class just to be sure that these ideas make sense for students.
Visit the class summary to read a student's perspective of the class and to view the lesson slides.
Wednesday, January 23, 2013
IPS - Day 8
Today we finished our exploration of simulations and experimental versus theoretical probability. We discussed the Sounding an Alarm worksheet. Most students did not have any idea how to list out the sample space. They also had no recall of calculating probabilities from previous classes they took.
We discussed the sample space and came up with the 8 possible outcomes. We then discussed how to calculate the probability for each outcome. As we calculated the probabilities, some students started to get a better sense of what they should be doing. The theoretical probability of having at least one alarm sound was calculated to be approximately 98%. This compared favorably to the 96% we calculated from a relatively small number of simulations. I discussed how sometimes we can calculate theoretical probabilities but that running enough simulations would get us close to the theoretical results.
The first portfolio problem was assigned today. Portfolio problems are designed to have students fully explain their reasoning and justify their results. I allow students to make revisions on their work until the get the results correct. Portfolio problems become mentor text that show students what they need to do when responding to questions. Students do not receive credit for a portfolio problem until it is 100% correct.
I also use a grading standard of Essential Correct (E), Partially Correct (P), and Incomplete (I). An E says that students "get it" and know what they are doing and can communicate their results and reasoning. There may be minor issues but that these are things that a student could, in essence, self-correct. A P indicates a student knows what they are doing but gets stuck, does not explain their reasoning, or would need some prodding or help through questioning to move on. An I indicates the student cannot proceed without a lot of assistance or provides an answer without any explanation of their thinking or justification of their results.
All graded work is scored using E, P, and I. On a test, a student getting all P's would receive a grade of C. A student getting all E's would receive an A. A student receiving a mix of E's and P's would fall somewhere in between.
The first portfolio problem asks students to assign random integers to simulation situations and explain why those assignments work. Since simulations will be used throughout the semester, I want to be sure students are comfortable with how to assign appropriate values for simulations. This is the first time I have used this particular version of the worksheet and I know that I want to modify it to reflect not just the assignment of digits but to include a description of the simulation process as well.
After a brief summary in their notes about what to remember when simulating situations we moved on to probability rules.
To get things going I have students work through an activity involving the Monty Hall problem. We use three cards (2 black and 1 red or vice versa) and have students work in groups of three. Each student rotates through a roll of player, host, and data recorder. The objective is to determine the probability of staying and winning versus the probability of switching and winning.
This took more time than expected as students did not understand the written instructions. I guess a whole class demonstration would help here. Once they understood what they were doing the activity moved along. We calculated probabilities of 36% and 55% for the two situations, which clearly show there is an advantage to switching. I use this to illustrate the probability and intuition do not mix well and that situations need to be thought through carefully to identify an appropriate sample space.
I asked students to consider any probability rules or properties that they remembered from previous classes. Very little came out of this discussion other than probabilities sum to 100%. We then went through some rules which I related to the Sounding an Alarm work. I asked students to review the rules in a textbook and then consider two examples and how the probability rules were applied to the situations. As one of the examples involved rolling a pair of dice, we discussed how the sample space was depicted and then I asked students to calculate some probabilities for some events: rolling a 7, rolling a pair, rolling a 3, and rolling a 4.
We'll work on applying probability rules next class.
Visit the class summary for a student's perspective of the class and to view the lesson slides.
We discussed the sample space and came up with the 8 possible outcomes. We then discussed how to calculate the probability for each outcome. As we calculated the probabilities, some students started to get a better sense of what they should be doing. The theoretical probability of having at least one alarm sound was calculated to be approximately 98%. This compared favorably to the 96% we calculated from a relatively small number of simulations. I discussed how sometimes we can calculate theoretical probabilities but that running enough simulations would get us close to the theoretical results.
The first portfolio problem was assigned today. Portfolio problems are designed to have students fully explain their reasoning and justify their results. I allow students to make revisions on their work until the get the results correct. Portfolio problems become mentor text that show students what they need to do when responding to questions. Students do not receive credit for a portfolio problem until it is 100% correct.
I also use a grading standard of Essential Correct (E), Partially Correct (P), and Incomplete (I). An E says that students "get it" and know what they are doing and can communicate their results and reasoning. There may be minor issues but that these are things that a student could, in essence, self-correct. A P indicates a student knows what they are doing but gets stuck, does not explain their reasoning, or would need some prodding or help through questioning to move on. An I indicates the student cannot proceed without a lot of assistance or provides an answer without any explanation of their thinking or justification of their results.
All graded work is scored using E, P, and I. On a test, a student getting all P's would receive a grade of C. A student getting all E's would receive an A. A student receiving a mix of E's and P's would fall somewhere in between.
The first portfolio problem asks students to assign random integers to simulation situations and explain why those assignments work. Since simulations will be used throughout the semester, I want to be sure students are comfortable with how to assign appropriate values for simulations. This is the first time I have used this particular version of the worksheet and I know that I want to modify it to reflect not just the assignment of digits but to include a description of the simulation process as well.
After a brief summary in their notes about what to remember when simulating situations we moved on to probability rules.
To get things going I have students work through an activity involving the Monty Hall problem. We use three cards (2 black and 1 red or vice versa) and have students work in groups of three. Each student rotates through a roll of player, host, and data recorder. The objective is to determine the probability of staying and winning versus the probability of switching and winning.
This took more time than expected as students did not understand the written instructions. I guess a whole class demonstration would help here. Once they understood what they were doing the activity moved along. We calculated probabilities of 36% and 55% for the two situations, which clearly show there is an advantage to switching. I use this to illustrate the probability and intuition do not mix well and that situations need to be thought through carefully to identify an appropriate sample space.
I asked students to consider any probability rules or properties that they remembered from previous classes. Very little came out of this discussion other than probabilities sum to 100%. We then went through some rules which I related to the Sounding an Alarm work. I asked students to review the rules in a textbook and then consider two examples and how the probability rules were applied to the situations. As one of the examples involved rolling a pair of dice, we discussed how the sample space was depicted and then I asked students to calculate some probabilities for some events: rolling a 7, rolling a pair, rolling a 3, and rolling a 4.
We'll work on applying probability rules next class.
Visit the class summary for a student's perspective of the class and to view the lesson slides.
Tuesday, January 22, 2013
IPS - Day 7
Today was a continuation of the lesson on simulations. The focus on these lessons is to help students build a foundation for how to use random integers to simulate a physical entity.
The first simulation problem was to simulate drawing cards with replacement from a standard deck of cards. After some discussion, students came up with using 0-12 with 0 representing an ace or 0-51 with 0-3 representing an ace. The question of interest was "What is the probability of drawing an ace within the first 10 draws?"
Students ran 3 simulations to gain some practice using their calculators. For students without calculators I had them use a random number table. We accumulated our results and found a probability of approximately 45%.
The next problem involved simulating rock-paper-scissors using random digits. Students readily came up with the idea to use either 0, 1, 2 or 1, 2, 3 to simulate RPS. They struggled with simulating the physical activity itself. I discussed how a simulation is recreating the physical process using random numbers. The challenge in this simulation is to simulate two people playing against one another.
Students then started to use the idea of generating two values and comparing them. One student determined that there were 9 possible outcomes to the game with 3 outcomes each of tie game, player 1 winning, player 2 winning. Using this idea, the student used 0,1,2 to represent ties, 3,4,5 to represent player 1 winning and 6,7,8 to represent player 2 winning.
Finally, we returned to Sounding the Alarm (from NCTM's Navigating through Probability in grades 9-12) and started to look at the theoretical probability. Students worked through an associated worksheet which they are to complete for homework. We'll discuss the results next class.
Visit the class summary for a student's perspective of today's class and to view the lesson slides.
The first simulation problem was to simulate drawing cards with replacement from a standard deck of cards. After some discussion, students came up with using 0-12 with 0 representing an ace or 0-51 with 0-3 representing an ace. The question of interest was "What is the probability of drawing an ace within the first 10 draws?"
Students ran 3 simulations to gain some practice using their calculators. For students without calculators I had them use a random number table. We accumulated our results and found a probability of approximately 45%.
The next problem involved simulating rock-paper-scissors using random digits. Students readily came up with the idea to use either 0, 1, 2 or 1, 2, 3 to simulate RPS. They struggled with simulating the physical activity itself. I discussed how a simulation is recreating the physical process using random numbers. The challenge in this simulation is to simulate two people playing against one another.
Students then started to use the idea of generating two values and comparing them. One student determined that there were 9 possible outcomes to the game with 3 outcomes each of tie game, player 1 winning, player 2 winning. Using this idea, the student used 0,1,2 to represent ties, 3,4,5 to represent player 1 winning and 6,7,8 to represent player 2 winning.
Finally, we returned to Sounding the Alarm (from NCTM's Navigating through Probability in grades 9-12) and started to look at the theoretical probability. Students worked through an associated worksheet which they are to complete for homework. We'll discuss the results next class.
Visit the class summary for a student's perspective of today's class and to view the lesson slides.
Discrete Math - Day 7
Today was a continuation of looking at permutation and combination problems. The basketball team problem is a nice follow-up to the egg problems from last class. The inclination is for students to consider that this problem is exactly the same.
Students made some nice visual representations for this problem and came to realize that things were different. For others, I asked if the teams of Fred, Jane and Joe and Jane, Joe, and Fred were the same. Once students realized they were the question was how do you account for the duplicate counting? Students worked through these issues with some interesting looks at the problem.
Before discussing their solutions I asked students to consider the similarities and differences between the basketball problem and the egg problem. Students came up with several ideas such as both problems involved 5 items and that factorials were involved. They realized the problems were different in that the egg problem involved arranging items while the basketball problem involved grouping items. I want students to consider these since most counting problems involve combinations of these things and they need to take into consideration what elements are at play in any given problem.
Afterward students presented representations of and thinking about the problem. In doing this I have students who took the wrong path but had some elements that could be built upon present first. One student considered the number of different ways that 3 people could form a team and realized there were 6 orders for the team. The student didn't know where to take that idea but it was an idea that tied into the solution and was worth bringing out. Other students showed the lists they created and this helped present a visual connection for what was happening in the problem. Finally, students talked about getting to their result. The final presenters showed how they started with 5! and wrote it out as 5 x 4 x 3 x 2 x 1. They then thought about the egg problem and realized they had 5 x 4 x 3 orderings of teams. They then found that each three person team had 3 x 2 x 1 orders and they needed to divided through to get a final count, they then wrote
(5 x 4 x 3) / (3 x 2 x 1) = 10
This was a nice representation that connected directly back to the egg problem but also differentiated what was happening with the duplicate teams. It also allowed a connection back to the first presenter's thinking.
The class then worked on the Pizza Problem. Again, visualizations were a key component of the thinking. Students created different list formats and also created tree diagrams. Students also explained their reasoning in calculating the number of pizza combinations with several different justifications for calculating 2 x 3 x 5.
In choosing students to present, I try to pick an order so that the presentations build upon one another and a story is created about this problem. This enables students at any level to contribute to the discussion and shows that their thinking is a viable way to approach a problem. I constantly remind students that I'm not looking for an answer but am interested in the journey they took to get to an answer.
After working through part (a) of the pizza problem I had students start on part (b). I asked students to think about what is happening in the problem. This was a homework assignment that we'll discuss next class. This problem presents new challenges as it is not simply one time of counting problem. It presages the type of problems that will be encountered shortly.
Visit the class summary for a look at a student's perspective on the lesson and to see the lesson slides.
Students made some nice visual representations for this problem and came to realize that things were different. For others, I asked if the teams of Fred, Jane and Joe and Jane, Joe, and Fred were the same. Once students realized they were the question was how do you account for the duplicate counting? Students worked through these issues with some interesting looks at the problem.
Before discussing their solutions I asked students to consider the similarities and differences between the basketball problem and the egg problem. Students came up with several ideas such as both problems involved 5 items and that factorials were involved. They realized the problems were different in that the egg problem involved arranging items while the basketball problem involved grouping items. I want students to consider these since most counting problems involve combinations of these things and they need to take into consideration what elements are at play in any given problem.
Afterward students presented representations of and thinking about the problem. In doing this I have students who took the wrong path but had some elements that could be built upon present first. One student considered the number of different ways that 3 people could form a team and realized there were 6 orders for the team. The student didn't know where to take that idea but it was an idea that tied into the solution and was worth bringing out. Other students showed the lists they created and this helped present a visual connection for what was happening in the problem. Finally, students talked about getting to their result. The final presenters showed how they started with 5! and wrote it out as 5 x 4 x 3 x 2 x 1. They then thought about the egg problem and realized they had 5 x 4 x 3 orderings of teams. They then found that each three person team had 3 x 2 x 1 orders and they needed to divided through to get a final count, they then wrote
(5 x 4 x 3) / (3 x 2 x 1) = 10
This was a nice representation that connected directly back to the egg problem but also differentiated what was happening with the duplicate teams. It also allowed a connection back to the first presenter's thinking.
The class then worked on the Pizza Problem. Again, visualizations were a key component of the thinking. Students created different list formats and also created tree diagrams. Students also explained their reasoning in calculating the number of pizza combinations with several different justifications for calculating 2 x 3 x 5.
In choosing students to present, I try to pick an order so that the presentations build upon one another and a story is created about this problem. This enables students at any level to contribute to the discussion and shows that their thinking is a viable way to approach a problem. I constantly remind students that I'm not looking for an answer but am interested in the journey they took to get to an answer.
After working through part (a) of the pizza problem I had students start on part (b). I asked students to think about what is happening in the problem. This was a homework assignment that we'll discuss next class. This problem presents new challenges as it is not simply one time of counting problem. It presages the type of problems that will be encountered shortly.
Visit the class summary for a look at a student's perspective on the lesson and to see the lesson slides.
Friday, January 18, 2013
IPS - Day 6
Today we continued to look at simulations. We started by estimating that at least one alarm would sound. We accumulated all of the class responses and found 539 out of 560 simulations had at least one alarm sound, resulting in a .96 probability.
We then discussed how the distribution of each individual's 20 simulations could be summarized. I do this as an assessment of what students already know. Students felt comfortable with creating bar and pie graphs for the categories of no alarms and at least one alarm. I briefly discussed how a histogram could be made from the data and explained the difference between a histogram and bar graph. By a raise of hands, most students did not feel they knew how to construct histograms. Students also mentioned calculating the average number of alarms that sounded.
The next task was developing a simple simulation. In this case, representing an 80% success rate and 20% failure rate. A trial consisted of generating 10 values. The question of interest was what would be the probability of seeing 10 successes.
I allowed time for students to consider how to simulate this situation using random numbers and to discuss their ideas in groups. About half the groups came up with ideas that reflected a 50% - 50% success/failure rate. I put these ideas on the board along with some ideas using either digits 0-4 or 0-9 with successes being the digits 1-4 and 2-9, respectively. In the discussion that ensued I focused students on how the 80% success rate was being modeled. Students realized they needed an 80-20 split on values and provided some additional examples.
I then had students run 5-10 simulations and had each group estimate the probability of seeing 10 successes in a row. Results varied from 0% to 20%. We gathered up the class data and had 6 out of 83 simulations show the 10 successes in a row, for an estimated probability of 7%.
I then had students write down things they wanted to remember when trying to simulate a situation.
Go to the day's summary page to see a student's perspective and the lesson slides that were used.
We then discussed how the distribution of each individual's 20 simulations could be summarized. I do this as an assessment of what students already know. Students felt comfortable with creating bar and pie graphs for the categories of no alarms and at least one alarm. I briefly discussed how a histogram could be made from the data and explained the difference between a histogram and bar graph. By a raise of hands, most students did not feel they knew how to construct histograms. Students also mentioned calculating the average number of alarms that sounded.
The next task was developing a simple simulation. In this case, representing an 80% success rate and 20% failure rate. A trial consisted of generating 10 values. The question of interest was what would be the probability of seeing 10 successes.
I allowed time for students to consider how to simulate this situation using random numbers and to discuss their ideas in groups. About half the groups came up with ideas that reflected a 50% - 50% success/failure rate. I put these ideas on the board along with some ideas using either digits 0-4 or 0-9 with successes being the digits 1-4 and 2-9, respectively. In the discussion that ensued I focused students on how the 80% success rate was being modeled. Students realized they needed an 80-20 split on values and provided some additional examples.
I then had students run 5-10 simulations and had each group estimate the probability of seeing 10 successes in a row. Results varied from 0% to 20%. We gathered up the class data and had 6 out of 83 simulations show the 10 successes in a row, for an estimated probability of 7%.
I then had students write down things they wanted to remember when trying to simulate a situation.
Go to the day's summary page to see a student's perspective and the lesson slides that were used.
Discrete Math - Day 6
Today's focus was on building conceptual understanding of permutations. Before diving into this I wanted to cover an open question from the last lesson; "For linear equations, why does taking the difference of successive y-values result in the coefficient of the x-term?" I briefly explained using the values of n and n + 1 along with the equation y = 3x + 7 to show what happens when the two y-values are subtracted. I did this more to show how generalized values can be used to explain results. This is starting to build exposure to proofs that will come later in the course.
The two permutation problems we tackled today focus on permutations. Some students may have exposure to these ideas but few if any have a real comfort level with the ideas.
I start the class picture problem by having 4 students stand in front of class. I proceed to rearrange them several times so the class has a clearer idea of what is happening. Students worked on the problem and several groups thought the answer would result in N x N. Since we were dealing with N=30 it was hard to verify if this was correct. I referenced the four students at the beginning and asked if they could verify their conjecture using 4 students. I tried to emphasize with the class that you need a way to check if your results are correct and working with smaller values is one way to do this.
A few students made lists for smaller numbers and arrived at the conclusion that factorials were being used. These visuals helped make it clearer for everyone in the class why factorial accurately represented the situation.
Part A of the next problem is a good check to see if students understand what is going on. Students quickly identified that there would be 10! arrangements. Part B of the second problem throws in a wrinkle by not using all of the items. You now are faced with having 5 items but only using 3 in the arrangement. This proved challenging for students. Several students came up with arguments as to why it could not be 5! or 3!, in effect creating bounds for the solution. A couple of students developed nice visual representations that made it clear the answer was 60.
One student represented the problem by making a list of all the arrangements for a fixed starting value (there were 12) and since any of the 5 items could be in the first position this would result in 60 total arrangements. A second student made a tree diagram and show there were five spokes to start, each of these had 4 spokes, and each of these had 3 spokes, resulting in 60 arrangements.
I had students conclude by writing down their thoughts about what to keep in mind when working with permutations and ways they can represent the problems.
Next class will dive into problems involving combinations.
Visit the class summary for a student's perspective and lesson slides.
The two permutation problems we tackled today focus on permutations. Some students may have exposure to these ideas but few if any have a real comfort level with the ideas.
I start the class picture problem by having 4 students stand in front of class. I proceed to rearrange them several times so the class has a clearer idea of what is happening. Students worked on the problem and several groups thought the answer would result in N x N. Since we were dealing with N=30 it was hard to verify if this was correct. I referenced the four students at the beginning and asked if they could verify their conjecture using 4 students. I tried to emphasize with the class that you need a way to check if your results are correct and working with smaller values is one way to do this.
A few students made lists for smaller numbers and arrived at the conclusion that factorials were being used. These visuals helped make it clearer for everyone in the class why factorial accurately represented the situation.
Part A of the next problem is a good check to see if students understand what is going on. Students quickly identified that there would be 10! arrangements. Part B of the second problem throws in a wrinkle by not using all of the items. You now are faced with having 5 items but only using 3 in the arrangement. This proved challenging for students. Several students came up with arguments as to why it could not be 5! or 3!, in effect creating bounds for the solution. A couple of students developed nice visual representations that made it clear the answer was 60.
One student represented the problem by making a list of all the arrangements for a fixed starting value (there were 12) and since any of the 5 items could be in the first position this would result in 60 total arrangements. A second student made a tree diagram and show there were five spokes to start, each of these had 4 spokes, and each of these had 3 spokes, resulting in 60 arrangements.
I had students conclude by writing down their thoughts about what to keep in mind when working with permutations and ways they can represent the problems.
Next class will dive into problems involving combinations.
Visit the class summary for a student's perspective and lesson slides.
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