Discrete Math - Day 12

Today was one of those classes that you want to catch on film because it went so well. Thankfully I was taping today's class; I can't wait to view the replay.

We looked at card-based problems today. I introduced the problems by briefly asking if students played card games and then if anyone had ever watched poker tournaments on TV. These tournaments will show probabilities of different hands being drawn. The question to the class was how could you go about calculating these probabilities?

To get things started, we looked at a hand of four cards. I related this to the draw of four cards in Texas Hold'em. The question was how many ways could you get two pair when drawing four cards. Students were off and running, making lists and other representations to get a sense of what was going on.

The solutions being developed presented two schools of thought. The first was thinking that there were 13 choices for the first pair. There would be 12 choices left for the second pair and therefore there would be
     13 x 12 = 156
ways to form two pair.

The second way that students were thinking of the problem was the first choice (say aces) had 12 pairings. Moving to kings, there would now be 11 pairings. The next rank would have only 10 pairings. Continuing down this chain you would finally reach 1 last pairing. The total pairing variations was then
     13 + 12 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 78
ways to form two pair.

Many groups were ready to stop there. For these groups I asked how they were accounting for the different suits? Of course this would elicit groans, glares, and other reactions. They would then dive back into it to add this additional wrinkle. Some students thought they could just double their value to account for the other suits that weren't being used. I asked these students if this would account for all of the pairings of the suits? Other groups found there were 6 ways to match suits. I then asked these students what about the suit pairings for the other two cards?

Work continued and students were starting to get results. I asked these students to consider how they could verify or confirm that their solution was correct. I also had students visit different groups to compare results. Many groups were getting 78 x 6 x 6 = 2808 for their answer. One group actually listed out the 36 suit arrangements and had 78 x 36 = 2808 for their answer.

I had students present their results. I started by asking students to show representations and discuss what they were thinking. I tell students I am not interested in an answer, we just want to focus on where they were trying to go. One student showed their list and why they thought there were 13 x 12 = 156 two pair matches. The next presenter showed why they thought there were 78 pairings using the sum referenced above. I asked students what connection there was between these two results since both started by looking at the number of rank pairings that could occur. One student mentioned that 78 x 2 = 156. There were no other thoughts forthcoming so I asked them to continue to consider this as we moved on.

Students next presented different ways that they accounted for suit variations and their final results. These groups consistently were showing a result of 2808.

After the presentations someone asked if there were a more direct way of getting to a result than making lists. I like students to make representations because it helps them to see patterns and structure. I told students this but then asked them to consider the problem. There are thirteen different ranks in a deck of cards. To form two pair we need to use two of these ranks. We are therefore choosing two items out of 13 and do not care about the order they are chosen, i.e. we have a combination.

     ₁₃C₂ = 78

I asked students to compare this value to ₁₃P₂ = 156. The discussion centered on the fact that using a permutation treats AAKK as being different from KKAA when from a card-hand perspective they are exactly the same.

To take into account the number of suit pairings, we have to choose two cards from four. This is another combination that yields

     ₄C₂ = 6

This is done for both ranks so the final solution is

     ₁₃C₂ x ₄C₂ x ₄C₂ = 2808

Students seemed to grasp the connections between what they had done and the use of combinations. To check this understanding we tackled a second problem of how many ways a full-house (3 cards of one rank and 2 cards of another rank) could be drawn in five cards.

I was amazed that one student got the answer in just a couple of minutes. I asked him how he could verify or check to see if his answer was correct. Many other groups were proceeding quickly as well. The main issue that came up was whether to use ₁₃P₂ or ₁₃C₂ to determine how many pairings were able to be created in full-house?

Most groups were correctly accounting for the suit variations using ₄C₃ and ₄C₂ to count the number of ways 3-of-a-kind and 2-of-a-kind can be formed.

I had some students present their thinking and we had results of 3744 or 1872. Students were asking which was right. I asked them to consider the two-pair problem we worked on versus the full-house problem. I listed out AA22 and AAA22 and asked what was a fundamental difference between the two problems?

Many students were stuck at this point. I asked what would happen if the roles of the A and 2 were flipped? The results would be 22AA and 222AA. In the first case we still have the same hand. In the second we have a completely different full-house. While we could use ₁₃C₂ for the two-pair problem, we need to use ₁₃P₂ for the full-house problem since reversing roles changes the outcome. Several students were telling their group that was what they were thinking in the first place but had then changed their minds. I told students to have conviction about their reasoning.

Students were mentally spent but feeling energized about the progress they were making. As one student said, she felt good that they were able to answer the first problem before the class was over, as well she should.

I had students write down their thoughts about the problems worked on today and to capture any take-aways that they had.

Visit the class summary for a student's perspective on the class and to view the lesson slides.

IPS - Day 12

Today was focused on practicing calculating probabilities. We finished working on probabilities using addition rules and then proceeded to working on probability problems using multiplication rules. I reviewed ways to represent probabilities (tables and Venn diagrams) and then introduced trees as another way to view probabilities. Students then worked on problems. Overall students struggled more with the idea of multiplying out probabilities. The tendency was for students to add probabilities rather than multiply them for consecutive events. Toward the end of class more students were picking up on the idea. I'll need to spend time next class with some additional practice problems.

View the class summary for a student's perspective of the class.

Discrete Math - Day 11

Today we worked on another flag problem. Before working on this problem I passed back their quizzes and talked about what I was looking for in responses. I emphasized that they needed to do more than simply write down a correct answer, which would receive an incomplete grade. There were a few students unhappy about this but in the grand scheme of things one score of 6 out of 10 in a category that accounts for only 10% of their grade will not make any significant difference in their grade.

I also had several students ask about making quiz corrections. My school has instituted quiz and test corrections in lower grades in the hope that students will learn and hopefully master content that they did not grasp at the time of taking the assessment. I have mixed feelings on this practice, since it tends to have students not study and prepare as much as they should ahead of time. Because my classes are composed primarily of seniors and since colleges and universities (I teach night courses at Metropolitan State University at Denver) do not typically have this policy, I do not allow quiz or test corrections and I do not provide extra credit.

A student's grade should reflect their knowledge of the content: concepts, skills, application, etc. At the same time, the tasks and assignments that students complete are scored in such a way that it is difficult for students to not receive a passing grade if they are doing the work they should. On the other side, students who receive an A grade should clearly demonstrate knowledge and performance well above average. Those students who receive an A grade should earn that grade.

The flag problem was a variation of the first problem. This problem asks how many flags can be formed with at least 2 blue stripes on it. We briefly revisited the results of the problem for at least 6 blue stripes and then students dove into the problem. I was encouraged to see students looking for patterns or making lists and other representations.

Some groups tried to focus on looking at flags with 5, 4, 3, and 2 blue stripes. Others initially started trying to figure out what was happening with the 2 blue stripe flag. These later groups quickly found themselves stuck. They correctly calculated the number of options for 2 blue stripes on the flag as 8C2 = 28. Some groups even used the successive values of combinations to determine how many ways 3, 4, and 5 blue stripes could be placed on the flag. What they struggled with was the remaining stripes.

I asked them questions about the remaining stripes. How could they be configured. What color options were available, etc. When they started listing these the students quickly gave up as the number of options seemed immense. I suggested that they reduce the size of the problem to get a feel for what might be happening. What if you only had a 3 or 4 stripe flag? Only one group pursued this investigation.

The two groups who started looking at flags with 5 blue stripes were much more successful. One group started listing what the other 3 stripes could look like and came up with 8 configurations. When they looked at flags with 4 blue stripes they listed out 16 configurations for the other 4 stripes. As they started listing out the options for 5 non-blue stripes they realized the options were doubling every time. Looking back they saw this doubling pattern start from the very beginning. Using this pattern they quickly found the remaining values and calculated the total flag options correctly.

The second group to start using the 5 blue-striped flag noticed a connection to the flag options and Pascal's triangle. They also realized that the options were doubling and calculated a correct total. Both groups came to their answers almost at the exact same time and were excited that the other had arrived at the same answer.

We had students discuss what they were trying, patterns that they saw, where they got stuck and why. The two groups that arrived at correct solutions then presented their results and the class discussed why these worked. When the patterns and connections to Pascal's triangle were presented several students were saying, "I knew there had to be a pattern connection" or "I told you there would be a connection."

Even though most groups got stuck I was pleased to see that students were attempting to look for these patterns and connections. It was also interesting to see how changing the starting point so radically changed the ability of students to complete the task. I'll need to remember that "try starting from the other end" should be another suggestion I give to students as they work through sticking points.

We'll be working on card-based problems next class. I think the students are ready to move on from the flags.

Visit the class summary to read a student's perspective on the class and view lesson slides.

IPS - Day 11

Today focused on practicing using the probability concepts and skills that have been developed. We started off by trying to construct a Venn diagram for the Sounding the Alarm problem that we had worked on previously. I like re-visiting problems as it helps students make connections and build knowledge bridges.

Students typically struggle with this as their inclination is to have on circle represent one alarm sounding, a second circle to represent two alarm sounding, and the intersection to represent three alarms sounding. They don't feel entirely comfortable with the representation but it seems to fit. Some questioning as to what distinguishes Alarm 1 from sounding versus Alarm 2 makes them realize that haven't accounted for this aspect of the problem. The next question I pose is whether two circles is enough to represent the situation?

One student actually came up with the idea of using 3 circles on his own. After discussing with the class the merits and deficiencies of various representations I had this student present his solution. The class could readily see how three overlapping circles represents the events of different alarms sounding.

We then spent the rest of the period working through a variety of problems that focused on representing problems through tables and Venn diagrams, calculating different probabilities based upon addition, and dealing with terminology, such as mutually exclusive, and symbolic representations for "and" and "or."

We'll continue practicing calculating probabilities next class. This is all in preparation for working with conditional probability and looking at the idea of independence.

Visit the class summary to read a student's perspective of the class.

IPS - Day 10

Today we looked at representing probability spaces using Venn diagrams. Most students have used Venn diagrams for classification purposes in science or social studies classes but very few have used them for probability. I spend a few minutes displaying a Venn diagram and discussing the components.

After this brief introduction, we started with a simple example for a deck of cards. I asked students to try and draw a Venn diagram that represented the two events of drawing a face card and drawing a heart. I allowed students to work on these and discuss in their groups. We presented some results on the board and discussed the representations.

Venn diagrams always pose a problem because sometimes information is presented where the values in the circles double count the overlap of the two events and sometimes the values do not include the overlap. I encourage students to write both values and distinguish the value that is not including the overlap as "Only."

I next provided a table of information for ear-piercing by gender. The table provided counts for those who did and did not have pierced ears. I asked students to represent this data in a Venn diagram. This problem always proves challenging because the inclination for students is to designate one circle for males and one for females. I then ask what the overlap of the two circles represent. They want it to represent either pierced ears or non-pierced ears.

It takes a few questions to get them to realize that their overlap represents individuals that are both male and female. It takes a little more time for them to realize (or from some coaching) that one circle needs to represent a gender and one piercing. At this point they can readily construct their Venn diagram and answer different probability questions using the diagram.

The class then worked on several practice problems. These problems allowed students to solidify the concepts for constructing and working with Venn diagrams.

I had students write in their notebooks things they wanted to remember when constructing and using Venn diagrams.

View the class summary for a student's take on the class and to see the lesson slides.

Discrete Math - Day 10

Today I handed back the first draft of portfolio problems. Although everyone had revisions to make, many students had a good foundation to build upon. I re-emphasized that the paper needs to communicate their reasoning and processes and should make sense to anyone else in the class who would read it. I asked students to re-write and submit both their corrected version and their initial version.

My intent is to copy several of the revised versions to use as a mentor text for future classes. This will allow students to better see what a portfolio problem write-up should look like. It is always a challenge when doing things for the first time to have decent mentor text.

Most of the remainder of the class was taken by a quiz on figurate numbers. The quiz was a single question that contained three parts. The quiz focused on hexagonal numbers and was designed to see how much students absorbed and could apply of their learning.

I allow students to use notes and other materials that they have worked on, such as portfolio problems. I do this since the quizzes and tests I give are not a simple regurgitation of information. The problems exam knowledge and skill levels at different levels.

As I walked around I could see that many students were on the right track with the questions. I did notice a couple of students who obviously did not know what they were doing and hadn't actual done work on the problems that we worked on in class.

After the quiz we revisited the flag problem from the previous class. I put the results we had found so far and mentioned that we had answered the question of how many flags with at least 7 blue stripes existed. The issue now was how many flags with 6 blue stripes existed. We knew that there were 28 ways to position 6 blue stripes through 8 positions. We also knew that for the remaining two stripes that we had the following configurations: red-red, red-green, green-red, green-green.

Students were still stumped on how to proceed. I asked them how many flags with 6 blue stripes and two stripes that were red-red existed. After a few moments of thought a couple of people responded, somewhat hesitantly, that there were 28 flags. I then asked how many red-green flags would exist; again the response was 28. We then proceeded through the other two permutations and the responses both times were 28. I then asked how many total 6 blue striped flags existed. Students said 4 x 28 = 112.

Adding 112 + 16 + 1 = 129 gave us the total number of flags with at least 6 blue stripes.

We'll work on another flag problem next class.

Visit the class summary to see a student's perspective on the class.

Discrete Math - Day 9

I started class with three pigeon-hole problems. I asked students to work on these on their own as this would be a good assessment for them as to how well they understood these types of problems. The first two problems were relatively straight-forward: 1) How many people do you need in a room to guarantee that at least two people were born on the same day of the week? and 2) How many people do you need in a room to guarantee that at least two people were born in the same month? Students did very well on these problems.

The third problem was a slight variation on the first question: How many people do you need in a room to guarantee that at least two people were born on Monday? It was interesting to hear how many students were troubled by this question. A few were able to articulate the troubling issue and why it was conceivable that you would never have this situation occur no matter how many people were in the room. As someone said, "No one likes Mondays."

I then had students consider the problems and what characteristics made the pigeon-hole principle applicable or not.

We then took on a more challenging counting problem. In fact, the next series of problems that we'll work on came from an article in the February, 2010 issue of Mathematics Teacher entitled "Common Errors in Counting Problems." This article provides a series of challenging problems that delve into the complexity of counting problems at an accessible level for students.

The first problem involves determining the number of possible flags that can be formed if a flag has eight horizontal stripes of colors red, green, or blue, and the flag must contain at least six blue stripes. Students worked on this problem for the rest of the period. There

IPS - Day 9

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.

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 6^th 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 _nC_r 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.

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.

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.

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.

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.

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.

Discrete Math - Day 5

This was an interesting class. I assigned the first portfolio problem. These 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. These portfolio problems become mentor text that shows 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.

After going through the portfolio problem requirements and grading, we started on the day's work. This involved using values in a table to explore functional relationship. We started with linear equations. I had students create linear tables and look at the difference in the y-value. Students quickly saw that the difference was equal to the coefficient. After confirming this worked for other linear equations I posed the question as to why this would happen. Some students related it to slope but didn't come up with any definitive response. I left this as an open question and then asked if this result would always work.

A student wondered if it would work with quadratic equations and so we explored the situation for two specific equations. They saw that the first difference wasn't a constant but that if you took the second difference the result was. They also noticed that it looked like you should multiply the coefficient times the exponent. We explored this for a couple of more quadratics.

The question again was would this hold and we proceeded to explore this for cubic equations. Students found that the third difference was constant and the third difference looked like (coefficient) x (exponent) x 2. Students were stumped about what there was a 2 being multiplied.

I recapped what we learned so far, that linear equations had a first difference that was constant, quadratic equations had a second difference that was constant, and that cubic equations had a third difference that was constant. We also knew that the difference took on the form of (coefficient) x (exponent) x (something) where the something was a value of 2 for cubic equations.

I asked the class what this would mean for a quartic equation and they said that the fourth difference should be a constant. We used y = x⁴ to verify this. Students confirmed that the fourth difference was constant and saw that its value was 24. I then referenced our other result of 24 = (coefficient) x (exponent) x (something) = 1 x 4 x (something) = 1 x 4 x 6.

Several students noticed that 6 was the (exponent) x (something) component of the cubic equation and suggested writing out the value as 24 = 1 x 4 x 3 x 2. At this point someone suggested to write a one at the end of the sequence and there were several aha's that the coefficient was being multiplied by the factorial of the exponent, i.e. 24 = 1 x 4!.

I then explained that if a polynomial equation had as its largest term a_nxⁿ then the n^th difference would be
a_n x n!.

We then looked at the first few pentagonal numbers to see what we could tell about the equation. The values we had were 1, 5, 12, and 22. Students saw that the second difference was a constant and its value was 3. There was some confusion as to whether that meant it was a quadratic or cubic. I said that the second difference was 2 and asked what this meant. Most came to the realization that this meant it had to be a quadratic equation. I then asked what the coefficient was. Again, the fact that the coefficient was a fraction threw them off a bit. I wrote out that a x 2! = 3 and asked what a had to equal. They said it had to be 1.5 or 3/2. I then said that this meant the equation for pentagonal numbers started with the term 1.5x².

I created a table and talked about the contribution that 1.5x² made to the y-value. We subtracted this contribution out and saw that the results decreased by .5 each time. This meant we had a linear component of the form -.5x. Repeating the process we saw that the result was always 0. The provided an equation for pentagonal numbers of: P_n = 1.5x² - .5x. This can be written as P_n = 3x(x-1)/2 the more traditional form of the equation.

I then had students summarize what they had learned about taking differences and how these results could be used to find equations.

Visit the summary page for a student's perspective on the class and to view the lesson slides.

IPS - Day 5

Today we transitioned from exploring random events to simulation.

To connect with the idea of what randomness looks like and the Law of Large numbers the class started with a couple of quick assessments of understanding. Students voted on which heads-tails sequence was more likely. Roughly half thought the probability was the same and about 25% went to each individual sequence. The thoughts around the "hot hand" concept was divided roughly the same, half thought there was no such thing, 25% felt like the person was likely to continue with their streak, and 25% thought the streak would end. We discussed the reasoning and students were able to articulate the idea of the Law of Large Numbers.

The One Boy Family Planning activity comes from NTCM's Navigating Through Probability in Grades 9-12. I find the activities in this book to be enlightening and accessible to all levels of students.

I start the activity by asking students to read through the scenario and then develop a hypothesis as to what they think will happen given their current knowledge. We then briefly discuss the simulation aspect of the problem and I ask them to consider other ways that the situation could be simulated. Students came up with flipping a coin, using the random number generator, and mentioned that any device that allowed for two equal outcomes would work.

Students then proceeded to simulate the situation and record their results. Some groups will have misunderstanding of the simulation process so it's a good idea to walk around and make sure everyone is on track.

I ask students to hold off completing the second table as this usually causes problems. I actually have students create two columns: Total Children and Cummulative Children. I give a brief example to illustrate what this looks like. Again, most will get it but some may wander off track. I have students the total number of children and divide it by the number of households and ask what this represents. After a little thought the come up with the average children per household. I then explain how to use the second table to calculate their probabilities.

We discussed their expectations and surprises that came out of the investigation. Many students were somewhat surprised by the average children per household but they did anticipate that roughly half the houses would have a single child since the probability of having a boy was 50%.

I gave students a few minutes to collect their thoughts and summarize their thinking about randomness and to write two questions that they have. I will start class the next class by asking some of their questions.

Next we explored Sounding the Alarm from the Navigating Through Probability in Grades 9-12. In this investigation, I start with providing the scenario and asking what assumptions we would have to make for our simulation. This is always an interesting discussion. I try to push students to consider what they are assuming about each smoke alarm. Some questioning may be necessary to push them to consider the independence of each alarm.

The question of interest is to determine the probability that at least one alarm sounds. I have students consider how the situation could be simulated. In their groups students are very good at determining how to simulate a single alarm. Walking around I ask what they are thinking. Invariably I ask how they would now simulate all three alarms. After additional thought most groups determine a viable way to simulate all three alarms.

Ideas typically include using 4-sided die, spinners, card decks, and random numbers. I explain how students can ignore two sides of a 6-sided die and provide them with their devices of choice to simulate the situation. Again, some students struggle to simulate the results properly but a little guidance quickly gets them on track.

I had each student run 20 simulations and then pool their results within their groups. The idea is to get a probability based upon the entire classes simulation, approximately 500 simulations.

We got as far as completing the simulations and will analyze the data and compare against the theoretical results next class.

Visit my web site to see a student's perspective of the class and to see the lesson slides.

IPS - Day 4

Today we continued examining the idea of random events with the goal to re-inforce that what we think looks random may not be and that true random events may appear to be not random.

To start this class I had columns that represented months of the year. Students wrote the day of their birth under the appropriate month. I then asked students to comment on the data. Their were different observations about the distribution. I mentioned we would need data on births by month to see if our data fit the overall patterns or not. I then asked about the probability of having two birthdays on the same day. Most students thought that it was unusual. I asked if we increased the number of people to fifty or so what they thought would happen. About a third of the class responded that they thought we would get at least one match.

The fun part for this lesson is going to different classrooms and seeing if we got matches. I ask different teachers if we can interrupt their class for a few minutes and then have students in the class call out their birthday. Any time there was a match the person with the matching birthday would call out match. In visiting 3 classes we had matches of 3, 2, and 1. In the post discussion, students were curious about why the matches were so frequent. I told them we would look at how to calculate the probabilities as we progressed forward.

We then explored random sequences of heads and tails. By this time students were engaged in the idea of randomness and actively trying to imaging random coin tosses. The length of sequences (consecutive heads or tails) were tallied. We then used a calculator to simulate coin tosses and counted the length of random sequences. These typically showed a wider dispersion of sequence length and longer sequences. I wrote two sequences on the board, one reflecting the imagined results and one reflecting the random results. In looking at these sequences, students agreed the imagined results looked random while the random result looked fake while in reality it was the other way around.

We'll continue exploring the idea of randomness before we start to move into a more formal look at simulations.

For a student's perspective on the lesson and slides for the day, visit the class summary page on the course web site.

Discrete Math - Day 4

Today we looked at the problem of adding the squares of two consecutive triangular numbers. Students were having trouble see patterns. I wrote out the first five sums that students found so that all of the students were clear on the values being squared and summed. I then asked students to look for patterns and connections in the first nine sums. This provides students with three values that connect to each other. Most students still struggled with making connections but one student noticed that the sum of T₁² and T₂² was the same as T₄ and that the sum of T₂² and T₃² was the same as T₉. Someone asked if the sum of T₃² and T₄² equaled a square number and if it was T₁₆. Students verified this and the next one by using the Gaussian summation formula to verify. The use of the Gaussian summation formula came from the students and it was encouraging to see them make use of this in reverse to validate the conjecture.

I briefly mentioned the idea of polygonal numbers and figurate numbers as being numbers that could be represented by dots arranged in the form of figures.

We then moved to pentagonal numbers. I asked students what a pentagonal number (see Wolfram Math - Pentagonal Number for more information) would look like and gave them a couple of minutes to think about how they might arrangement dots. Students were reluctant to share so I said that I'd get things going. I drew a single dot, labeled it as P₁ and set it equal to 1. I said that now that I got things going someone could draw the next figure. Someone came up and drew five dots in the shape of a pentagon, labeled it as P₂ and set it equal to five. A student had used their smartphone to google the next result of P₃ = 12. We were a bit more challenged to draw the figure but finally had it drawn correctly. We also managed to get P₄ drawn and get its value of 22.

The challenge was to find a general formula for P_n, the n^th pentagonal number. I asked students to look for patterns and try to make connections. A student found a relationship between the value of n and the difference between P_n=1 and P_n. Specifically, he found P_n+1 = P_n + 3n + 1. This provided an opportunity to discuss recursive formulas. I related this back to the triangular numbers and the idea that most had found that T_n+1 = T_n + n + 1. I also mentioned that theoretically every recursive formula could be converted to a closed formula, such as T_n = n(n+1)/2. The next challenge was to find the closed formula for P_n.

Students have little experience converting non-linear relationships into formulas. In addition, even with linear expressions, many students are unsure and hesitant about creating a formula. I use this as a stepping off point to explore finite differences. It leads to many aha moments and provides a foundation for students to develop formulas from their table values.

Today was the first day that a student asked about when figurate numbers are used. I told them that the purpose of use working with these is for them to become better at reasoning and problem solving. These are unfamiliar items and they need to be able to work through the uncertainty, making connections to things they already know and to find patterns that they can explore and try to make sense of in the context of the problem. They seemed comfortable with the response. What they will come to realize is that this material is some of the easier material they will face and that building their confidence and abilities to explore, reason, and persist in problem solving now will serve them well as we dive into more complex problems.

Visit the class summary for a student's perspective of the day's lesson.

IPS - Day 3

Today we wrapped up our look into theoretical and experimental probability. We opened with a discussion of the similarities and differences between the rolling die and tossing hair clip experiments. This lead to a discussion of the "Law of Large Numbers" and the fallacious "Law of Averages."  Examples were provided, such as an athlete having a hot hand or being due to score. An explicit connection was made between experimental and theoretical probability. For students who had pre-calculus or calculus, you can relate this to a limiting process. A few additional examples were provided to help reinforce the idea.

Next, another experiment was conducted by tossing a coin 100 times. As before, students were asked to consider what they thought would happen. The number of heads minus the number of tails becomes the response variable that is measured. I had students put these values up on the board along with the longest streak of heads or tails that was tossed. Most students expected to see heads and tails within 4 or 5 of each other. They were surprised to see one student had tossed 38 heads and 62 tails. Another student had 58 heads and 42 tails. These exceeded what students thought might happen. The length of streaks was also surprising to students. Many thought 3-5 would be the length of the longest streak but there were many that were 10 or more, including one that was 15 and another that was 13.

This was an opportunity to introduce the idea that the data we gathered was random data and that randomness often looks quite different from what our mind conceives. I used this as an opportunity to mention that statistics looks at random data that we generate and compares it to our understanding of what random data should look like. If the two mesh then everything is fine, but if the two don't mesh we need to understand what is at play to cause the difference.

We finished with playing a coin tossing game. Again, students set down expectations before playing. They were surprised at the percentage of games that were won by a 4-1 split versus a 3-2 split.

For reflection, students considered connections between theoretical probability, experimental probability, and the "Law of Large Numbers." They were also asked to state the law in their own words and provide an example.

Visit the course notes for the day to read a student's perspective about the class and see the day's slides.

Discrete Math - Day 3

We started the day summarizing what was learned about triangular and square numbers from last class. Since this was a Monday morning class I thought it would be a good idea to have students re-engage in their thinking and notes before tackling the next set of problems.

We wrote out the formulas for T_n and S_n, the values of the nth triangular and square numbers, respectively. It was interesting that students struggled with the formula for S_n = n², since we didn't explicitly discuss this last week. They were much more comfortable reciting the Gaussian summation formula for triangular numbers, T_n = n(n+1)/2.

Students tackled the next three problems related to triangular numbers. The purpose of these are to get students comfortable working through problems. I warned students that part e was a tougher pattern to see. Students made slow progress on these but did identify the patterns for d and f. For part d, we have

         T_n-1 + T_n = n².

For part f we have T²_n-1 + T²_n = n³.

After having students present results I wrote out the associated formulas for both problems. There wasn't much progress on part e, so I asked students to write out the first eight terms and to look for a pattern.

Visit the class summary page to get a student's perspective on the day's activities and to view slides of the lesson.

We'll work with these tomorrow, along with looking at pentagonal numbers. I intend on the pentagonal numbers to be the first portfolio problem.

IPS - Day 2

The next two days focuses on understanding theoretical and experimental probability. We were actually able to start this lesson on the first day and had students complete the 40 die rolls. They completed the results and calculations as homework.

I then had students record their graph results on the board. I think that next year I'll pass around a graph that each group can plot on and then project the results. This will prevent students from waiting around while graphs are plotted on the board.

We discussed what they saw on the graphs (a convergence of the results) and the asked what would happen if we continued rolling the die more and more. I accumulated total counts for all the groups to calculate a value off a larger number.We compared this result to the theoretical result.

We then moved to tossing a hair clip which does not have equal probability. You can use any six-sided, asymmetrical object. I ask students to develop a hypothesis, their guess based on their current understand, as to what the probability might be. We then conducted the experiment and saw comparable graph convergence. We calculated a probability using the cumulative class data. This becomes our best estimate as to what the theoretical value would be. I also convey that we would need millions of tosses to hone in on a value.

We concluded class with students recording their thoughts about the two experiments: similarities, differences, insights, surprises, and understandings.

All of this is building to understanding the connection between experimental and theoretical probability and the Law of Large Numbers.

Visit the class summary page to get a student's perspective on the day's activities and to view slides of the lesson.

Discrete Math - Day 2

Today we continued to build norms for the classroom by working with figurate numbers. Figurate numbers come from ancient Greece and were an attempt to connect geometric figures to numbers that would somehow unlock secrets of nature and the universe.

The first problems looked at square numbers. The class was asked to look at these without using calculators and to try to make direct connections between the figures and the values. I used the physical connections made between Pascal's triangle and the number of names said in the Circle-Name game as an example.

The square problems are hard for students to think of from a fresh perspective since they are so familiar with square numbers. I push them to look for patterns and make connections to the figures. Typical patterns include that the number of dots in a side match the figure number and the difference in square values are odd numbers that increase by two. The discussion around the first square number shows that it is a logical extension of the patterns but in fact creates degenerate square.

Next came triangular numbers. This forces students to look at patterns and make connections. Most realize that you need to sum up values. You may see other interesting patterns and recursive formulas that crop up. Student presentations are always interesting because you typically can have students present in such a way that the story about triangular numbers builds upon itself. The discussion of what the first triangular number should be is similar to the first square number.

Most students will not be familiar with the Gaussian summation formula and this is a good time to tell an anecdote, introduce the Greek capital letter sigma as a symbol for summation, and to connect the formula to the work that was done.

The presentations of these simpler problems help develop classroom community and establish the norms and expectations as to how the classroom operates. Students are starting to realize that solutions and insights come from them.

Visit the class summary for a student's perspective of the class and to view a copy of the day's lesson slides.

Discrete Math - Day 1

This blog will discuss my course in Discrete Math. My Discrete Math web site explains the orientation of the course, provides summaries of daily lessons and slides for those lessons. This class is an inquiry-based course. The purpose of this is to have students make sense and connections with and between the mathematical concepts presented. Students are encouraged to explore, question, and investigate the mathematics with an eye toward how mathematicians work. This is much different than learning procedures or problem solving techniques that most students experience in their math courses.

The first day is always interesting. I start with the Circle-Name game. It provides an opportunity to learn student names and build classroom community. I then assign students randomly to groups of three or four.

The first task is for students to determine how many names said during the Circle-Name game. I have students work on their own (individual think time based on brain research) that allows students to engage in the mathematics. I then have students discuss their results at their tables and then have a whole class discussion. I don't push for formulas at this time. I am trying to establish class norms on working through problems, sharing results, and discussing methods.

We then explored Pascal's triangle (see below). I like to start with Pascal's triangle because the patterns are observed in so many situations. Depending on what topics you want to cover, Pascal's triangle will show up in combinatorics, number theory, and graph theory. The NCTM's Navigating Through Discrete Mathematics in Grades 6-12 provides several Pascal triangle explorations and is a good resource for other Discrete Math topics.



I gave students five minutes to work on the problems on their own before working in their groups. Not every group came to resolution on the results. We shared partial solutions and results that differed. This gave an opportunity to show how thinking from different groups can build upon each other and how do to go about determining which of many offered solutions is correct. For patterns in the triangle, students made observations about row content, diagonal patterns, and the general structure of the triangle. One student noted that the triangle had a line of symmetry. I was able to illustrate mathematical thinking as one student pointed out that every other row had a middle value. I suggested that given all the patterns in the triangle I wondered if the middle values followed a pattern and whether that pattern connected to some known entities.

After discussing course policies and course topics covered in the semester, we dove back into looking at Pascal's triangle with an eye toward making connections to the number of names said in the circle. This discussion provided some nice connections. One student said she saw the ones on the outside of the triangle as the people standing in the circle. She said she didn't feel it was all that insightful but it was what she saw. Another student made a connection with the third diagonal, noting that these represented the total names said for any given number of students. Students did not have any other connections. As I was listening to the class discussion I thought about the first diagonal representing the individuals and then thought of the second diagonal as representing the number of names an individual would say. The third diagonal represented the total names said. This made a nice connection between the observations and the reality of the situation. I didn't pursue it further but it would be interesting to see if there are any connections between the fourth diagonal and the Circle-Name game.

The class wrap-up was for students to summarize connections between Pascal's triangle and the number of names stated in their own words. This summary is based upon brain research that says that giving students time to think about their learning at the end of class and writing down their thoughts helps to embed the learning in long-term memory.

IPS - Day 1

The posts referencing IPS refer to my Inferential Probability and Statistics (IPS) class. My IPS web site describes the course and provides summaries daily lessons with lesson slides.

In this blog I will be re-capping some of the days activities and my reflections on the lessons.

The first day was devoted to conveying a sense of what statistics is about, what will be covered in class, and how the class will work. The Ted video I show lets students know that seemingly simple probability problems do not result in the obvious answer. The disease example and the trial evidence example get students thinking about the ideas about what happens randomly. I am actually able to relate a personal experience similar to the video's trial example.

The idea of what do random events look like and how does actual data compare to expectations of what randomness looks like will be a recurring theme throughout the course. The first unit covers probability and simulations, building the foundation of what randomness should look like.