Lesson Theme: Introduction to Probability and Counting Prerequisite Knowledge: Permutations and Combinations Here's an activity that I introduced to four groups of tenth graders in a recent unit on probability:
Provide minimal guidance as the groups decide on the number of combinations for each scenario. I made an exception for students who asked clarifying questions, such as:  Are there any repeating digits?  How many digits repeat? The hints are open to interpretation on purpose in order to get students thinking about the sorts of constraints they would need to consider when calculating the total number of outcomes. The discussion phase of the activity provided a rich opportunity to address student misconceptions about permutations and combinations, as well as the importance of reasoning, i.e. Does this number make sense? Is this estimate too high or too low? How does this number compare with my initial guess (intuition)? What if there were no constraints, what would the answer be? The Solutions: TEAM GARRY  3 digit code, repetitions allowed. This hint is not much of a hint at all. "Repetitions allowed" could mean that there may be or may not be any repetitions in the code. So, one possible answer is simply 10 x 10 x 10 = 1000. There are ten ways of picking the first, second, and third digits. TEAM ORVILLE  3 digit code, no repetitions. This narrows the playing field a bit. 10 x 9 x 8 = 720. Ten ways to pick the first digit, only 9 choices left for the second, then 8 for the third. TEAM APRIL  4 digit code, numbers 2 and 3 appear in the code. (a) This would be significantly easier if the ONLY numbers in the code were 2 and 3. There are only 4 possible combinations, which are easy to figure out by hand: {2323, 3232, 2233, 3322} or 4!/ (2! x 2!) = 4. (b) With repetition. There are 4 ways of positioning the "2" in a four digit combination. For each way that the "2" has been positioned, there are 3 ways to position the "3." In total, we can arrange the digits "2" and "3" in 4 x 3 or 12 ways. If repetition is allowed, the total number of combinations would be 12 x 10 x 10 = 1200. (c) Without repetition. Similar to above, except that once we have found a placement for the "2" and "3", there are only 8 and 7 digits left to choose from respectively. The final answer would be 2 x 8 x 7 = 672 My notes and observations:
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My lesson on permutations was inspired by this post by Dan Meyer. I teach a group of very bright and mathematically inclined students at our school. While my students' computational abilities and mathematical knowledge are almost second to none, a majority of them tend to lack skills of inquiry and critical thinking. This is due to the fact that the curriculum is dense and completely knowledge driven, which leaves little opportunities for linking (making connections) or creativity. I believe the second culprit of this phenomena is the postSoviet style culture and traditions in which the Kazakh education system is rooted. I see that my colleagues are under constant pressure to deliver heaps of content from a prescribed curriculum which is flawed to begin with. To provide some context, one term is roughly 6 weeks long. In those six weeks, we typically cover three units of work, with a prescribed 1216 hours of teaching per unit. The end of each unit is followed by a formative assessment (12 hours). This leaves 1014 hours of actual instructional time, of which is no where near enough time to cover all the topics we need to cover AND engage students in meaningful inquirybased projects. As a result of the limited time constraints, we are basically teaching one new concept a day. The students, meanwhile, are left to soak up as much as they can (like sponges that retain very little water, or knockoff expandable water toys that actually stay the same size) before the end of term summative. Add to this the other 710 classes the students are required to take, and you can see the students have an enormous workload. They are in school six days a week, at least 9 hours a day. There's just no time! The default solution? LECTURES. There aren't many opportunities for engaging students in rich learning tasks, but I try to squish in bits of it whenever I can. As I said, my students are extremely gifted but are used to thinking in terms of algorithms and formulas, so I often get a lot of blank stares and a lot of "Why are we doing this?" when I engage them in conceptual thinking  which is exactly the point! "Why ARE we doing this?" I ask, and that really grinds their gears! Slowly but surely, the students are getting used to this rather "oddball" tendency of mine (in their p.o.v.) to turn things around put the onus of learning on them. And golly I think it's working! So anyway, here's how I began our unit on Combinations: I placed students into groups and organized a placemat activity. Instead of me asking the questions, I wanted to know what opportunities the students saw when looking at this picture. Some answers they came up with (no modifications made):  How much combinations can be made sum of digits in each number is 53?  How much combinations do we have if key consist of 3 digits?  What is the possible length of the hardest password?  How many possible variation of making code with all numbers (all numbers can be used one time, and must be used)?  How many numbers that password include?  What is the probability of randomly unlocking the lock?  How many explosive charges are required to blow it? As I ponder this list I see a rich minefield of opportunities before me. Within a five minute brainstorming session, my students touched on permutations, probability, and of course, "reallife" problems. I put "reallife" in quotations here because I believe the relevance of math to everyday life is relative. My version of "reallife" is different from my students, and I certainly don't expect all my students to be making calculations with factorials on an everyday basis. I was blunt with my students, though I certainly didn't mean to be... it sort of just slipped out. "Some of you might never use this again in your lives," [cue snickering, whoops], "but..." There's always a "but" of course, and I'll leave it to you and your imagination to fill in the rest of the sentence. Once the snickering subsided, I proceeded to introduce factorial notation. The sequencing worked out beautifully because once the students were familiar with factorial notation, we revisited the lock picture and the students were able to derive the formula for permutations themselves! All I did was ask "How many possible combinations are there?" and students were quickly able to discover that we needed to define more parameters in order to answer the question. E.g. How many digits are there in the code? Can numbers be repeated? Beautiful! At the end, I gave them the following exit questions: 1. Define permutation, in your own words. 2. Give an example (not used in class) of a permutation problem. I was surprised at some of the responses I got. First, I learned that the class had a diverse understanding of the word "permutation" (I never actually gave them a definition), so now I know where the gaps in my teaching were. Second, not only did their examples show me the depth at which they understood the topic, but some students were able to accurately predict the types of questions we would be covering in future lessons. So instead of using textbook questions, we can explore the ones they came up with in class. Brilliant! 
April SooInternational math educator who writes, occasionally. Archives
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