You can demonstrate possible combinations by actually pushing the buttons on
the lock youve drawn. Make sure students understand the task (to count the number
of combinations and see if it lives up to the billing of thousands) and the rules for
the presses which are:
Depending on the level of you class, you may want to introduce the two fundamental rules of counting: One way to accomplish rules (1) and (2) is to overcount and adjust. You can also, as a class, list all the combinations for a 2-button lock. There is a small enough number to list them all out, but there are enough to get at some common points of confusion:
Allow the students to spend the majority of the period (at least 30 minutes) exploring ways to find the number of combinations for a 5-button . You can set the students to work as individuals or in small groups of 2 or 3. Usually, students will start out by trying to write down all the combinations for a 5-button lock. Soon, they will discover they need some systematic way of counting the combinations. As you walk around and observe students work, pay attention to students with creative, elegant, or clear systemmatic approaches, and be sure to ask them to share their work at the end of class. Also, be prepared for some groups or students to be stuck, or to think theyre done quickly. (Often, theyll shout out 152! as an answer without any explanation.) It can be challenging to convince them to keep working without saying theyre wrong or giving away the answer. For these groups of students, it helps to start them on a strategy, especially one that will help them see any combinations theyve missed. See the Hints document for more details on each of these:
If a group thinks they are done, but have only counted the combinations that use all five buttons, for example, you might say, Some combinations dont use all 5 buttons. So there are two kinds of combinations, and youve counted all of one kind. How can you count the others? If a group is completely stuck, working with a smaller number of buttons is helpful. You can also encourage them to fill in a table like the one here, and to look for patterns.
You should judge how much your students need in the way of hints or ideas. Try a table! might be enough for a group that has already been looking at locks with different numbers of buttons. Other groups may need more structure to get started. At the end of the first day, ask several groups to share strategies but no answers. Ask how they organized their work, what questions they asked themselves, and so on. If you have one or two strategies that you want to be sure students see, you can ask groups to present them or you can explain them yourself. You might also spend some time talking about approaches that are similar to each other, to get students thinking more broadly about kinds of approaches. You might decide to encourage students to pursue one of the five main approaches:
2 After Day 1Where you go after the first day of the project is up to you. Here are some approaches teachers have used in the past: 2.1 Finish IndividuallyAfter the first day of group work, students are asked to complete the project individually. Encourage the students to ask questions as they arise. They usually turn in a rough draft - explaining their approach, their answer, and any questions that havent been addressed - a week or so after the initial work. This keeps the students working on the problem and prevents losing momentum from the first day.The rough drafts can help you see if there are major misunderstandings or common mistakes students are making. You can spend some class time, if necessary, going over these issues when students get comments on their rough drafts. They then finish up the projects on their own. 2.2 Continuing Group WorkYou may want to devote a second day of class time to students working in their groups. At the end of the first day, they may see connections to other stragies and get ideas for how to continue their work. On the second day, encourage each group to pick one of the strategies that was discussed and pursue it to get an answer for the number of combinations of a 5-button lock.Many groups will arrive at a numerical answer by the end of the first day, or early on during the second. Often, they do this by a brute-force (though clever) listing of the combinations. Allow students with different answers to compare methods to see where under and over counts may have happened. Once the students agree on the numerical solution, you can ask them to start making some connections and following up on other questions:
For students who know about elementary combinatorics and/or students who have worked on the trains project (see Warm Up Problems), you might ask them directly how they could use the formula to help them organize their work and solve the problem. For more advanced groups, you can ask them to come up with a closed form or recursive formula for the number of combinations on an n-button lock. You can also encourage them to come up with their own extensions to the problem by changing the question in some way (perhaps changing one or more of the rules about pressing buttons). 2.3 Filling in and Coming BackIf your students dont yet have the background to tackle some of the more general solutions, you can interrupt work on the project (make sure it is well-documented first) for a day or more of background building. You might want to cover some of these topics:
When you come back to the problem, ask students to find a way to use one or more of these tools as they work on it. |
Translations of mathematical formulas for web display were created by tex4ht. |