Mthed 377 Lesson Plan

MthEd 377 Lesson Plan Cover Sheet

Name: Karl Arthur Date: 29 November 2005 Section Title: Optimization Station: COMAP Mathematics: Lesson Five, Activity Six Big Mathematical Idea(s): For p houses and n mailboxes: -If n > p or if the number of clumps (counting houses all-in-a-row as one house) is > p then the position of mailboxes is arbitrary after each house or clump receives one mailbox. -When generalizing the case of placing multiple mailboxes, after n clumps have been determined the position of the mailbox for each clump will be just as in the case for a firehouse, where the median or minimax position will be where the mailbox is placed - The position that minimizes max, total, or both max and total distances traveled when placing mailboxes determines which houses will belong to which clump. Why are these BMIs important mathematically? These BMI's are important in generalizing for finding multiple medians and means. While in the mathematics community there is usually only one mean or median taught, in statistics different samples typically have different means and medians. Also, the students are being exposed to more than one variable which will help them make the transition back into two-dimensional Gridville. How does this lesson fit in to the overall unit? The previous activity focuses on finding the minimax (the position that minimizes the maximum distance traveled in Linear Village). All lessons before this time were spent in finding and justifying rules of optimization in Linear Village, a town that is a simplified view of Gridville, which, in turn, is a simplified view of actual cities. Students discover the “best” position for a firehouse in these towns. The students are on their way back to thinking about two dimensional space, but first, we investigate what the best positions would be if the town were to build two or more firehouses, (or in this case two mailboxes, generalized to several mailboxes). The students by this time have explored the concept of absolute value, and in the previous lesson have seen how this function makes the optimization easy to visualize. This lesson is meant to again challenge the students mathematically. Grading rubric (for Keith’s use)

5 The Big Mathematical Idea addresses core mathematical concepts and is clearly articulated 5 Description of the importance of the topic is well thought out and relevant 5 There is a clear, insightful discussion of how this lesson fits in to the mathematical content of the overall unit 5 Lesson sequence is well thought out and detailed 5 Students' thinking is anticipated with forethought and detail 5 Reactions to students' thinking is mathematically oriented, insightful and detailed

10 3-5 reflection paragraphs 10 Met with Dr. Leatham and made demonstrate thoughtful appropriate revisions based on this reflection and are clearly OR discussion articulated 30 3-5 page reflection paper demonstrates thoughtful reflection and is clearly articulated Lesson Sequence: Learning Formative activities, tasks and key Anticipated Student Your response to Assessment, questions (what you will do Time Thinking and student responses Miscellaneous and say, what you will ask Responses and thinking things to the students to do) remember Launching the Lesson Over the last couple of Students may still We are not going Rules before days and weeks we have be wondering what to limit our were: helped both Gridville and the “best” criteria conversation right 1. Nothing is Linear Village to plan where is, or have in mind now what the placed where a they should place their that they need to “best” criterion is. house already firehouse. As you might focus on what we (Carly may have is, and have noticed already, there decided the already decided 2. We want to are plenty of factors that go previous class what the “best” minimize both into planning the position of period (i.e. Carly’s criterion is by max and total a firehouse. Today we are lesson). consensus, but distances going to discuss another today I will be important consideration of Students may not asking questions Today: town development while comment on minimizing We will ask following the same rules as both total and max what happens before, in the placing of distances.) when the mailboxes. Through this mailboxes are activity we will be able to allowed justify the places we put anywhere on these mailboxes. the street. Orchestrating the Task Activity: Say you have two I would place a Does it matter In my opinion houses, 3 and 24, and two mailbox on the lot which side it goes doesn’t matter mailboxes to install in to the side of each on? which side of Linear Village (LV). Where house. house it goes would you place the two No: it just needs to on. mailboxes in order to be next to it. minimize the maximum Yes: if the distance traveled? mailman wants to travel less distance, could place both on inside Okay, so this (23 This is to or 25) mailbox make sure services address 3, students are No! and . . . (students paying The dist to both stop me before I attention and houses is minimal finish) realize why when right next to Why not? they chose the the house it services Why do you not spot next to put the mailboxes the houses. To minimize total distance (same answers for next to just one (Do not have traveled? To minimize total total and max) house or at 13 or to use if dist? 14? students don’t get humor) Lesson Sequence: Learning Formative activities, tasks and key Anticipated Student Your response to Assessment, questions (what you will do Time Thinking and student responses Miscellaneous and say, what you will ask Responses and thinking things to the students to do) remember Facilitating the Discussion If we were in Hermitville (a On both sides of the Is there a need for town with only one house) house. the second where would you place the mailbox? (at least two mailboxes? someone will say “no” to this) Only need one Good. Why would Students may mailbox. there be no need be bothered One next to house, for more? Can we that one the other anywhere. apply this to any house's size town, say mailbox never Yes, only need as with p houses and changes many mailboxes as n mailboxes? position: they have houses. Good. So, if n > p can generalize we can ignore the or reference extra houses. previous Along these same lines, Put on either side, That’s good, and answer if want what would you do if there so people have to if there were still to were a clump of houses with walk half as far. an outlier? no place to put the mailbox Minimax: leave Would it be safe Other answers or boxes except at the ends? with outlier. to say that once similar to their Total dist: if # we have found a findings from houses less than clump of houses previous days distance traveled by we can treat these are expected, outlier, can leave houses like their but not to be both mailboxes own town with focused on. with the clump. one mailbox? In Linear Village what was I don’t know what What were you your main method of you mean. trying to minimize determining the “best” in placing your positions for the mailboxes? houses? How did you clump the The mailbox houses? positions depended on the distance After you found traveled and minimizing this distance Houses were (For these two be clumped around sure the other is mailboxes. mentioned also) Mailboxes were positioned near clumps of houses Lesson Sequence: Learning Formative activities, tasks and key Anticipated Student Your response to Assessment, questions (what you will do Time Thinking and student responses Miscellaneous and say, what you will ask Responses and thinking things to the students to do) remember Debriefing the Lesson -So, for n mailboxes and p 2min No comment These are the houses, we found that when BMI's n > p or if the number of paraphrased clumps (counting houses all- in-a-row as one house) is > Look to see p then the position of that the mailboxes is arbitrary after students are each house or clump writing this receives one mailbox. down or at -We also found that after n least nodding clumps had been determined their head in each clump was treated as a acknowledg- single town where the ment median or minimax position determined where the mailbox is placed for that clump. -And, the position that minimizes max, total, or both max and total distances traveled when placing mailboxes determines which houses will belong to which clump. Lesson Sequence: Learning Formative activities, tasks and key Anticipated Student Your response to Assessment, questions (what you will do Time Thinking and student responses Miscellaneous and say, what you will ask Responses and thinking things to the students to do) remember Where would you put the Across from the mailboxes if you could houses since they place them anywhere to will not have so far minimize max dist, and to walk why? At 3 or 4 since same minimax here What if there was a house at (and at 23 or 25 for 6 also? Where would you same reasons as put the mailboxes to before) minimize max dist traveled (similar reasons as and why? previous) To minimize total distance At position 4.5 traveled? because it is right If can place anywhere to between 3 and 6, (Since average minimize max dist, and at 24 because it is dist traveled why? only house that Is there a fairer will be same needs it. place to put it? for any A position on a lot position from next to a house. 3 to 6) To minimize total dist? (same answers as previous) Q7: Suppose a neighboring I would place the Is there any other Next question linear town has houses with mailboxes at 6 place you would addresses the addresses 2, 3, 4, 5, 7, 8, and because it is close put the mailboxes? issue I am 9 (notice there is no 6). to more houses and concerned Which lots would you place therefore better with. This is the mailboxes? Explain how than 10 or 1 intermediate you chose which house goes At 1 because brings step to bring with which mailbox. max dist from four minimizing down to two. out Q8: Where would you place At 6 again and at 1 Why 6? (lowers the mailboxes if another or 11 since there max and total dist house at 10? will be two houses traveled) brought closer. Now, is this position fair? Can You could and it you place the would be fair, but it mailboxes at just 1 would not minimize and 11? either distance