An Approach to Introducing Ehrhart Theory in High-School Algebra
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1 An Approach to Introducing Ehrhart Theory in High-School Algebra Prepared by: Steven Collazos Fall 2012 Contents 1 Introduction Motivation Pre-requisites Investigation: Many Ways to Meet the Demand Materials Outline Handout for the Activity Comments and Solution Commentary about the Investigation How Lesson Went When It Was Delivered Feedback Received and Suggestions for Future Presentations Appendix and References Discrete Geometry List of Related Readings
2 1.1 Motivation 1 Introduction Let P be a polygon superimposed on a grid with points. In geometry, students are taught to compute the area of a polygon by using a variety of methods. What they might not be aware of is that one could also compute the area by counting the grid points. The fact that this methods always works is the substance of Pick s Theorem. In algebra, students are taught to solve systems of linear equations and to plot systems of linear inequalities. In paper, a lesson will be introduced where a problem can be solved by counting the grid points inside a region determined by a system a linear inequalities. The goal of this set of lessons is two-fold: Provide an alternative method of introducing systems of linear equations of inequalities. The second is to show the relevance of these methods to solve a real-world problems. To achieve this, Ehrhart Theory is used as a motivation. This paper is organized as follows. We begin the lesson by giving an outline with tasks that should be performed and ideas students should take home. Then we provide the content of what we will refer to as an investigation. After the investigation, we will provide some commentary about the reaction of students and issues that arise during the lesson. We end our paper with an appendix about the mathematics that lies behind the scenes. We will provide references and pointers to other writings. 1.2 Pre-requisites For the Investigation, students should be comfortable with plotting regions arising from systems of linear equations and inequalities. They should also be comfortable with modeling word problems involving linear equations and inequalities. For example, one of the steps in this investigation is graphing a system of linear inequalities consisting of five inequalities. In order to complete this task in a reasonable amount of time, it is necessary for students to have the aforementioned skills. While knowledge of geometry is not required, familiarity with polygons is desirable. This would enable the classroom instructor to discuss with the students the object that will arise upon modeling the word problem given in this lesson. If there are deficiencies on graphing linear equations, modeling word problems with equations and rates of change, I recommend that these topics be covered first or reviewed at the very least. Refer to the warm up problem in the Commentary about the Investigation subsection to have an idea for what kind of word problems or exercises could be included in such a review. 2
3 2 Investigation: Many Ways to Meet the Demand 2.1 Materials Rulers Graph paper As will be noted below, it is essential that these materials be available to present the activity because it is necessary to draw an accurate graph of the region determined by system of inequalities. 2.2 Outline In the given word problem, the goal is to model the situation with linear inequalities, graph the region defined by the system of linear inequalities and figure out how this region can be used to answer the question. It turns out that this can be done by counting the number of points in the region whose coordinates are integers. Alternatively, a problem where in addition an inequality can be used to solve the problem can be presented as an gentle introduction to the activity. In this particular word problem, there are four groups in a bakery. These groups produce two types of bread at different rates. Furthermore, there are constraints as to how much bread each group can produce. As the problem is set up, the situation can be modeled by a system consisting of four linear inequalities. The rates at which each group produces each type of bread is given. The constraints are also given. (For example, it is given that Group 1 can produce at most 38 units of bread.) After the linear inequalities have been constructed, proceed to plot the region created by this system of linear inequalities. In order to eliminate inaccuracies in the number of desired solutions, precision in the drawings needs to be emphasized. Otherwise, counting the points that can be used to answer the question will not yield the right answer. After the students have student have plotted the region, the question is how this picture can be used to answer the question. It is essential that they first realize that the every single point in the region satisfies the system of linear inequalities. However, the problem further states that the number of hours the groups work are integer-valued. It is this constraint that says that one should count the number of points that have integer coordinates in the region. In other words, the number of grid points, or integer points, tell us how many different ways the groups can work to meet the demand for bread. The lesson is intended to be around an hour long. 3
4 2.3 Handout for the Activity Sofia has a bakery where she produces bread for the city of Lattiseau. She wants to meet Lattiseau s demand for bread, but she has to make sure her four groups of bakers produce a certain number of units of bread. There are two kinds of bread made every day, namely whole wheat and rye. Below is a description of each group and how many units of bread they must produce. These groups must work the same number of hours producing each kind of bread. Additionally, hours can only take whole number values. (So groups can work, say, 2 hours, but not 1.5 hours.): Group 1: It produces 2 units of whole wheat bread per hour. It also produces 3 units of rye bread every hour. This group must produce at most 38 units of bread. Group 2: It produces 1 unit of whole wheat bread per hour. It also produces 1 unit of rye bread per hour; however, the difference between the number of units of the former type of bread with the latter is at most 4 units. Group 3: It produces 1 unit of whole wheat bread per hour. It also produces 1 unit of rye bread per hour; however, the difference between the number of units of the latter type of bread with the former is at most 4 units. Group 4: It produces 1 unit of whole wheat bread per hour. It also produces 1 unit of rye bread per hour. The total number of units produced must be at least 6. Sofia wants to know in how many different ways her bakery can produce bread in such a way that it satisfies the stated constraints. Questions Model the production constraints of each group using a linear inequality. Graph the system of linear inequalities you came up. accurate. Make sure your drawing is Can you answer Sofia s question using the plot you just made? 4
5 2.4 Comments and Solution The situation given in the word problem can be modeled as a system of linear inequalities. Let x represent the number of units of wheat bread and y represent the number units of rye bread. After determining the constraints, the system of linear inequalities students should arrive at is the following: 3x + 2y 38 x y 4 y x 4 x + y 6 The resulting region should look like this: Note that we only need to consider the integer points inside the region, which are now highlighted: 5
6 Hence, in order to know how many different ways the demand can be met, it suffices to count the the highlighted points above. Note that physically speaking, these points correspond to the different number of combinations of hours the groups can work. Since there are 45 points The number of ways these groups can satisfy the production of bread is Commentary about the Investigation 3.1 How Lesson Went When It Was Delivered The lesson was delivered to a pre-calculus classroom of about 10 students. While the class is one hour and 30 minutes long, I spent about an hour covering the lesson. A teacher and another graduate student were assisting students during the lesson. Before introducing the problem in the handout, I introduced a warm up problem. This warm up consisted of a word problem where the students were asked to come up with an equation that could be used to answer a question involving rates. The problem is given below to give an idea: There is a vehicle that costs $3, 000 per mile to be displaced. Regardless of how many miles is displaced, there is a fixed cost of $6, 000 for moving the vehicle. Up to how many miles can the vehicle be displaced if there are $10, 000 available? Note that a linear equation needs to be formulated in order to arrive at a solution, and it is helpful to sketch the relevant region. It was helpful to remind students that is not necessary to work with $3, 000, $6, 000 and $10, 000, but 3, 6 and 10 for the purposes of constructing a linear equation and then graphing it. It took a total of about 15 minutes to go over this exercise. After completing it, I began to hand out the worksheet to the students. After about 10 minutes, it seemed clear that the 6
7 most, if not all, students did not know how to get started. First, I read the problem on the worksheet and went over how to come up with an inequality modeling Group 1. Working as a group, we wrote how many units of bread would be produced if Group 1 worked a specific number of hours say, 1 hour producing whole wheat bread and 4 hours producing rye bread. This group exercise was enough to get the students started on formulating the rest of linear inequalities. After the explanation, students proceeded to work as teams on different descriptions of the groups in the bakery. After that, students moved on to graph the region. After realizing that every point in the region satisfies the system of linear inequalities, students realized that the integer points were the points of interest. Unfortunately, there was not enough time to realize how this could be used to answer the question in the worksheet. There was not enough time to count the points either. 3.2 Feedback Received and Suggestions for Future Presentations In order to run the lesson comfortably, more than one hour should be set aside. If this is not possible, the lesson should be split into two. If the lesson is to be presented over two lessons, I would recommend modeling the linear inequalities and possibly getting started with graphing the solutions to the systems of linear inequalities. For the second lesson, I would bring up the inequalities that were derived in the previous lesson and finish plotting the feasible region. After this has been done, then the students should proceed to trying to answer the main question in the handout, namely how many different ways there are of meeting the demand for bread. A suggestion that was made to me was to avoid discussion about units. While the units in the equations certainly make sense, students will in general have an intuitive understanding of the units involved. For example, consider the inequality 3x + 2y 38. I explained to the students that the units of the coefficients 3 and 2 were bread/hour, so x and y need to be in units of hour. This implies that the 38 must be in units of bread. However, this seemed to slow down the progress of the presentation and did not clarify the content of the inequality. (I would recommend to go over such a discussion only as a sanity check, but not as part of the scaffolding to answer the main question of the activity.) I also recommend having ready a fast way of reproducing the region determined by the system of linear inequalities. This can be accomplished by remembering the vertices of the region. These can be found by computing the points where some pairs of lines intersect. 7
8 4 Appendix and References 4.1 Discrete Geometry Let P = {x R d : Ax b}, where A is a k d matrix with integer entries and b Z k. The notation Ax b is a compact version of expressing a system of linear inequalities. To illustrate this, consider the system of linear equations that appear in the activity. If P is a bounded region, we call P a rational convex polytope. An example of such a polytope arises in the activity. If P is not a bounded region, we call it a polyhedron. A simple example of a polyhedron is the region determined by the inequality y 0. The examples considered so far are geometric objects of two dimensions. However, these geometric object can exist in higher dimensions. Here are some pictures of polytopes in three dimensions: These polytopes are called the cube, the octahedron and the dodecahedron, respectively. A question one could ask is how many grid points there are in the polygon and its integral dilates. Ehrhart Theory can be used to develop methods to count the number of these points. A question that is not pursued in the exercises is that of what the best integer solution is. In other words, given a model and number of integer solutions, how do we decide which one to pick? Answering this question is one of the goals of the field called integer programming, which is part of the important subject of linear programming. 4.2 List of Related Readings Counting integer points, also known as lattice points, comes up in a variety of fields: Computer Science, statistics, representation theory, algebraic geometry, commutative algebra and optimization [3]. A lot more can be said about Ehrhart Theory and the geometry behind systems of linear inequalities [1]. Polytope theory is a multifaceted area of research, and Ehrhart theory is just one part of it. There are expositions that discuss other aspects of this polytope theory [4]. Linear programming is an active field of mathematics. We refer the interested reader to an introduction to this area that has many applications to solve real-world problems [2]. 8
9 References [1] Matthias Beck and Sinai Robins, Computing the Continuous Discretely, Undergraduate Texts in Mathematics, Springer, New York, MR (2007h:11119) [2] V. Chvatal, Linear Programming, WH Freeman, [3] Jesús A. De Loera, The Many Aspects of Counting Lattice Points in Polytopes, Math. Semesterber. 52 (2005), no. 2, MR (2006c:52015) [4] G.M. Ziegler, Lectures on polytopes, vol. 152, Springer,
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