Systems of Inequalities & GeoGebra Ellie Scheiber MATH150 Lesson Plan

Transcription

1 Systems of Inequalities & GeoGebra Ellie Scheiber MATH150 Lesson Plan Stage 1 Desired Results Core Standard(s): CCSS.MATH.CONTENT.HSA.REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane, and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Objective: After completing this lesson, the student will be able to solve a system of inequalities using GeoGebra. Understanding(s): The students will understand that GeoGebra is an effective and beneficial tool to use for graphing and solving systems of inequalities. Essential Question(s): How can we use GeoGebra to find the solution to a set of inequalities? Students should already know How to graph a linear function How to substitute coordinates into an equation How to solve an equation for y General meanings of less than, greater than, and less/greater than or equal to How to set up and solve a system of equations to find x and y Students will learn. How to set up a system of linear inequalities from a word problem How to graph the solution to a linear inequality using GeoGebra How to graph the solution to a set of linear inequalities using GeoGebra How to use the maximizing/minimizing equation to find the actual solution to the problem Students will be able to Correctly graph a system of linear inequalities Find the solution to the system of inequalities Stage 2 Assessment Evidence What students will be assessed on: Ability to use the word problem to create a set of inequalities Ability to correctly solve/graph the system of inequalities and find the bounded region Ability to find the coordinates to the unknown vertex/vertices Ability to identify the solution to the system of inequalities using vertices and the maximizing/minimizing equation Tools used for assessment: GeoGebra (making sure they used the correct inequalities and graphed them correctly) Worksheet (to see their written work and how they found the answer)

2 Tools used by students: GeoGebra Worksheet Stage 3 Learning Plan Tools used by instructor: GeoGebra Launch: Students will have about five minutes (or less, depending on when they finish) to complete the worksheet titled Introduction to Systems of Linear Inequalities. After five minutes, we will briefly discuss the answers. Activity/Exploration: After reviewing the worksheet, we will begin with the examples. The first example will be demonstrated on my laptop while the students follow along. The second example will be for them to do. They can work with a partner, but each person must complete the activity on their own laptop. The third and fourth examples are independent. Closure/Summary: Ask the students what the steps were to find the solution. The steps are as follows: identify the constraints and maximizing/minimizing equation, graph the constraints, graph the constraints as linear functions, identify the vertices, input the maximizing/minimizing equation, substitute the vertices, identify the solution. Students are assessed on correct completion of the worksheets (including those with the examples). Important Terms and Definitions: Inequality: A mathematical sentence built from expressions using one or more of the symbols <, >,, or o Constraints: A system of linear inequalities o Bounded Region: The solution to the system of linear inequalities; that is, the set of all points that satisfy the constraints. Only points in this region can be used. A bounded region will have maximum/minimum values. o Vertex: A point where two or more straight lines intersect; a corner o Vertices: Plural of vertex o System of Equations: Two or more equations working together o Answers for Worksheet: 1. System of Equations 2. Vertex 3. Constraints 4. Vertices 5. Inequality 6. Bounded Region A. Constraints B. System of Equations C. Vertex D. Bounded Region8

3 Name Date A B Introduction to Systems of Linear Inequalities Fill in the blank with the appropriate term. Terms will be used only once, and some will not be used at all. Choose from the following terms: inequality, constraints, bounded region, vertex, vertices, system of equations 1. A(n) is two or more equations working together. 2. The point where two or more straight lines intersect is called the. 3. is/are a system of inequalities. 4. The plural of vertex is. 5. A mathematical sentence built from expressions using one or more of the symbols <, >,, or is called a(n). 6. The is the set of all points that satisfy all of the constraints. Label the diagram below using the terms above. Not all terms will be used. D C A. C. B. D.

4 Name Date Use GeoGebra to find the solutions to the following three problems. Show any written work. Example 1: A rancher is mixing two types of food, Brand X and Brand Y, for his cattle. Brand X contains 15 grams of protein and 10 grams of fat and costs $0.80 per unit. Brand Y contains 20 grams of protein and 5 grams of fat and costs $0.50 per unit. If each serving is required to have at least 60 grams of protein and 30 grams of fat, how many units of each type of food should be used to minimize cost to the rancher? Step 1: Identify Constraints and Maximizing/Minimizing Equation Step 2: Graph Constraints Step 3: Graph the Constraints as Lines Step 4: Identify Vertices Step 5: Input Maximizing/Minimizing Equation Step 6: Substitute Vertices and Identify Answers

5 Example 2: An organization decides they will make T-shirts and posters to raise money. There are 20 hours and 600 people available. It takes 6 minutes to make a T-shirt and 3 minutes to make a poster. Each T-shirt requires 2 people, and each poster requires 3 people. By request, they need at least 50 posters. Each T-shirt makes a profit of $3.00, and each poster makes a profit of $1.50. How many T-shirts and posters should they make to maximize profit? Let x = the number of posters and y = the number of T-shirts.

6 Example 3: Maximize C = 3x + 7y given the following vertices: A: 0.0 B: 6.17,0 C: 6.42,2.29 D: (0,4) Example 4: Identify the bounded region given the following constraints: x 16, y 3, and 12x 25y 13.

7 Example 1: A rancher is mixing two types of food, Brand X and Brand Y, for his cattle. Brand X contains 15 grams of protein and 10 grams of fat and costs $0.80 per unit. Brand Y contains 20 grams of protein and 5 grams of fat and costs $0.50 per unit. If each serving is required to have at least 60 grams of protein and 30 grams of fat, how many units of each type of food should be used to minimize cost to the rancher? Step 1: Identify the constraints and the maximizing/minimizing equation. Constraints: o x 0 (because we cannot have a negative amount of Brand X) o y 0 (because we cannot have a negative amount of Brand Y) o 15x + 20y 60 (amount of protein) o 10x + 5y 30 (amount of fat) Minimizing Equation: o C = $0.80x + $0.50y Step 2: Graph the constraints on GeoGebra. Darkest shade (highlighted blue) on the graph is the bounded region. x 0 Added y 0 Added 15x + 20y 60 Added 10x + 5y 30

8 Step 3: Reduce the inequalities and graph them as lines so we can solve for the vertex/vertices. Step 4: Identify the vertices of the bounded region and create points in GeoGebra. A: The y-intercept of f à Coordinate can be clearly seen on the graph, so you can directly type it into the Input box à (0,6) B: Intersection of functions e and f à Type intersect into Input box à Select Intersect[<Object>,<Object>] à Type e and f into Object spaces à Enter C: The x-intercept of e à Coordinate can be clearly seen on the graph, so you can directly type it into the Input box à (4,0)

9 Step 5: Type the minimizing/maximizing equation into the Input box. Only type the side of the equation with the variables/constants, not the C= side. Type 0.80x y Step 6: Substitute points into the minimizing/maximizing equation by typing g(a), g(b), and so on, until all of the vertices have been substituted. Step 7: Using the answers in the Number section on the left, find your solution. They are listed in the order they were typed. The vertex used to find that number is your solution. To minimize cost, use 2.4 units of Brand X and 1.2 units of Brand Y. o Point B (2.4,1.2) = $2.52 Example from

10 Example 2: An organization decides they will make T-shirts and posters to raise money. There are 20 hours and 600 people available. It takes 6 minutes to make a T-shirt and 3 minutes to make a poster. Each T-shirt requires 2 people, and each poster requires 3 people. By request, they need at least 50 posters. Each T-shirt makes a profit of $3.00, and each poster makes a profit of $1.50. How many T-shirts and posters should they make to maximize profit? Let x = the number of posters and y = the number of T-shirts. Step 1: Identify the constraints and the maximizing/minimizing equation. Constraints: o x 0 (can t have a negative number of posters) o y 0 (can t have a negative number of T-shirts) o x 50 (required to have at least 50 T-shirts) o 3x + 6y 1200 (time in minutes available) o 3x + 2y 600 (number of people available) Maximizing Equation: o P = $1.50x + $3.00y

11 Step 2: Graph the constraints on GeoGebra. Darkest shade (highlighted blue) on the graph is the bounded region. x 0 Added y 0 Added x 50 Added 3x + 6y 1200 Added 3x + 2y 600

12 Step 3: Reduce the inequalities and graph them as lines so we can solve for the vertex/vertices.

13 Step 4: Identify the vertices of the bounded region and create points in GeoGebra. A: The x-intercept of f à Directly type it into the Input box à (50,0) B: The x-intercept of h à Directly type it into the Input box à (200,0) C: Intersection of functions g and h à Select Intersect[<Object>,<Object>] à Type g and h into Object spaces à Enter D: Intersection of functions f and g à Select Intersect[<Object>,<Object>] à Type f and g into Object spaces à Enter Step 5: Type the minimizing/maximizing equation into the Input box. Only type the side of the equation with the variables/constants, not the P= side. Type 1.50x y

14 Step 6: Substitute points into the minimizing/maximizing equation by typing i(a), i(b), and so on, until all of the vertices have been substituted. Step 7: Using the answers in the Number section on the left, find your solution. They are listed in the order they were typed. The vertex used to find that number is your solution. Points C and D both create a profit of $600. Any point on function g between these two points will create an equal profit, so any of those combinations can be made. Example from

15 Example 3: Maximize C = 3x + 7y given the following vertices: A: 6.42,2.29 B: 6.17,0 C: 15,0 Step 1: Type the maximizing equation into the input box. Step 2: Substitute the points to find the solution. To substitute the points, type them as shown below. The highest value is the solution. We can see that point C maximizes the equation. The original graph made on GeoGebra to create this problem is given below. This will not be needed for the example, but I wanted to include it to show how I created the problem.

16 Example 4: Identify the bounded region given the following constraints: x 16, y 3, and 12x 25y 13. Step 1: Graph the three constraints. Step 2: Find the darkest shaded region. This is the bounded region.