4.3 Quantitative Methods for Decision Making: Math 115

Quantitative Methods for Decision Making (Math 115) is a Liberal Arts Mathematics course; therefore it cannot be credited toward fulfillment of Mathematics Majors requirements. This course is designed to help students understand the basic properties and applications of equations, inequalities, functions, and their graphs. Students will also  discover how exponential and logarithmic functions are used to model real life situations in physics, e.g. radioactive decay, in biology, e.g. growth in bacteria, and in finance, e.g. simple interest and compound interest, annuities, and loans. Students will learn the fundamentals of linear algebra and solve systems of linear equations using matrix techniques; apply the basics of linear programming and the simplex method.

OER-Enabled Pedagogy: Prompts and Examples for Quantitative Methods for Decision Making (Math115)

The three OER activity prompts listed below may be easily adapted for some mathematics courses.The first activity may be implemented as early as the end of each lesson or at the end of the first chapter covered in the semester. The strategy described in the first activity prompt may be employed throughout the semester to reinforce content and to test understanding of the material. The second activity may be done at the end of the first test. As students are engaged in creating their own test questions and solutions, they will be involved in active recalling and processing of the content, leading to a deeper understanding and better retention of the material. The third activity may be done towards the end of the semester, by which time students will have acquired sufficient knowledge of how lessons are connected to the real world.

OER Activity Prompt #1: Students Generated Tutorials, Mini-Lesson Templates and/or Introductory Videos

1) Students will be randomly placed in groups of 3–5 students depending on the class size.

2) To demonstrate how lessons are related to real-life situations, the instructor will upload several real-world word problems that connect a particular lesson to the real world. This activity may be done on a weekly basis, at the end of each lesson or at the end of each chapter.

3) Students will be prompted to do the following:

1. Each group will choose one problem from the instructor’s pool of word problems.
2. No two groups may choose the same problem.
3. Each student will rewrite the problem in his or her own words.
4. Students will then meet in their respective groups to discuss the problem, share ideas amongst themselves, and develop a strategy.
5. Afterward, each group will create a step-by-step tutorial outlining how they approached the problem, list the steps taken to solve the problem and note the final solution. Students may refer to their class notes or any other resource that they may find helpful.
6. Students may be challenged to create a mini-lesson template on how they will teach the lesson to a student who was absent. They will be required to utilize a different and innovative method to explain the problem.
7. Students may also be challenged to create an introductory video on the topic they are working on so that future students can watch ahead of the class and therefore be better prepared.

4) Each group will then upload their final solution with the supporting work and their mini-lesson template or introductory video to all groups in the class by the due date posted by the instructor.

5) Groups will submit a peer review providing positive and/or constructive feedback on each group’s solution by the due date posted by the instructor.    

6) An opportunity will then be given for each group to modify their contribution to this active learning assignment by the due date posted by the instructor.

7) Finally, groups will evaluate the final step by step templates of each group’s work.

OER Activity Prompt #2: Student-Generated Test Questions and Solutions

1) Students will be randomly placed in groups of 3 – 5 students depending on the class size.

2)  Each group of students will use an old exam provided by the instructor to create likely test questions and/or revise existing test questions and their solutions to be used for the current class and for future classes. Alternatively, groups may choose to create their own likely test questions on specific content topics. For this assignment, students may refer to their class notes or any other materials that may be helpful.

3)  Two due dates will be given by the instructor. Each group will submit their completed assignment to all groups on or by the first due date. Each group will then be required to review and submit their positive and constructive feedback to all groups on or by the second due date.

4)  The instructor will define the types of questions that students are expected to create.

5)  Each group will be required to create four questions and a worked example, one from   each category listed below:

          a) A matching question

           b) A true/false question

           c) A multiple-choice question

           d) A fill in the blank question

           e) A worked example

6)  For (a – d), students will be required to provide an answer key and a thorough explanation of the answer chosen. For (e), students will create a step-by-step tutorial on the worked example, using graphs, images and/or diagrams, when appropriate.  

7)  After completion of the assignment, students will submit their possible test questions and solutions for peer review and feedback on or by the first due date.

8)  Groups will review and submit their positive and constructive feedback to each group’s test questions on or by the second due date. Groups may even be invited to edit the possible test questions each group has submitted.

9)  Finally, the instructor will review the possible test questions, make any necessary corrections and develop exams and possibly quizzes based on students’ test questions.

OER Activity Prompt #3: Student-Created Word Problems

Towards the end of the semester, each student or each group will be required to create at least one word problem and its solution that demonstrates how any lesson taught during the semester is connected to the real world. It is assumed that students are familiar with how lessons are connected to the real world. It is also assumed that students know how to create step-by-step templates on how to solve word problems.  

1)  Students will be randomly placed in groups of 3 – 5 students depending on class size.

2)  Student groups will choose a leader, and a scribe.

     a) The group leader will be responsible for the following:

1. Keeping the group on task.
2. Setting timelines.
3. Scheduling meetings.
4. Ensuring that every member of the group participates, is actively involved, and willing to help others.

          b) The scribe will be responsible for keeping detailed notes of all discussions.

3)  Each member of the group will be responsible for submitting one word problem to the group. Students may only submit their own original version of a word problem.

 4)  Each group will then cast a vote to determine which word problem to use for this   active learning activity.

5)  Students will be provided with different options by the instructor – one group may choose to create a PPT slide, another group may choose to create a tutorial video, another group may choose to develop a lesson plan or another group may choose to create a step-by-step tutorial.                                            

a) Initially, each student will attempt to solve the problem independently.    

b) Then each group will discuss the problem, share ideas amongst themselves, develop a strategy and decide on the format they will use to showcase their problem.                                                                                                                        

c) Regardless of the option chosen, students will be required to submit a detailed explanation of how they approached the problem, list the steps taken to solve the problem and note the final solution. Graphs, diagrams, and/or images may be used, when appropriate.

6)  Upon completion of this assignment, groups will upload either their step by step tutorial with the supporting work, their PPT slide, their tutorial video or their lesson plan by the due date posted by the instructor.

7)  Groups will review and be invited to provide positive or constructive feedback to each group’s work regardless of the format by the due date posted by the instructor.  

8)  Groups will then be given the opportunity to fine-tune their contribution to this active learning activity by the due date posted by the instructor.

9)  Finally, groups will submit the edited version of their work for grading by the instructor.

Worked Examples for OER Activity Prompt #3: Creating Real-Life Linear Models

Objective: Each student will create at least one-word problem that demonstrates how any lesson taught during the semester is connected to the real-world. Then, each group will solve the word problem and use one of the following options to showcase their work: a PowerPoint slide, a tutorial video, a lesson plan or a step-by-step tutorial.  

Example #1: A Jamaica Avenue vendor discovers that when a t-shirt was charged for  $15, 40 t-shirts per week were sold. However, when a t-shirt was charged for $10, 70 t-shirts per week were sold. Find a linear model for a correlation between the price of a t-shirt and the number of t-shirts sold.

Objective:  Students will be able to use this linear model to do the following:

1. Identify the two given points.
2. Find the slope of the linear equation.
3. Write the equation in slope-intercept form.
4. Predict how many t-shirts will be sold at a given price.

I will now use this linear model to create a step-by-step template below:

Solution:

1) Read the question carefully to determine what facts are given.

2) Let’s identify the input (independent variable) and output (dependent variable)  

        The number of t-shirts sold depends on the price → input: price, output→ t-shirts

3) Now let’s convert the data into x and y values and write as two ordered pairs

        → ($15, 40), ($10, 70)

4) Find the average rate of change per week (slope) → m = − 6  

5) Write the linear model in slope-intercept form → y = mx + b

6) We have two ordered pairs.

        What is the next step? → Yes, use any one of the two ordered pairs to find b                                

                  → 70 = − 6(10) + b → b = 130 or 40 = − 6(15) + b → b = 130

    (Regardless of which ordered pairs we use, the value of b is always the same).

7) We are now ready to write our equation → y = − 6x + 130

8) We will now use the point-slope form → y – y1 = m (x – x1)

        → y – 40 = − 6(x – 15) → y = − 6x + 130  

9) Let’s take this problem one step further, can you predict how many t-shirts will be sold

    if the price drops to $8? → y = − 6(8) + 130 → y = 82 → 82 t-shirts will be sold at $8.

10) You are now ready to create your word problem as modeled in this example.

Example #2: A Jamaica Avenue vendor discovers that the maximum revenue of $700 per week occurs when he charges $10 per t-shirt. When he charges $15 per t-shirt, the revenue drops to $600 per week. Create a quadratic model to illustrate the relationship between the selling price per item and the number of items sold.  [Revenue = (selling price of an item) x (# of items sold)]

Objective: Students will be able to use this quadratic model to do the following:

1. Determine the two ordered pairs.
2. Identify which of the two points represent the vertex of the quadratic equation.
3. Determine whether to use the vertex form or the standard form to write an equation for this quadratic model.
4. Transform the equation into another form.
5. Graph the quadratic function.

I will now use this quadratic model to create a step-by-step template below:

Solution:

1) Read the question carefully to determine what facts are given.

2) Let’s identify the input (independent variable) and output (dependent variable)                            

    Revenue depends on the price → input: price, output→ revenue

3) Now let’s convert the data into x and y values and write as two ordered pairs

        → (10, 700), (15, 600)

4) Identity the maximum value → (10, 700)

5) Let’s sketch a parabola that models this situation.              

               ↓ maximum revenue                                                                            

6) Let’s look at the information again and determine whether we use the standard form or the vertex form of the quadratic equation. Yes, we use the vertex form because the maximum value represents the vertex. So we are given the vertex and one point.

7) We are ready to make our substitution into the equation: f(x) = a(x – h)2 + k        vertex: (h, k) = (10, 700) and point: (x, y) = (15, 600)  

        → 600 = a(15 – 10)2 + 700 → 600 = a(5)2 + 700 → − 100 = 25a → a = − 4

   (Notice that since we have a maximum value, a is negative)

8) We are now ready to write our quadratic equation in vertex form:

        → f(x) = – 4(x – 10)2 + 700

9) Let’s transform this equation into standard form:

   Expanding (x – 10)2 → x2 – 20x + 100

   Substituting → f(x) = – 4(x2 – 20x + 100) + 700

   Using distributive property → f(x) = – 4x2 + 80x – 400 + 700

   Combining like terms → f(x) = – 4x2 + 80x + 300

10) Let’s graph the parabola that models this situation

Figure 10

Quadratic model graph                                                                                                                                                      

10) You are now ready to create your word problem as modeled in this example.

By Ilene Ahamad