Systems of Equations – Sample Lessons and Resources

For this months lesson feature, I am going to focus on Systems of Equations. I chose this topic because I just did a workshop with Algebra 1 teachers in NJ, and this is where they were in their pacing guide, so I am making an assumption that many algebra teachers might also be focusing on this content as well this time of year. I am using a problem from Fostering Algebraic Thinking with Casio Technology in order to provide a real-world problem-solving experience (and I have the resource), but I have altered the problem so that I can utilize the all-in-one capabilities of (tables, graphs, equations, geometry, text).

The Problem

In 2010, there were approximately 950,000 doctors in the United States, and approximately 350,000 of them were primary care doctors. It was estimated that more than 45,000 new primary care doctors will be needed by 2020, but the number of medical school students entering family practice decreased by more than 25 percent from 2002 to 2007. With laws reforming health care, many more people will be insured in the United States. 

For many reasons, including a growing and aging population, the demand for doctors will likely increase in future years. The number of doctors available is also expected to increase. But, due to the high cost of insurance and the fear of malpractice lawsuits, many have predicted that the increase in the number of practicing doctors will not keep up with the increase in demand for doctors.

The table to the right provides data from a study conducted in the state of Michigan. These data approximate the number of doctors that were or will actually be licensed and practicing in Michigan, called the supply, and the number of doctors that were or will be needed by the people of Michigan, called demand.

The question is, will there be enough doctors to provide all the services? The shortage of doctors is a problem that challenges the entire country, not just Michigan.

The Lesson

A shared paper has been created in called Systems of Equations Help! Not Enough Doctors, which you can access by clicking on the title. The idea behind this problem is to provide a real-world context where students can use tables, graphs, and equations (along with calculations) to create a system of equations. They can solve these using methods such as substitution, elimination, and graphing. Students will also be practicing how to model with mathematics, applying what they know about relationships and being able to create a system of equations that fits the context of the situation in order to find a reasonable solution.

In the activity, there is obviously some focus first on getting students to really understand the problem and what the numbers represent, and then the idea is to have them look for patterns and relationships as they look for a solution. First in the table, then by looking at a scatter plot of the data, where they again try to determine a solution based on a visual. Continuing to look for trends, they use prior knowledge to recognize linear relationships, create equations that model the data, and then graph those equations to find a more precise solution. Then, as a check, they solve their system of equations algebraically. It’s all about multiple representations and helping students see the connections between all the representations, and depending on whether you want a specific, precise answer or just a generalized answer, you might choose a different representation. – Lesson In Action

The video below shows the activity and does a brief walk through of some of the components and what it would be like doing the activity from a student perspective. I am a big believer in the think-pair-share approach, so I would suggest having students do the Notice and Wonder individually first, then pair up, then share so that you can make sure that any misunderstandings about the context, and clarification about the numbers is figured out before students start solving. Then I would suggest small groups for working on the problem itself.

Other System of Equation Activities and/or video links



More than Calculators – Teacher Support & Resources

I received a message the other day from a reader who commented on how much he liked the Prizm, but because 2016-05-13_12-52-22Casio didn’t have any resources to support the learning of the Prizm, he was a little reluctant to try it.  My first reaction was “What?!! We have a TON of resources!!”  My second reaction was to ask myself why might he think this? I was able to answer my own question when I searched for our resources – the issue being they are a bit hidden among all of Casio’s other products, (which, just so you know, is of course in the process of changing as we create a more user-friendly web-page).

In the meantime, I want you to see the great teacher/student resources we have! Let me share with you the resources we have that supports teachers (and students), from complete subject-specific or grade-specific resource books (i.e. complete lessons), so sample lessons and activities (free), to online course for Prizm (free) to webinars (free).  There are teacher-created resources and quick-start guides.  Casio WANTS teachers and students to use their calculators and get the help and support they need to use them appropriately.

  1. Free online activities and sample questions:
    • These include grade-level activities and specific Casio Prizm-vs-TI 84 activities
    • Scrolling down the page you will find sports activities for use with five different calculators
    • Keep scrolling to our Quick Start Guides for 6 of our calculators (including Prizm)
    • Keep scrolling to Subject-specific Teacher Resource Guides and Calculator Tips
    • Scroll further to see all our grade-level and subject-level resource books that contain complete lessons
  2. If you look at our products page, under Software & Additional Products, you will be able to scroll through all our grade-specific/subject-specific resource books:
  3. Here’s a short-link to our Casio Lesson Library (with teacher created activities):
  4. Short-link to Guided tours for the Prizm:
  5. 2016-05-13_12-56-04If you are interested in the Prizm, we have a whole webpage dedicated to Prizm activities and support, which includes lessons, videos, and also has the OS updates. 
  6. We have a free online course for the Prizm (self-paced).  If you complete the course, you get the Prizm (fx-CG) emulator software for free.
  7. Free webinars on many math topics (statistics, geometry, algebra, calculus, etc.)(you do have to register your email to view these, but they are free):
  8. Links to manuals for specific calculators:
  9. And let’s not forget the videos showing you how-to’s and comparisons!  and

As you can see, we have a ton of support for teachers and students wanting to use and learn-to-use Casio calculators to support their instruction and/or math learning. We hope those of you out there excited to start working with Casio calculators start using these supports. We are educators here at Casio and want you to love the calculators as much as we do!!

Linear Programming – Great Real-world Applications

I’ve been playing around with graphing inequalities on the Casio Prizm (and fx-9860GII & fx-9750GII), realizing 2016-02-11_11-54-52how easy they are to graph and how easy it is to find the area of intersection. In doing so, trying to find some relevant problems that would be applications of graphing inequalities, I was reminded how relevant linear programming problems are to both graphing inequalities and finding intersections, but more importantly, to real-world situations.

I spent the majority of my teaching in K-12 with Algebra and Geometry, so linear programming was never on the required curriculum back when I was teaching.  It’s usually in an Algebra II course other ‘higher level’ mathematics courses.  And, if I remember, it is often a content that is ‘skipped’ because it isn’t on standardized tests. In looking at some of the activities in Fostering Algebraic Thinking with Casio Prizm (Goerdt, Horton), trying to find some inequality problems, I found some activities under Linear Programming that made me stop and think – these problems are much more relevant and connect to actual situations than those “naked math” inequality problems we tend to give our students. Find the intersection of these two inequalities is not very engaging, but find out number of servings of potato chips and peanuts you can have while staying within your recommended daily allowance (I know I am asking this question!!). Or, if a company wants to advertise in the SuperBowl, based on their budget and cost of an ad during the first half or the second half, when should they advertise? Still looking for intersections of inequalities but now making a connection in context where there is a purpose for using the math.

These type of activities require a lot more math than just graphing inequalities and finding intersections.  You have to translate the situation into inequalities. You have to convert these inequalities to a form that  will allow you to graph them (i.e. y=). You have to consider ‘implied’ constraints (if you are dealing with money, you can’t have x or y less than zero, for example). There’s a lot of sense-making about what is being asked and the constraints given, modeling a situation with the correct mathematical equation, seeing if your answer makes sense, understanding what the graphical representation of the intersection means, and what do the intersection points mean in the context of the situation.  Linear Programming problems are really robust applications. There’s a lot of Common Core content and more importantly, Mathematical Practices going on here.  Students can then apply this type of thinking to situations that impact them – for example, selling girl scout cookies….if we have to make a certain amount, the cookies cost this amount, the troop gets this percentage, how many do we need to sell to make our profit?  I think these are a lot more interesting than ‘find the intersections of these inequalities’.

I have attached an activity  from Fostering Algebraic Thinking called “The Snack Problem” (Download: Linear Program Snack Prob).  You can look at it in depth by downloading it.  I am just going to state the problem here, identify the inequalities, and then do a quick video on how to graph and find the intersection.  There is a lot more to the activity then whats below so I encourage you to take a look and try it with your students.

Problem:  Assume you like snacks and insist on having at least one serving of dry roasted peanuts and one serving of potato chips each day.  Each serving of the peanuts contains 15% of the recommended daily allowance of saturated fat; each serving of potato chips contains 10%.  Each serving of the peanuts contains 12% of the recommended amount of dietary fiber; each serving of potato chips contains 5%. You determine you want to consume no more than 60% of the recommended allowance of dietary fat from these two snacks, but you want to get at least 30% of the recommended allowance of fiber from them.  Sketch the feasible region.

(please refer to the handout for the other questions).2016-02-11_12-39-57

Inequalities: x represents dry roasted peanuts, y represents potato chips

  • 0.15x + 0.10y ≤ 0.60 (saturated fats) –> y≤(0.60-0.15x)/0.10
  • 0.21x + 0.05y ≥ 0.30 (fiber)  –>                        y ≥ (0.30-0.21x)/0.05
  • x ≥ 1 (at least 1 serving peanuts a day)
  • y ≥ 1 (at least 1 serving potato chips a day)

The video below shows how to graph all four of these inequalities on one graph (yes, even x ≥ 1)! And how to construct just the intersection (vs. the union of these four inequalities) and find specific intersection points.  The attached activity includes all these steps as well.