Mini-Math Lessons – Non-Linear Inequalities

I have really been trying to find some examples of non-linear inequality situations, along the lines of linear programming. Not as easy. But, you can have inequalities that are NOT linear, so I thought I would explore these today just to make sure we are well-rounded in our inequalities experiences this week.

Obviously, there are the bare-naked types of situations, where students are given linear and non-linear inequalities to graph and then describe the solution set (or solve algebraically). The first activity is really just exploring what inequalities are and where the solution sets lie. There will be a look at the graphs and tables and just making sense of the different regions when you have a system of inequalities.

The second activity is a application problem involving a system of inequalities that uses both a quadratic and a linear relationship. Students use the graphs and intersection areas to describe the possible solutions to a construction problem. They use perimeter and area to come up with the length and width ranges to build a dog park.

Here are the links to the two activities and the video overview that walks through each activity briefly:

  1. Inequalities Graphing – Linear and Non-linear
  2. Systems of Inequalities – Going to the Dogs

The tool being used in these mini-math lessons is the FREE web-based math software,

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Mini-Math Lessons – Graphing Systems of Inequalities with

Yesterday I shared how to actually graph inequalities and look at either the union or intersection of those graphs. We also explored how to limit the domain/range in order to only see the viable regions for solutions. Today I want to use those skills to actually solve systems of inequalities in two-variables. I am using two tasks from open-source curriculum created by Illustrative Math. Task 1  and Task 2 come from HS Modeling.

The first activity starts with a graph and asks students to interpret the graph and create the inequalities from the graph. This requires the ability to recognize that dotted lines represent inequalities with no equal sign, solid lines include the equal, and that the shaded regions represent the solution areas and help determine whether it’s greater-than or less-than. Obviously, students also need to be able to find slope and write equations of lines (we are only dealing with linear inequalities right now).

The second activity presents a real-world situation involving determining how many adults and children can fit on a boat safely, based on weight. It requires students to develop the inequality equations from the description, graph them, and then answer questions about the areas of the graph. They are then asked to determine whether different combinations of adults and children can feasibly rent boats safely, requiring them to understand where the solution region lies in their graphs.

Here are the links to the activities and the video overview that explains and demonstrates some of each activity:

  1. Solution Sets
  2. Fishing Adventures


The tool being used in these mini-math lessons is the FREE web-based math software,

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Math Hardware versus software – Similarities & Differences with Casio

Students using technology as part of learning math is important because of the extension of learning that is possible, the visual connections, and explorations that become possible as a result of technology. The most common technology students use these days are their phones, tablets, computers, and of course, hand-held devices such as calculators. It all depends where you live, what schools you attend, what’s allowed or not allowed, and also what resources are actually available and understood by both teachers and students. From my own research, some schools/teachers have a multitude of resources, but most schools have limited options. And – even if there are many technology tools available, teachers tend to utilize the tool (s) they are most comfortable with, and that the majority of students have access to. Basically, it comes down to choosing a technology that is going to support the learning and that students and teachers can use relatively efficiently, so that time is not lost to ‘tool logistics’. Often times, again, based on my own research (dissertation), teachers choose tools that may NOT be the best choice for learning because they know how to use it over a much better, more appropriate tool, that they are unfamiliar with or uncomfortable with, so many times better technology tools go unused because of the ‘learning curve’.

What I wanted to use this post for today was to show how Casio has really recognized the ‘learning curve’ issue and tried to keep functionality consistent across handheld models and even in their software, providing intuitive steps and menu options right within the graphing menu itself that alleviate some of that ‘learning new tool functionality’ concerns that teachers and students often face when using technology. Our graphing calculators basically use the same steps, buttons, layout, even from the very basic ones (fx9750) (fx9860), to the more advanced ones (CG50), so if you know one, you know them all. And, even the new software,, is built along the same lines, though obviously with more features and capabilities.  But there is no ‘searching for menus’ – relatively intuitive no matter the tool. Obviously, as you get into the newer models and then into the software, the functionality and options increase – we go from black-and-white displays to color, we go from intersection points on the graphing calculators to union/intersections on the software. But knowing how to use one tool makes transitioning easy, and if you had students with several different models of the handhelds, you could still be talking about the same steps and keystrokes.

The best way to compare and demo is to show you how to do the same thing on the different models. I’ve chosen to show graphing two inequalities, so that you can see, even on the older models, that shading and intersections occur. But also to show that as you progress into the newer and more powerful tools (i.e. memory capacity, color, larger screens, resolution, etc), allowing for more options and learning extensions.

Here are the two inequalities that are being graphed in each of these short GIF’s:

Each GIF below graphs the two inequalities and finds intersection points of the two graphs. The software extends that to allow for finding the Union and the Intersection of all points.





















































Be sure to check out the free software that does calculating, graphing, statistics and geometry:































































































































































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.