Solving and Visualizing Solutions to Simultaneous Equations/Polynomial Functions

I decided to keep focusing on the fx-991EX Classwiz Scientific Calculator this week, since it’s one of my favorite calculators, in particular because of the added capability to show a visual representation through the QR code, which is something most scientific calculators can’t provide. Multiple representations are so important to helping students see relationships, recognize patterns, understand solutions, etc. Particularly to understand what it means to ‘solve’ an equation or system of equations – i.e. what does the ‘solution’ represent? Being able to see it numerically and visually is important for overall understanding. Especially if you are not just doing ‘naked math’, but have provided a context (problem-solving) for the functions/equations you are solving, where students really need to explain the solution in relation to the context.

With that in mind, I wanted to share a system of equations activity from the open-source curriculum, Illustrative Math, which has some great problem-solving tasks that allow students to apply mathematical skills, explain their thinking, and think beyond just skill-based practice.  For this particular posting, to show how the fx-991EX can help support finding solutions and visualization, I searched the Illustrative Math tasks for simultaneous equations, and found the task Kimi and Jordan. This is a grade-8 task that provides information about two kids, Kimi and Jordan, who both earn weekly allowances and also work jobs where they get paid a certain amount per hour. The problem asks students to create a table that matches total earned to hours worked each week, and also to compare who is saving more money if they work the same number of hours. This last question really is the ‘solve the system’ question, but involves more than just a single answer, since there can be several answers dependent on the number of hours. This is where having the ability to both visualize the equations and look at the graphs and really explain what is happening in the context of the problem. What does the intersection of the graphs represent? When, if at all, would Kimi earn more? When, if at all would Jordan earn more. So, adding the context of the situation makes this a beyond skill-practice problem, since they have to explain their thinking and there isn’t just one solution (i.e. the intersection point). This a nice problem to use with a graphing calculator or dynamic math software, but also if you have the fx-991EX as your technology tool, you can do the multiple representations as well, including a visualization.  Below are some images from the fx-991EX on this specific task.

This is the two equations, in standard form, for Kimi (top) and Jordan (bottom)

This is the ‘solution’ (intersection point) for the simultaneous equations. However – what does that mean in context? Does it answer the question who will earn more for a certain number of hourse?

This is the visualization of the equations and solution (graph) – notice you see the equations in standard form (and the decimal is in standard notation). Students now can look at the graphs and talk about when Jordan might earn more, when Kimi might earn more, and when they earn exactly the same amount….all based on hours. So three answers to the question “who earns more…”















I like this problem because it goes beyond a single solution for (x, y), and instead forces students to explain the different possible solutions based on hours worked. Real applications of mathematical skills require students to analyze, justify and understand that mathematics is just a way to model situations and help you make decisions.

Here’s the link for the activity on the Illustrative Math website, which also includes a discussion of the solution, and also a link to the Youtube video that shows you how to use the fx-991EX Classwiz to solve simultaneous equations and look at the visualization with the QR code option.

Be sure to visit Casio Cares:

Here are quick links:

STEM – Inertia, Force and Velocity – Newton Knew Inertia (Mini-Math Lesson)

I wanted to focus on some STEM lessons this week, using, since it is so great for collecting data, showing statistical plots, and it’s ability to quickly change things to see the impact and more importantly, to do everything in the one activity (i.e. calculations, data collection, graphs, and explanations). So, I am going to share a different STEM focused lesson each day this week.

STEM is an acronym that stands for Science, Technology, Engineering, and Mathematics. It’s really about ensuring educational experiences that blend these four areas, so that learning is not in isolation, but rather a connected learning on real-world concepts, that require students to problem-solve, collect data and analyze results, and communicate their findings and use critical thinking. There are many definitions and reasoning behind the push for STEM in education – here is one article that I think gives a good overview if you feel you need more information.

For me, STEM means learning and problems that are real-world, that students can actually do or experiment or relate to, that require them to use science, math, technology and engineering in realistic ways or situations. It’s looking for and collecting evidence, and then modeling this information, and applying understandings of the subjects to make sense or make decisions. The activities I am going to focus on this week come from Fostering STEM Education with Casio Technology, Casio 2013. It is a resource with some really great real-world problems and explorations, and these can be done by students or there is also sample data provided if you don’t have the materials needed to do the experiments yourself. So you can make these activities as hands-on as you want, but if not possible, still have the great discovery and conversations and critical-thinking experiences needed for deep learning and application of STEM concepts.

Today’s activity is one that you could very easily do with students (and it even adheres to social distancing rules!).  In the image above, you will see it involves six students – five to stand at designated and one to push the object in a straight path.  This activity explores how the amount of push impacts inertia and acceleration. There is an object that starts from static position (so a disc (like a frisbee) or ball), and then a student pushes it in a same line with as much consistent push as possible, and as it passes student at the set up positions, they time when the object passes.  The force is changed for three trials and then students compare the data in several ways. So, a fun activity, but, if you are unable to do this with all the materials, or students, then you can use the sample data provided.

Here is the link to the activity, both the version and the PDF that can be used and a video overview:

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 – Quadratic Equations & Multiple Representations

I am cheating a bit today and using a lesson I created a while ago, along with the accompanying video. You will note that this video support is a bit longer than usual for these mini lessons because I am really using many different aspects of and going into a bit more depth with them as I talk through this activity. We will explore data, scatter plots, fitting curves, using the connected functionality of to do column calculations quickly. There’s graphing, equations, calculations, curve fitting and exploring of real-world context, i.e. finding the dimensions of a garden, which means thinking about dimensions. This lesson has a lot!

Here are the links to the activity itself as well as the support video:

  1. Quadratic Functions – Area of A Garden
  2. Video Overview of Quadratic Functions – Area of A Garden


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 – Multiple Topics (Calculus, Trig, Geometry….)

2020-03-25_13-41-35I am re-sharing some lessons/videos created by Ish Zamora (@seemathrun) because they are so good and if you hadn’t seen them before, I am hoping that you explore them now. (Also, because I have run out of time to create some new ones today, due to some other responsibilities!)  I will be back tomorrow continuing my transformation theme for the week with some new mini-math lessons focused on Dilations.

For today, I am sharing a several activities created by Ish, along with accompanying videos where he talks about the activities. Some of these have been shared in previous blog posts and/or on our FB and Twitter and Youtube accounts, but here they are in one place for you to explore and choose from!

Links to some of Ish’s FREE activity papers and video support:

  1. Exponential Decay – Bouncing Ball  and Calculus Bouncing Ball (Parabolic/derivative)
  2. Proportions & Scale – Dr. Evil   and Trigonometry Mini Me 2 (Trig Intro)
  3. Taxi Cab Geometry – Circles and Distance and Uber/Taxi Cab Geometry
  4. Multiple Representations – A Gutter Problem
  5. The Soggy Grasshopper – Zeno’s Paradox

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:


Weather and Integers – The Importance of Real World Connections

A lot of my math teacher friends have been posted images from weather reports on FaceBook and Twitter, like this one to the left from @seemathrun, showing the real-world application of integers due to the extreme weather conditions that are happening across the country right now. It really is a perfect opportunity to show a true application of mathematics that students can definitely relate to, especially if they are in those freezing climates. Add in the wind chill, and you have some interesting data and comparisons and a chance to talk about the relevance of math and understanding numbers.  Here’s an image to the left showing wind chill, temperature, and frost bite times someone else shared that could help explain why so many schools are closed, even though there may not be any snow on the ground, (which is usually the reason behind winter closures). I know one of my colleagues and friends, @ClassPadnut, was sharing with me yesterday that with the wind chill, it was -60 where he lives.  Yikes!!!

There is obviously a lot of different math concepts you could explore with students, dependent on grade level and questions asked. I find the wind chill graph the most interesting. Looking at the wind chill chart, the drop in temperature is almost, but not quite, constant, like you would think – i.e. You will note that there is an equation for the calculation of wind chill at the bottom of the image. I was  curious about whether students could find that connection from the data alone -something to challenge students with. How would they graph this data? Could they? Thinking of statistical tables, what would they enter and what statistical plots would be appropriate? If students are in areas where schools actually closed, you could talk about how the data supports the decisions, and what is the ‘cut-off’ temperature/wind speed that might influence the decision? Lots of things to explore.

I found another image that showed the lowest temperatures reported in each state, so you could do a comparison across states. Even Hawaii is cold!!!  Crazy.  Below is the image, which I then used to enter the data in a table in, and then make two different plots to represent the data – a histogram and a box-plot. You can see from the box plot five-number summary that the median temperature in the U.S. for this day in January is -40.  Wow!!! (And boy, don’t want to be in Alaska at -80!) Again – think of the interesting class discussions about integers, about how these temperatures will impact things such as the orange crops in Florida or the tourism in Hawaii or California. (Here’s a link to the paper that has the image, table, and graphs shown below:

As you can see, using what is actually happening right now in our country, i.e. REAL world connections of weather (temperature, wind speed, wind chill), is an amazing opportunity to help students see the relevance of integers and statistics and how this data is being used to make important decisions, such as do we close schools? Who should not venture outside? How long before you get frostbite? The visuals help students ‘see’ mathematics in action, and particularly if we focus on the integer aspect, provide a clear connection to integer addition (and subtraction, depending on the questions asked), something many students struggle with.

Whenever possible, we should be trying to connect the math concepts students are learning and using to a real-world application. Here’s a perfect opportunity, no matter the grade level, to have some great class discussions about the impact of weather on our world, about the relevance of integers, and about how statistical information is important to decision making.