From Tables to Tangibles: Making Math Concepts Stick

As a math teacher for grades 11 and 12, I use words like “exponential growth” or “constant change” all the time in and outside the classroom. Reading and interpreting graphs is almost second nature to me.

However, I realized that translating information from a table to a graph isn’t as straightforward for students.

After a couple of “oooooohhhh”s from different students, I knew this activity had been successful. Now they truly understand functions and their graphs.

Students came up with a bunch of creative graphs and I am very happy to share them with you! :)

Starter: 5,4,3,2,1 — grounding exercise

This is a useful hack that I use when I don’t want to meditate.

But more importantly, it is used by those who want to manage stress and anxiety by grounding themselves in their surroundings.

It involves identifying 5 things you can see, 4 things you can touch, 3 things you can hear, 2 things you can smell, and 1 thing you can taste.

I had the students do this one to calm them down and get them in the mood for stress-busting craftwork ( drawing the graphs! :D )

Group Discussion: Atomic Habits — the 1% better each day mentality.

I haven’t read the book, but I know the gist of it.

It is to develop a mindset that empowers you to make small changes every day to achieve big results. it essentially employs the power of compounding: a small percentage increase every day, can result in huge gains in the long run. The focus being consistent effort and tiny gains every day.

This is where we as a class calculate the “level” each day starting from Level 1 on Day 1.

It helped them recall their knowledge of compound interest and its link to geometric progression.( something we had already done last year in class ).

Then I showed them this iconic graph used in the book :

Here, the discussion on growth mindset and atomic habits leads to a discussion on showing non-quantifiable things such as feelings on a graph!

Main Activity: Make Your Own Graph

Now that they have an idea of how creative one can be by showing things graphically, let them explore.

1. Ask them to decide on two variables which are related to each other. And classify them as dependent and independent variables.
2. Ask them to describe the change they anticipate.

• Is the change going to be the same every time ( linear )?
• Is it going to increase or decrease rapidly ( exponential )?
• Is it going to peak, reach a max value and then come back the same way it went up ( parabolic )?

3. Ask them to sketch their graphs on a chart paper and tell them it will be displayed in the classroom.

I know it sounds silly and not as dense or “mathy” as a high school math class should be. However, the lesson intends to strengthen conceptual understanding. And nothing works better when students are in their own domain of expertise, and expressing themselves.

Here are a few examples of what my students made :

Re-collect and Re-flect

The activity above will take up a good amount of time. So you might have the urge to end the class, but this is the most important part.

You must have them present their graphs to the class and explain the nature of the relationship between the two variables.

Later, the discussion can address these questions :

1. What kind of functions did we explore today? : You can list them on the board, and draw a sample graph next to it.
2. How can you identify which graph to use when? : focus on the terminology they might encounter in the questions. For eg: “constant increase”, “fixed percentage change” etc.
3. What are the standard forms of writing the equations of those functions?

Here’s an important link that you can make: Law of Diminishing Returns.

And here’s a great video that uses graphs to express these abstract concepts: