Note: Some math equations and tables on this webpage may not be fully accessible with a screen reader. For full accessibility, download the Word document version of this post.
Mathematical modeling is like building a bridge between real-life situations and math. Sometimes, we may have a real-world problem where it’s not easy to come up with an equation immediately. Instead, we can create a table of values, and using this table, create a graph to show how two things are related. For example, if we know how the temperature in Celsius relates to the temperature in Fahrenheit but don’t know how they are related, we can make a table with different temperatures and then plot the points on a graph. This graph helps us see the pattern and figure out how the two temperatures are connected without having to guess the equation right away. A graphing tool such as the Desmos Graphing Calculator can then be used to help us figure out an algebraic equation that best matches the graph.
The Desmos graphing calculator is a powerful and accessible tool that helps students, including those with visual impairments, explore mathematical modeling in a hands-on way. Instead of drawing graphs by hand, students can enter tables of values, create graphs and listen to descriptions of the graphs using screen readers or audio tracing features. This makes it easier to understand how changing one value affects another, even if it’s hard to picture it visually.
In this lesson, let us start by working with a real-life situation that is familiar. Temperature can be measured in both degrees Celsius as well as Fahrenheit. You might already be aware of a formula that helps you convert from one temperature unit to another. Let us try to come up with this formula with the help of the Desmos Graphing Calculator. Let us say that we measure water at different temperatures, using both a Celsius and a Fahrenheit thermometer. This helps us come up with a table of values such as the one shown below:
Let us start our modeling using this table of values.
There are four main steps in creating and interpreting a mathematical model in Desmos.
The first step is to create a table and then import the table with the two quantities to Desmos. For example, a table with two columns: one for Celsius (°C) and one for Fahrenheit (°F). You can make this table in a spreadsheet like Excel. Note that the table values can also be entered directly into Desmos. However, in this tutorial, we will learn to copy a table from Excel onto Desmos. For this step, download the attached Excel Document, “Temperature.xls”. Watch the video “Importing a Table into Desmos with JAWS”.
Copying and Importing a Table into Desmos video:
Desmos will automatically plot the points from the imported table. The next step is to choose the appropriate “regression” model for the points plotted. When you add a regression model in Desmos, it means you are drawing a graph and finding an equation that best fits the points you have plotted. Imagine you have a bunch of points from real-world data (like Celsius and Fahrenheit temperatures). Those points might line up perfectly or they might be a little scattered. A regression model draws the best possible line (or curve) through those points to show the overall trend. This helps you create a mathematical model that can predict other values and better understand the relationship between the two variables you are studying. By Default, Desmos tries to create a linear graph. However, if a linear graph does not fit perfectly, we need to choose other appropriate regression models which is a topic that can be learned in another tutorial.
Watch the video “Choosing and Applying a Regression Model in Desmos with JAWS” to add a regression model for our temperature data.
Choosing and Applying a Regression Model in Desmos with JAWS video:
Step 3: Interpreting the Pattern Using JAWS or the Audio Tracing Feature in Desmos
After adding the regression model, you can explore and interpret the pattern that emerges. With JAWS, Desmos can read out the points and graph details to you. You can also use Desmos’ audio tracing feature, which plays sounds that change as you move along the graph, helping you hear the shape and slope of the line. A steady rise in pitch, for example, would indicate a line that is moving upwards consistently, which matches the pattern you would expect when converting Celsius to Fahrenheit. The audio tracing feature in Desmos is available to all users and not only to screen reader users. This interpretation step helps you better understand the nature of the relationship without relying on visual information. Watch the video “Interpreting the Pattern with JAWS and Audio Tracing.”
Interpreting the Patterns with JAWS and Audio Tracing video:
You will notice that a straight-line graph best explains the relationship between the two units of temperature – Celsius and Fahrenheit. In other words, the two units of temperature have a linear relationship.
Step 4: Getting the Equation and Understanding Its Use
Desmos automatically calculates the best values for constants, like the slope and intercept in a linear equation, and displays the equation for the line graph on the screen. The slope tells you how much the second variable (Fahrenheit) changes every time the first variable (Celsius) increases by one unit. The intercept is the value where the line crosses the y-axis — it shows what the second variable would be when the first variable is zero.
In our case, Desmos would display the following equation. To go to the equation with JAWS, simply use TAB on your keyboard until you hear JAWS saying “Equation”.
y = 1.8x + 32
Where y is the temperature in degrees Fahrenheit and x is the temperature in degrees Celsius. Is this formula familiar? You might have learned this formula in a slightly different form as follows:
F = 9/5 * C + 32
This equation is useful because it allows you to predict new values that aren’t already in your table. It helps you understand how one variable changes compared to the other and describe the relationship using math instead of just pictures or words. For example, if you create a model for Celsius and Fahrenheit, the equation lets you easily find the Fahrenheit temperature for any Celsius value, even if you didn’t originally plot that point.
From our lesson, you can see how mathematical modeling bridges real-world data with mathematical understanding. By creating and importing/copying a table of values, you start with actual data points that represent a real-life relationship—in this case, between Celsius and Fahrenheit. Using Desmos to plot these points and apply a regression model helps us establish a graphical and algebraic relationship between these variables – converting real-world data into mathematical language. The process of interpreting the pattern using tools like JAWS or audio tracing deepens your understanding by allowing you to engage with the graph through sound or speech, making the mathematical relationships more tangible. Finally, the regression equation that Desmos provides gives you a way to predict future values and better understand how one variable affects another. This whole process demonstrates how mathematical models are not just abstract concepts, but practical tools that allow us to describe, predict, and analyze real-world phenomena.
Cab Fares.xls – accessible version of the Cab Fares Table
Creating and Analyzing Challenging Data Patterns series:
Additional Resources:
This post was created as part of the TEAM Initiative to support Teachers of Students with Visual Impairments (TSVIs) in teaching foundational technology skills—such as using Excel and other digital tools—through math lessons and video tutorials designed to help students access and succeed in digital math along side their peers.
Written by TSVI Anitha Muthukumaran and content expert, Kanchana Suryakumar. If you would like more information about the TEAM Initiative, contact Leslie Thatcher at [email protected].
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