Think of the circumference of a circle like a piece of string. If we could unroll it, it would look like any other straight line. That's the beauty of understanding circumference. It's not just about a formula – it’s about seeing how we can measure something that looks complicated in a straightforward way. While some concepts, like circumference, are abstract and difficult for students to understand, presenting them in an engaging, hands-on manner can significantly help student understanding. 

Connection to Standards:

While these concepts can be explored across multiple grades, this article focuses on Grade 7, where students begin to explore the formulas for area and circumference of a circle. These concepts align to standards, like the following 7th grade Common Core State Standard (CCSS): 

Why not use VEX IQ to bring context and meaning to common abstract mathematical concepts like circumference and Pi? Giving students opportunities to make connections between formulas, calculations, and real-world applications (such as competition robots) helps not only with understanding, but also with students appreciating these concepts and how they can be used outside of the classroom.

The VEX IQ Wheel Turns Activity provides an opportunity for students to explore circumference, not by memorizing a formula, but by deriving it, and hopefully internalizing it. In this Activity, students measure the distance a VEX IQ wheel travels in one rotation by rolling the wheel along a piece of paper, as shown in this animation using the VEX IQ Parts Ruler. 

They then cut the length of paper, and wrap it around the outside of the wheel, or the wheel’s circumference. This is a great way for students to visualize that a circle’s circumference is really just a curved line, or a distance. After students have measured the circumference, they will calculate it using the formula: Circumference = Pi times diameter. This enables them to compare the results between the formula and the measurement, and to begin to understand how and why a formula can be more accurate than measuring by hand. 

Knowing the distance that a wheel travels in one turn, or its circumference, can be valuable information when coding a robot, particularly one with a custom drivetrain for competition. This real world robotics application is explained well in this Live Session recording.

Take learning further by investigating Pi

Once students have a grasp of the concept of circumference, it opens the door to looking more closely at the formula – naturally asking the question of “What is Pi?” Students may know that Pi is infinite, or remember the numerical value to plug into a calculation (3.14), but helping them to understand the meaning of Pi is powerful learning. 

To investigate Pi, students can measure both the circumference and diameter of other round VEX IQ parts, like the Wheel Hub or 2x2 Center Offset Round Lock Beam. Then, use their measurements as coordinates (diameter, circumference), and graph them. You can use a graphing program like GeoGebra, as shown in the image here, or have students graph and calculate slope by hand.

Here is where the magic happens! Have students find the slope of the line created in the graph. The slope of the line is close to 3.14 – or Pi! Why is that? 

This is because slope is calculated as ‘Rise over Run’ or change in Y divided by the change in X. In this case, the change in Y is the change in the circumference and the change in X is the change in diameter. Pi represents the relationship (or ratio) between a circle’s outer-distance (circumference), and its across-distance (diameter). 

When students start graphing, they’ll begin to see Pi as the constant that is the same for all circles – big or small. This activity lets students figure out the formula for circumference on their own. Most will see that: Pi = circumference / diameter. If they rearrange it, they get the formula for circumference.

Educators can aim to make mathematical concepts like circumference and Pi tangible through hands-on activities, such as the Wheel Turns Activity. When students see how they can measure a more complicated shape by building on their prior knowledge of measuring linear distances, they start to see that math is not just a list of formulas that have no connection, but mathematics is a beautifully connected series of patterns. By viewing math in this way, students can predict a robot's movement based on its wheel's circumference. This knowledge is valuable everywhere from the classroom to competitive robotics, allowing for precise movement and navigation. 

How do you plan on using this activity or an adapted version with your students? Please share in the PD+ Community, or schedule a 1-on-1 Session with a VEX Expert to discuss these concepts further.