By: Nick Spencer, Elizabeth Englehart, Sam Marcoe, and Grayson Windle

Introduction:

In Dan Meyer’s “Circles and Squares” video found at http://www.101qs.com/2859, we find a square and a circle on a number line ranging from 0 to 40.  We also notice that the sizes of the two geometrical figures change depending on their position on the number line, indicating some sort of relationship between the two shapes and their respective positions.

 

Group Findings:

The video raises several questions, including the obvious “What is this relationship?”.  Our group first began by trying to answer another question, “When are the areas of the two shapes equal?”  In the above screenshot of the video, we see that the circle and the square seem nearly equal in size in this position.  After some investigation, we found that the areas of the two shapes are equal when the distance from where the shapes touch and the end of the number line is 21.2 to the left and 18.8 to the right.  We were able to test this by visually seeing they are nearly the same in size at 20, and then by measuring the side length of the square and the radius of the circle with some guess-and-check methods.

To find the precise answer, however, we found that you could use the quadratic formula, as well as a couple of other ways through investigating.

Classroom Adaptation:

This problem can be used in a variety of grade levels, particularly in a middle-school classroom.  This problem utilizes algebraic, geometric, conceptual and problem-solving skills to solve.  Teachers can aim for their students to approach this problem in a variety of ways, depending on whether they want this to be an algebra-heavy problem or more of a conceptual experience.  This is a great example of a problem for students to practice the quadratic formula, but even better a problem for students to be able to find connections in mathematics, and use those connections to derive explanations for their findings.

CCSS:

CCSS.MATH.CONTENT.7.G.B.4

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.