Common Core Standard: 7.G.B.4
I have to confess that while I have done a lot of conceptual work with students around circumference and area of a circle (see some of it by checking out my blogposts on each subject), I have not done much to tackle the latter part of this standard, 7.G.B.4, “give an informal derivation of the relationship between the circumference and area of a circle.” I have done those problems where you have to find the area given the circumference and not the radius, but I believe that is not is meant here.
In fact, I have truly neither considered the relationship between the circumference and the area of a circle nor do I know of any teachers who have talked about doing the same. They may have; they just never told me about it. This blogpost represents my first shot at understanding it. Like relating the area of a rectangle/ parallelogram to the area of a circle, I challenged myself to think of it visually each step of the way during successive manipulations to the formula. I have to admit at the onset that I have only been partially successful at this point. Some relationships I could understand, but I had a difficult time expressing them visually. Others I could not visualize at all after a first pass. I’ll have to presevere in my problem solving.
In part one, I am able to provide some Sticky Math Triads to put the process before students and teachers. I am also able to hopefully shed some light on the resulting relationship through the use of visuals in part two. So, in a break from tradition, I will actually start with the symbolic, algebraic formulas and finish with the visuals. There is always an exception to the rule isn’t there? What follows could be a single lesson; although, you might be better off focusing on the manipulation of formulas for one lesson and understanding the relationship in the following lesson both have valuable connections to other concepts you could explore in the process.
WHERE DO WE BEGIN?
We begin by looking at what we know about circumference & area. Students could brainstorm this before you unveil the Sticky Math Pair below, which incidentally might be my warm up for this lesson. The ultimate goal is for students to realize both formulas share pi and r, the radius, in common. We will focus on the radius as pi is a constant. The first method employs substitution while the second method employs the Transitive Property.
This Sticky Math Pair might seem overly simplistic and not worthy of the time; however, it provides valuable repetition, a chance to utilize vocabulary, and formative assessment as well as a chance to explore some other foundational concepts. Similarities (with purposeful questions to pose) could include:
- Both stickies have a pi (Why?)
- Both stickies have a r (How might I use this fact?)
- Both are equations and have an equal sign (Why?)
- Both have capital and lower case variables (Why?)
- both utilize the digit two (how is it used differently? Why?)
Differences (with purposeful questions to pose) could include:
- the left sticky defines circumference in terms of r while the right sticky defines area in terms of r
- the left sticky, describes the relationship between circumference and radius while the right sticky, describes the relationship between area and the radius
- the radius is to the first power in the circumference formula while it is to the second power in the area formula (Why? How does the power of r relate to circumference? How does the power of r relate to area?)
- the digit two is used differently in each case (What is the effect of the two in the case of circumference? Why is it there? What is the effect of the two in the case of area? Why is it there?)
The bolded bullets above are the primary goal of the comparison; however, there are some other fundamental concepts present. Besides being a formative assessment of understanding each formula and its related vocabulary, there is a chance to connect linear and quadratics to their meanings in context. The term 2πr is linear, which is a chance to connect the “invisible,” understood exponent of one to the fact that measuring around a circle is done in a single, linear dimension; whereas, the term πr^2 is quadratic, which is a chance to connect the exponent of two to the fact that measuring the area of a circle is done in a square units, area has elements of length and width to it.
DEFINING RADIUS IN TERMS OF CIRCUMFERENCE
In order to examine the relationship between circumference and area, we need them both in the same equation. We can solve for r in one equation and substitute that value in to the other equation. In this case, we will solve for r in the circumference equation, which avoids solving a simple quadratic in the area equation – a task for higher grade levels:
Connections could include:
- it is the same formula just in different forms or they each have the same components
- in a previous lesson, students learned that pi diameters make up the circumference of a circle. They may apply that here and see the 2r as being a diameter even though they are “split” by the pi.
- the circumference is made up of six radii plus a little bit more (Where is that found in the equations?)
- two radii, pi times make up the circumference of the circle
- the middle sticky defines circumference, while the right sticky defines the radius
- in all cases, there seems to be a relationship of two pi between the radius and the circumference
The key here is to see that if you divide the circumference by a little more than 6 or two pi; then you get the radius. We will need the definition of the radius in the next step.
INTRODUCING CIRCUMFERENCE BY SUBSTITUTING FOR THE RADIUS
The point of the following Sticky Math Triad is to see the substitution of the value of r into the area formula:
Connections could include:
- any of the connections found in previous Sticky Math activities
- the equivalent value for r from the middle sticky has been substituted for the value of r in the left sticky to create the right sticky.
- C/(2π) is found in the middle and right stickies
- The left and middle stickies have pi listed only once, while the right sticky has it twice (Why?)
- r is not found in the right sticky (Why?)
- two of the stickies contain a square (Why?)
- the middle sticky does not contain an A (Why?)
- the left sticky does not contain a C (Why?)
- the left and right stickies have the same structure [SMP 2, 7 and/or 8]
- the left sticky uses the 2 as an exponent, the middle sticky uses it as a constant, and the right sticky does both (Why?)
Once again, the key here is to see the substitution of the value of r into the area formula. Notice the fact that the same variables are included in two of three formulas is a clue to that happening. Another clue that students may not pick up on is in the final bullet above, which could be shared as, “someone in my other period noted that…” (you don’t have to tell them it was you.). The presence of pi twice in the right sticky is yet another clue.
Here is a quick note about a powerful three letter word, why. Students might make & state a connection; however, that is just the beginning of the journey. Part of the purpose of Sticky Math is to promote student discourse. Ask all students to engage with one another in pairs or small groups around one student’s connection by posing a purposeful question. Anticipating these types of advancing questions (Smith & Stein, 2011) is best; however, an easy three letter fallback is “Why?” Repeating it often would not be as powerful as mixing it up with some other well anticipated questions; yet, it is sometimes preferred and can help build a culture and expectation of understanding in the classroom. Incidentally, by stopping the class to consider one student’s connection, you are promoting mathematical agency, authority & identity in that student and in others as you ask them to consider your purposeful questioning rather than simply illuminating the connection yourself via direct instruction.
SQUARING THE RADIUS
The point of the following Sticky Math Triad is to see the two new, equivalent representations of r squared that lead to simplifying the expression. Once again, this is a break from the usual multiple representation of a single concept in different ways; however, it remains a good comparison to realize how r squared can be represented in different ways:
Connections could include:
- any of the connections found in previous Sticky Math activities
- the square on the outside of the parentheses has moved to each term on the inside of the parentheses [this connection helps students build procedural fluency explicitly]
- the first factor of pi has “vanished” in the right sticky (Why? How?)
- the square has “vanished” from the pi in the denominator on the right sticky (Why? How?)
- 2 squared has been simplified to 4
- students may notice a change in the number of pi’s present between stickies (How many pi’s are present in each sticky? Why?)
- all stickies define area and include the variable A
- the expression on the right side of the equation becomes more simplified as you view it from left to right
We often jump right into simplification of expressions. By providing the mathematics here, we can examine what is happening to increase procedural proficiency before asking students to engage in the process themselves. It also provides a chance to examine the reasoning behind what is being done so students can learn when and why to do such things rather than simply how.
CLEARING THE DENOMINATOR
A Sticky Math Quad is a rare sighting here on the website when it is not a part of a comparison between two Pairs. One of the connections we hope students might make might help you understand it as we move forward here. It is organized clockwise from the top left sticky:
Connections could include:
- any of the connections found in previous Sticky Math activities
- it seems to be organized clockwise from the top left sticky (You may choose to give this fact away or better yet start with a purposeful question of, “How do you think they are organized?”)
- 4π is being introduced to both sides via multiplication (What property allows us to do this? Why is it being done?)
- the two denominators seemed to have been switched or transposed in the two right stickies (What property allows us to do this? Why is it being done?)
- the denominator of one “disappears” between the two bottom stickies (What property allows us to do this? Why is it being done?)
- the fraction 4π/4π “disappears” between the two bottom stickies (What is happening here mathematically? What property allows us to do this? Why is it being done?)
- the 4π has “moved” between the two left stickies
We have arrived at our goal of defining the relationship between the circumference and area of a circle, but I doubt many people, especially our students, might realize it. The relationship is not as simplistic as students are used to finding, and it is certainly not easily understood by them. While this has been a valuable exploration in developing procedural fluency in manipulating formulas, I suggest seeing it through to fully understanding the relationship exposed by this new equation, which is found in part two.
Common Core Standards: 7.G.B.4
What connections are your students making? What modifications are you making to use this with students? We would love to hear your feedback; please submit a comment below or consider submitting your own Sticky Math activity here.