This lesson focuses on the formula for the perimeter of a circle in terms of its radius, C = 2πr, which the teacher calls Archimedes’ formula. It is the second lesson in a unit focused on the circumference of a circle. The lesson is 45 minutes long.
Lesson outline
| 00:13–01:31 |
(Whole class) The teacher introduces the day’s topic: circumference (perimeter) of circle. |
| 01:31–04:37 |
(Whole class) Students volunteer formulas during a review of perimeter formulas for square, rectangle, and triangle. |
| 04:37–26:44 |
(Whole class) Students volunteer ideas and discuss how to calculate approximations to the circumference of a circle in terms of its diameter or its radius. |
| 5:40–6:03 |
A student comes to the board and circumscribes a square about the circle. |
| 8:34–9:58 |
Another student suggests inscribing a square in the circle. |
| 9:58–10:37 |
Students suggest an octagon and a triangle. |
| 10:37–11:32 |
A student suggests inscribing a hexagon in the circle. |
| 11:32–13:11 |
Students suggest a 12-sided polygon and a 40-sided polygon. |
| 13:11–26:44 |
The teacher discusses Archimedes’ formula. |
| 26:44–37:49 |
Students work individually on problems that require use of Archimedes’ formula. |
| 37:49–44:17 |
Students come to the board to show their work on more problems requiring use of Archimedes’ formula. |
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Mathematical and pedagogical comments
by TIMSS work session team members. [Note: the team discussions and comments are based upon the original translation provided on the CD, and not the new translation available with this resource guide.]
Attention to language
The teacher mentions that "length of circumference" and "perimeter of a circle" mean the same thing (1:00–1:41).
Preserving integrity of the mathematics while maintaining the appropriate level
(14:58–18:47) The teacher deals with an idea at the appropriate developmental level, without providing false or misleading information for future mathematical topics (as opposed to something like, "you always subtract the smaller number from the larger number," which is a non-example). The idea of showing that π is between 3 and 4 by using the perimeter of familiar shapes (square and hexagon) may be new to many U.S. teachers.
Management of discussion
(8:34–10:37) The teacher needs to make choices about which student suggestions to discuss in depth.
Meaning of "derive"
The lesson description on the CD indicates that this lesson focuses on the derivation of the formula for the perimeter of a circle, which it does not accomplish. What it does is to begin a derivation of this formula, which can be completed in a future mathematics class; and, given the empirically derived fact that the circumference of a circle is a constant time the diameter, shows that the constant is between 3 and 4.
Making connections
Finding an approximation of the perimeter of the circle was coded as "making connections". See the researcher’s comment (17:38) in the lesson commentary section on the CD.
Motivating need for a different version of the formula
(39:51) The teacher assigns problems that ask students to find the radius and diameter of a circle with a given circumference.
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Technical details
File name: NB11121452
The lesson graph, subtitles, and transcript have errors. See the new translation. The textbook pages are not available on the CD.
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Further Discussion
Comments on specific segments
| Time |
Comments |
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0:41–1:31
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Language
Mentioning that “perimeter and circumference describe the same thing” connects the meaning of circumference to other lengths. U.S. teachers traditionally use the phrase “distance around the circumference,” which may not help students to envision the “stretching out” of the shape into a linear form.
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3:28–4:20
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Time
The connection made between 2a + 2b and 2(a+b) is important, and the teacher remarks that this will be useful when studying algebra. It is important to make this connection explicit and discuss why it is true. From perimeters and rectangles, one can get the above relationship. From areas and rectangles one gets a more general relationship, t(a+b) = ta + tb.
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5:27
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Approximation
The teacher says that Archimedes figured out the perimeter of a circle formula and asks, "Do you have an idea?" A student suggests finding the perimeter of the inscribed square.
Questions to ask teachers after viewing the video:
- What’s often done in the U.S. to show that the circumference of a circle is a constant times the diameter is to have students measure the circumference and diameter of circular objects. What are the relative strengths and weaknesses of each of these methods? [Measuring gives students a feel for why this is true, but one cannot make a proof out of it, so one needs another method when students should be engaging in mathematical reasoning. Eighth grade is in the middle of these two periods. This lesson leads toward a method that can be carried out in more detail in a few years.]
- One student suggests inscribing a square, but the teacher may have felt this was too hard, although he says that they could find the area of this square. What was it that might have led the teacher to suggest not using this construction?
- A student suggests using a triangle before another student recommends a hexagon. The teacher does not accept the suggestion of a triangle but accepts the suggestion of a hexagon. What does the teacher have to know to make this decision? Another student suggests using an octagon. Would this have been possible based upon what students have learned as shown in the video and in the information accompanying the video?
- Pi can be introduced either through suggesting that the length of the circumference of a circle is a constant times the diameter (see #1) or by computing an approximation to the area of a circle. How does one sketch an argument that suggests that the constant in the formula for the circumference of a circle (constant x diameter) and the constant in the formula for the area (constant x the radius squared) are the same? See the figures below:
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15:35
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Independent and dependent variable
The class connects the formula for the circumference of a circle to the other mathematical ideas: "dependency" and "direct proportion."
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20:20–22:28
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History
There is an interesting reference to a Dutch contribution: Ludolph van Ceulen’s calculation of π. There also are two references to Archimedes (12:33, 15:09, 21:30).
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Other Issues
Management of discussion
Different formats are associated with different kinds of choices the teacher makes. In J3, for example, the teacher can monitor student work, choose students to present their work, and choose the order in which students present their work. In C2, the teacher does not have these options during the whole-class discussion.
Meaning of "derive"
Two missing derivations in this lesson (summarized here) are discussed below.
(00:00–26:44) The teacher draws a circle of diameter D. One student shows that a hexagon with sides of length R (that is D/2) can be inscribed in the circle. Another student shows that a square with sides of length D can be circumscribed about the circle. Thus the perimeter of the circle is shorter than the perimeter of the square but longer than the perimeter of the hexagon:
(1) perimeter of hexagon < circumference of circle < perimeter of square
3D < circumference of circle < 4D
The teacher notes that in grade 7 the students learned that the circumference C of a circle was proportional to its diameter:
(2) If a circle has diameter D and circumference C, then C = kD.
Putting the inequality (1) together with the proportion (2):
3D < kD < 4D.
This shows that k is between 3 and 4.
The teacher says (at 18:47) that k is approximately 22/7 or 3.14, and later (19:49), that k is called π. The formulas that the students use are thus:
C 
3.14D
C (22/7)D
Gap 1: We don’t know how the students learned the formula (2). They may have done so by measuring circumferences and diameters of different circles, and assuming that the pattern that they saw in their measurements held for all circles. This is appropriate for this grade level, but is not a derivation of the formula (2).
Gap 2: The teacher says that they do not know the value of k (17:22), but that it is approximately 22/7 or 3.14 (18:47). This is correct, but it is not derived in the lesson, although the teacher has sketched ideas (11:44–12:35) that could be used in a derivation.
In the lesson, π is said to be the constant k that is the ratio of the circumference of a circle to its diameter. This implies a definition of π: that it is a name for the constant k in C = kD.
The teacher mentions that π is irrational (22:38–23:51), i.e., it cannot be represented as a fraction that is the quotient of two integers. (For a "(relatively) simple proof that π is irrational," requiring only knowledge of calculus, see Niven, 1947.)
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Relationship with previous learning
Students know that the perimeter of a polygon is the sum of the lengths of the sides of the polygon, and this is used for a circumscribed square and an inscribed hexagon to get upper and lower bounds on the value of pi. They do not know which regular polygons would have sides whose lengths could be found by them since they suggest polygons which have sides which would have been hard for them to compute given their current knowledge. This knowledge might have included the Pythagorean theorem but not trigonometry. The teacher has to make a decision about which of the suggested figures are appropriate and which should not be followed up on, so he needs to know what would be easy enough and what would be too hard.
The teacher remarks that in seventh grade the students learned that the circumference of a circle is proportional to the diameter of a circle (15:58). This lesson builds on this knowledge to start seeing how to mathematically approximate the number π that occurs in this proportion.
In the discussion of values to use for π, the teacher (31:13) tells the class: "To indicate that the result is an approximation, we should place a dot above the equal sign. The equal sign is only used when the two sides are exactly equal to each other." Nothing is said about whether this notation is new or not, but the use of the equal sign as restricted to when two things are exactly equal to each other seems to have been taught earlier.
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Relationship with future learning
Proportions show up in many mathematics problems. Similarity of triangles is another example which will be studied in school mathematics. After the students learn the Pythagorean theorem and some basic trigonometry, it will be possible to derive formulas for the perimeter of regular polygons, and so obtain better estimates for π, and show mathematically that there is a constant k which is the proportionality constant for C/D where C is the circumference of a circle and D is its diameter. Similar arguments using inscribed and circumscribed regular polygons can be used to show that A/r² is a constant, where A is the area of a circle and r is the radius of this circle. In calculus, both areas and arc lengths are studied. Areas can be approximated for simple figures, usually using rectangles, and arc length can be found by approximating a curve using line segments connecting nearby points.
The identity 2a + 2b = 2(a + b) will be generalized by replacing 2 by an arbitrary real number, to get what is called the distributive property. At 4:07 the teacher says: "Please take a look at how it's written. It's a neat algebraic trick to know. Very soon, we will start with algebra, and you will learn all about it. You will learn how to simplify algebraic expressions. For example, you will learn how to distribute the two, which will get rid of the parentheses. You will also learn how to work in the opposite direction. In words, how to factor the expression so it contains parentheses. Everything is useful. What you have learned today, you will need tomorrow."
Starting at about 18 minutes the teacher describes what they have done, asks how the estimates they found for π could be improved, and mentions that the method they finally come up with of taking the inscribed hexagon and doubling the number of sides a few times would do what Archimedes did; in his case, going to a regular 96-gon to get a lower estimate of 3 1/7 or 22/7. The other approximation the students are to use for numerical work is 3.14. Both of these are just approximations. At 23:51, the teacher describes how to get π on a line by marking a point on a wheel and roll the wheel until the marked point has gone around once. He then talks about measuring the distance with a ruler whose length is that of the radius of the wheel, and claims this cannot be done exactly.
In discussion about the decimal expansion of π, the teacher claims it does not repeat, so π cannot be expressed as a fraction (23:38). In his commentary, the teacher writes that he tried to give students a hint that the number π is irrational. In ninth grade, when irrational numbers will be discussed directly, the teacher says that the letter π will be used.
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Historical notes
The lesson contains two references to Archimedes (12:33, 15:09). Archimedes's work in "Measurement of a Circle" is accessible once high school geometry has been learned. It contains approximations to pi from above, 3 1/7, and from below, 3 1/71, which are obtained by circumscribing and inscribing regular polygons. 3 1/71 comes from first inscribing a regular hexagon, as in this lesson, and then successively doubling the number of sides up to 96. 3 1/7 comes from circumscribing a regular hexagon rather than a square as in this lesson, and then doubling the number of sides up to 96. The Pythagorean theorem is needed to get started in this case, which is likely why the teacher is willing to start with a circumscribed square, but not an inscribed square, which also would have used a special case of the Pythagorean theorem.
A very readable account about Archimedes is provided by Stein (1999). A new translation of this work is available from Jones (2005). There also is "The Story of Pi", from Project Mathematics!, which includes a full set of videos and workbooks.
Calculations of approximations of π have been done in many cultures and continue at the present time; some calculations used are summarized by Joseph (2000). There is a New Yorker article by Preston (1992) dealing with calculations of π by David and Gregory Chudnovsky. There are many articles posted on the web dealing with π, including one by O'Connor and Robertson. They wrote that Ludolph van Ceule’s calculations involved polygons with about 262 sides and took most of his life to get to 35 digits. More effective methods have been developed, and now more than 1 trillion digits have been computed. The MacTutor History of Mathematics Archive has a chronological table of more current approximations and a history of π.
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Lesson graph errors
Segment 1: "plane of a line" should be "relationships of possible alignments of a line and a circle."
Segment 2:
Section I: The first figure shows only a square, but it should show the square inscribed in a circle. (John circumscribes a square about the circle that the teacher has previously drawn.) The last sentence should be, "A student suggests inscribing a square within in the circle" (see 8:34).
Section II: The second figure should show a hexagon inscribed in a circle, which is inscribed in a square.
Section III: "4th diameter and 3rd diameter" should be "4D and 3D."
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References
Jones, A. (2005). The works of Archimedes: Translation and commentary. Vol. 1: The two books on the sphere and the cylinder. Notices of the American Mathematical Society, 52(5), 520–525. This is a review of a new translation, which also gives an account of recent findings about Archimedes. This is a review of a new translation, which also gives an account of recent findings about Archimedes.
Joseph, G. (2000). The crest of the peacock: Non-european roots of mathematics (revised edition). Princeton: Princeton University Press.
Niven, I. (1947). A simple proof that π is irrational. Bulletin of the American Mathematical Society, 53, 509.
O'Connor, J. and E. Robertson, Article on Ludolph van Ceulen.
Preston, R., The Mountains of Pi, The New Yorker, March 2, 1992, 36-67.
Project Mathematics! The Story of Pi.
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