This unit was designed to be a project-based inquiry combining mathematics, science, and hand's-on experience. One of my motivations as a mathematics teacher is to help students understand that, while modern mathematics tends to be taught within the realm of the ideal and the abstract, it evolved historically out of people looking to solve practical problems, such as how to measure long distances using a wheel.
Additionally, I wanted to include some foundation concepts of limits to prepare students for calculus. Rather than presenting these as solely calculus concepts, it is my hope that exposing students to limits will empower them, so that they are undaunted later in their academic careers.
The purpose of this lesson plan is to explore the nature of the value of π and how wheels can be used to measure distance. This unit is intended for high school sophomores in an advanced geometry class or for high school juniors and seniors in a pre-calculus class. It is designed to allow students to use items in their own context and community; the final “hands-on” artifact involves bicycles.
It is assumed that, as members of an urban community, some portion of the students will have at least some experience with bikes. Another option would be to use the wheels used to demarcate baseball diamonds, another possible presence of wheels in these students’ lives.
Prior knowledge of wheels would naturally be car wheels; while these move too quickly to be something that students can actively use to measure, they can be an important part of the benchmark discussions. Specifically, the introductory benchmark lesson can discuss how both speedometer and odometer readings in a car can be tied to the number of wheel rotations, on the assumption that the wheel has a particular diameter.
Class-based prior knowledge would include previous experience in geometry, including basic trigonometry, and algebra, as well as of measurement and data collection methods.
For those activities that require working directly with Excel data, GeoGebra and Mathematica, the ideal is that students (each or in groups) will have access to computers that they can work on. Barring that, those portions can be done as demonstrations.
Class 1: Benchmark – Introduction to Pi in its Historical Context
Class 2: Benchmark – Exploring the Relationship between Diameter and Circumference
Class 3: Benchmark – Estimating Pi using Inscribed and Circumscribed Polygons (Limits)
Class 4: Benchmark – Estimating Pi using Circumscribed Rectangles (Leftmost Riemann Sums)
Class 5: Workday – Group exploration: Using bicycles to estimate distances
Class 6: Workday – Groups create summative exercise
L1.1.6 Explain the importance of the irrational numbers √2 and √3 in basic right triangle trigonometry, and the importance of π because of its role in circle relationships.
L1.2.4 Organize and summarize a data set in a table, plot, chart, or spreadsheet; find patterns in a display of data; understand and critique data displays in the media.
L2.3.1 Convert units of measurement within and between systems; explain how arithmetic operations on measurements affect units, and carry units through calculations correctly.
L2.4.2 Describe and explain round-off error, rounding, and truncating.
L1.2.5 Read and interpret representations from various technological sources, such as contour or isobar diagrams.
G1.6.1 Solve multistep problems involving circumference and area of circles.
P1.1B Evaluate the uncertainties or validity of scientific conclusions using an understanding of sources of measurement error, the challenges of controlling variables, accuracy of data analysis, logic of argument, logic of experimental design, and/or the dependence on underlying assumptions.
P1.1C Conduct scientific investigations using appropriate tools and techniques (e.g., selecting an instrument that measures the desired quantity—length, volume, weight, time interval, temperature—with the appropriate level of precision).
P1.1D Identify patterns in data and relate them to theoretical models.
P1.1E Describe a reason for a given conclusion using evidence from an investigation.
P2.1E Describe and classify various motions in a plane as one dimensional, two dimensional, circular, or periodic.
Class 1 rationale and structure
Students in most modern mathematics classrooms tend to be presented with the topic as primarily an abstract field, but all mathematics of the sort encountered in high school began in order to answer real world problems. In the case of the geometry of circles, a primary driving force behind the mathematics involved a need to measure distances using circular wheels. The purpose of this unit is to come to an understanding of the importance of being able to measure a wheel (and hence of π, roughly 22/7 or 3.14159) in order to then use a count of rotations of a wheel to measure distances, thread lengths, and other measurements.
Before beginning this discussion, students will complete a “current knowledge” exercise asking them to describe what they know about the geometry of circles, in particular, diameters and how to measure them, the nature of the circumference, and the significance of π. It should be stressed to them that this is not a graded assignment, but rather that the instructor is seeking to determine each student’s current understanding.
Next, there will be an exploration between the instructor and the class concerning how wheels might be used for measurement, and what would need to be known. After students have been given the opportunity to present their own theories and previous knowledge, the instructor should fill in the historical information as appropriate to form a more complete picture.
As an exit ticket, students are asked to summarize what they learned, if anything, beyond their initial knowledge. The initial and final summaries are then turned in, and are used formatively.
Class 2 rationale and structure
Now that the students have a common base of knowledge and the instructor has an understanding of the breadth of knowledge among the students, the students will measure a variety of circular objects. Groups (of 4-5 students) should be constructed based on student strengths and weaknesses, so that groups are balanced.
Each group will measure the same set of circular objects and gather data. They will measure the diameters and circumferences of each object, and explore the relationship between the two. This is meant to be entirely constructive: The instructor should not offer the formula, or guide student thinking in a particular direction. The instructor should, however, monitor student measurements to make sure they understand how to measure, particularly diameters.
Students will complete a worksheet collecting their data, and once all groups have completed their measurements and had some time to reflect on the relationship, the teacher should summarize all data. Invariably, there will be some discrepancies in measurement. Reflect as a group on why there are discrepancies. Small discrepancies can most likely be attributed to differences in reading; larger discrepancies are likely due to someone’s error.
Class 3 rationale and structure
Now that students have been given the opportunity for concrete exploration, students will explore numbers in more abstract fashion. Hopefully students will have come up with a number close to π as part of their hands-on work, and are now curious about how the exact number was arrived at. After all, π is an irrational number; how can mathematicians be so confident about such a strange number?
Students will first complete the worksheet “The perimeter of polygons” alone or in groups. This represents the perimeter of polygons inscribed into a circle. Students should discuss their finding, particularly the significance of the last column as the number of sides increase.
The instructor will discuss this finding by showing an Excel table and graph on the “Inscribed” tab showing the value of the half-perimeter as the number of sides increase, as well as the GeoGebra “Inscribed Polygons” file. The instructor can also discuss the notion of limits.
Students will then complete the worksheet “The perimeter of polygons revisited”, again alone or in groups. This represents the perimeter of circumscribed polygons. This will act as a formative assessment for the period.
Students will discuss how the two worksheets differ, and how it can be that they come to the same final value. Because students will usually round the values in each column, their final column answers are likely to be slightly inaccurate: Have them discuss why this is. Which of the two methods do they think is better, and why? As part of this, the instructor can provide the “Circumscribed” tab of the Excel file and GeoGebra’s “Circumscribed Polygons” file.
At this point, you may wish to assign the reading on π for further reinforcement. It should not be assigned earlier than this because it contains much of the information they’ve constructed here.
Class 4 rationale and structure
This is an advanced topic. The instructor may choose to extend class 3 into two days and skip this entirely. However, if time and situation allows, this topic represents an opportunity to develop π based on area, as well as to introduce another aspect of calculus (along with limits).
This will involve Riemann sums, but I do not advise using this term, at least not in the beginning portion of the class, in order to prevent defensive mathematics anxiety from obstructing learning. This should instead be presented as another approach.
Begin by comparing the area of a square with a width of 1 unit to that of an inscribed quarter circle. How is the area of the square visually comparable to that of the quarter circle? Ask students how we might improve this estimate. Expect answers such as breaking it up into smaller squares, breaking it up into rectangles, using trapezoids or triangles, moving the top bar lower, and so on.
Open the Excel file to the “Area Worksheet” tab. This can be done either as a class or (if the students have access) in groups or alone. Enter various, increasing side counts for the polygons and see how it affects the area estimates. The instructor can either modify this file to only show a single Riemann sum (presumably the leftmost, since that’s where the square would start), or leave it be and discuss why some of the estimates are better than others.
Have students fill in the “Estimating Pi” worksheet and discuss the results, also references the “Area of Rectangles” tab in the Excel worksheet and, if possible, the “Riemann Sums – Circle” Mathematica notebook.
The formative assessment for this exercise should be the completed worksheet, which can be reviewed overnight and returned for the next class.
Class 5 rationale and structure
Now that students have an understanding of the meaning of π and have done some work measuring the circumference of circular objects, they will be gathering more real-world data, this time with the goal of measuring distances.
Instruct students beforehand to bring bicycles. This class can take place in the school parking lot or some other large space. Alternatively, take the students to a park or other outside arena where bicycles are available for rent. The goal at this point is to have at least one bicycle for each group of students (using the same groups as in class 2).
You will also need some ribbons, playing or baseball cards, and/or some chalk. Have students discuss how they might determine how many times a wheel has gone around. Suggestions will hopefully include putting a card in the spokes, tying a ribbon on the spokes, or marking the tire with chalk: You’ll be ready for at least some of these suggestions. Other suggestions might include looking for some sort of salient marker already on the tire, such as the valve or a broken spoke.
Have students measure the radius of the tire, and then have them measure some distance (such as between ends of the parking lot, or between two trees). Each group should measure the same length so that they can explore the different measures. As before, make sure to discuss why there are differences in the estimates; for this experience, the tolerance is likely to be higher than for class 2.
Possible causes of measurement error: Error in measuring the radius, error in counting rotations, failure to follow a straight line, difference in radius caused by putting weight on the bicycle, and so on.
Class 6 rationale and structure
For this class, students will be presenting their measurement of some larger distance. Allow students to choose a route for which they can verify the length through other sources. For instance, groups of students could choose to take a straight route between mile roads, or they could enter their route into Google Maps or a GPS to determine the distance travelled.
Students will compare that distance to the distance they measure through the bicycle method, and give a presentation on how they arrived at this, including appropriate discussion of the relationship between radius and circumference.
This presentation will represent their final assessment for this unit.
This section connects how this unit plan correlates to each of the Wayne State University College of Eduation core competencies.
An Initial Certification Completer…
- Knows the subject area content and best practices in those areas.
- This lesson plan balances solid mathematical theory with real world, hand’s on inquiry. Indeed, the first uses of pi as a constant were tied to the very sort of practical measurement questions explored in this lesson.
- Organizes and implements effective instruction including the integration of content across curriculum areas.
- While the basic lesson is designed for a geometry class, the lesson plans include other areas of mathematics (primarily algebra and calculus) as well as other fields (particularly scientific data collection and historical contexts).
- Utilizes appropriate classroom organization and management techniques to ensure a safe and orderly environment conducive to learning.
- Student groups are designed to encourage role-taking and distributing responsibilities, maximizing expectations of inclusion. At the same time, it is expected that the teacher will be sensitive to severe political clashes within the student population and avoid building groups very likely to cause friction. For the hand’s-on activities, particularly when bicycles are involved, the teacher should be mindful of safety concerns.
- Provides a learning environment that engages students’ creative and critical thinking.
- Bicycles are likely the part of the lives of at least some of the students, either directly or indirectly. This lesson creates a problematic project, that is, a project based on an interesting problem, rather than based solely on theory. Scaffolding should be designed in the direction of encouraging students towards independent thought, rather than as a way to let students avoid deeper levels of thought.
- Demonstrates knowledge of human growth and is committed to all students and their learning
- As mentioned above, groups should be structured to allow for role-taking, and to allow students with differing strengths to each excel to the best of their current ability and desire. Additionally, during the in-class portions, material is provided through visual, kinesthetic, and analytic channels, in order to maximize the accessibility of the information to individual students.
- Exhibits Professional Dispositions: Behaves in an ethical, reflective and professional manner participates regularly in College of Education sponsored learning communities and is committed to all students and their learning.
- At all points in this unit plan, the teacher should be mindful of and adjusting for unexpected problems. For the portions of this unit that I did actually teach (day 3, which I taught in more detail over two days), I reflected between my two periods teaching it and adjusted as needed, as well as reflecting on the overall experience.
- Understands and integrates varying perspectives to enhance students’ awareness, respect and appreciation of diverse populations.
- This is also accomplished through the construction of groups, as well as through presenting the theoretical data in multiple formats.
- Selects appropriately from a variety of assessment strategies to evaluate student learning and uses this information to make informed curriculum decisions.
- In addition to the various worksheets and final assessment, the teacher should monitor student interactions and impromptu feedback for understanding and lapses thereof. Overall, student mastery should be assessed based on all information.
- Utilizes school/district/community resources.
- Ideally, each student (or at least student group) will have a computer made available to them by the school. Additionally, all or some of the cylindrical material might be materials found in the school. Ideally, class 5 should take place in a local park.
- Communicates and interacts with parents/guardians/families to enhance student success.
- Any field trip would involve a parental permission slip.