Gold Seal Lesson:
Going Around In Circles

Copernicus Education Gateway

Subject:

Mathematics

Grade:

9-12

ICLE Standards:

Mathematics as Communication: Reflect on and clarify thinking about mathematical ideas and situations.

Mathematical Connections: Explore problems and describe results using graphic, numerical, physical, algebraic, and verbal mathematical models or representations.

Patterns and Functions: Describe and represent relationships with tables, graphs, and rules.

Geometry: Use circle concepts and measurement to understand the relationship between direct variation and slope.

Performance Task:

Note to the teacher: For this activity, it is best that students work in groups of 2 or 3.
Your task is to investigate the relationship between the diameter and circumference of a circle. Select 10 circular shapes of different sizes. Some samples of circular shapes are dowel rods, tin cans, round clocks, etc. Using metric rulers and tape measures, measure, to the nearest millimeter, the diameter and the circumference of each circular shape you choose. Record your measures in the first two columns of the table below.


Using a calculator, compute the value of C/d for each circle measured. Record your answer in the third column of the table. In your math journal, explain why the relationship between the diameter and circumference appears to be an example of direct variation. State the approximate constant of variation and write an equation to model the relationship between circumference and diameter using C, d, and your constant of variation.

On a piece of graph paper, graph the function showing the relationship between the diameters and the circumferences for the 10 circles. Use the x-axis for diameters and the y-axis for circumferences. Draw the line of best fit for the points you plot. Choose two points from your fitted line whose values you can easily read and find the slope of your fitted line using these two points. Repeat this for two other pairs of points on your line. In your math journal, compare this value, which is the slope of your line, with the constant of variation found earlier in the activity. Write a summary of your findings in your journal.

Knowledge / Skills:

Understand the properties of circles (e.g., radius, arc, diameter, chord, secant, tangent, etc.).  (m10)

Know the components and properties of the rectangular coordinate system, (i.e., x - y axis, origin, quadrants, abscissa (x-coordinate) and ordinate (y-coordinate), and the general representation of a point (x,y)).  (m23)

Know how to measure circle quantities (e.g., area, angle formed by two secants, circumference, length of segments, etc.).  (m30)

Use the technique of dimensional analysis to convert units of measure (e.g., convert km/hr to m/min).  (m33)

Understand the concepts and uses of matrices in modeling (i.e., finite graphs (structures) can be represented geometrically and interpreted algebraically in the form of a matrix).  (m51)

Understand the concepts and uses of matrices in modeling (i.e., finite graphs (structures) can be represented geometrically and interpreted algebraically in the form of a matrix).  (ela4)

Rubric:

4 Points =  The students complete the entire task without any assistance from others. The students identify 10 circular shapes and correctly measure their diameters and circumferences. Their work and journal entries show a thorough understanding of direct variation and how the ratio of the circumference and diameter of a circle is an example of direct variation. The students show a full understanding of the connection between direct variation and the graphic model of the data gathered in the activity.

3 Points  =  The students need some assistance to complete the task. They identify 10 circular shapes, but are a bit inaccurate with their measurements, resulting in difficulty identifying the relationship as a direct variation. The students' work and journal entries show a fairly good understanding of how the activity relates to direct variation. The students' graph is not totally accurate and they demonstrate only a partial understanding of the connection between direct variation and the graphic model of the data gathered in the activity.

2 Points  =  The students need much assistance to complete the task. They identify fewer than the required 10 circular shapes and have difficulty measuring the diameters and circumferences of these. As a result, they have much difficulty seeing any of the relationships in the activity. Their graphs are inaccurate and do not lead to any reasonable conclusions. The students appear to have little understanding of the concepts involved in the activity.

1 Point  =  Even with assistance from others, the students are unable to complete the task. They only identify a few circular figures and have great difficulty measuring the circumference and diameters of them. the students' work and journal entries indicate little, if any, understanding of direct variation and its relation to the circle activity. the students' graphs are meaningless. It is evident they do not understand slope and, therefore, are unable to make any connections in the activity.

Keywords:

ALGEBRA
COORDINATES
DIRECT VARIATION
SLOPE GEOMETRY
CIRCLES
CIRCUMFERENCE 
DATA COLLECTION
GRAPHS
MEASUREMENT

Grades:

Kg [] - 1 [] - 2 [] - 3 [] - 4 [] - 5 [] - 6 [] - 7 [] - 8 [] - 9 [X] - 10 [X] - 11 [X] - 12 [X]

ICLE Application:

D

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