options: Prove geometric theorems.

Solve real-world and mathematical problems involving area, surface area, and volume.

Classify two-dimensional figures into categories based on their properties.

Define trigonometric ratios and solve problems involving right triangles.

Reason with shapes and their attributes.

Apply geometric concepts in modeling situations.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Make geometric constructions.

Understand similarity in terms of similarity transformations.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Understand and apply the Pythagorean Theorem.

Find arc lengths and areas of sectors of circles.

Apply trigonometry to general triangles.

Translate between the geometric description and the equation for a conic section.

Use coordinates to prove simple geometric theorems algebraically.

Understand and apply theorems about circles.

Understand congruence in terms of rigid motions.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Prove theorems involving similarity.

Analyze, compare, create, and compose shapes.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Explain volume formulas and use them to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Experiment with transformations in the plane.

Draw, construct, and describe geometrical figures and describe the relationships between them.
problem_activity: Activity
In this task, students use the constant of proportionality they estimated in the previous task to calculate circumferences of circles given the diameter and vice versa. The purpose is to reinforce the proportional relationship between circumference
\(c\)
and diameter
\(d\)
and use the formula
\(c=kd\)
, where
\(k\)
is about 3 or 3.1 (the value agreed upon in the previous task).
If there is not sufficient time to allow students to work through all of the calculations, consider dividing up the work, assigning one circle to each student or group of students. It is important to save enough time for the discussion about the meaning and value of
\(\pi\)
at the end of this activity.
Before students begin the task, a digital applet found at
http://ggbm.at/H8UuD96V
can be used to reinforce the constant of proportionality.
Launch
Ask students to label these measurements given in the table on the picture of the circles. Instruct students to use the constant of proportionality the class estimated in the previous activity. Give students quiet work time followed by whole-class discussion.
Student Facing
Here are five circles. One measurement for each circle is given in the table.
[IMG]
Use the constant of proportionality estimated in the previous activity to complete the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "diameter (cm)"}, {"_attributes": {"scope": "col"}, "_value": "circumference (cm)"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "circle A"}], "td": [{"_value": "3"}, {}]}, {"th": [{"_value": "circle B"}], "td": [{"_value": "10"}, {}]}, {"th": [{"_value": "circle C"}], "td": [{}, {"_value": "24"}]}, {"th": [{"_value": "circle D"}], "td": [{}, {"_value": "18"}]}, {"th": [{"_value": "circle E"}], "td": [{"_value": "1"}, {}]}]}]}]}
Student Response
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Student Facing
Are you ready for more?
The circumference of Earth is approximately 40,000 km. If you made a circle of wire around the globe, that is only 10 meters (0.01 km) longer than the circumference of the globe, could a flea, a mouse, or even a person creep under it?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Some students may multiply the circumference by the constant of proportionality instead of dividing by it. Prompt them to consider whether the diameter can be longer than the circumference of a circle.
Some students may struggle to divide by 3.1 if that is the constant of proportionality decided on in the previous lesson. Ask these students if they could use an easier number as their constant, and allow them to divide by 3 instead. Then ask them how their answer would have changed if they divided by 3.1.
Activity Synthesis
Display the table from the task statement for all to see. Ask students to share their answers and write these in the table, resolving any discrepancies.
Ask students if they noticed any connections between the tables and the graphs from the previous activity. Students will have used the equation
\(C=kd\)
, possibly without realizing it, to fill in the missing information so it is important for them to make this connection. To reinforce this connection, add the points from the table they just completed to the graph from the previous activity. Point out that the constant of proportionality appears in the table as the unit rate in the row with diameter 1 cm.
Ask if students have heard of the number
pi
or seen the symbol
\(\pi\)
. Define
\(\pi\)
as the exact value of the constant of proportionality for this relationship. Explain that the exact value of
\(\pi\)
is a decimal with infinitely many digits and no repeating pattern, so an approximation is often used. Frequently used approximations for
\(\pi\)
include
\(\frac{22} 7\)
, 3.14, and 3.14159, but none of these are exactly equal to
\(\pi\)
.


correct_option_index: 10