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Checkpoint
- Course 2, Unit 6
Geometric Form and Its Function
In each unit, there is a final lesson and Checkpoint that helps students summarize the key ideas in the unit. The final Checkpoint will generally be discussed in class, with the teacher facilitating the summarizing, and students making notes in their Math Toolkits (teachers may just refer to this as "notes") of any points they need to remember, adding illustrative examples as needed. If your student is having difficulty with any investigation in this unit, this Checkpoint and the accompanying answers may help you recall the concepts involved, and give you the big picture of what the entire unit is about. If your student has completed the unit, then a version of this should be in his or her notes or toolkit. Students should also have Technology Tips in their toolkits, which may be useful for this unit.
Possible Responses to Unit Summary Checkpoint
| In this unit, students studied how mechanisms work and how their function is related directly to the form or shape of the mechanism. They also investigated how some patterns of periodic change could be modeled. The bolded words are vocabulary and concepts your student should be familiar with. | |
| a. | What characteristics of a
parallelogram make the shape widely useful as a linkage? A parallelogram is flexible, the opposite sides remain parallel and congruent, but the angles can change. This permits a parallelogram linkage to drive useful objects like wheels, which make complete rotations, or wipers, which turn through a prescribed angle. |
|---|---|
| b. | Two plane shapes are similar
with a scale factor k. How are the lengths of corresponding
segments related? How are measures of corresponding angles related?
How are areas of corresponding regions related? The lengths of the segments of the second shape are k times the lengths of the segments of the first plane shape. Measures of corresponding angles are equal. The area of the second shape is k2 times the area of the first shape. (See Quick Summary - Course 2, Unit 6 for an example.) |
| c. | Define the sine, cosine,
and tangent of the acute angle of a right triangle. How can these
ratios be used to determine lengths that can not be measured directly?
How can these ratios be used to determine angle measures that cannot
be measured directly? If C is an acute angle in a right angled triangle, then sin C = (length of leg opposite angle C)/(length of hypotenuse); cos C = (length of leg adjacent to angle C)/(length of hypotenuse); tan C = (length of leg opposite angle C)/(length of leg adjacent to angle C). (See Quick Summary - Course 2, Unit 6 for specific example.)
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| d. | How does the angular velocity
of one rotating pulley or gear affect the angular velocity of a second
pulley or gear connected to it by a belt, if the radii are r1 and r2 and
the following is true:
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| e. | Although often measured in
degrees, angles can also be measured in radians. Describe what it
means to say an angle has radian measure 2. Describe another use
for radians. One radian is the amount of rotation or angle size that results when a radius of any size sweeps out an arc equal to the radius. Thus, an angle that measures two radians would be as below: 
Since we need 2π radii to wrap around a circle, we know that 2π radians = 360°. One radian is approximately 60°. Students used radians to model time measurements on a sine graph, y = sin x, where y tracks height above the vertical axis as time passes and x changes, so the y values repeat after 2π or approximately 6 time units. An example of this was a graph showing hours of daylight as the months change. If we want the period to be different from 2π, we have to introduce a factor into the equation. For example, y = sin 0.5x would have period 4π, or approximately 12. This would be a more convenient model for the daylight hours problem if we want a change of 1 unit in x to represent 1 month change, but the y values will start at 0 and increase and decrease periodically between a maximum of 1 and a minimum of -1. (If students use y = cos x, they get the same answers for y, but the starting y value is 1 instead of zero.) If we want the y values to better represent the actual number of daylight hours, then we need to introduce an amplitude into the equation. y = 16 sin 0.5x would vary between 16 and -16. |
| f. | Why are variations of trigonometric
rules such as y = a sin bx or y = a cos bx often
used to model periodic change? What does the value of a tell
you about the situation being modeled? What does the value of b tell
you? Periodic motion is a change of position that follows a repetitive pattern. y = a sin bx and y = a cos bx are used to model periodic motion because these functions are periodic. The value of a, the amplitude, is the measure of the maximum displacement of the motion being modeled from its middle position. The value of b represents the angular velocity of a rotating object if x is a measure of time. The value of b gives the period: period = 2π/b (if the unit of measure is radians) or 360°/b. |
If you would like to see specific problems from Course 2, Unit 6, a link is provided to Examples of Tasks from Course 2, Unit 6. If you are interested in following up on the Geometry strand in general, then the Scope and Sequence will help you see where different concepts are introduced. On the Geometry page, you can read an explanation of the main geometry concepts as they are developed in all four courses.
- Tips for Helping a Student
- List of Topics
- Information on the Geometry strand in general
- Using the Math Toolkit