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

Explain volume formulas and use them to solve problems.

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

Use coordinates to prove simple geometric theorems algebraically.

Experiment with transformations in the plane.

Make geometric constructions.

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

Understand and apply theorems about circles.

Analyze, compare, create, and compose shapes.

Define trigonometric ratios and solve problems involving right triangles.

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

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

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

Understand and apply the Pythagorean Theorem.

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

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

Find arc lengths and areas of sectors of circles.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Reason with shapes and their attributes.

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

Understand similarity in terms of similarity transformations.

Prove geometric theorems.

Understand congruence in terms of rigid motions.

Apply trigonometry to general triangles.

Apply geometric concepts in modeling situations.

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

Prove theorems involving similarity.
problem_activity: Activity
Students observe that the angle formed by connecting endpoints of a diameter to a third point on the circle appears to be a right angle. They confirm that this is true for a few particular points, then write a conjecture. This conjecture will be generalized in an upcoming unit.
Monitor for different ways that students word the conjecture to highlight during the discussion. Listen for terms such as
diameter
,
endpoints
,
right triangle
, and
right angle
. The term
chord
will be defined in a subsequent unit. It is not necessary to define it here.
This activity works best when each student has access to devices that can use the embedded applet because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, projecting the applet would be helpful during the synthesis.
Digital
Launch
Arrange students in groups of 4. Provide access to internet-enabled devices, ideally one per group. Distribute one index card with each internet-enabled device.
Conversing: MLR2 Collect and Display.
As students work on this activity, listen for and collect the language students use to justify why the angle formed by segments
\(BD\)
and
\(CD\)
is a right angle. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a phrase such as, “The lines make 90 degrees because they have opposite slopes” can be improved by restating it as “Segments
\(BD\)
and
\(CD\)
are perpendicular because their slopes are opposite reciprocals.” This will help students use the mathematical language necessary to precisely justify why the measure of the angle formed by segments
\(BD\)
and
\(CD\)
is 90 degrees.
Design Principle(s): Optimize output (for justification); Maximize meta-awareness
Student Facing
Use the applet to answer the questions.
First, observe and describe the image:
What kind of line is
\(BC\)
in reference to the circle?
What does the measurement of angle
\(BDC\)
appear to be?
Move point
\(D\)
around. How does the measurement of angle
\(BDC\)
appear to change? Use a corner of the index card to compare the size of the angle as it moves.
Move point
\(D\)
to a location with integer coordinates. Each student in the group should choose a different spot for
\(D\)
. Calculate the slopes of segments
\(BC\)
and
\(BD\)
. What do your results tell you?
Now select the button labeled “slopes.” This will show the slopes of
\(BD\)
and
\(BC\)
and their product. Move point
\(D\)
around. What do you notice about the product of the slopes?
Based on the results, write a conjecture that captures what you are seeing.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 4.
Conversing: MLR2 Collect and Display.
As students work on this activity, listen for and collect the language students use to justify why the angle formed by segments
\(BD\)
and
\(CD\)
is a right angle. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a phrase such as, “The lines make 90 degrees because they have opposite slopes” can be improved by restating it as “Segments
\(BD\)
and
\(CD\)
are perpendicular because their slopes are opposite reciprocals.” This will help students use the mathematical language necessary to precisely justify why the measure of the angle formed by segments
\(BD\)
and
\(CD\)
is 90 degrees.
Design Principle(s): Optimize output (for justification); Maximize meta-awareness
Student Facing
The image shows a circle with several points plotted on the circle.
[IMG]
What kind of segment is
\(BC\)
in reference to the circle?
Choose one of the plotted points on the circle and call it
\(D\)
. Each student in the group should choose a different point. Draw segments
\(BD\)
and
\(DC\)
. What does the measure of angle
\(BDC\)
appear to be?
Calculate the slopes of segments
\(BD\)
and
\(DC\)
. What do your results tell you?
Compare your results to those of others in your group. What did they find?
Based on your group’s results, write a conjecture that captures what you are seeing.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Ask previously identified groups to share their conjectures. Challenge students to decide whether all the conjectures are saying the same thing, and if any of the conjectures need to be refined. Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
There are many ways to correctly word the conjecture. As students work to refine the wording, be sure that the word
diameter
is included, that it’s clear that 2 points are at the endpoints of the diameter, and that the third point is clearly somewhere else on the circle. Students may describe the result as 2 segments that form a right angle or 3 segments that form a right triangle.
Once the conjecture is finalized, ask students if they have proven it (they have not; they’ve just shown it’s true for a few particular cases). Tell students that we will look at a more general case of this assertion in an upcoming unit.
Action and Expression: Develop Expression and Communication.
Invite students to talk about their ideas with a partner. Display sentence frames, such as: “How did you develop your conjecture?” and “I chose this conjecture because. . . .” to support students when they explain their ideas.
Supports accessibility for: Language; Organization


correct_option_index: 3

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