options: Use coordinates to prove simple geometric theorems algebraically.

Understand and apply theorems about circles.

Explain volume formulas and use them to solve problems.

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

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

Apply trigonometry to general triangles.

Experiment with transformations in the plane.

Understand and apply the Pythagorean Theorem.

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

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

Prove geometric theorems.

Understand congruence in terms of rigid motions.

Apply geometric concepts in modeling situations.

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

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

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

Understand similarity in terms of similarity transformations.

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

Analyze, compare, create, and compose shapes.

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

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

Prove theorems involving similarity.

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

Reason with shapes and their attributes.

Find arc lengths and areas of sectors of circles.

Define trigonometric ratios and solve problems involving right triangles.

Make geometric constructions.
problem_activity: Activity
In this activity, students work together to investigate rotation symmetry and communicate their thinking in a visual display. As you monitor, ask groups if they have found all possible angles of rotation that produce symmetry and to explain how they know these angles apply to all shapes of that type.
Launch
Arrange students in the same groups as the previous class. Provide each group with tools for creating a visual display. Assign each group a different shape from Self Reflection and provide enough copies of the shape for everyone in the group.
Give students 5 minutes of work time followed by 5 minutes to put together their visual display. Explain that they will not have enough time to make the visual display perfect, so the purpose is just to get their rough ideas down in an organized way.
Student Facing
Determine all the angles of rotation that create symmetry for the shape your teacher assigns you. Create a visual display about your shape. Include these parts in your display:
the name of your shape
the definition of your shape
drawings of each rotation that creates symmetry
a description in words of each rotation that creates symmetry, including the center, angle, and direction of rotation
one non-example (a description and drawing of a rotation that does
not
result in symmetry)
Student Response
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Student Facing
Are you ready for more?
Finite figures, like the shapes we have looked at in class, cannot have translation symmetry. But with a pattern that continues on forever, it is possible. Patterns like this one that have translation symmetry in only one direction are called
frieze patterns
.
[IMG]
What are the lines of symmetry for this pattern?
What angles of rotation produce symmetry for this pattern?
What translations produce symmetry for this pattern if we imagine it extending horizontally forever?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of this discussion is to identify what types of shapes have rotation symmetry.
Ask groups to display their visual displays in the classroom for all to see in order of the number of angles of rotation that create symmetry for their shape. They will have to communicate with other groups to accomplish this. If possible, display the reflection symmetry visual displays from the previous class for comparison.
Invite students to do a “gallery walk” in which they leave written feedback on sticky notes for the other groups. Here is guidance for the kind of feedback students should aim to give each other:
Was there anything about the organization of the visual display that made the ideas especially clear? Was there anything about the organization that could be improved?
Was there anything about the way the ideas are explained that made the ideas especially clear? Was there anything about the explanations that could be improved?
Are there any angles of rotation creating symmetry that the group missed?
Are there any angles of rotation that don’t create symmetry for
all
shapes of the given type?
After the gallery walk, ask students to share their observations about rotation symmetry. Consider asking:
“Which shapes had the fewest angles of rotation that created symmetry? Which had the most?” (The isosceles trapezoid does not have rotation symmetry because the only rotations that take it to itself are 0 degrees or 360 degrees. For the circle, any angle of rotation creates symmetry.)
“What do you notice about the order of the shapes in terms of reflection symmetry versus the order of the shapes in terms of rotation symmetry?” (The circle has the most lines of symmetry and the most angles of rotation that create symmetry. It seems to be the most symmetric shape overall. The parallelogram that is not a rhombus has 180 degree rotation symmetry, but no lines of symmetry; but the rhombus has both 180 degree rotation symmetry and 2 lines of symmetry.)
If questions about 0 or 360 degree rotations come up, explain that all shapes, no matter how strange, look the same when rotated by 0 degrees or 360 degrees. If these rotations are the only ones that take the shape to itself, then it is
not
considered to have rotation symmetry.
Speaking: MLR 7 Compare and Connect.
Use this routine when students present their categories and their sequence of rotations, translations, and reflections to get from the original figure to the image. Ask students to consider what is the same and what is different about the sequences of transformations. Draw students' attention to the way that transformations are described precisely (i.e., order, numbers, words). These exchanges strengthen students' mathematical language use and reasoning about transformations.
Design Principle(s): Maximize meta-awareness
Engagement: Develop Effort and Persistence.
Break the class into small group discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention


correct_option_index: 6