few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Concept/skill: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Concept/skill: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Answer: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

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im_task_000870_neg_0

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im_task_001160_pos_0

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standard_description: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
problem_activity: Activity
In this activity, students work together to investigate lines of symmetry and communicate their thinking in a visual display. As you monitor, ask groups if they have found all possible lines of symmetry and to explain how they know all shapes of that type have that same symmetry.
Launch
Arrange students in groups of 2–4. Provide each group with tools for creating a visual display.
Assign a different shape from the blackline master to each group and provide enough copies of that shape for each student in each group. Note that the cool-down uses a rectangle, so you may choose to assign the rectangle to a group of students who need more processing time.
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 to get their ideas down in an organized way.
Student Facing
Determine all the
lines of 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 line of symmetry
a description in words of each line of symmetry
one non-example in a different color (a description and drawing of a reflection
not
over a line of symmetry)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Look at all of the shapes the class explored and focus on those which had more than one line of symmetry.
What is true for all the lines of symmetry in these shapes?
Give an example of a shape that has two or more lines of symmetry that do not intersect at the same point.
What would happen if you did a sequence of two different reflections across lines of symmetry for the shapes you explored in class?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students might think that any line that divides the area of a figure in half, like a diagonal of a non-square rectangle, is a line of symmetry. Ask students to reflect across these lines to see that the shapes are not taken onto themselves.
Activity Synthesis
The purpose of this discussion is to identify what types of shapes have reflection symmetry.
Ask groups to display their visual displays in the classroom for all to see in order of the number of lines of symmetry their shape has. They will have to communicate with other groups to accomplish this.
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 lines of symmetry that the group missed?
Are there any lines of symmetry that don’t work for all shapes of the given type?
After the gallery walk, ask students to share their observations about reflection symmetries for the different shapes. Consider asking:
“Which shapes had the fewest lines of symmetry? Which had the most?” (The parallelogram that is not a rhombus didn’t have any lines of symmetry. The circle has the most since it has infinite.)
“Were there any lines that you thought would be lines of symmetry, but when you tried to reflect, they turned out not to be?” (The diagonal of a parallelogram that is not a rhombus seemed like it would be a line of symmetry, but it’s not.)
“Describe a shape that would have no lines of symmetry.” (A scalene triangle has no lines of symmetry.)
Save the visual displays if possible for comparison with displays of angles of rotation that create symmetry in a subsequent lesson.
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


aligns: no