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: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with 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

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
problem_activity: Activity
The goal of this activity is for students to use the definition of similarity, and to practice applying some of the skills they learned when studying congruent figures to similar figures. Students practice writing a sequence of rigid motions and dilations to take one figure onto another, write accurate similarity statements about the figures, and reason about what must be true about corresponding side lengths.
Launch
Ask students to add this definition to their reference charts as you add it to the class reference chart:
One figure is
similar
to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second. (Definition)
Translation and dilation takes
\(\triangle ABC\)
onto
\(\triangle FDE\)
so
\(\triangle ABC \sim \triangle FDE\)
[IMG]
Tell students
\(\triangle ABC \sim \triangle FDE\)
is a similarity statement, and we read
\(\sim\)
as “is similar to.”
Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses for the sequence of transformations that takes one triangle to the other. Give students time to meet with 2–3 partners to share and get feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What is the directed line segment of the translation?” and “What is the center and scale factor of the dilation?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students use precise language to describe the sequence of transformations that takes one triangle to the other.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Representation: Internalize Comprehension.
Activate or supply background knowledge about symbolic notation. Provide students with access to explanations of symbols used in the caption for parallel and perpendicular, and connect to the right angle symbol in the image which shows that two segments are perpendicular.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Are the triangles
similar
?
\(\overline{AB}\parallel\overline{QR},\overline{AB}\perp\overline{AE},\overline{QR}\perp\overline{QT}\)
[IMG]
Write a sequence of transformations (dilation, translation, rotation, reflection) to take one triangle to the other.
Write a similarity statement about the 2 figures, and explain how you know they are similar.
Compare your statement with your partner’s statement. Is there more than one correct way to write a similarity statement? Is there a wrong way to write a similarity statement?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If students struggle to define sequences of rigid motions and dilations, review an example from the card sort.
Activity Synthesis
Ask students to determine if their sequence of rigid motions and dilations will always work for any pair of triangles that have all pairs of corresponding angles congruent and all pairs of corresponding sides lengths in the same proportion. (Probably not, some figures would require rotations and dilations. But the general method of dilating and using rigid motions will always work.)
For students who dilated first, ask:
How do you choose the dilation to use? (Dilate from any vertex, by the scale factor given by the ratio of a pair of corresponding sides.)
Once you dilate, how do you know you’ll be able to use rigid motions to take one triangle to the other? (The triangles will be congruent, because their corresponding angles will remain congruent, and all their side lengths will be multiplied by the value
\(k\)
that you need to make them the same length as the corresponding side. Since all the corresponding angles and all the corresponding sides will match, you’ll be able to take one triangle to the other.)
For students who used rigid motions first, ask:
How do you know rigid motion will line up parts of the figures? (Because corresponding angles are congruent, we’ll be able to line up both of the rays that meet at the vertex by translating and rotating, because you can always take an angle onto another congruent angle.)
How do you know that dilation will then line up the remaining two points? (The sides lengths will be in the same proportion, because we’re given that information, and the angles all line up, so the triangles are dilations of each other.)


aligns: no