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


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: 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. Solving this problem does not require use of the Pythagorean identity.
Answer: B

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


Standard options: 
A. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
B. Solve quadratic equations in one variable.


Thought: This task shows one quadratic equation with one variable, x.
Answer: B

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 $P$.


Standard options: 
A. 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.
B. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.


Thought: 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.
Answer: A

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Verify experimentally the properties of dilations given by a center and a scale factor:
- A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
- The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
problem_activity: Activity
Students dilated a single point in a previous activity. In this activity, they dilate a figure with multiple points and multiple shapes. The goal is for them to practice dilation and begin to confirm some properties of dilation. Students will have additional opportunities to formalize properties of dilation in subsequent lessons (A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor.)
Monitor for students who use the following (incomplete) strategies:
use the two given lines to place
\(D’\)
and
\(B’\)
, and then estimate all of the other shapes and points
measure distances within the figure and try to replicate them, for example calculating the radii of the two circles and trying to place them without scaling
\(A\)
and
\(C\)
along rays through
\(P\)
Monitor for students who use the precise strategy of drawing rays from
\(P\)
through each point in the original figure and placing points in the image halfway between
\(P\)
and the corresponding point.
Launch
It's okay if students do not have time to dilate every point, encourage them to record things they notice and wonder as they draw the dilation.
Engagement: Internalize Self Regulation.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Assign students an appropriate number of points to choose and dilate.
Supports accessibility for: Organization; Attention
Student Facing
Dilate the figure using center
\(P\)
and scale factor
\(\frac12\)
.
[IMG]
What do you notice? What do you wonder?
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?
[IMG]
Dilate segment
\(AB\)
using center
\(P\)
by scale factor
\(\frac12 \)
. Label the result
\(A'B'\)
.
Dilate the segment
\(AB\)
using center
\(Q\)
by scale factor
\(\frac12\)
.
How does the length of
\(A''B''\)
compare to
\(A'B\)
? How would the length of
\(A''B''\)
change if
\(Q\)
was infinitely far away? Explain or show your answer.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The goal of this discussion is to surface some observations. If most students were comfortable drawing a dilation move directly to that. If not, select multiple students who used different strategies (both precise and incomplete) to share their work. Ask students to explain what was
precise
in each student’s method. The key point is that the most precise way to draw a dilation is to draw rays from the center of the dilation through each point of the original figure, and then measure to figure out where the point goes in the image.
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the relative length of segments, or measure of angles do not come up during the conversation, ask students to discuss these ideas.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. As students share their strategies for drawing the dilation of the figure, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped clarify the original explanation such as, “draw rays from the center of dilation through each point of the original figure.” This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 0