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: Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
problem_activity: Narrative
The purpose of this activity is to deepen students’ understanding that a shape can belong to multiple categories because of its attributes. Students analyze shapes and determine all the ways that each one could be named. The names may refer to a broad category such as triangle or quadrilateral, or a narrower subcategory such as rhombus or rectangle. As they name the different categories students need to be precise both about the meaning of the categories and verifying the properties of the different shapes (MP6).
MLR8 Discussion Supports.
Synthesis: For each observation that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language.
Advances: Listening, Speaking
Engagement: Provide Access by Recruiting Interest.
Leverage choice around perceived challenge. Invite students to select at least 4 of the 6 problems.
Supports accessibility for: Organization, Attention, Social-Emotional Skills
Launch
Groups of 2
“Look at the quadrilateral in the first problem. Work independently to circle all the names that you could use to describe the quadrilateral. Be prepared to share your reasoning.”
1 minute: independent work time
“Discuss your responses and reasoning with your partner.”
2 minutes: partner discussion
Share responses.
Activity
“Complete the rest of the problems independently. Be prepared to explain your reasoning.”
3–5 minutes: independent work time
“Now, discuss your answers with your partner. Be sure to explain your reasoning for each way you described the shape. Also, be sure to ask your partner if you have any questions about their reasoning.”
5–7 minutes: partner discussion
Monitor for students who notice that some shapes can be described using multiple terms.
Student Facing
Select
all
the ways you could describe each shape. Be prepared to explain your reasoning.
1
[IMG]
triangle
quadrilateral
square
rhombus
rectangle
2
[IMG]
triangle
quadrilateral
hexagon
rhombus
rectangle
square
3
[IMG]
triangle
quadrilateral
pentagon
rhombus
rectangle
square
4
[IMG]
triangle
quadrilateral
hexagon
rhombus
rectangle
square
5
[IMG]
hexagon
quadrilateral
triangle
square
rectangle
rhombus
6
[IMG]
hexagon
quadrilateral
triangle
rhombus
rectangle
square
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students use only one name for a shape that can be named in multiple ways, consider asking:
“How did you describe the shape?”
“Are there any other names that could be used to describe the shape?”
Activity Synthesis
Select 1–2 students to share the terms they selected for each of the last four quadrilaterals and their reasoning.
Consider asking:
“Who can restate _____’s reasoning in a different way?”
“Does anyone want to add on to _____’s reasoning?”
“Do you agree or disagree? Why?”
“The last shape can be described with 4 of the choices. How is it possible that it can be described in so many ways?” (It is a quadrilateral because it has 4 sides. It is a rhombus because it’s a quadrilateral with 4 sides that are the same length. It is a rectangle because it has 4 right angles and 2 pairs of sides that are the same length. It is a square because it has 4 sides that are the same length and 4 right angles. The last three are more specific descriptions of a quadrilateral.)


correct_option_index: 1