few_shots: 
Example 1:
Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

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


Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

Example 2:
Concept/skill: 
Solve quadratic equations in one variable.

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


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

Example 3:
Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

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 $P$.


Answer: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
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.)


aligns: no