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: 
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.

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: 
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, 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_pos_0

HS_2
standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Activity
This activity presents students with two graphs without numerical values assigned to any points on the graphs. The only labeled point is where the two graphs intersect. Students will need to use their understanding that tripling each day is a greater growth factor than doubling each day. This is their only means to decide which function represents which situation. Students then interpret the graphs in terms of the situations that they represent.
Launch
Display the graph for all to see. Give students a moment to observe the graph and ask, "What do you notice? What do you wonder?" Invite students to share their observations and questions.
Student Facing
Here are graphs representing two functions, and descriptions of two functions.
[IMG]
Function
\(f\)
: The area of a wall that is covered by Mold A, in square inches, doubling every month.
Function
\(g\)
: The area of a wall that is covered by Mold B, in square inches, tripling every month.
Which graph represents each function? Label the graphs accordingly and explain your reasoning.
When the mold was first spotted and measured, was there more of Mold A or Mold B? Explain how you know.
What does the point
\((p, q)\)
tell us in this situation?
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
Some students may think that the dashed graph represents the mold that is tripling because it has a greater vertical intercept. Ask these students to think carefully about the second question.
Activity Synthesis
Select students to share their responses and explanations. Focus on how they deciphered the meanings of the graphs. Discuss questions such as:
The solid graph started off lower than the dashed graph. What does that mean in this context? (The mold population represented by the solid graph is initially smaller or covers a smaller area on the wall than the one represented by the dashed graph.)
After some point, the solid graph passes the dashed one. What does that mean in this context? (The mold population that started out covering a greater area on the wall is surpassed by the other population, so it is growing at a slower rate.)
What does the intersection of the two graphs mean? What do
\(p\)
and
\(q\)
represent? (The two mold populations are the same at time
\(p\)
with population covering an area of
\(q\)
square inches.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each explanation that is shared, 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 statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


aligns: no