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


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.

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:
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_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.
problem_activity: Warm-up
The mathematical purpose of this lesson is for students to motivate a conceptual understanding of the
addition rule
. Monitor for students discussing counting people twice.
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Student Facing
The table displays information about people at a neighborhood park.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "wearing sneakers"}, {"_attributes": {"scope": "col"}, "_value": "not wearing sneakers"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "wearing a hat"}], "td": [{"_value": "8"}, {"_value": "2"}, {"_value": "10"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "not wearing a hat"}], "td": [{"_value": "3"}, {"_value": "12"}, {"_value": "15"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "total"}], "td": [{"_value": "11"}, {"_value": "14"}, {"_value": "25"}]}]}]}]}
Andre says the number of people wearing sneakers or wearing a hat is 21, because there are a total of 10 people wearing a hat and a total of 11 people wearing sneakers. Is Andre correct? Explain your reasoning.
What is the probability that a person selected at random from those in the park is wearing sneakers or wearing a hat?
Student Response
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Activity Synthesis
The goal of this discussion is to use the context as a model to introduce a conceptual understanding of the addition rule. Ask previously identified students to explain why Andre counted some people twice when he stated that the number of people wearing sneakers or wearing a hat is 21.
Here are some questions for discussion.
“Why did Andre need to subtract 8 from 21 to get the correct answer?” (It needs to be subtracted because the 8 people that were wearing sneakers and wearing a hat were counted twice. We counted them twice when adding the column total for wearing sneakers and the row total for wearing a hat.)
“How could the number of people wearing sneakers or wearing a hat be represented using a Venn diagram?” (A Venn diagram could be made using two overlapping circles, one for wearing sneakers and one for wearing a hat. There would be 8 people in the middle where the two circles overlap. There would be an additional 3 people in the wearing sneakers circle representing those wearing sneakers but not a hat. There would also be an additional 2 people in the circle representing those wearing a hat but not sneakers.)
“What is the probability that a person selected at random is not wearing sneakers and not wearing a hat?” (
\(\frac{12}{25}\)
)


aligns: no