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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. 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.

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_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
problem_activity: Narrative
In this warm-up, students practice estimating a reasonable answer using known information, rounding, and multiplicative reasoning strategies. For example, students may create a diagram and arrive at
\(7 \times 7 \times 7 \times 7\)
as an estimate of the number of fish. Then, they may approximate
\((7 \times 7) \times (7 \times 7)\)
with
\(50 \times 50\)
, or estimate
\((7 \times 7 \times 7) \times 7\)
with
\((50 \times 7) \times 7\)
or
\(350 \times 7\)
. Some students may notice that the question is vague and asks, “How many are going to the park?” rather than “How many people are going to the park?” and account only for the number of children and teachers in their estimation. If students ask for clarification, ask them to make their own assumptions and explain why they made those assumptions when they share their estimates.
Launch
Groups of 2
Display the description.
Activity
“What is an estimate that’s too high? Too low? About right?”
2 minutes: quiet think time
1 minute: partner discussion
Record responses.
Student Facing
Seven teachers are going to the park.
Each teacher is taking 7 students.
Each student is bringing 7 fishbowls.
Each fishbowl has 7 fish.
[IMG]
How many are going to the park?
Record an estimate that is:
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col", "style": "text-align:center"}, "_value": "too low"}, {"_attributes": {"scope": "col", "style": "text-align:center"}, "_value": "about right"}, {"_attributes": {"scope": "col", "style": "text-align:center"}, "_value": "too high"}]}]}, {"tr": [{"td": [{"span": [{"_attributes": {"class": ["math"]}, "_value": "\\(\\phantom{\\hspace{2.5cm} \\\\ \\hspace{2.5cm}}\\)"}]}, {"span": [{"_attributes": {"class": ["math"]}, "_value": "\\(\\phantom{\\hspace{2.5cm} \\\\ \\hspace{2.5cm}}\\)"}]}, {"span": [{"_attributes": {"class": ["math"]}, "_value": "\\(\\phantom{\\hspace{2.5cm} \\\\ \\hspace{2.5cm}}\\)"}]}]}]}]}]}
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Consider asking:
“Is anyone’s estimate less than 500? Greater than 5,000?”
“Did anyone include the number of teachers and students in their estimate?”
Invite students to share their estimation strategies. After each explanation, ask if others reasoned the same way.
Consider revealing the actual value of 2,457 teachers, students, and fish.


aligns: no