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: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Answer: 
No. This task does not involve an equation in two variables. There is only 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: 
Construct a tangent line from a point outside a given circle to the circle.

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, 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_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
problem_activity: Narrative
In this activity, students analyze three different ways to subtract. They see that taking away is one way to find the difference, but that you can also count on or use known addition facts. Students further solidify their understanding that addition and subtraction are related, which sets the groundwork for a later activity when students solve subtraction problems within 10.
Required Materials
Materials to Gather
Connecting cubes or two-color counters
Launch
Groups of 2
Give students access to connecting cubes or two-color counters.
Activity
Read the task statement.
“First you will work on your own. Think about what each student means and be ready to explain your thinking in a way that others will understand.”
5 minutes: independent work time
4 minutes: partner discussion
Monitor for students who can use the 10-frame with six red counters to explain the relationship between
\(10 - 6\)
and
\(6 + \boxed{\phantom3} = 10\)
.
Student Facing
Mai, Diego, and Noah find the value of
\(10 - 6\)
.
Diego says, “I can take away.”
[IMG]
What does Diego mean?
Be ready to explain your thinking in a way that others will understand.
Mai says, “I can count on.”
[IMG]
What does Mai mean?
Be ready to explain your thinking in a way that others will understand.
Noah says, “I can use what I know about
\(6 + 4\)
to help me.”
What does Noah mean?
Be ready to explain your thinking in a way that others will understand.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Activity Synthesis
Invite previously identified students to share.
“Who can restate what _____ just showed us?” (Diego subtracted by taking away 6 counters one at a time and saw that there were 4 counters left. Mai subtracted by thinking about addition. She counted on from 6 until she got to 10 and noticed she counted up 4. Noah knows his sums of 10. He knows 10 can be made by 6 and 4, so
\(10 - 6 = 4\)
.)
“Which method do you like best?” (I know my sums to 10 so I would use that. I like counting on because I like to add more than take away.)


aligns: yes