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 following problem or activity 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

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im_task_001160_neg_0

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standard_description: Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?
problem_activity: Task
Materials
ruler, meter sticks, yard sticks, measuring tape
paper
crayons
Actions
Part 1
Explain that students will be working in pairs to determine the length of each partner’s foot. Ask what tools would be appropriate for determining length.  Chart student thinking about appropriate tools, and today’s goal of  using standard measurement tools such as rulers, yard sticks, meter sticks, or measuring tape.
Ask student pairs to consider the unit(s) that would give the most precise measure of the length of a student’s foot.  If students struggle with this idea, the teacher might model using his/her foot as an example considering yards, meters, feet, inches and centimeters.
Ask student pairs to identify the unit they will work with and predict the length of each partner’s foot and record their estimations.  Estimates should be recorded in units.
Part 2
Have one student from each pair stand on a blank sheet of paper and have his/her partner make a mark at the student’s heel and another at the toe.
Then, together, have the pair measure the distance between the two marks to determine the length of the first student’s foot and record the length.  Students may have to find the closest unit.
Have students then find the difference between the estimation and actual length.
Repeat with the second student, using a different color to make the marks.  The second student should stand at the same endpoint as the first student so that students can visually compare the two representations and compare the number of the distance in units in part three.
Part 3
Working in pairs, have students compare the distances between the lengths of each student’s foot.
Class Discussion
Engaging in a class discussion will support students in thinking about the mathematical ideas embedded within the task.
What tool did you use and why?
How was that tool helpful?
If you were going to measure _____ (a desk, a whiteboard, an eraser), would you use the same tool? Why?
How long was your foot? What was your estimate? What was the difference?
What unit did you use to measure? How did you decide to use ______? Why is the unit important?
How long was your partner’s foot? What was his/her estimate? What was the difference?
What was the difference between the length of your foot and the length of your partner’s foot? How did you find the difference?
What did you notice about finding the difference? Is there more than one way to find the difference? (This might be an opportunity to discuss finding the difference by measuring how much longer one length is than the other or by using an operation such as addition to add up from the smallest length to the larger length or by subtracting the smaller length from the larger length.)


aligns: no