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: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
problem_activity: Optional activity
This activity gives students a chance to recall and use various ratio strategies in the context of a voting problem. Two classes voted on a yes or no question. Both classes voted yes. Students are asked to determine which class was more in favor (“yessier”). Students need to make sense of the invented word “yessier” by thinking about how it might be quantified (MP1, MP2).
Monitor for students who use different ways to make sense of the problem.
Launch
Arrange students in groups of 2–4. Ask students to use the mathematical tools they know to answer the question.
Student Facing
Two sixth-grade classes, A and B, voted on whether to give the answers to their math problems in poetry. The “yes” choice was more popular in both classes.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "yes"}, {"_attributes": {"scope": "col"}, "_value": "no"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "class A"}], "td": [{"_value": "24"}, {"_value": "16"}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "class B"}], "td": [{"_value": "18"}, {"_value": "9"}]}]}]}]}
Was one class more in favor of math poetry, or were they equally in favor? Find three or more ways to answer the question.
Student Response
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Anticipated Misconceptions
Students may not understand the question. The word “yessy” was invented by other students solving a similar problem. It is not standard English.
If students do not understand what comparing the two classes means, give a more extreme example, such as comparing 20 to 1 with 11 to 10. The 20 to 1 class is much more yessy because almost everyone said yes. The other class had almost equal yesses and nos.
Activity Synthesis
The situation is mathematically the same as other rate comparison problems, such as comparing the tastes or colors of two mixtures.
Invite several students to present different methods, at least one who used a ratio of yesses to nos, and another who used a ratio of yesses to all students. Make sure to present a solution using percentages.
If more than a few students did not use multiplicative techniques (for example, if they compared only the yesses, or subtracted to find how many more yesses than nos) remind them of the rate comparisons they did earlier in the year, in which they could check by tasting a drink mixture or looking at the color of a paint mixture.
Speaking, Listening, Representing: MLR2 Collect and Display.
Use this routine to display examples of the different methods students used to determine which class was “yessier." Circulate and collect examples of formal and informal language, calculations, tables, or other writting. Ask students to compare and contrast percentage calculations with fractions and equivalent ratios.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


aligns: yes