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 the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
problem_activity: Activity
In this activity, students continue to identify points on a scatter plot as representatives of a single month when two variables are measured. Students are also asked to explain the meaning of some abstract points in the context of the problem and identify when it does not make sense to extrapolate information from the graph based on the context (MP2).
Monitor for students who understand that 60 degrees Celsius is unreasonable for an average monthly temperature in most parts of the world, as well as those who identify the trend to predict negative sales which does not make sense in this context either.
Launch
Arrange students in groups of 2. Allow students 5 minutes quiet work time followed by 5 minutes partner discussion and whole-class discussion.
Ask students to predict what a scatter plot might look like when the temperature is along the
\(x\)
-axis and the sales of coats at a store is along the
\(y\)
-axis.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, ask students to use the different colors to represent and annotate temperature and coat sales in the graph and table.
Supports accessibility for: Visual-spatial processing
Student Facing
A clothing store keeps track of the average monthly temperature in degrees Celsius and coat sales in dollars.
| temperature (degrees Celsius) | coat sales (dollars) |
|-------------------------------|----------------------|
|              -5               |        1,550         |
|              -3               |        1,340         |
|               3               |        1,060         |
|               8               |        1,070         |
|              15               |         680          |
|              21               |         490          |
|              23               |         410          |
|              21               |         510          |
|              17               |         600          |
|              11               |         740          |
|               6               |         940          |
|              -2               |        1,390         |
[IMG]
What does the point
\((15, 680)\)
represent?
For the month with the lowest average temperature, estimate the total amount made from coat sales. Explain how you used the table to find this information.
For the month with the smallest coat sales, estimate the average monthly temperature. Explain how you used the scatter plot to find this information.
If there were a point at
\((0, A)\)
what would it represent? Use the scatter plot to estimate a value for
\(A\)
.
What would a point at
\((B, 0)\)
represent? Use the scatter plot to estimate a value for
\(B\)
.
Would it make sense to use this trend to estimate the value of sales when the average monthly temperature is 60 degrees Celsius? Explain your reasoning.
Student Response
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Activity Synthesis
​The purpose of the discussion is for students to understand that each point in the graph represents a single month in which two measurements are made. Select students to share their responses, including those identified for the last problem.
Consider asking some of the following questions:
“When discussing functions, what is a point of the form
\((0,A)\)
called? What about a point of the form
\((B,0)\)
?” (The
\(y\)
-intercept. The
\(x\)
-intercept.)
“In a scatter plot is there a
\(y\)
-intercept?” (It depends on the context. For some situations, it may not make sense for the variable represented along the
\(x\)
-axis to be zero. For example if the variable is the height of a building. On other situations, there may be more than one point along the
\(y\)
-axis. For example, in a situation like the one in the task, there may be multiple months where the average temperature is 0 degree Celsius.)
“When the average monthly temperature goes up, what seems to be happening to sales?” (They are decreasing.)
Writing, Speaking: MLR1 Stronger and Clearer Each Time.
Use this routine to give students a structured opportunity to revise and refine their response to the last question. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help students strengthen their ideas and clarify their language (e.g., “Can you give an example?”, “Why do you think…?”, “How do you know…?”, etc.). Students can borrow ideas and language from each partner to strengthen their final version.
Design Principle(s): Optimize output (for explanation)


aligns: yes