few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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:
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).

Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

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 $P$.


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, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

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 the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.  For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
problem_activity: Activity
In this task, students are presented with a situation that leads to a linear relationship that is not proportional because there is a non-zero starting amount. By trying to answer the question, “How many cups are needed to get to a height of 50 cm?” the students explore the rate of change, which is the increase per cup after the first cup. The rate of change can be seen in the graph as the slope. Students use representations and ideas from previous lessons on proportional relationships. They use tables and graphs to represent the situation and make deductions by generalizing from repeated reasoning (MP8), arguing that each additional cup increases the height of the stack by the same amount. As students are working, suggest that students make a graph or a table if they are stuck or if they have trouble explaining their reasoning. Students should be prepared to share their strategies with the class.
Students can attack this problem using any method. If they get stuck, here are some things it would be useful to figure out:
How much height does each cup add?
Is the first cup different from the others?
Create a graph or table to help you reason about this problem.
As students work, identify students who use different strategies, i.e. graphs, tables, equations.
This activity and the next were written using a particular type of cup. Photos are included of all measurements needed, so these activities can be used without any additional preparation. However, if desired, the lesson could be modified so that students are measuring stacks of actual cups. The numbers then may have to be adjusted since the cups come in different sizes. Teachers who want to use real cups should measure the height of two stacks ahead of time, find the rate of change and make sure that it is approximately constant. Note that rounding error will likely play a role unless the number of cups in the stacks is chosen carefully, so some flexibility in answers may be necessary if students measure actual stacks of cups.
Launch
Before students look at the picture with all of the given information, display just this part of the picture.
[IMG]
Ask students how many cups they see, and how tall the stack is. (There are 6 cups, and the stack is 15 cm tall.) Then, ask students to quietly make a prediction about the height of 12 cups and give you a silent signal when they have a response. When everyone has come up with an answer, ask a few students to share their responses and write these on the board. It is likely that some students will say 30 centimeters, and some might have an argument for why the height of 12 cups must be less than 30 centimeters. This question will be resolved as soon as students look at the photo in the task. Tell students that their job is to figure out how many cups would be needed in order to stack them to a height of 50 centimeters.
Arrange students in groups of 2–4. Give access to rulers and graph paper. Allow individual think time of 3 minutes before students work together in groups.
Representation: Internalize Comprehension.
Begin the activity with concrete or familiar contexts. Set up a display of foam cups next to a ruler, as depicted. Encourage students to consider the rate of change and whether or not there is a proportional relationship.
Supports accessibility for: Conceptual processing; Memory
Conversing, Representing, Writing: MLR2 Collect and Display.
As students discuss, listen for and collect the language students use to describe whether they think the relationship is proportional. Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their paired and whole-group discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
We have two stacks of styrofoam cups.
One stack has 6 cups, and its height is 15 cm.
The other stack has 12 cups, and its height is 23 cm.
How many cups are needed for a stack with a height of 50 cm?
[IMG]
Student Response
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Anticipated Misconceptions
Students compute
\(\frac{15}{6}=2.5\)
or
\(\frac{23}{12}=1.9\)
and use this as the increase per cup. Measurement is approximate; students may be looking for exact, nice numbers. Make sure numbers agree approximately between students, but reassure them that they are close enough.
Activity Synthesis
Poll the class about the number of cups they came up with to reach a total height of 50 cm.
If the class used real cups, stack enough of them to get to a height of 50 cm, and let students check if they were correct. Otherwise display this photo:
[IMG]
Ask students who used different strategies to share their reasoning with the class.
Ask students if they think the relationship is proportional. If there is a disagreement, ask one person per opinion to share their reasons. Encourage them to come to the conclusion that it is not proportional since doubling the number of cups from 6 to 12 did not double the height. Students can see that when stacking the cups only the rim adds to the height. The first cup adds more height since there are additionally 7 cm from the bottom of the cup to the bottom of the rim.
Tell students that even though the relationship is not proportional, this relationship has things in common with proportional relationships, and that they will explore this in the next activity.


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