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: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with 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_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
problem_activity: Activity
The goal of this activity, and the two that follow, is for students to solve an equation in a real-world context while previewing some future work solving systems of equations. Here, students first make sense of the situation using a table of values describing the water heights of two tanks and then use the table to estimate when the water heights are equal. A key point in this activity is the next step: taking two expressions representing the water heights in two different tanks for a given time and recognizing that the equation created by setting the two expressions equal to one another has a solution that is the value for time,
\(t\)
, when the water heights are equal.
Launch
Give students 2–3 minutes to read the context and answer the first problem. Select students to share their answer with the class, choosing students with different representations of the situation if possible. Give 3–4 minutes for the remaining problems followed by a whole-class discussion.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using diagrams. If students are unsure where to begin, suggest that they draw a diagram to help illustrate the information provided.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Representing, Conversing: MLR2 Collect and Display.
To begin the whole-class discussion, give students the opportunity to discuss their solutions to the first question in groups of 3–4. Circulate through the groups and record language students use to describe what is happening in each tank. Listen for language related to rate of change, differences between rates, initial amounts of water, etc. If groups are stuck, consider asking, “What happens to the amount of water in Tank 1 (or 2) as time goes on?” and “Can you draw a picture of Tank 1 (or 2) at 5 minutes and then another picture at 10 minutes? What do you notice?” Post the collected language in the front of the room so that students can refer to it throughout the rest of the activity and lesson. This will help students talk about the relationship between the two tanks prior to being asked to find the time when they are equal.
Design Principle(s); Maximize meta-awareness
Student Facing
The amount of water in two tanks every 5 minutes is shown in the table.
| time (minutes) | tank 1 (liters) | tank 2 (liters) |
|----------------|-----------------|-----------------|
|       0        |       25        |      1000       |
|       5        |       175       |       900       |
|       10       |       325       |       800       |
|       15       |       475       |       700       |
|       20       |       625       |       600       |
|       25       |       775       |       500       |
|       30       |       925       |       400       |
|       35       |      1075       |       300       |
|       40       |      1225       |       200       |
|       45       |      1375       |       100       |
|       50       |      1525       |        0        |
Describe what is happening in each tank. Either draw a picture, say it verbally, or write a few sentences.
Use the table to estimate when the tanks will have the same amount of water.
The amount of water (in liters) in tank 1 after
\(t\)
minutes is
\(30t + 25\)
. The amount of water (in liters) in tank 2 after
\(t\)
minutes is
\( \text-20t + 1000\)
. Find the time when the amount of water will be equal.
Student Response
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Activity Synthesis
The purpose of this discussion is to elicit student thinking about why setting the two expressions in the task statement equal to one another is both possible and a way to solve the final problem.
Consider asking the following questions:
“What does
\(t\)
represent in the first expression? The second?” (In each expression,
\(t\)
is the time in minutes since the tank’s water level started being recorded.)
“After we substitute a time in for
\(t\)
and simplify one of the expressions to be a single number, what does that number represent? What units does it have?” (The number represents the amount of liters in the water tank.)
“How accurate was your estimate about the water heights using the table?” (My estimate was within a few minutes of the actual answer.)
“If you didn’t know which expression in the last problem belonged to which tank, how could you figure it out?” (One of the expressions is increasing as
\(t\)
increased, which means it must be Tank 1. The other is decreasing as
\(t\)
increases, so it must be Tank 2.)
“How did you find the time the two water heights were equal using the expressions?” (Since each expression gives the height for a specific time,
\(t\)
, and we want to know when the heights are equal, I set the two expressions equal to each other and then solved for the
\(t\)
-value that made the new equation true.)


aligns: no