few_shots: 
Example 1:
Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

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


Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
problem_activity: Activity
The purpose of this activity is to connect features of an equation representing a function to what that means in a context.
Students start with two functions that represent a tank being filled up and another being drained out and are asked to determine which equations represent which situation. This gives students the opportunity to connect initial value and slope, which they learned about in a previous unit, to the general form of the linear equation and to the fact that linear relationships are functions.
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time and then time to share their responses with their partner and reach agreement on their answers. Encourage partners to talk about specific parts of the graph and equation that indicate whether the tank is filling up or draining out. Follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example present one question at a time.
Supports accessibility for: Organization; Attention
Writing, Conversing: MLR5 Co-Craft Questions.
Display the task statement without the questions that follow. Ask pairs of students to write possible mathematical questions about the situation. Then, invite pairs to share their questions with the class. Look for questions that ask students to determine which tank is increasing or decreasing in volume. This helps students produce the language of mathematical questions and talk about the relationship between slope and the situation of water flowing in or out of a tank prior to solving the questions in this activity.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
There are four tanks of water.
The amount of water in gallons,
\(A\)
, in Tank A is given by the function
\(A = 200 + 8t\)
, where
\(t\)
is in minutes.
The amount of water in gallons,
\(B\)
, in Tank B starts at 400 gallons and is decreasing at 5 gallons per minute. These functions work when
\(t \geq 0\)
and
\(t \leq 80\)
.
Which tank started out with more water?
Write an equation representing the relationship between
\(B\)
and
\(t\)
.
One tank is filling up. The other is draining out. Which is which? How can you tell?
The amount of water in gallons,
\(C\)
, in Tank C is given by the function
\(C = 800 - 7t\)
. Is it filling up or draining out? Can you tell just by looking at the equation?
The graph of the function for the amount of water in gallons,
\(D\)
, in Tank D at time
\(t\)
is shown. Is it filling up or draining out? How do you know?
[IMG]
Student Response
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click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Pick a tank that was draining out. How long did it take for that tank to drain? What percent full was the tank when 30% of that time had elapsed? When 70% of the time had elapsed?
What point in the plane is 30% of the way from
\((0,15)\)
to
\((5,0)\)
? 70% of the way?
What point in the plane is 30% of the way from
\((3,5)\)
to
\((8,6)\)
? 70% of the way?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Consider asking some of the following questions to begin the discussion:
“For the second problem, what in the equation tells you that the slope is decreasing? Increasing?” (Decreasing: the slope in the equation is negative; Increasing: the slope in the equation is positive.)
“For the third problem, what is similar between the equation in this problem and the decreasing equation in the previous problem?”
(Both slopes are negative.)
“For the last problem, what in the graph tells you that Tank D is draining out? What would a graph that has a tank filling up look like? What would be different?” (I know it is draining out because the graph is going down from left to right. If the tank were filling up, the graph would be going up from left to right.)
Tell students that a linear function can always be represented with an equation of the form
\(y = mx+b\)
. The slope of the line,
\(m\)
, is the rate of the change of the function and the initial value of the function is
\(b\)
.
If time allows, give students the following scenario to come up with a possible equation for Tank D:
“Tank D started out with more water than Tank B but less water than Tank C. The water is draining from Tank D faster than from Tank B but slower than Tank C. What is a possible equation for the graph of the function for the amount of water
\(D\)
in Tank D over time
\(t\)
?” (Students should choose an initial value between 400 and 800, and a constant rate of change between -7 and -5. One possible such equation might be
\(D=600-6t\)
.)


aligns: yes