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: Recognize and represent proportional relationships between quantities.
problem_activity: Optional activity
This info gap activity uses a quality control situation working with percent error. It is very common that products in a factory are checked to make sure that they meet certain specifications. In this case the odometer of a car is tested and the amount of liquid in a bottle that is automatically filled is checked. It makes sense that the decision should be based on percent error and not on absolute error.
The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).
Here is the text of the cards for reference and planning:
[IMG]
Launch
Provide access to calculators. Tell students they will continue to work with percent errors in realistic scenarios. Explain the Info Gap structure and consider demonstrating the protocol if students are unfamiliar with it. There are step-by-step instructions in the student task statement.
Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles.
Engagement: Develop Effort and Persistence.
Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Conversing
: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems involving percent error. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation
Student Facing
Your teacher will give you either a
problem card
or a
data card
. Do not show or read your card to your partner.
If your teacher gives you the
problem card
:
Silently read your card and think about what information you need to be able to answer the question.
Ask your partner for the specific information that you need.
Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem.
Share the
problem card
and solve the problem independently.
Read the
data card
and discuss your reasoning.
If your teacher gives you the
data card
:
Silently read your card.
Ask your partner
“What specific information do you need?”
and wait for them to
ask
for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.
Before sharing the information, ask “
Why do you need that information?
” Listen to your partner’s reasoning and ask clarifying questions.
Read the
problem card
and solve the problem independently.
Share the
data card
and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The purpose of the discussion is to recognize what information is needed when dealing with tolerances based on percent error.
After students have completed their work, share the correct answers and ask students to discuss the different ways they solved this problem. Some guiding questions:
"What information did you and your partner have to figure out?"
"What different calculations did you have to make for the two situations?"
"Why might a car manufacturer not want to sell a car that shows a speed lower than what the car is actually going?" (It could be dangerous for the driver if they think they are going slower than they actually are. They are also more prone to getting speeding tickets if they think they are going under the speed limit based on the speedometer, but are actually going faster.)
"Why might a juice manufacturer not want to have too much more juice in the bottle than it says on the label?" (If too many bottles are overfilled, the company may be losing money by giving away extra juice without charging the customer more.)
Conversing:
This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems involving equivalent ratios.
Design Principle(s): Cultivate Conversation


aligns: no