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: Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
problem_activity: Narrative
The purpose of this activity is for students to use what they’ve learned about multiplication to solve and represent situations that involve equal groups. Students now have experience with multiple representations and have the opportunity to choose which representation is most helpful to represent multiplication situations.
The launch of the activity is an opportunity for students to share their experiences and ask questions about the objects to ensure each student has access to the context. If it is helpful, display images of the items for students to reference.
MLR8 Discussion Supports.
Synthesis: As students describe their representations of the problems, use gestures to emphasize the number of groups and how many are in each group. For example, trace with your finger around each group, and point to each object in each group to show how many there are.
Advances: Listening, Representing
Representation: Access for Perception
. Provide access to connecting cubes. Ask students to identify correspondences between the concrete and visual representation used.
Supports accessibility for: Conceptual Processing
Launch
Groups of 2
Write the list of objects (“teams, earrings, pencils, pieces of glass, carrots”) on a display for all students to see.
“Take a minute to read this list. When you are done, discuss the objects you know or have questions about.”
3 minutes: partner discussion
Share a few responses.
“Now we’re going to represent and solve more problems with equal groups. Take a couple of minutes to begin working on the problems before you work in partners.”
2–3 minutes: independent work time
Activity
“Work with your partner to solve each problem.”
5–7 minutes: partner work
Identify students who use different representations like drawings of equal groups or tape diagrams as they solve the problems.
Student Facing
Solve each problem. Show your thinking using diagrams, symbols, or other representations.
There are 4 soccer fields. Two teams are on each field. How many teams are there altogether?
There are 7 windows. Each window has 2 pieces of glass. How many pieces of glass are there in the windows?
Jada has 5 bags. Each bag has 10 earrings. How many earrings does Jada have?
Kiran has 4 boxes. Each box has 5 pencils in it. How many pencils does Kiran have?
Andre has 3 bags of carrots. Each bag has 10 carrots. How many carrots does Andre have?
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Advancing Student Thinking
If students add or subtract instead of multiply to solve the problems, consider asking:
“Tell me about how you solved this problem.”
“How does the problem involve equal groups?”
Activity Synthesis
For each problem, display different representations, one at a time.
“How does this representation help us see what’s happening in the problem?” (You can see there are 4 groups for the fields and 2 dots in each group for the teams. The diagram is split into 5 parts for the 5 bags, and each section has a 10 in it for the number of earrings.)
“How could each representation help us solve the problem?” (Counting the dots. Counting by 10.)
If needed, “What expression could we write to represent this situation?”


aligns: no