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: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
problem_activity: Activity
Following the concept developed in the warm-up, students will continue to use the idea that outcomes from previous experiments can help inform the likelihood of an outcome when the experiment is repeated. In this activity, students play a game while collecting data. Bags of colored blocks are set up with different probabilities to win and students use random chance to collect points when the color of the block they choose matches the color written on the bag. In the discussion, students will use the data they have collected in three rounds of the game to make a decision about which bag might be most likely to produce a winning block. As an opening activity, this is meant to motivate students into thinking about likelihood of events. As such, it is not important to resolve the questions in the discussion into perfect agreement.
Students are arranged in groups of 4 and given one of 3 different bags labeled with colors. Create enough bags so that each group will have a turn with each of the different color bags after three rounds.
Launch
Arrange students in groups of 4. Explain to students how to play the game:
There will be 3 rounds of the game. Their group will get a different bag of blocks for each round.
They are not allowed to look into the bag or take more than one block out of the bag at a time.
During a round, each person in the group gets 4 turns to take a block out of the bag.
If the block matches the color that is written on the outside of the bag, the person scores (1 point during round 1, 2 points during round 2, and 3 points during round 3). When the block is any other color, they do not earn any points.
They put the block back in the bag, shake the bag to mix up the blocks, and pass the bag to the next person in between each turn.
At the end of a round, they record everyone's points from that round and wait for a new bag of blocks.
Distribute bags of blocks. Each group gets one of these bags:
A bag labeled "Green" that contains 9 green blocks and 3 of another color.
A bag labeled "Blue" that contains 8 blue blocks and 8 of another color.
A bag labeled "Red" that contains 4 red blocks and 10 of another color.
Engagement: Provide Access by Recruiting Interest.
Begin with a small-group or whole-class demonstration of how to play the game.  Check for understanding by inviting students to rephrase directions in their own words.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Your teacher will give your group a bag of colored blocks.
Follow these instructions to play one round of the game:
Everyone in the group records the color written on the bag in the first column of the table.
​Without looking in the bag, one person takes out one of the blocks and shows it to the group.
If they get a block that is the same color as the bag, they earn:
1 point during round 1
2 points during round 2
3 points during round 3
Next, they put the block back into the bag, shake the bag to mix up the blocks, and pass the bag to the next person in the group.
Repeat these steps until everyone in your group has had 4 turns.
At the end of the round, record each person's score in the table.
|         | What color bag? | person 1's score | person 2's score | person 3's score | person 4's score |
|---------|-----------------|------------------|------------------|------------------|------------------|
| round 1 |                 |                  |                  |                  |                  |
| round 2 |                 |                  |                  |                  |                  |
| round 3 |                 |                  |                  |                  |                  |
Pause here so your teacher can give you a new bag of blocks for the next round.
Repeat the previous steps to play rounds 2 and 3 of the game.
After you finish playing all 3 rounds, calculate the total score for each person in your group.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Tyler’s class played the block game using purple, orange, and yellow bags of blocks.
During round 1, Tyler’s group picked 4 purple blocks and 12 blocks of other colors.
During round 2, Tyler’s group picked 11 orange blocks and 5 blocks of other colors.
During round 3, Tyler forgot to record how many yellow blocks his group picked.
For a final round, Tyler’s group can pick one block from any of the three bags. Tyler’s group decides that picking from the orange bag would give them the best chance of winning, and that picking from the purple bag would give them the worst chance of winning. What results from the yellow bag could have lead Tyler’s group to this conclusion? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some groups may choose a bag for the bonus round based on the number of points they got in the earlier rounds. Since points are equal in the bonus round, help them see that the blocks picked out is a more important thing to consider than the points for the round.
Activity Synthesis
Tell students that there is a bonus, fourth round. In this round, a block that matches the color on the bag is worth 25 points. Each person will only have one chance to draw a bonus block. In this round, each group will get to choose which bag they would like to draw from. At least 3 people in the group must agree on the bag they will use.
Give students 5 minutes of small-group discussion to agree on which bag the group will use. Tell students that, after the bonus round, they will be asked to explain their reasoning for choosing the bag. It is ok for more than one group to select the same color bag. Identify groups that use the results from the previous rounds to determine which bag will be more likely to draw a winning block.
Allow each group to use the bag they have selected to play the bonus round and total their points.
After all students have played the bonus round, select groups to share their reasoning for choosing which bag to use. Select previously identified students to share their reasoning as well.
Ask students, "Was one bag more likely to give you a winning cube than the others? Explain your reasoning."
Open one of each color bag to reveal the contents. Ask students, "Based on what you see now, does this change your answer? Explain your reasoning."
The green bag should provide the best chance to win points, but it is not essential for the class to come to this understanding at this point.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support whole-class discussion as groups share which bag they would choose during the bonus round. Display sentence frames for students to use as they respond to each group: "I agree because ….” or "I disagree because ….” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.
Design Principle(s): Support sense-making


aligns: no