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: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

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


Answer: 
No. This task does not involve an equation in two variables. There is only 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_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
problem_activity: Optional activity
Since this activity is mainly included for practice and may take some additional time to complete, it is included as an optional task and including it is up to the teacher’s discretion.
In this activity, students practice doing many trials of multi-step situations to estimate the probability of an event. In the discussion following the activity, students construct arguments (MP3) about how changes to the game might affect the probability of winning.
Launch
Arrange students in groups of 3. Provide each group with a paper bag containing 2 black blocks and 2 white blocks inside. If black and white blocks are not available, instruct students on their color equivalents. Give students 5 minutes to run the simulation, 5 minutes for partner discussion, then have a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer for recording their results and calculations of probability of winning Kiran’s game.
Supports accessibility for: Language; Organization
Student Facing
Kiran invents a game that uses a board with alternating black and white squares. A playing piece starts on a white square and must advance 4 squares to the other side of the board within 5 turns to win the game.
[IMG]
For each turn, the player draws a block from a bag containing 2 black blocks and 2 white blocks. If the block color matches the color of the next square on the board, the playing piece moves onto it. If it does not match, the playing piece stays on its current square.
Take turns playing the game until each person in your group has played the game twice.
Use the results from all the games your group played to estimate the probability of winning Kiran’s game.
Do you think your estimate of the probability of winning is a good estimate? How could it be improved?
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?
How would each of these changes, on its own, affect the probability of winning the game?
Change the rules so that the playing piece must move 7 spaces within 8 moves.
Change the board so that all the spaces are black.
Change the blocks in the bag to 3 black blocks and 1 white block.
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
The purpose of the discussion is for students to think about how changing the rules of the game might change the probability of winning.
Collect the data from the class for the number of wins and display the results for all to see.
Consider asking these discussion questions:
“Based on the class’s data, estimate the probability of winning the game.” (The theoretical probability of winning is
\(\frac{3}{16} \approx 0.19\)
.)
“Does the game seem too easy or hard to win? If so, how could Kiran change the game slightly to make it harder or easier?” (If it is too hard, move the pawn closer to the end or allow more than 5 moves to win.)
“The bag contained 2 black and 2 white blocks. If the bag had 4 blocks of each color, would that make it easier or harder to win?” (Neither. It would be the same difficulty since there is still an equal chance to get each color.)
“Do you think the estimate from the class’s data is a better estimate than the one you got on your own?” (Since there is more data from the entire class, it should be a better estimate than one individual’s.)
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Before students share their estimated probability of winning Kiran’s game, present an incorrect solution based on a misconception that arises when conducting a few trial simulations. For example: “The probability of winning Kiran’s game is zero because nobody in our group won in all of the six games played.” Ask students to identify the error, critique the reasoning, and revise the original statement. As students discuss in partners, listen for students who state that estimating the probability of winning based on only six trials will not result in an accurate estimate. Therefore, more trials must be conducted in order to improve the estimate of the probability of winning. This routine will engage students in meta-awareness as they critique and correct the language used to estimate the probability of winning a game.
Design Principles(s): Cultivate conversation; Maximize meta-awareness


aligns: no