few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with one variable, x.

Example 3:
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 $P$.


Concept/skill: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Answer: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
problem_activity: Activity
In this activity, students have the opportunity to design their own simulations that could be used to estimate probabilities of real-life events (MP4).  Students attend to precision (MP6) by assigning each possible outcome for the real-life experiment to a corresponding outcome in their simulation in such a way that the pair of outcomes have the same probability. In the discussion following the activity, students are asked to articulate how these simulations could be used to estimate probabilities of certain events.
Launch
Keep students in groups of 3. Give students 5 minutes quiet work time to design their own experiments, followed by small-group discussion to compare answers for the situations and whole-class discussion.
As students work, monitor for students who are using the same chance events for multiple scenarios (for example, always using a spinner) and encourage them to think about other ways to simulate the event.
Engagement: Provide Access by Recruiting Interest.
Leverage choice around perceived challenge. Invite students to select 2–3 of the situations to complete.
Supports accessibility for: Organization; Attention; Social-emotional skills
Student Facing
For each situation, describe a chance experiment that would fairly represent it.
Six people are going out to lunch together. One of them will be selected at random to choose which restaurant to go to. Who gets to choose?
After a robot stands up, it is equally likely to step forward with its left foot or its right foot. Which foot will it use for its first step?
In a computer game, there are three tunnels. Each time the level loads, the computer randomly selects one of the tunnels to lead to the castle. Which tunnel is it?
Your school is taking 4 buses of students on a field trip. Will you be assigned to the same bus that your math teacher is riding on?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may think that the number of outcomes in the sample space must be the same in the simulation as in the real-life situation. Ask students how we could use the results from the roll of a standard number cube to represent a situation with only two equally likely outcomes. (By making use of some extra options to count as "roll again.")
Activity Synthesis
The purpose of this discussion is for students to think more deeply about the connections between the real-life experiment and the simulation.
Select partners to share the simulations they designed for each of the situations.
Some questions for discussion:
"How could a standard number cube be used to simulate the situation with the buses?" (Each bus is assigned a number 1 through 4. If the cube ends on 5 or 6, roll again.)
"If one of the buses was numbered with your math teacher's favorite number and you wanted to increase the probability of that bus being selected, how could you change the simulation to do this?" (Add more of the related outcome. For example, using the standard number cube as in the previous discussion question, the bus with the favorite number could be assigned numbers 4 and 5 while the other buses are still 1 through 3.)
Two of the tunnels in the video game lead to a swamp that ends the game. How could you use your simulation to estimate the probability of choosing one of those two tunnels? (Since all of the tunnels are equally likely to lead to the swamp, it can be assumed that "left" and "right" lead to the swamp. Spin the spinner many times and use the fraction of times it ends on "left" or "right" to estimate the probability of ending the game. It should happen
\(\frac{2}{3}\)
or about 67% of the time.)
"You and a friend are among the people going to lunch. How could you use the simulation you designed to estimate the probability that you or your friend will be the one to choose the restaurant?" (My friend and I will be represented by 1 and 2 on a number cube. Roll the number cube a lot of times and find the fraction of times 1 or 2 appear, then estimate the probability that we will be the ones selected.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. After students share the simulations they designed, display the following sentence frames to help students respond: "I agree because ….” or "I disagree because ….” Encourage students to use mathematical language to support their response. This will support rich and inclusive discussion about how to simulate a real-world situation using a simple experiment that reflects the probability of the actual event.
Design Principle(s): Support sense-making, Cultivate conversation


aligns: no