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: 
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.

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: 
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, 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_pos_0

HS_2
standard_description: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
problem_activity: Activity
The mathematical purpose of this activity is for students to create and analyze a distribution and estimate the mean for rolls of a number cube. Students should recognize the distribution of sample means to be approximately normal which will be important in understanding the margin of error attached to estimates.
Digital
Launch
Arrange students in groups of 4. Collect the data from each student or group to create a dot plot for the class data.
Provide access to devices that can run GeoGebra or other statistical technology.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Students can work with a partner to complete the trials by taking turns rolling the number cube and recording the results.
Supports accessibility for: Language; Social-emotional skills
Student Facing
Roll your number cube applet 35 times, recording the values as you do so.
Every 5 values, find the mean.
| rolls | 1 through 5 | 6 through 10 | 11 through 15 | 16 through 20 | 21 through 25 | 26 through 30 | 31 through 35 |
|-------|-------------|--------------|---------------|---------------|---------------|---------------|---------------|
| mean  |             |              |               |               |               |               |               |
Share your means with your group and create a dot plot of all the means from your group.
What do you notice about the shape of the distribution of means?
Using the dot plot of means, what do you think is a good estimate for the mean of all 140 rolls from your group? How does this value compare to your estimate from the warm-up?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 4. Distribute one number cube to each student. Collect the data from each student or group to create a dot plot for the class data.
Provide access to devices that can run GeoGebra or other statistical technology.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Students can work with a partner to complete the trials by taking turns rolling the number cube and recording the results.
Supports accessibility for: Language; Social-emotional skills
Student Facing
Roll your number cube 35 times, recording the values as you do so.
Every 5 values, find the mean.
| rolls | 1 through 5 | 6 through 10 | 11 through 15 | 16 through 20 | 21 through 25 | 26 through 30 | 31 through 35 |
|-------|-------------|--------------|---------------|---------------|---------------|---------------|---------------|
| mean  |             |              |               |               |               |               |               |
Share your means with your group and create a dot plot of all the means from your group.
[IMG]
What do you notice about the shape of the distribution of means?
Using the dot plot of means, what do you think is a good estimate for the mean of all 140 rolls from your group? How does this value compare to your estimate from the warm-up?
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
It can be easy for students to get confused about find the standard deviation and mean of mean values. Encourage students to slow down in their thinking and examine one level at a time. The dot plot consists of mean values and the distribution can be described by its center and spread by using the mean and standard deviation.
Activity Synthesis
The goal of this discussion is for students to think about how sample means usually produce an approximately normal distribution with some variability. Collect all of the means from the class and create a dot plot. If it does not come up, ask students to describe the distribution shape. It should be approximately normal (bell-shaped and symmetric around 3.5).
Here are some questions for discussion:
“Why do you think the mean is approximately 3.5?” (The center of the distribution is near 3.5.)
“Which interval contains most of the means for your group’s data?” (Most of our means were between 2.8 and 4.2.)
“What percentage of the means are in that interval?” (21 out 28 or approximately 78%)
“Estimate the percentage of the class data that is in this interval?” (It looks like close to 90% of the means are in this interval.)
“How does the class distribution compare to your group’s distribution?” (The class distribution is more bell-shaped and more symmetric.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


aligns: yes