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: 
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).

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

HS_1

im_task_001160_pos_0

HS_2
standard_description: Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
problem_activity: Activity
In this activity, students calculate the mean of a data set and interpret it in the context of the given situation. The first data set students see here has a dozen values, discouraging students from redistributing the values incrementally and encouraging them to use a more efficient method. In the second question, students analyze the values in data sets and use the structure (MP7) to decide whether or not it makes sense that a given mean would match the data set.
As students work and discuss, notice the reasons they give for why the data sets in the second question could or could not be Tyler's data set. Identify students who recognize that the mean of a data set cannot be expected to be higher or lower than most of the values of the data set, and that a fair-share value would have a value that is roughly in the middle of data values.
Launch
Keep students in groups of 2. Give them 6–7 minutes of quiet work time, and then time to discuss their responses with their partner.
Representation: Internalize Comprehension.
Activate or supply background knowledge about determining means of data sets. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
For the past 12 school days, Mai has recorded how long her bus rides to school take in minutes. The times she recorded are shown in the table.
9
8
6
9
10
7
6
12
9
8
10
8
Find the mean for Mai’s data. Show your reasoning.
In this situation, what does the mean tell us about Mai’s trip to school?
For 5 days, Tyler has recorded how long his walks to school take in minutes. The mean for his data is 11 minutes. Without calculating, predict if each of the data sets shown could be Tyler’s. Explain your reasoning.
data set A: 11, 8, 7, 9, 8
data set B: 12, 7, 13, 9, 14
data set C: 11, 20, 6, 9, 10
data set D: 8, 10, 9, 11, 11
Determine which data set is Tyler’s. Explain how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Select a couple of students to share how they found the mean of Mai's travel times. Poll the class briefly to see if others in the class found the mean the same way.
Then, focus the discussion on the second task and on what values could be reasonably expected of a data set with a particular mean. Ask students how they decided to rule out or keep certain sets of data as potentially belonging to Tyler. If not mentioned by students, highlight that the mean of a data set would be a value in the middle of the range of numbers in order for it to be a fair-share value.
Point out that, unlike hours of work or cats in crates, the times of travel here cannot actually be redistributed. The interpretation of mean translates into a thought experiment:
The mean is the travel time each day if all the travel times in the set were the same such that the combined travel time for this data set and the original data set was the same.
Speaking, Listening: MLR8 Discussion Supports.
Use this routine to support students’ explanations of which data set is Tyler’s. Invite students to use the sentence frame “Data set ___ is Tyler’s because . . . .” Invite listeners to use the sentence frame “I agree/disagree, because . . . .” This will help students explain their own reasoning and make the sense of the reasoning of others.
Design Principle(s): Support sense-making; Optimize output (for justification


aligns: no