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


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: 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. Solving this problem does not require use of the Pythagorean identity.
Answer: B

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


Standard options: 
A. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
B. Solve quadratic equations in one variable.


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

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 $P$.


Standard options: 
A. 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.
B. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.


Thought: 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.
Answer: A

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

Summarize numerical data sets in relation to their context, such as by:
- Reporting the number of observations.
- Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
- Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
- Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
problem_activity: Activity
In the previous activity, students evaluated the performance of three students based on the mean and variability. Here they learn the term
mean absolute deviation (MAD)
as a way to quantify variability and calculate it by finding distances between the mean and each data value. Students compare data sets with the same mean but different MADs and interpret the variations in context.
While this process of calculating MAD involves taking the absolute value of the difference between each data point and the mean, this formal language is downplayed here. Instead, the idea of “finding the distance,” which is always positive, is used. This is done for a couple of reasons. One reason is to focus students' attention on the statistical work rather than on terminology or symbolic work. Another reason is that finding these differences may involve operations with signed numbers, which are not expected in grade 6.
Teacher Notes for IM 6–8 Accelerated
The activity Shooting Hoops (Part 2) and the activity referred to in the launch are not included in IM 6–8 Math Accelerated.
During the launch, ask students what they think will be true about distances between data points and the mean, knowing that the mean can be considered a balance point. (The sum of the distances to the left of the mean is the same as the sum of the distances to the right of the mean.) Then tell students that the distances can also tell us something else about a distribution that will be explored in this activity.
Launch
Remind students in the last lesson they found distance between each data point and the mean, and found that the sum of those distances on the left and the sum on the right were equal, which allows us to think of the mean as the balancing point or the center of the data. Explain that the distance between each point and the mean can tell us something else about a distribution.
Arrange students in groups of 2. Give students 4–5 minutes to complete the first set of questions with their partner, and then 4–5 minutes of quiet time to complete the remaining questions. Follow with a whole-class discussion.
Student Facing
The tables show Elena, Jada, and Lin’s basketball data from an earlier activity. Recall that the mean of Elena’s data, as well as that of Jada and Lin’s data, was 5.
Record the distance between each of Elena’s scores and the mean.
|      **            Elena           **      | 4 | 5 | 1 | 6 | 9 | 7 | 2 | 8 | 3 | 3 | 5 | 7 |
|--------------------------------------------|---|---|---|---|---|---|---|---|---|---|---|---|
| **            distance from 5           ** | 1 |   |   | 1 |   |   |   |   |   |   |   |   |
Now find
the average of the distances
in the table. Show your reasoning and round your answer to the nearest tenth.
This value is the
mean absolute deviation (MAD)
of Elena’s data.
Elena’s MAD: _________
Find the mean absolute deviation of Jada’s data. Round it to the nearest tenth.
|      **            Jada           **       | 2 | 4 | 5 | 4 | 6 | 6 | 4 | 7 | 3 | 4 | 8 | 7 |
|--------------------------------------------|---|---|---|---|---|---|---|---|---|---|---|---|
| **            distance from 5           ** |   |   |   |   |   |   |   |   |   |   |   |   |
Jada’s MAD: _________
Find the mean absolute deviation of Lin’s data. Round it to the nearest tenth.
|       **            Lin           **       | 3 | 6 | 6 | 4 | 5 | 5 | 3 | 5 | 4 | 6 | 6 | 7 |
|--------------------------------------------|---|---|---|---|---|---|---|---|---|---|---|---|
| **            distance from 5           ** |   |   |   |   |   |   |   |   |   |   |   |   |
Lin’s MAD: _________
Compare the MADs and dot plots of the three students’ data. Do you see a relationship between each student’s MAD and the distribution on her dot plot? Explain your reasoning.
[IMG]
Student Response
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Student Facing
Are you ready for more?
Invent another data set that also has a mean of 5 but has a MAD greater than 2. Remember, the values in the data set must be whole numbers from 0 to 10.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may recall the previous lesson about thinking of the mean as a balance point and think that the MAD should always be zero since the left and right distances should be equal. Remind them that distances are always positive, so finding the average of these distances to the mean can only be zero if all the data points are exactly at the mean.
Activity Synthesis
During discussion, highlight that finding how far away, on average, the data points are from the mean is a way to describe the variability of a distribution. Discuss:
“What can we say about a data set whose data points have very small distances from the mean?”
“What about a data set with points that show large distances from the mean?”
“Does a data set with smaller distances (and therefore smaller average distances) show less or more variability?”
“What do MAD values of 2, 1.5, and 1 mean in this context?”
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary.  Include the following term and maintain the display for reference throughout the unit: mean absolute deviation (MAD). Invite students to suggest language or diagrams to include on the display that will support their understanding of this term.
Supports accessibility for: Memory; Language
Listening, Speaking, Representing: MLR8 Discussion Supports.
To provide a support as students explain their reasoning, invite students to use the sentence frame: “The largest (or smallest) mean absolute deviation value matches with the dot plot with the _____ spread because . . .”. This will help students produce explanations connecting visual and numeric representations of data distribution.
Design Principle(s): Support sense-making


correct_option_index: 1