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


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Thought: This problem applies trigonometric identities, including the angle addition formula for cosine.
Answer: A

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


Mathematical concept/skill options: 
A. Apply and extend previous understandings of arithmetic to algebraic expressions.
B. Write expressions in equivalent forms to solve problems.
C. Understand solving equations as a process of reasoning and explain the reasoning.
D. Use the four operations with whole numbers to solve problems.
E. Generate and analyze patterns.
F. Understand and apply properties of operations and the relationship between addition and subtraction.
G. Use properties of operations to generate equivalent expressions.
H. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
I. Represent and analyze quantitative relationships between dependent and independent variables.
J. Understand properties of multiplication and the relationship between multiplication and division.
K. Work with equal groups of objects to gain foundations for multiplication.
L. Represent and solve problems involving multiplication and division.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Work with radicals and integer exponents.
O. Interpret the structure of expressions.
P. Solve equations and inequalities in one variable.
Q. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
R. Multiply and divide within 100.
S. Represent and solve problems involving addition and subtraction.
T. Use polynomial identities to solve problems.
U. Represent and solve equations and inequalities graphically.
V. Understand the connections between proportional relationships, lines, and linear equations.
W. Reason about and solve one-variable equations and inequalities.
X. Rewrite rational expressions.
Y. Understand the relationship between zeros and factors of polynomials.
Z. Perform arithmetic operations on polynomials.
AA. Create equations that describe numbers or relationships
AB. Analyze and solve linear equations and pairs of simultaneous linear equations.
AC. Gain familiarity with factors and multiples.
AD. Write and interpret numerical expressions.
AE. Add and subtract within 20.
AF. Work with addition and subtraction equations.
AG. Solve systems of equations.
AH. Analyze patterns and relationships.


Thought: This task asks students to solve an equation in one variable. Since the problem asks for many different solution methods, students are also encouraged to solve the problem graphically.
Answer: P, U

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 $P$.


Mathematical concept/skill options: 
A. Apply trigonometry to general triangles.
B. Understand similarity in terms of similarity transformations.
C. Analyze, compare, create, and compose shapes.
D. Prove geometric theorems.
E. Make geometric constructions.
F. Graph points on the coordinate plane to solve real-world and mathematical problems.
G. Understand congruence and similarity using physical models, transparencies, or geometry software.
H. Understand congruence in terms of rigid motions.
I. Draw, construct, and describe geometrical figures and describe the relationships between them.
J. Translate between the geometric description and the equation for a conic section.
K. Reason with shapes and their attributes.
L. Prove theorems involving similarity.
M. Explain volume formulas and use them to solve problems.
N. Find arc lengths and areas of sectors of circles.
O. Define trigonometric ratios and solve problems involving right triangles.
P. Use coordinates to prove simple geometric theorems algebraically.
Q. Visualize relationships between two-dimensional and three-dimensional objects.
R. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
S. Understand and apply theorems about circles.
T. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
U. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
V. Classify two-dimensional figures into categories based on their properties.
W. Apply geometric concepts in modeling situations.
X. Solve real-world and mathematical problems involving area, surface area, and volume.
Y. Experiment with transformations in the plane.
Z. Understand and apply the Pythagorean Theorem.
AA. Draw and identify lines and angles, and classify shapes by properties of their lines and angles.


Thought: This task asks students to make geometric constructions, such as bisecting an angle.
Answer: E

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Use probability to evaluate outcomes of decisions.

Interpret linear models.

Use random sampling to draw inferences about a population.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Understand and evaluate random processes underlying statistical experiments.

Investigate chance processes and develop, use, and evaluate probability models.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Develop understanding of statistical variability.

Understand independence and conditional probability and use them to interpret data.

Calculate expected values and use them to solve problems.

Summarize, represent, and interpret data on a single count or measurement variable.

Summarize and describe distributions.

Investigate patterns of association in bivariate data.

Draw informal comparative inferences about two populations.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.
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
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?
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
Teachers with a valid work email address can
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: 11