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

Use probability to evaluate outcomes of decisions.

Use random sampling to draw inferences about a population.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

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

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

Investigate patterns of association in bivariate data.

Draw informal comparative inferences about two populations.

Understand and evaluate random processes underlying statistical experiments.

Summarize and describe distributions.

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

Interpret linear models.

Develop understanding of statistical variability.

Calculate expected values and use them to solve problems.

Investigate chance processes and develop, use, and evaluate probability models.
problem_activity: Optional activity
In this activity, students compare and contrast different measures of center and variability for data sets that have gaps and are not symmetrical. They interpret mean, MAD, median, and IQR in the context of a situation. Unlike many of the data sets students have seen so far, this one shows values that could roughly divide into three parts: the days when there is little or no homework, the days when there is a moderate number of homework problems, and the days when the assignment is relatively large. Because of this distribution, finding a typical number of homework problems (or whether it would be helpful to identify a typical number) is not obvious, prompting students to interpret measures of center and spread more carefully (MP2).
As students work and discuss, identify at least one student or group that decides that the mean and MAD are appropriate measures of center and spread and can explain their reasoning, and another that decides to go with the median and IQR and could support their choice. Invite them to share during whole-class discussion.
Digital
Launch
Keep students in groups of 2. Give students a moment of quiet time to look at the data on homework problems and identify at least one thing they notice and one thing they wonder. Give them another brief moment to share their observation and question with their partner. Then, ask a few students to share their responses with the class.
Students are likely to notice that the data values are quite different, that there are some days with no homework and others with quite a few problems, that there is not an obvious cluster, or that the number of problems could be roughly grouped into three kinds (a little, moderate, and a lot). They are likely to wonder why the numbers are so spread out and varied.
Briefly discuss the following questions to encourage students to think about the data contextually:
“Why might the homework assignment data show this distribution? What are some possible explanations?” (When only one problem was assigned, the problem might be particularly challenging or might require considerable work or collaboration. Another possibility: there might be an upcoming exam, so the homework load was reduced. When many problems were assigned, the problems might be quick exercises with short answers, or the assignment might be review materials for an entire chapter.)
“How might we describe ‘a typical number of homework problems’ in this case?”
“Which do you predict would be higher: the mean or the median number of problems? Why?”
Next, give students 8–10 minutes to complete the task, either independently or collaboratively. Ask students to think quietly about the last question before discussing their response with their partner.
If students are using the digital activities, they will need to enter the data points in the column “A” for the applet to “list”, “sort”, etc. The applet allows for students to populate their own mean, Q1 and Q3 values.
Student Facing
Over a two-week period, Mai recorded the number of math homework problems she had each school day.
2
15
20
0
5
25
1
0
10
12
Calculate the following. Show your reasoning.
The mean number of math homework problems.
The mean absolute deviation (MAD).
Interpret the mean and MAD. What do they tell you about the number of homework problems Mai had over these two weeks?
Find or calculate the following values and show your reasoning.
The median, quartile, maximum, and minimum of the same data on Mai’s math homework problems.
The interquartile range (IQR).
Which pair of measures of center and variability—mean and MAD, or median and IQR—do you think summarize the distribution of Mai’s math homework assignments better? Explain your reasoning.
You may use the applet below to help if you choose to. Begin by dragging the left edge across the screen until you see only one column in the spreadsheet. Enter the values needed to calculate the IQR and the mean when prompted.
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
Keep students in groups of 2. Give students a moment of quiet time to look at the data on homework problems and identify at least one thing they notice and one thing they wonder. Give them another brief moment to share their observation and question with their partner. Then, ask a few students to share their responses with the class.
Students are likely to notice that the data values are quite different, that there are some days with no homework and others with quite a few problems, that there is not an obvious cluster, or that the number of problems could be roughly grouped into three kinds (a little, moderate, and a lot). They are likely to wonder why the numbers are so spread out and varied.
Briefly discuss the following questions to encourage students to think about the data contextually:
“Why might the homework assignment data show this distribution? What are some possible explanations?” (When only one problem was assigned, the problem might be particularly challenging or might require considerable work or collaboration. Another possibility: there might be an upcoming exam, so the homework load was reduced. When many problems were assigned, the problems might be quick exercises with short answers, or the assignment might be review materials for an entire chapter.)
“How might we describe ‘a typical number of homework problems’ in this case?”
“Which do you predict would be higher: the mean or the median number of problems? Why?”
Next, give students 8–10 minutes to complete the task, either independently or collaboratively. Ask students to think quietly about the last question before discussing their response with their partner.
If students are using the digital activities, they will need to enter the data points in the column “A” for the applet to “list”, “sort”, etc. The applet allows for students to populate their own mean, Q1 and Q3 values.
Student Facing
Over a two-week period, Mai recorded the number of math homework problems she had each school day.
2
15
20
0
5
25
1
0
10
12
Calculate the following. Show your reasoning.
The mean number of math homework problems
The mean absolute deviation (MAD)
Interpret the mean and MAD. What do they tell you about the number of homework problems Mai had over these two weeks?
Find or calculate the following values and show your reasoning.
The median, quartiles, maximum, and minimum of Mai’s data
The interquartile range (IQR)
Which pair of measures of center and variability—mean and MAD, or median and IQR—do you think summarizes the distribution of Mai’s math homework assignments better? Explain your reasoning.
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
Briefly discuss students’ interpretations of the measures they just calculated:
“What do the mean of 9 and MAD of 7.4 tell us? How can we interpret them in this context?”
“What do the median of 7.5 and IQR of 14 tell us?”
Then, select two or more previously identified students to share their responses about which measures of center and spread are appropriate for summarizing the data set. After each person shares, briefly poll the class to see if others reasoned about the measures the same way. Sum up by asking:
“Now that you have two pairs of measures of center and spread, how would you respond if someone asked you, ‘What is a typical number of homework problems for Mai’s class?’ Is the question easier to answer now?”
Students should walk away with increased awareness that, in some cases, measures of center and spread do not always paint a full picture of what the actual data set entails, and that the measures should be interpreted with care.


correct_option_index: 9