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.

Develop understanding of statistical variability.

Draw informal comparative inferences about two populations.

Understand and evaluate random processes underlying statistical experiments.

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

Use random sampling to draw inferences about a population.

Calculate expected values and use them to solve problems.

Use probability to evaluate outcomes of decisions.

Summarize and describe distributions.

Investigate patterns of association in bivariate data.

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

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

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

Interpret linear models.

Understand independence and conditional probability and use them to interpret data.
problem_activity: Activity
In this activity, students are asked to interpret in context the slope and vertical intercept of a linear model given a scatter plot and the equation for a linear model that fits the data well. The linear model is also used to interpolate and extrapolate information about the data in context.
Monitor for students who:
estimate values based on the graph
use the equation for the linear model to find the values
Launch
Arrange students in groups of 2.
Student Facing
The scatter plot shows the sale price of several food items,
\(y\)
, and the cost of the ingredients used to produce those items,
\(x\)
, as well as a line that models the data. The line is also represented by the equation
\(y = 3.48x + 0.76\)
.
[IMG]
What is the predicted sale price of an item that has ingredients that cost \$1.50? Explain or show your reasoning.
What is the predicted ingredient cost of an item that has a sale price of \$7? Explain or show your reasoning.
What is the slope of the linear model? What does that mean in this situation?
What is the
\(y\)
-intercept of the linear model? What does this mean in this situation? Does this make sense?
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
Although the grid in the picture does not look like squares, each line in both the horizontal and vertical directions are 0.5 apart. Tell students who are confused to look at the values listed on the axes and identify the values attached to a few of the grid lines.
Activity Synthesis
The purpose of this discussion is for students to describe where to find the slope and vertical intercept from a scatter plot and interpret the values in terms of the context.
Select previously identified students to share in the order listed in the narrative. Ask students:
“Are the values obtained from looking at the graph close to the values calculated using the equation?” (Yes, the values are close.)
“Which method will you use to predict values based on the model?” (Maybe a mixture of both methods. Using the equation produces a more precise value from the model, but it is also important to look at the graph to get a sense of how well the model fits the data where I am predicting.)
“Why do you think the intercept for the model is not
\((0,0)\)
?” (Companies may still try to charge money for items that cost nothing to produce. They need to pay for the salaries of their workers, research and development, as well as other things that require the company to make money more than just the cost of making the item. It may also be the case that it does not make sense to interpret the
\(y\)
-intercept, since we have no evidence to believe that the same linear relationship will hold there.)
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Before students share their interpretations of the slope, present an ambiguous explanation. For example: “The slope means that as sales increase, the cost of ingredients increases.” Ask students to identify inaccuracies, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe what happens to the sale price when the cost of ingredients increases by
$
1. This helps students evaluate, and improve on, the written mathematical arguments of others, as they clarify the meaning of slope in context.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Action and Expression: Internalize Executive Functions.
Provide students with a two-column graphic organizer. Label one column “graph” and another column “equation.” Ask students to transfer their answers into the corresponding column on their organizer. When the whole group shares, they can continue to use the organizer capturing the answer if anyone used the alternative method for each question.
Supports accessibility for: Language; Organization


correct_option_index: 13