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: Analyze proportional relationships and use them to solve real-world and mathematical problems.

Understand ratio concepts and use ratio reasoning to solve problems.
problem_activity: Activity
This activity prompts students to compare and contrast two representations of equivalent ratios. Students work collaboratively to observe similarities and differences of using a double number line and using a table to express the same situation. Below are some key distinctions:
|                   double number line                   |                                                table                                                 |
|--------------------------------------------------------|------------------------------------------------------------------------------------------------------|
| Distances between numbers and lengths of lines matter. |                   Distances and lengths do not matter because there are no lines.                    |
|       The numbers on each line must be in order.       | Rows of ratios can be out of order; within a column, numbers can go in any order that is convenient. |
|       Each value of a ratio is shown on a line.        |                             Each value of a ratio is shown in a column.                              |
|   Pairs of values of a ratio are aligned vertically.   |                          Pairs of values of a ratio appear in the same row.                          |
You will need The International Space Station blackline master for this activity.
Launch
To help students build some intuition about kilometers, begin by connecting it with contexts that are familiar to them. Tell students that “kilometer” is a unit used in the problem. Then ask a few guiding questions.
“Can you name two things in our town (or city) that are about 1 kilometer apart?” (Consider finding some examples of 1-kilometer distances near your school ahead of time.)
“How long do you think it would take you to walk 1 kilometer?” (Typical human walking speed is about 5 kilometers per hour, so it takes a person about 12 minutes to walk 1 kilometer.)
“What might be a typical speed limit on a highway, in kilometers per hour?” (100 kilometers per hour is a typical highway speed limit. Students might be more familiar with a speed limit such as 65 miles per hour. Since there are about 1.6 kilometers in every mile, the same speed will be expressed as a higher number in kilometers per hour than in miles per hour.)
Arrange students in groups of 2. Give one person a slip with the table and the other a slip with a double number line (shown below). Ask students to first do what they can independently, and then to obtain information from their partners to fill in all the blanks. Explain that when the blanks are filled, the two representations will show the same information.
[IMG]
| distance traveled (km) | elapsed time (sec) |
|------------------------|--------------------|
|           0            |         0          |
|           80           |         10         |
|                        |         1          |
|                        |                    |
|                        |                    |
Student Facing
The International Space Station orbits around the Earth at a constant speed. Your teacher will give you either a double number line or a table that represents this situation. Your partner will get the other representation.
[IMG]
Complete the parts of your representation that you can figure out for sure.
Share information with your partner, and use the information that your partner shares to complete your representation.
What is the speed of the International Space Station?
Place the two completed representations side by side. Discuss with your partner some ways in which they are the same and some ways in which they are different.
Record at least one way that they are the same and one way they are different.
Student Response
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Student Facing
Are you ready for more?
Earth’s circumference is about 40,000 kilometers and the orbit of the International Space Station is just a bit more than this. About how long does it take for the International Space Station to orbit Earth?
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 with the double number line representation may decide to label every tick mark instead of just the ones indicated with dotted rectangles. This is fine. Make sure they understand that the tick marks with dotted rectangles are the ones they are supposed to record in the table.
Activity Synthesis
Display completed versions of both representations for all to see. Invite students to share the ways the representations are alike and different. Consider writing some of these on the board, or this could just be a verbal discussion. Highlight the distinctions in terms of distances between numbers, order of numbers, and the vertical or horizontal orientations of the representations.
Although it is not a structural distinction, students might describe the direction in which multiplying happens as a difference between the two representations. They might say that we “multiply up or down” to find equivalent ratios in a table, and we “multiply across” to do the same on a double number line. You could draw arrows to illustrate this fact:
[IMG]
The vertical orientation of tables and the horizontal orientation of double number lines are conventions we decided to consistently use in these materials. Mathematically, there is nothing wrong with orienting each representation the other way. Students may encounter tables oriented horizontally in a later course. Later in this course, they will encounter number lines oriented vertically.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, highlight the distinctions in terms of distances between numbers, order of numbers, and the vertical or horizontal orientations of the representations.
Supports accessibility for: Conceptual processing; Organization


correct_option_index: 1