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: Reason about and solve one-variable equations and inequalities.

Add and subtract within 20.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Create equations that describe numbers or relationships

Work with equal groups of objects to gain foundations for multiplication.

Rewrite rational expressions.

Understand the connections between proportional relationships, lines, and linear equations.

Represent and solve problems involving addition and subtraction.

Interpret the structure of expressions.

Multiply and divide within 100.

Use the four operations with whole numbers to solve problems.

Write and interpret numerical expressions.

Work with radicals and integer exponents.

Understand solving equations as a process of reasoning and explain the reasoning.

Generate and analyze patterns.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Understand the relationship between zeros and factors of polynomials.

Gain familiarity with factors and multiples.

Represent and analyze quantitative relationships between dependent and independent variables.

Understand properties of multiplication and the relationship between multiplication and division.

Represent and solve equations and inequalities graphically.

Understand and apply properties of operations and the relationship between addition and subtraction.

Write expressions in equivalent forms to solve problems.

Use properties of operations to generate equivalent expressions.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Solve systems of equations.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Solve equations and inequalities in one variable.

Use polynomial identities to solve problems.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Represent and solve problems involving multiplication and division.

Analyze patterns and relationships.

Work with addition and subtraction equations.

Perform arithmetic operations on polynomials.
problem_activity: Narrative
The purpose of this activity is for students to practice interpreting relationships between patterns generated from two different rules. Students may need to generate patterns beyond the boxes provided. Encourage them to continue the pattern as needed. Students may describe the patterns and relationships in different, but accurate ways. Encourage them to notice as many relationships as they can and describe the relationships in whatever way makes sense to them. This is the first time students see two patterns where the numbers in one pattern are not multiples of the numbers in the other pattern.
This activity uses
MLR7 Compare and Connect
. Advances: Representing, Conversing.
Launch
Groups of 2
Activity
5 minutes: independent work time
2 minutes: partner discussion
MLR7 Compare and Connect
“Create a visual display that shows your thinking about the relationships between each set of patterns. Include details such as notes, diagrams, and drawings to help others understand your thinking.”
2–5 minutes: independent or group work
3–5 minutes: gallery walk
Student Facing
Complete the patterns for each set of rules.
What are some relationships between the patterns for each set of rules? Be prepared to explain your thinking.
Set A
Rule 1: Start at 0. Keep adding 3.
[IMG]
Rule 2: Start at 0. Keep adding 6.
[IMG]
Set B
Rule 1: Start at 4. Keep adding 3.
[IMG]
Rule 2: Start at 9. Keep adding 6.
[IMG]
Set C
Rule 1: Start at 0. Keep adding 5.
[IMG]
Rule 2: Start at 0. Keep adding 3.
[IMG]
Student Response
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Activity Synthesis
“What is the same and what is different in the way we represented the relationships between the patterns?” (Some of us used numbers and symbols, some of us wrote sentences.)
30 seconds: quiet think time
1 minute: partner discussion
“What is the same and what is different about Set A and Set B?” (They both add the same amount, but one set starts at 0. Set B has a lot more odd numbers in it. In set A, each number in rule 2 is double the number in rule 1, but in set B, each number in rule 2 is 1 more than double the number in rule 1.)
“For which pair of rules was it most challenging to notice and describe relationships?” (Set B because there was no multiple that worked to get from one pattern to the other. Set C because there weren’t doubles or halves.)


correct_option_index: 31