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: Write and interpret numerical expressions.

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

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

Write expressions in equivalent forms to solve problems.

Reason about and solve one-variable equations and inequalities.

Rewrite rational expressions.

Analyze patterns and relationships.

Generate and analyze patterns.

Represent and solve problems involving multiplication and division.

Solve systems of equations.

Multiply and divide within 100.

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

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

Represent and solve equations and inequalities graphically.

Interpret the structure of expressions.

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

Work with addition and subtraction equations.

Use properties of operations to generate equivalent expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Gain familiarity with factors and multiples.

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

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

Perform arithmetic operations on polynomials.

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

Work with radicals and integer exponents.

Use polynomial identities to solve problems.

Use the four operations with whole numbers to solve problems.

Add and subtract within 20.

Create equations that describe numbers or relationships

Understand the relationship between zeros and factors of polynomials.

Solve equations and inequalities in one variable.

Represent and solve problems involving addition and subtraction.
problem_activity: Narrative
The purpose of this activity is for students to solve two-step word problems using all four operations. Students should be encouraged to solve the problem first or write the equation first, depending on their preference. Encourage students to use parentheses if needed to show what is being done first in their equations.
When students make sense of situations to solve two-step problems they reason abstractly and quantitatively (MP2).
MLR5 Co-Craft Questions.
Keep books or devices closed. Display only the problem stem, without revealing the question. Give students 2–3 minutes to write a list of mathematical questions that could be asked about this situation, before comparing their questions with a partner. Invite each group to contribute one written question to a whole-class display. Ask the class to make comparisons among the shared questions and their own. Reveal the intended questions for this task and invite additional connections.
Advances: Reading, Writing
Engagement: Develop Effort and Persistence.
Some students may benefit from feedback that emphasizes effort and time on task. For example, check in and give feedback after each party planning problem.
Supports accessibility for: Attention
Required Materials
Materials to Gather
Base-ten blocks
Materials to Copy
Centimeter Grid Paper - Standard
Launch
Groups of 2
Give students access to grid paper and base-ten blocks.
Activity
“Work independently to solve these problems and write an equation with a letter for the unknown quantity to represent each situation. You can choose to solve the problem first or write the equation first.”
5–7 minutes: independent work time
“Share your solutions and your equations with your partner. Also, tell your partner if you think their solutions and equations make sense or why not.”
5–7 minutes: partner discussion
Student Facing
For each problem:
a. Write an equation to represent the situation. Use a letter for the unknown quantity.
b. Solve the problem. Explain or show your reasoning.
[IMG]
Kiran is making paper rings each day to decorate for a party. From Monday to Thursday he was able to complete 156 rings. Friday, Kiran and 2 friends worked on making more rings. Each of them made 9 more rings. How many rings did they make over the week?
Mai has 168 muffins. She put 104 of the muffins in a basket. She packed the rest of the muffins into 8 boxes with the same number of muffins. How many muffins were in each box?
There are 184 cups on a table. Three tables with 8 people at each table come up to get drinks and each use a cup. How many cups are on the table now?
Student Response
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Advancing Student Thinking
If students don't write a single equation to represent both steps of the problem, consider asking:
“What equations did you write for each part of the problem?”
“How could you combine your equations into one equation that would represent the problem?”
Activity Synthesis
For each problem have a student share their equation and discuss how it represents the problem.
Consider asking:
“Where do we see ______ from the problem in the equation?”
“What information from the situation did we need to solve and write our equation?”
“How are parentheses used in the equation?”


correct_option_index: 23