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.


Your response:
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.


Your response:
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.


Your response:
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 expressions in equivalent forms to solve problems.

Interpret the structure of expressions.

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

Reason about and solve one-variable equations and inequalities.

Create equations that describe numbers or relationships

Analyze patterns and relationships.

Represent and solve problems involving multiplication and division.

Represent and solve problems involving addition and subtraction.

Generate and analyze patterns.

Represent and analyze quantitative relationships between dependent and independent variables.

Understand the relationship between zeros and factors of polynomials.

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

Solve equations and inequalities in one variable.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

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

Perform arithmetic operations on polynomials.

Multiply and divide within 100.

Gain familiarity with factors and multiples.

Solve systems of equations.

Work with radicals and integer exponents.

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

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

Work with addition and subtraction equations.

Use polynomial identities to solve problems.

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

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

Represent and solve equations and inequalities graphically.

Add and subtract within 20.

Write and interpret numerical expressions.

Use the four operations with whole numbers to solve problems.

Rewrite rational expressions.

Use properties of operations to generate equivalent expressions.

Understand and apply properties of operations and the relationship between addition and subtraction.
problem_activity: Narrative
In this activity, students analyze three different ways to subtract. They see that taking away is one way to find the difference, but that you can also count on or use known addition facts. Students further solidify their understanding that addition and subtraction are related, which sets the groundwork for a later activity when students solve subtraction problems within 10.
Required Materials
Materials to Gather
Connecting cubes or two-color counters
Launch
Groups of 2
Give students access to connecting cubes or two-color counters.
Activity
Read the task statement.
“First you will work on your own. Think about what each student means and be ready to explain your thinking in a way that others will understand.”
5 minutes: independent work time
4 minutes: partner discussion
Monitor for students who can use the 10-frame with six red counters to explain the relationship between
\(10 - 6\)
and
\(6 + \boxed{\phantom3} = 10\)
.
Student Facing
Mai, Diego, and Noah find the value of
\(10 - 6\)
.
Diego says, “I can take away.”
[IMG]
What does Diego mean?
Be ready to explain your thinking in a way that others will understand.
Mai says, “I can count on.”
[IMG]
What does Mai mean?
Be ready to explain your thinking in a way that others will understand.
Noah says, “I can use what I know about
\(6 + 4\)
to help me.”
What does Noah mean?
Be ready to explain your thinking in a way that others will understand.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Activity Synthesis
Invite previously identified students to share.
“Who can restate what _____ just showed us?” (Diego subtracted by taking away 6 counters one at a time and saw that there were 4 counters left. Mai subtracted by thinking about addition. She counted on from 6 until she got to 10 and noticed she counted up 4. Noah knows his sums of 10. He knows 10 can be made by 6 and 4, so
\(10 - 6 = 4\)
.)
“Which method do you like best?” (I know my sums to 10 so I would use that. I like counting on because I like to add more than take away.)


correct_option_index: 28

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