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: Geometric measurement: understand concepts of angle and measure angles.

Relate addition and subtraction to length.

Represent and interpret data.

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Tell and write time.

Work with time and money.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Convert like measurement units within a given measurement system.

Measure and estimate lengths in standard units.

Describe and compare measurable attributes.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Classify objects and count the number of objects in each category.

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

Measure lengths indirectly and by iterating length units.
problem_activity: Narrative
The purpose of this activity is for students to build any object they want to. In the next activity, students focus only ;on rectangular prisms. As students count, they will need to make sure to count each cube once and only once. Monitor for these strategies:
Touch each cube as it is counted
Count the cubes by layer (including when there are different numbers of cubes in each layer)
Use addition or multiplication when the cubes are organized in layers
Required Preparation
Each group of 2 needs 24 connecting cubes.
Launch
Groups of  2
Give 24 connecting cubes to each group.
“In this activity, you will use unit cubes to build objects and describe how you would measure the volume.”
10 minutes: partner work time
Activity
As students work, monitor for students who build rectangular prisms to share during the synthesis.
Student Facing
Partner A: Build an object using 8–12 cubes and give the object to Partner B.
Partner B: Explain how you would count the number of cubes in the object.
Partner A: Explain if you would count the cubes in the same way or in a different way.
Switch roles and repeat.
Which objects were easiest to count? Why?
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students are counting all of the cubes one at a time, ask, “where do you see cubes that could be counted as a group?”
Activity Synthesis
Discuss responses to the last problem.
“Which objects were most challenging to count?” (Objects where I had to count each individual cube because it was a weird shape.)
“Which objects were the simplest to count?” (Organized shapes like towers, rectangles, cubes, or boxes.)
Invite selected students to share the rectangular prisms they built and how they counted the number of cubes.
“How did you count the cubes?” (I knew that there were 4 cubes in each layer and 2 layers so that’s 8 cubes.)
“We call these objects
rectangular prisms
. What were some ways you used to count the cubes in rectangular prisms?” (I counted the cubes in groups and then skip counted. I used multiplication to find the total.)


correct_option_index: 4