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: Measure and estimate lengths in standard units.

Convert like measurement units within a given measurement system.

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

Describe and compare measurable attributes.

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

Represent and interpret data.

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

Work with time and money.

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

Geometric measurement: understand concepts of angle and measure angles.

Measure lengths indirectly and by iterating length units.

Relate addition and subtraction to length.

Tell and write time.

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

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
problem_activity: Narrative
The purpose of this activity is for students to learn stage 2 of the Estimate and Measure center. Students estimate and measure the length of objects. This stage includes centimeters, inches, and feet. Students should only measure in centimeters at this point in the unit. In this activity, students also find the difference between their estimate and their measurement for extra practice.
Required Materials
Materials to Gather
Metersticks
Objects of various lengths
Rulers (centimeters)
Materials to Copy
Estimate and Measure Stage 2 Recording Sheet
Required Preparation
Gather or identify objects of various lengths (pencils, markers, books, glue, scissors, shoe, tape dispenser, side of desk, length of bulletin board).
Launch
Groups of 2
Give students access to centimeter rulers and meter sticks.
Give each student a recording sheet.
“We are going to learn a new center called Estimate and Measure. Today we are going to estimate and measure the length of different objects.”
“First you and your partner will decide on an object in the room to measure. Then you will decide what tool you will use to measure the object.”
Choose an object to measure and hold it up for students to see.
“What tool would you use to measure this object?”
“Before measuring the object, you and your partner will both estimate the length of the object using the unit you chose. Your estimates do not have to be the same. Record your estimate.”
Demonstrate estimating the length of your object and recording it on the recording sheet.
“Now work with your partner to measure the object and record the actual measurement.”
Demonstrate measuring the object, thinking aloud as you line up the endpoints and use the number on the tool to determine the length.
Record the actual measurement on the recording sheet.
“If you have time: Once you have estimated and measured the object, find the difference between your estimate and the actual length.”
Demonstrate finding the difference.
“Put your object back and choose another object to measure.”
Activity
10 minutes: partner work time
Activity Synthesis
Display a row of the recording sheet with this information:
Object: notebook
Estimate: 20 cm
Actual Measurement: 31 cm
“How can we find the difference?” (We can add on to 20 until we get to 31. We can subtract 20 from 31.)


correct_option_index: 0