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: Represent and analyze quantitative relationships between dependent and independent variables.

Perform arithmetic operations on polynomials.

Solve systems of equations.

Analyze patterns and relationships.

Create equations that describe numbers or relationships

Write expressions in equivalent forms to solve problems.

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

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

Reason about and solve one-variable equations and inequalities.

Add and subtract within 20.

Represent and solve problems involving multiplication and division.

Interpret the structure of expressions.

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

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

Generate and analyze patterns.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Rewrite rational expressions.

Multiply and divide within 100.

Gain familiarity with factors and multiples.

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

Work with addition and subtraction equations.

Represent and solve problems involving addition and subtraction.

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.

Use properties of operations to generate equivalent expressions.

Use polynomial identities to solve problems.

Write and interpret numerical expressions.

Represent and solve equations and inequalities graphically.

Understand the relationship between zeros and factors of polynomials.

Work with radicals and integer exponents.

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

Use the four operations with whole numbers to solve problems.

Solve equations and inequalities in one variable.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
problem_activity: Narrative
The purpose of this activity is for students to apply conservation of length by measuring with a tool that does not start at zero. Until now, students have focused on measuring by lining up the edge of an object with zero. In this activity, students consider how the units on a ruler, and in later lessons, a number line, can help us solve problems and measure even when we are not starting at zero. It is important for students to recognize that the tick marks show a consistent distance between numbers, so any part of the ruler can show a specific number of units. The synthesis focuses on how addition (counting on from the starting number) and subtraction can be used to find the length of an object. The teacher records students’ thinking with an equation with a question mark for the unknown.
It is recommended that students use rulers to measure the length of smaller objects (2
–
10 inches) in this activity. This smaller number range helps students connect what they notice about measuring starting at different numbers on the ruler to familiar addition and subtraction equations within 20. However, this activity can be adapted to use measuring tapes or yardsticks to measure longer objects if they are available.
When students measure objects starting at different non-zero spots on the ruler and observe what stays the same (the difference) and what changes (the starting and ending spots) they observe a structure which repeats itself no matter which spot they on the ruler they begin the measurement (MP7, MP8).
Action and Expression: Develop Expression and Communication
. Give students access to inch tiles to double check their measurement. Reiterate how the measurement is the same regardless of the whole inch they start at on the ruler.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Required Materials
Materials to Gather
Objects of various lengths
Rulers (inches)
Required Preparation
Each group of 2 needs access to several objects between 2
–10 inches long. Consider using classroom objects such as markers, colored pencils, paperclips, or other objects or images used in previous activities.
Launch
Groups of 2
Give each student a ruler.
Display the image.
Activity
“Han measured the same notebook as Jada. Jada’s measuring tape was torn, but she got the same answer. How did that happen?”
1 minute: quiet think time
“Work with a partner to figure out how Jada got the same answer as Han. Be prepared to explain your thinking.”
3–4 minutes: partner discussion
Monitor for a student who explains the problem as
\(10 - 2 = {?}\)
and one who explains the problem as
\(2 + {?} = 10\)
.
Share responses.
“Now, using your ruler, you and your partner should find 1 thing in the classroom to measure without starting at zero. Each of you should start at a different number and compare your measurements.”
“Don’t forget to include the unit with your measurement and to use a question mark for the unknown in your equation.”
5 minutes: partner work time
Monitor for students who discuss:
that the length must be the same because the object doesn’t change
the equations they write have the same value or show the same relationship
the relationship between the numbers in their different expressions (for example, “the numbers I subtracted are two more than yours because I moved the object 2 inches over on the ruler. The difference is the same.”)
Student Facing
Jada and Han used an inch ruler to measure the short side of a notebook.
[IMG]
Han says 8 inches.
[IMG]
Jada says 8 inches.
How did Han and Jada get the same measurement?
Write an equation that could show Jada’s thinking.
Measure an object using Jada’s method.
I measured a
________________________________
.
I started with the number
____________
.
I ended with the number
____________
.
Equation: ________________________________.
The length of my object is
____________
.
What do you notice about you and your partner’s measurements?
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
Students may find the correct length when counting length units on the measuring tool, but identify the length as the total number if they represent their measurement with an addition equation. (For example, they count on from 2 inches to 10 inches on the ruler and write
\(2 + 8 = 10\)
, but say the length of the object is 10). Consider asking:
“How did you measure the ___?”
“What do the numbers in your equation represent?”
Activity Synthesis
Invite selected students to share their measurements and their reasoning.
Write equations that correspond to the students’ thinking using a question mark to represent the unknown value.
“Why do you think you get the same measurement each time, even if you measure starting from a different place?” (The object didn't change, so the length has to stay the same. The total length in inches is still the same between the number that you start with and end with.)


correct_option_index: 9