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: Classify two-dimensional figures into categories based on their properties.

Understand similarity in terms of similarity transformations.

Find arc lengths and areas of sectors of circles.

Analyze, compare, create, and compose shapes.

Apply trigonometry to general triangles.

Define trigonometric ratios and solve problems involving right triangles.

Understand and apply the Pythagorean Theorem.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Prove theorems involving similarity.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Understand congruence in terms of rigid motions.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Visualize relationships between two-dimensional and three-dimensional objects.

Translate between the geometric description and the equation for a conic section.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Understand and apply theorems about circles.

Explain volume formulas and use them to solve problems.

Make geometric constructions.

Reason with shapes and their attributes.

Use coordinates to prove simple geometric theorems algebraically.

Experiment with transformations in the plane.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Apply geometric concepts in modeling situations.

Prove geometric theorems.

Solve real-world and mathematical problems involving area, surface area, and volume.
problem_activity: Activity
This activity is a culmination of all the work students have done with angles in this unit. With less support than in previous activities, students come up with equations that represent the relationships between angles in a figure. Then, students solve their equation to find each unknown angle measure.
Launch
Encourage students to write an equation for each problem. Give students 2–3 minutes of quiet work time followed by a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge by asking students to start by looking for any vertical, complementary and supplementary angles. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Find the unknown angle measures. Show your thinking. Organize it so it can be followed by others.
[IMG]
[IMG]
Lines
\(\ell\)
and
\(m\)
are perpendicular.
[IMG]
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The diagram contains three squares. Three additional segments have been drawn that connect corners of the squares. We want to find the exact value of
\(a+b+c\)
.
[IMG]
Use a protractor to measure the three angles. Use your measurements to conjecture about the value of
\(a+b+c\)
.
Find the exact value of
\(a+b+c\)
by reasoning about the diagram.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students struggle to see the angle relationships in the figures, prompt them to look for any angles that are vertical, complementary, or supplementary to get them started.
Activity Synthesis
The goal of this discussion is for students to see different equations that can be used to represent and solve for the same unknown angle measures.
Select students to share their answers to each problem. Consider asking some of the following questions:
“Did anyone use a different equation for this same problem? If so, did you get the same answer?”
“Were any of the questions harder than others? Why?”
“Were there any questions you used a strategy that was new to you?”
Speaking: MLR8 Discussion Supports.
When selected students share how they calculated the angle measurements, invite other students to challenge an idea, elaborate on an idea, or clarify the idea using improved mathematical language. Encourage students to demonstrate central concepts of the angle relationships (e.g., complementary, supplementary, vertical) multi-modally by explaining their reasoning using images of the angles as well as gestures. This will support student understanding about how to write equations that represent the relationships between angles in a figure.
Design Principle(s): Optimize output (for explanation); Support sense-making


correct_option_index: 16