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: Solve real-world and mathematical problems involving area, surface area, and volume.

Prove geometric theorems.

Experiment with transformations in the plane.

Explain volume formulas and use them to solve problems.

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

Apply geometric concepts in modeling situations.

Understand similarity in terms of similarity transformations.

Use coordinates to prove simple geometric theorems algebraically.

Reason with shapes and their attributes.

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

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

Understand and apply the Pythagorean Theorem.

Prove theorems involving similarity.

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

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

Find arc lengths and areas of sectors of circles.

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

Apply trigonometry to general triangles.

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

Analyze, compare, create, and compose shapes.

Make geometric constructions.

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

Understand congruence in terms of rigid motions.

Understand and apply theorems about circles.

Define trigonometric ratios and solve problems involving right triangles.

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

Translate between the geometric description and the equation for a conic section.
problem_activity: Activity
In this activity students will apply their knowledge of trigonometry to a real-world context. They will use the safe ladder ratio to determine the safe ladder angle. Then they can use this angle to decide if a ladder is long enough for the given scenario.
Choose if students will use the ratio
\(4:1\)
or measure it themselves. Note that answers will vary if students measure. The image in the task is closer to
\(2:1\)
than
\(4:1\)
, so be prepared to discuss if that much variation is safe.
Launch
Tell students a ladder could tip over backwards if the angle is too large or slide down the wall if the angle is too small.
Provide students the safe ladder ratio by telling them: “A good rule of thumb for a safe angle to lean a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground. On most people that ratio is approximately
\(4:1\)
.”
Or if students will estimate the safe ladder ratio themselves, tell them: “A good rule of thumb for a safe angle to lean a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground. Form groups and measure the triangle one person’s body forms. That is, measure the distance from the floor to their shoulder. Then measure the distance from the front of their chest to their wrist.”
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to help students interpret the language of ratios and angles in context. Display the image and the first sentence of this problem, without revealing the questions that follow. Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions with a partner before selecting 1–2 students to share their questions with the class. Listen for and amplify any questions that connect the ratios and angles to the context.
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer collaboration. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills
Student Facing
A good rule of thumb for a safe angle to use when leaning a ladder is the angle formed by your body when you stand on the ground and hold your arms out parallel to the ground.
What are the angles in the triangle formed by your body and the ladder?
What are the angles in the triangle formed by the ladder, the ground, and the railing? Explain or show your reasoning.
You have a 13 foot long ladder and need to climb to a 12 foot tall roof.
If you put the top of the ladder at the top of the wall, what angle is formed between the ladder and the ground?
Is it possible to adjust the ladder to a safe angle? If so, give someone instructions to do so. If not, explain why not.
[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?
People have various proportions to their body. Suppose that someone’s height to arm ratio is
\(5:1\)
.
What are the angles in the triangle formed by their body and the ladder?
How far off is this from the
\(4:1\)
safe angle?
What could this person do to make the ladder closer to the safe ladder angle?
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
Some students might struggle to determine if the ladder can reach the roof. Inform them that the roof height has to be 12 feet but we can use as much of the ladder length as we want up to 13 feet. Invite them to draw a diagram where the ladder extends beyond the top of the building.
Activity Synthesis
Display the image for all to see and invite a student to demonstrate their explanation that the triangles are similar by annotating the image. Emphasize that all of trigonometry is based on similar right triangles. These are just two of the infinite set of right triangles with a
\(4:1\)
ratio and an angle measure of 76 degrees.


correct_option_index: 24