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: Analyze proportional relationships and use them to solve real-world and mathematical problems.

Understand ratio concepts and use ratio reasoning to solve problems.
problem_activity: Optional activity
In this activity, students explore percentages that describe parts of a whole. They find both
\(B\)
and
\(C\)
, where
\(A\%\)
of
\(B\)
is
\(C\)
, in the context of available and consumed water on a hike.
Students who use a double number line may notice that the value of
\(B\)
is the same in both questions, so the same double number line can be used to solve both parts of the problem. To solve the second question, however, the diagram needs to be partitioned with more tick marks.
As in the previous task, students may solve using other strategies, including by simply multiplying or dividing, i.e.,
\((1.5) \boldcdot 2 = 3.0\)
and
\(\frac{80}{100} \boldcdot (3.0) = 2.4\)
. Encourage them to also explain their reasoning with a double number line, table, or tape diagram. Monitor for at least one student using each of these representations.
Launch
Give students quiet think time to complete the activity and then time to share their explanation with a partner. Follow with a whole-class discussion.
Student Facing
During the first part of a hike, Andre drank 1.5 liters of the water he brought.
If this is 50% of the water he brought, how much water did he bring?
If he drank 80% of his water on his entire hike, how much did he drink?
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?
Decide if each scenario is possible.
Andre plans to bring his dog on his next hike, along with 150% as much water as he brought on this hike.
Andre plans to drink 150% of the water he brought on his hike.
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
After students create a double number line diagram with tick marks at 50% and 100%, some may struggle to know how to fit 80% in between. Encourage them to draw and label tick marks at 10% increments or work with a table instead. Some students may think that the second question is asking for the amount of water Andre drank on the second part of the hike. Clarify that it is asking for his total water consumption on the entire hike.
Activity Synthesis
Select 1–2 students who used a double number line, a table of equivalent ratios, and a tape diagram to share their strategies. As students explain, illustrate and display those representations for all to see.
Ask students how they knew what 100% means in the context. At this point it is not necessary for students to formally conceptualize the two ways percentages are used (to describe parts of whole, and to describe comparative relationships). Drawing their attention to concrete and contextualized examples of both, however, serves to build this understanding intuitively.
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Before students share their answers for the second question, present an incorrect answer and explanation. For example, “If Andre drank 80% of his water on his entire hike, then he drank 1.2 liters of water because 0.8 times 1.5 liters is 1.2 liters.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss in partners, listen for students who identify and clarify the assumption in the statement. For example, the author assumed that Andre brought 1.5 liters for his entire hike; however, the problem states that Andre drank 1.5 liters of the water he brought. This will remind students to carefully identify the quantity they are finding a percentage of.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness


correct_option_index: 1