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: Understand ratio concepts and use ratio reasoning to solve problems.

Analyze proportional relationships and use them to solve real-world and mathematical problems.
problem_activity: Activity
The purpose of this task is to introduce students to the idea of a proportional relationship. From previous work, students should be familiar with the idea of equivalent ratios, and they may very well recognize the table as a set of equivalent ratios. Here, we are starting to expand this concept and the language associated with it to say that there is a proportional relationship between the cups of rice and number of people as well as the number of spring rolls and number of people. More generally, there is a proportional relationship between two quantities when the quantities are characterized by a set of equivalent ratios.
The context in this activity and the numbers used are intended to be accessible to all students so that they can focus on its mathematical structure (MP7) and the new terms introduced in the lesson without being distracted.
While students are working, monitor for groups using each of these approaches:
Drawing that depicts 15 cups of rice and 45 people, organized into three people per cup. This representation of the problem can support all learners in moving forward. The teacher should highlight the organization of three people per cup, and identify correspondences between parts of the drawing and numbers in the table.
Moving down the table, multiplying both numbers in a row by the same number. “Since I multiply 2 by 5 to get 10, I will multiply 6 by 5 and get 30. Since 9 times 5 is 45, I will also multiply 3 by 5 to get 15.” If it comes up, the most appropriate name for the multiplier would be
scale factor
, but it is not necessary for students to call it this.
Calculation and use of a unit rate. “I reasoned that 1 cup of rice must serve 3 people. So 10 cups of rice must serve 30, and 15 cups must serve 45.”
Moving across the table. “It looks like we always multiply number of cups of rice by 3 to get the number of people it serves, so I will multiply 10 cups by 3 to get 30, and divide the 45 by 3 to get 15.”
Launch
Say, “Rice is a big part of the traditions and cultures of many families. Does your family cook rice, and if so, how?" Invite a student to describe the process (measure rice, measure water, simmer for a while). You use more rice for more people and less rice for fewer people. If students have trouble understanding or representing the context, show them the measuring cup so that they have a sense of its size, or draw a literal diagram that looks something like this:
[IMG]
Similarly, ask students if they have ever eaten a spring roll and invite them to describe what they are. While some spring rolls can be very large, the ones referred to in this activity are smaller.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because _____. Then, I….,” “I noticed _____ so I….,” and “I tried _____ and what happened was….”
Supports accessibility for: Language; Social-emotional skills
Student Facing
A recipe says that 2 cups of dry rice will serve 6 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
How many people will 10 cups of rice serve?
How many cups of rice are needed to serve 45 people?
| cups of rice | number of people |
|--------------|------------------|
|      2       |        6         |
|      3       |        9         |
|      10      |                  |
|              |        45        |
A recipe says that 6 spring rolls will serve 3 people. Complete the table.
| number of spring rolls | number of people |
|------------------------|------------------|
|           6            |        3         |
|           30           |                  |
|           40           |                  |
|                        |        28        |
Student Response
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Activity Synthesis
Select students to share their approaches using the sequence listed in the Activity Narrative.
After students have shared their reasoning and connections have been drawn between the different approaches, introduce the term
proportional relationship
. For example, “Whenever we have a situation like this where two quantities are always in the same ratio, we say there is a proportional relationship between the quantities. So the relationship between the number of cups of rice and the number of people is a proportional relationship. The number of cups of rice is proportional to the number of people.” Write out some of these statements for all to see (in the next activity, students will need to write a statement like these about a different relationship). If the class has a word wall or students keep track of mathematical vocabulary in their notebooks, these new terms can be included there.
Notice that methods 2 and 3 involve identifying relationships of column entries, namely that the entry in the right-hand column is three times the corresponding entry in the left-hand column. This pattern can be explained in terms of the
unit rate
(people per cup of rice), which tells us how many people we can serve with a given amount of rice. The next activity is intended to build the understanding underlying this explanation.
After discussing the rice context thoroughly, ask students to share their solution approaches to the spring roll context. Ask students to identify and interpret the unit rate in this situation (0.5 people per spring roll may sound strange, but it means that 1 spring roll only halfway satisfies a person.)


correct_option_index: 1