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: Create equations that describe numbers or relationships

Write and interpret numerical expressions.

Reason about and solve one-variable equations and inequalities.

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

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

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

Perform arithmetic operations on polynomials.

Represent and solve problems involving multiplication and division.

Understand the relationship between zeros and factors of polynomials.

Use properties of operations to generate equivalent expressions.

Use the four operations with whole numbers to solve problems.

Rewrite rational expressions.

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

Write expressions in equivalent forms to solve problems.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Add and subtract within 20.

Work with radicals and integer exponents.

Use polynomial identities to solve problems.

Generate and analyze patterns.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Represent and analyze quantitative relationships between dependent and independent variables.

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

Solve equations and inequalities in one variable.

Work with addition and subtraction equations.

Gain familiarity with factors and multiples.

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

Represent and solve equations and inequalities graphically.

Analyze patterns and relationships.

Represent and solve problems involving addition and subtraction.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Interpret the structure of expressions.

Multiply and divide within 100.

Solve systems of equations.
problem_activity: Activity
The purpose of this activity is to give students an opportunity to practice checking whether a particular value is a solution to an equation, and to recall properties of operations and equality that preserve the solution set of an equation.
This will be useful when students consider when and why equations have the same solution in the associated Algebra 1 lesson.
Monitor for students who:
check the value of
\(x\)
in only one expression and reason about the second expression
identify properties by name such as commutative, distributive, and associative
identify properties informally such as “changing the order doesn’t matter” or “you can add first and then multiply or multiply first and then add” or “you can multiply three or more terms in any order” or “you can add the same thing to each side”
manipulate the expressions
graph one or both expressions
Making graphing technology available gives students an opportunity to choose appropriate tools strategically.
Student Facing
For each pair of equations, decide whether the given value of
\(x\)
is a solution to one or both equations:
Is
\(x = 2\)
a solution to:
\(x(2 + 3) = 10\)
\(2x + 3x = 10\)
Is
\(x = 3\)
a solution to:
\(x - 4 = 1\)
\(4 - x = 1\)
Is
\(x = \text{-}2\)
a solution to:
\(7x = \text{-}14\)
\(x \boldcdot 14 = \text{-}28\)
Is
\(x = \text{-}1\)
a solution to:
\(x + 3 = 2\)
\(3 + x = 2\)
Is
\(x = \text{-5}\)
a solution to:
\(3 - x = 8\)
\(5 - x = 10\)
Is
\(x = (8 + 1)+3\)
a solution to:
\(\frac{12}{2} = \frac12(x)\)
\(18 = 2x\)
Is
\(x = 2\)
a solution to:
\(\frac{12}{x} = 6\)
\(6x = 12\)
Is
\(x = \frac{10}{3}\)
a solution to:
\(\text{-}1 + 3x = 9\)
\(9 = 3x - 1\)
Is
\(x = \frac12\)
a solution to:
\(5(x + 1) = \frac{15}{2}\)
\(5x + 1 = \frac{15}{2}\)
Student Response
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Activity Synthesis
Ask students to talk to their neighbor about some things they can look for, to see if two equations are likely to have the same solution or different solutions. If needed, draw students’ attention to which equations had the same solution and which had different solutions, perhaps by displaying them for all to see, grouped by whether the equations were equivalent or not.
Call on students to share their thinking and make an informal list displayed for all to see. It’s fine if students do not name all possible properties, or only describe them informally. They will have more opportunities to encounter them formally in their Algebra 1 course.
Remind students that
equivalent
equations have the same solution set. Use the language of
equivalent
interchangeably with “have the same solution,” and monitor for students to pick up on this vocabulary.


correct_option_index: 30