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: Use complex numbers in polynomial identities and equations.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Reason quantitatively and use units to solve problems.

Extend the properties of exponents to rational exponents.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Apply and extend previous understandings of numbers to the system of rational numbers.

Perform operations on vectors.

Perform arithmetic operations with complex numbers.

Represent and model with vector quantities.

Compute fluently with multi-digit numbers and find common factors and multiples.

Represent complex numbers and their operations on the complex plane.

Perform operations on matrices and use matrices in applications.

Use properties of rational and irrational numbers.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
problem_activity: Activity
The purpose of this activity is to understand that the division facts for rational numbers are simply a consequence of the multiplication done previously. Students work several numerical examples relating multiplication to division, and then articulate a rule for the sign of a quotient based on the signs of the dividend and divisor (MP8).
Monitor for students who identify and describe the rule clearly.
Launch
Remind students that we can rearrange division equations to be multiplication equations, and vice versa. It may be useful to demonstrate with positive numbers if students struggle to recall this. For example, ask how we could rewrite
\(10 \div 2 = 5\)
as a multiplication equation. Students can also reference the multiplication table from a previous lesson, if needed.
Arrange students in groups of 2. Give students 4 minute of quiet work time followed by 2 minutes of partner discussion, then follow with whole-class discussion.
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, reference examples from the previous lesson on signed multiplication to provide an entry point into this activity.
Supports accessibility for: Social-emotional skills; Conceptual processing
Reading: MLR6 Three Reads.
Use this routine to support students’ reading comprehension of the final question. In the first read, students read the text with the goal of understanding what the situation is about (e.g., Han and Clare are walking toward each other). In the second read, ask students to name the important quantities (e.g., Han’s velocity, Clare’s velocity, each person's distance from 0 in feet, elapsed time in seconds). After the third read, ask students to brainstorm possible strategies to determine where each person will be 10 seconds before they meet up, and when each person will be
\(-10\)
feet from the meeting place. This will help students connect the language in the word problem with the reasoning needed to solve the problem.
Design Principle(s): Support sense-making
Student Facing
Find the missing values in the equations
\(\text-3 \boldcdot 4 = \text{?}\)
\(\text-3 \boldcdot \text{?} = 12\)
\(3 \boldcdot \text{?} = 12\)
\(\text{?} \boldcdot \text-4 = 12\)
\(\text{?} \boldcdot 4 = \text-12\)
Rewrite the unknown factor problems as division problems.
Complete the sentences. Be prepared to explain your reasoning.
The sign of a positive number divided by a positive number is always:
The sign of a positive number divided by a negative number is always:
The sign of a negative number divided by a positive number is always:
The sign of a negative number divided by a negative number is always:
Han and Clare walk towards each other at a constant rate, meet up, and then continue past each other in opposite directions. We will call the position where they meet up 0 feet and the time when they meet up 0 seconds.
Han's velocity is 4 feet per second.
Clare's velocity is -5 feet per second.
Where is each person 10 seconds before they meet up?
When is each person at the position -10 feet from the meeting place?
Student Response
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Student Facing
Are you ready for more?
It is possible to make a new number system using
only
the numbers 0, 1, 2, and 3. We will write the symbols for multiplying in this system like this:
\(1 \otimes 2 = 2\)
. The table shows some of the products.
| \(\otimes\) | 0 | 1 | 2 | 3 |
|-------------|---|---|---|---|
|      0      |   |   |   |   |
|      1      |   |   |   |   |
|      2      |   |   |   |   |
|      3      |   |   |   |   |
In this system,
\(1 \otimes 3 = 3\)
and
\(2 \otimes 3 = 2\)
. How can you see that in the table?
What do you think
\(2 \otimes 1\)
is?
What about
\(3\otimes 3\)
?
What do you think the solution to
\(3\otimes n = 2\)
is?
What about
\(2\otimes n = 3\)
?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Have students refer to their previous work with signed multiplication to help them with the division statements.
Activity Synthesis
Select students who can articulate and explain what they have learned about the sign of a quotient based on the signs of the dividend and divisor (note that students do not need to use these words, they can just give examples). Highlight that the division facts are simply a consequence of the multiplication done previously. Moving forward, we are going to take these facts to be true for all rational numbers.


correct_option_index: 1