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 decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Use equivalent fractions as a strategy to add and subtract fractions.

Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
problem_activity: Narrative
The purpose of this activity is for students to compare two numbers in context, to explain or show their reasoning, and record the results of the comparisons with the symbols >, =, or < (MP2). The numbers may be fractions with the same numerator or the same denominator, or a fraction and a whole number.
Students are likely to generate different comparison statements for the same situation. For example, they may write
\(\frac{5}{8} > \frac{3}{8}\)
or
\(\frac{3}{8}<\frac{5}{8}\)
to represent
\(\frac{5}{8}\)
being the greater fraction. During synthesis, discuss how both statements capture the comparison and are valid.
MLR8 Discussion Supports.
Synthesis: As students share the similarities and differences between the representations and comparison statements, use gestures to emphasize what is being described. For example, show with your fingers the partitions such as fourths or eighths that are the same in the representations that are being compared.
Advances: Listening, Representing
Engagement: Provide Access by Recruiting Interest.
Synthesis: Optimize meaning and value. Invite students to share a connection between activity content and their own lives. Ask “How can I use this in my own life?”
Supports accessibility for: Conceptual Processing
Launch
Groups of 2
“Let’s use what we learned about comparing fractions and recognizing equivalent fractions to solve problems about lengths.”
Activity
“Work independently to solve the problems. For each one, be sure to show your thinking and to write a comparison statement.”
6–8 minutes: independent work time
“Share your responses and reasoning with your partner.”
2–3 minutes: partner discussion
Monitor for:
different representations or reasoning strategies used for the same problem, such as diagrams, fraction strips, number lines, or explanations in words
different statements written for the same problem, such as
\(4=\frac{12}{3}\)
and
\(\frac{12}{3}=4\)
, or
\(\frac{2}{3} > \frac{2}{8}\)
and
\(\frac{2}{8} < \frac{2}{3}\)
Student Facing
For each problem:
Answer the question and explain or show your reasoning.
Represent your answer with a statement that uses the symbols >, <, or =.
A beetle crawled
\(\frac{2}{8}\)
of the length of a log. A caterpillar crawled
\(\frac{2}{3}\)
of the length of the same log. Which insect crawled farther?
A grasshopper is 4 centimeters long. A caterpillar is
\(\frac{12}{3}\)
centimeters long. Which insect is longer?
[IMG]
A ladybug crawled
\(\frac{3}{8}\)
the length of a branch. An ant crawled
\(\frac{5}{8}\)
the length of the same branch. Which insect crawled farther?
A grasshopper jumped
\(\frac{5}{8}\)
the width of the sidewalk. A frog jumped
\(\frac{5}{6}\)
the width of the same sidewalk. Which jumped a longer distance?
Student Response
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for free access to Student Response.
Activity Synthesis
Select 2–3 students to share their reasoning strategies or representations for at least one of the situations.
“How did you use what you’ve learned in earlier lessons to compare fractions?” (If the fractions had the same numerator, I thought about the size of the denominators. If they had the same denominator, I compared the numerators.)
Select students who wrote different but equally valid comparison statements (for instance,
\(\frac{2}{8}<\frac{2}{3}\)
and
\(\frac{2}{3} > \frac{2}{8}\)
) to share.
Discuss how to read each statement and ask students whether both accurately represent the comparison.
Emphasize that we can write comparison statements in more than one way, but we need to check that the statements make sense given the numbers we write and the symbols we use.


correct_option_index: 1