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: Geometric measurement: understand concepts of angle and measure angles.

Classify objects and count the number of objects in each category.

Tell and write time.

Measure and estimate lengths in standard units.

Relate addition and subtraction to length.

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Describe and compare measurable attributes.

Represent and interpret data.

Convert like measurement units within a given measurement system.

Measure lengths indirectly and by iterating length units.

Work with time and money.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
problem_activity: Narrative
The purpose of this activity is for students to learn about the meter as a longer metric unit of length. Students measure longer lengths with a new tool, the meter stick. Students measure strips of tape of different lengths and choose between a centimeter ruler and a meter stick in order to measure each. In the synthesis, students share their measurements of each line and discuss how they chose which tool to use to measure (MP5). Students measure Strips D
–
G in this activity. Students measure the komodo dragon (Strip D) twice to experience measuring the same length with a ruler and a meter stick. They discuss and compare their measurements in the lesson synthesis.
MLR2 Collect and Display.
Direct attention to words collected and displayed from the previous lesson. Add “meter stick” to the collection. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading
Engagement: Provide Access by Recruiting Interest.
Synthesis: Optimize meaning and value. Invite students to share people they know or specific jobs they recognize that may use the measuring tools they have been exposed to. “When and why might someone use these measuring tools?”
Supports accessibility for: Conceptual Processing
Required Materials
Materials to Gather
Metersticks
Rulers (centimeters)
Launch
Groups of 3–4
Give each group a meter stick.
“We saw that measuring longer lengths with centimeters can be challenging. Luckily, there is a standard unit that is much larger than a centimeter that we can use. It’s called a
meter
.”
Show students a meter stick.
“This is called a meter stick. The length of one end to the other is 1 meter long. Just like we used centimeter cubes at first to measure centimeters. We can use the length of a meter stick to measure length in meters.”
“What do you notice about the meter stick?” (There are lots of little lines; the numbers go up to 100; every tenth number is bold)
30 seconds: quiet think time
Share responses.
“The meter stick makes it easier to measure longer lengths. We can use the meter stick to measure in meters or centimeters.”
“Find 50 cm on the meter stick.”
If time permits, find a few other markings.
Activity
“Now, let’s go back to our job as zookeepers and measure some even larger reptiles.”
“You are going to measure the komodo dragon again and 3 new reptiles."
“You may use any of the measuring tools you have used today. Be prepared to explain why you chose each tool.”
12 minutes: group work time
Monitor for the different ways students measure the ribbon snake and adult alligator including how they record their measurements in meters.
Student Facing
D: komodo dragon
[IMG]
E: adult alligator
[IMG]
F: adult cobra
[IMG]
G: ribbon snake
[IMG]
Measure the length of the komodo dragon in meters.
What is the same or different about your measurements for the komodo dragon from Activity 1 and Activity 2?
Measure each reptile in centimeters or meters. Don’t forget the units.
How long is an adult alligator?
How long is an adult cobra?
How long is a ribbon snake?
Student Response
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Advancing Student Thinking
If students do not measure using the meter stick, or only report their measurements in centimeters, consider asking:
“About how many meters long is the __?”
“What is different about measuring the __ in meters (or meters and centimeters) rather than only centimeters?”
Activity Synthesis
“Which tools did you use to measure Strip G, the ribbon snake? Why did you choose this tool?” (We used the meter stick and took away 10 because it was so close.)
“What did you do when you found that Strip E, the adult alligator, was longer than 3 meters but shorter than 4 meters?” (We saw it was closer to 3 meters so we said it was about 3 meters. We used the meter stick 3 times and then the cm ruler for the rest.)


correct_option_index: 3