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
In this activity, students graph three time-distance relationships along with the one from the previous lesson, “Tyler’s Walk.” One of these is not a proportional relationship, so students must pay close attention to the quantities represented. The purpose of this activity is to give students many opportunities to connect the different features of a graph with parts of the situation it represents. In particular, they attach meaning to any point that is on a graph, and they interpret the meaning of the distance when the time is 1 second as both the constant of proportionality of the relationship and the person’s speed in the context in meters per second. Comparing different but related situations and their graphs supports students as they make sense of the situation.
Digital
Launch
Arrange students in groups of 2–3. Provide access to colored pencils and rulers.
Tell students that this activity is tied to the activity titled "Tyler's Walk" from the previous lesson. All references to Tyler going to the bumper cars come from the statements in that activity.
The digital version has an applet with options to change line colors and hide points. You may want to demonstrate the applet before students use it, perhaps graphing Tyler's data from the previous activity together. Note: the applet can graph lines, rays, or segments. Your class can decide how to represent the data.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use a different color for each person to highlight the connection between the table, graph, and constant of proportionality.
Supports accessibility for: Visual-spatial processing
Student Facing
Diego, Lin, and Mai went from the ticket booth to the bumper cars.
1. Use each description to complete the table representing that person’s journey.
Diego left the ticket booth at the same time as Tyler. Diego jogged ahead at a steady pace and reached the bumper cars in 30 seconds.
Lin left the ticket booth at the same time as Tyler. She ran at a steady pace and arrived at the bumper cars in 20 seconds.
Mai left the booth 10 seconds later than Tyler. Her steady jog enabled her to catch up with Tyler just as he arrived at the bumper cars.
| Diego's time (seconds) | Diego's distance (meters) |
|------------------------|---------------------------|
|           0            |                           |
|           15           |                           |
|           30           |            50             |
|           1            |                           |
| Lin's time (seconds) | Lin's distance (meters) |
|----------------------|-------------------------|
|                      |            0            |
|                      |           25            |
|          20          |           50            |
|          1           |                         |
| Mai's time (seconds) | Mai's distance (meters) |
|----------------------|-------------------------|
|                      |            0            |
|                      |           25            |
|          40          |           50            |
|          1           |                         |
2. Using a different color for each person, draw a graph of all four people’s journeys (including Tyler's from the other day).
Drag the names to the correct lines to label them.
If you choose to, you can use the Paint Brush tool to change the color of each line. Select the tool, click on a color in the palette below the graph, and then click on a line. Click on the Move tool (the arrow) before changing to a new paint brush color.
You can hide any points you create with the checkbox below the graph.
3. Which person is moving the most quickly? How is that reflected in the graph?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 2–3. Provide access to colored pencils and rulers.
Tell students that this activity is tied to the activity titled "Tyler's Walk" from the previous lesson. All references to Tyler going to the bumper cars come from the statements in that activity.
The digital version has an applet with options to change line colors and hide points. You may want to demonstrate the applet before students use it, perhaps graphing Tyler's data from the previous activity together. Note: the applet can graph lines, rays, or segments. Your class can decide how to represent the data.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use a different color for each person to highlight the connection between the table, graph, and constant of proportionality.
Supports accessibility for: Visual-spatial processing
Student Facing
Diego, Lin, and Mai went from the ticket booth to the bumper cars.
Use each description to complete the table representing that person’s journey.
Diego left the ticket booth at the same time as Tyler. Diego jogged ahead at a steady pace and reached the bumper cars in 30 seconds.
Lin left the ticket booth at the same time as Tyler. She ran at a steady pace and arrived at the bumper cars in 20 seconds.
Mai left the booth 10 seconds later than Tyler. Her steady jog enabled her to catch up with Tyler just as he arrived at the bumper cars.
| Diego’s                                     time                                     (seconds) | Diego’s                                     distance                                     (meters) |
|------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------|
|                                               0                                                |                                                                                                   |
|                                               15                                               |                                                                                                   |
|                                               30                                               |                                                50                                                 |
|                                               1                                                |                                                                                                   |
| Lin’s                                     time                                     (seconds) | Lin’s                                     distance                                     (meters) |
|----------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------|
|                                                                                              |                                                0                                                |
|                                                                                              |                                               25                                                |
|                                              20                                              |                                               50                                                |
|                                              1                                               |                                                                                                 |
| Mai’s                                     time                                     (seconds) | Mai’s                                     distance                                     (meters) |
|----------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------|
|                                                                                              |                                                0                                                |
|                                                                                              |                                               25                                                |
|                                              40                                              |                                               50                                                |
|                                              1                                               |                                                                                                 |
Using a different color for each person, draw a graph of all four people’s journeys (including Tyler's from the other day).
[IMG]
Which person is moving the most quickly? How is that reflected in the graph?
Student Response
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click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Write equations to represent each person’s relationship between time and distance.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Students may expect the graphs to intersect because everyone arrives at the same location. However, they did not arrive there at the same time (with the exception of Tyler and Mai). Because all characters traveled the same distance from the ticket booth and no further, the endpoints of their graphs lie on the same horizontal line
\(y = 50\)
, that is, they have the same
\(y\)
-coordinate. The points will vary in position from right to left depending on the number of seconds after Tyler left the ticket booth it took each person to arrive at the bumper cars. Note this features in a whole-class discussion.
The most important goals of the discussion are to attach meaning to any point that is on a graph, and to interpret the meaning of the distance when the time is 1 second as both the constant of proportionality of the relationship and the person’s speed in the context in meters per second.
These questions may be used to facilitate the class discussion:
"For each graph that shows a proportional relationship, what is the constant of proportionality?" (Tyler's was 1.25, Diego’s was
\(1\frac23\)
, Lin’s was 2.5.)
"How did you find them?" (Answers will vary, but students could have divided a
\(y\)
-coordinate by its associated
\(x\)
-coordinate.)
"Where do constants of proportionality occur in the tables, and where do they occur on the graphs?" (They occur in the tables as the values in the second column that correspond to the value of 1 in the first column; on the graphs as the
\(y\)
-coordinate of points where the
\(x\)
-coordinate is 1.)
"Which is the only graph that does not represent a proportional relationship?"
"A classmate argues that Mai's graph must represent a proportional relationship, because she jogged at a steady rate. How do you answer?" (Mai's graph does not pass through the origin, so it does not represent a proportional relationship. That is, the distance she traveled is not proportional to the time elapsed. What does “time elapsed” mean? It is time elapsed since Tyler left the ticket booth. However, distance vs. time elapsed since
Mai
left the ticket booth until she arrived at the bumper cars is a proportional relationship.)


correct_option_index: 1