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
Note: if it is possible in your local environment, we recommend using the digital version of this activity.
This activity is intended to help students identify correspondences between parts of a table, graph, and an equation of a line through the origin in the first quadrant. Students are guided to notice:
Any pair of positive values
\((a, b)\)
determine a proportional relationship.
Given a point
\((a,b)\)
other than the origin on the graph of a line through the origin, the constant of proportionality is always
\(\frac{b}{a}\)
.
In an equation
\(y=\frac{b}{a} x\)
that represents the relationship, the constant of proportionality appears as the coefficient of
\(x\)
.
The constant of proportionality is the
\(y\)
-coordinate when
\(x\)
is 1, that is,
\((1,\frac{b}{a})\)
is a point on the graph.
In the print version, students plot one point and draw a ray that starts at the origin and passes through their point. They also create a table and an equation to represent the relationship. They respond to a series of questions about their representations. Then, they compare their representations of their relationship with the representations of other relationships created by members of their group.
In the digital version (recommended), students interact with a dynamic sketch while recording observations and responding to prompts. In the sketch, students are able to manipulate the graph of a proportional relationship, while changes to an associated table and equation automatically update.
Digital
Launch
For the print version: Arrange students in groups of 3. Assign each student in each group a letter: A, B, or C. Provide access to rulers.
For the digital version: Arrange students in groups of 2–3 and have them complete the task.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Speaking, Writing: MLR2 Collect and Display.
While pairs or groups are working, circulate and listen to student talk about the connections they see between the tables, characteristics of the graphs, and the equations. Write down common or important phrases you hear students say about the connections onto a visual display. Throughout the remainder of the lesson, continue to update collected student language and remind students to borrow language from the display as needed. This will help students read and use mathematical language during paired and whole-class discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
Explore the graph. Start by dragging the gray bar on the left across the screen until you can see both the table and the graph. Notice the values in the table and the coordinates of the labeled point. Grab the point and move it around.
What stays the same and what changes in the table? in the equation? on the graph?
Choose one row in the table and write it here. To what does this row correspond on the graph?
| \(x\) | \(y\) |
|-------|-------|
|       |       |
Grab and drag the point until you see the equation
\(y=\frac 32 x\)
.
Do not move the point. Choose three rows from the table, other than the origin. Record
\(x\)
and
\(y\)
, and compute
\(\frac yx\)
.
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|-------|-------|-----------------|
|       |       |                 |
|       |       |                 |
|       |       |                 |
What do you notice? What does this have to do with the equation of the line?
Do not move the point. Check the box to view the coordinates
\((1, ?)\)
. What are the coordinates of this point? What does this correspond to in the table? What does this correspond to in the equation?
Drag the point to a different location. Record the equation of the line, the coordinates of three points, and the value of
\(\frac yx\)
.
Equation of the line: _______________________________
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|-------|-------|-----------------|
|       |       |                 |
|       |       |                 |
|       |       |                 |
Based on your observations, summarize any connections you see between the table, characteristics of the graph, and the equation.
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
For the print version: Arrange students in groups of 3. Assign each student in each group a letter: A, B, or C. Provide access to rulers.
For the digital version: Arrange students in groups of 2–3 and have them complete the task.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Speaking, Writing: MLR2 Collect and Display.
While pairs or groups are working, circulate and listen to student talk about the connections they see between the tables, characteristics of the graphs, and the equations. Write down common or important phrases you hear students say about the connections onto a visual display. Throughout the remainder of the lesson, continue to update collected student language and remind students to borrow language from the display as needed. This will help students read and use mathematical language during paired and whole-class discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
Your teacher will assign you
one
of these three points:
\(A=(10,4)\)
,
\(B=(4,5)\)
,
\(C=(8,5)\)
.
[IMG]
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|-------|-------|-----------------|
|   0   |       |       NA        |
|   1   |       |                 |
|   2   |       |                 |
|   3   |       |                 |
|   4   |       |                 |
|   5   |       |                 |
|   6   |       |                 |
|   7   |       |                 |
|   8   |       |                 |
|   9   |       |                 |
|  10   |       |                 |
On the graph, plot and label
only
your assigned point.
Use a ruler to line up your point with the origin,
\((0,0)\)
. Draw a line that starts at the origin, goes through your point, and continues to the edge of the graph.
Complete the table with the coordinates of points on your graph. Use a fraction to represent any value that is not a whole number.
Write an equation that represents the relationship between
\(x\)
and
\(y\)
defined by your point.
Compare your graph and table with the rest of your group. What is the same and what is different about:
your tables?
your equations?
your graphs?
What is the
\(y\)
-coordinate of your graph when the
\(x\)
-coordinate is 1? Plot and label this point on your graph. Where do you see this value in the table? Where do you see this value in your equation?
Describe any connections you see between the table, characteristics of the graph, and the equation.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The graph of an equation of the form
\(y = kx\)
, where
\(k\)
is a positive number, is a line through
\((0,0)\)
and the point
\((1,k)\)
.
Name at least one line through
\((0,0)\)
that cannot be represented by an equation like this.
If you could draw the graphs of
all
of the equations of this form in the same coordinate plane, what would it look like?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
When all students have completed the activity, ask them to share their responses with a partner or small group. After a minute, go around the room asking each group to share one thing from the last question, and display these for all to see. When each group has shared, ask if there were any important observations that were missed.
Ensure that all the important connections are highlighted:
A graph of a line through the origin and passing through the first quadrant represents a proportional relationship.
The value of
\(\frac ba\)
computed from any point
\((a,b)\)
on that line (other than the origin) is the constant of proportionality.
An equation of the relationship is given by
\(y = kx\)
where
\(k\)
is
\(\frac ba\)
for any point
\((a,b)\)
on the graph other than the origin.


correct_option_index: 1