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: Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Multiply and divide within 100.

Understand the relationship between zeros and factors of polynomials.

Understand solving equations as a process of reasoning and explain the reasoning.

Use the four operations with whole numbers to solve problems.

Understand the connections between proportional relationships, lines, and linear equations.

Reason about and solve one-variable equations and inequalities.

Represent and analyze quantitative relationships between dependent and independent variables.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Perform arithmetic operations on polynomials.

Represent and solve problems involving addition and subtraction.

Solve systems of equations.

Represent and solve problems involving multiplication and division.

Use polynomial identities to solve problems.

Represent and solve equations and inequalities graphically.

Interpret the structure of expressions.

Analyze patterns and relationships.

Use properties of operations to generate equivalent expressions.

Work with equal groups of objects to gain foundations for multiplication.

Generate and analyze patterns.

Write and interpret numerical expressions.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Solve equations and inequalities in one variable.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Create equations that describe numbers or relationships

Write expressions in equivalent forms to solve problems.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Rewrite rational expressions.

Gain familiarity with factors and multiples.

Understand properties of multiplication and the relationship between multiplication and division.

Work with addition and subtraction equations.

Work with radicals and integer exponents.

Add and subtract within 20.

Understand and apply properties of operations and the relationship between addition and subtraction.
problem_activity: Activity
In the previous activity students looked at 9 different equations. In this activity, each student focuses on just one of those equations to learn about the real-world situation it represents. They interpret the variables and values in terms of the situation (MP7). Adding to the table they saw in the previous activity gives students an opportunity to practice both evaluating expressions and solving equations.
This activity also asks students to create a graph of their assigned relationship on graph paper. In previous activities, students were given a grid with the axes already labeled and numbered whenever they were asked to create a graph of a relationship. Here they must decide for themselves how to scale each axis.
Launch
Decide whether you want to assign the equations or allow students to select which equation they will work with. Make sure there is at least one student working with each equation.
Give students 5–10 minutes of quiet work time followed by partner and whole-class discussion. If there is more than one student working with the same equation, consider letting them share their answers and reasoning with each other, after they have had some time to work on their own.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. Check in with students within the first 2-3 minutes of work time. Look for students who need additional support deciding how to scale each axis for table. Invite students to share strategies they have used so far, as well as any questions they have before continuing.
Supports accessibility for: Memory; Organization
Student Facing
The equations in the previous activity represent situations.
\(S - 2 = T\)
where
\(S\)
is the number of sides on a polygon and
\(T\)
is the number of triangles you can draw inside it (from one vertex to the others, without overlapping)
\(G = J + 13\)
where
\(G\)
is a day in the Gregorian calendar and
\(J\)
is the same day in the Julian calendar
\(P = I - 47.50\)
where
\(I\)
is the amount of income and
\(P\)
is the profit after
$
47.50 in expenses
\(C + 273.15 = K\)
where
\(C\)
is a temperature in degrees Celsius and
\(K\)
is the same temperature in Kelvin
\(e = 6s\)
where
\(e\)
is the total edge length of a regular tetrahedron and
\(s\)
is the length of one side
\(m = 8.96V\)
where
\(V\)
is the volume of a piece of copper and
\(m\)
is its mass
\(y = \frac{1}{12}x\)
where
\(x\)
is the number of eggs and
\(y\)
is how many dozens that makes
\(t = \frac{d}{2.5}\)
where
\(t\)
is the amount of time it takes in seconds to jog a distance of
\(d\)
meters at a constant speed of 2.5 meters per second
\(g = 28.35z\)
where
\(g\)
is the mass in grams and
\(z\)
is the same amount in ounces
Your teacher will assign you one of these equations to examine more closely.
Rewrite your equation using words. Use words like product, sum, difference, quotient, and term.
In the previous activity, you matched equations and tables. Copy the values from the table that matched your assigned equation into the first 3 rows of this table. Make sure to label what each column represents.
| independent variable:                             ________________________ | dependent variable:                             ________________________ |
|----------------------------------------------------------------------------|--------------------------------------------------------------------------|
|                                                                            |                                                                          |
|                                                                            |                                                                          |
|                                                                            |                                                                          |
|                                     60                                     |                                                                          |
|                                                                            |                                   300                                    |
Select one of the first 3 rows of the table and explain what those values mean in this situation.
Use your equation to find the values that complete the last 2 rows of the table. Explain your reasoning.
On graph paper, create a graph that represents this relationship. Make sure to label your axes.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may struggle to understand what each step is asking them to do. Consider preparing an example to share with them, such as this for the equation
\(2.54i = c\)
, and point out which part of the example corresponds to the question they are on.
The number of centimeters is the product of the number of inches and 2.54.
| independent variable:                       length (inches) | dependent variable:                       length (centimeters) |
|-------------------------------------------------------------|----------------------------------------------------------------|
|                              5                              |                              12.7                              |
|                             36                              |                             91.44                              |
|                             75                              |                             190.5                              |
|                             60                              |                                                                |
|                                                             |                              300                               |
If something is 36 inches long, this is equivalent to 91.44 when measured in centimeters.
In the table above, fill in 152.4 in the right column, next to the 60, because
\(2.54 \boldcdot 60 = 152.4\)
. Fill in 118.11 in the left column, next to the 300, because
\(2.54 \boldcdot 118.11 \approx 300\)
A graph with “length (inches)” on the horizontal axis, “length (centimeters)” on the vertical axis, and points at
\((5, 12.7), (36, 91.44), (75, 190.5), (60, 152.4)\)
, and
\((118.11, 300)\)
Some students may struggle with setting up their graph from scratch on graph paper. Prompt them to think about the maximum value they want to represent on each axis and what number they could count by to get to that maximum value in the amount of space they have.
Activity Synthesis
Continue to the next activity. The discussion of these relationships occurs after students have created their visual displays.


correct_option_index: 6

7

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