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.


Your response:
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.


Your response:
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.


Your response:
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: Work with radicals and integer exponents.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Use the four operations with whole numbers to solve problems.

Gain familiarity with factors and multiples.

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

Reason about and solve one-variable equations and inequalities.

Understand the relationship between zeros and factors of polynomials.

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

Multiply and divide within 100.

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

Solve equations and inequalities in one variable.

Add and subtract within 20.

Represent and solve equations and inequalities graphically.

Use polynomial identities to solve problems.

Interpret the structure of expressions.

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

Solve systems of equations.

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

Work with addition and subtraction equations.

Write expressions in equivalent forms to solve problems.

Write and interpret numerical expressions.

Perform arithmetic operations on polynomials.

Represent and analyze quantitative relationships between dependent and independent variables.

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

Represent and solve problems involving addition and subtraction.

Create equations that describe numbers or relationships

Generate and analyze patterns.

Rewrite rational expressions.

Analyze patterns and relationships.

Understand and apply properties of operations and the relationship between addition and subtraction.

Use properties of operations to generate equivalent expressions.

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

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

Represent and solve problems involving multiplication and division.
problem_activity: In Class Launch
Use after Unit 2, Lesson 3
Elicit students’ prior knowledge about the solar system and what it looks like. Sketch a picture like this of the sun and a few planets with their orbital paths.
[IMG]
Using student input and making corrections if necessary, write a list of the planets in order from nearest to farthest away from the sun (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune). Ask students to estimate how long it takes each planet to go around the sun, and write the estimates next to each planet.
Tell students that there is a relationship between a planet’s distance from the sun and its orbital period (the time it takes to complete one revolution around the sun). In this activity, they will figure out how to describe that relationship using polynomials.
Teacher Instructions
Although students saw how to find equations to fit data in an earlier course, this is the first time they fit polynomial equations to data. If students need a hint about how to find a polynomial equation that fits the data better, it may help them to know that polynomials of higher degree can fit data points more closely. One good strategy is to start with a polynomial of degree 1 or 2 and then add more terms. For example, students could start with an equation of the form
\(y=nx^2\)
and then move to
\(y=nx^2+mx\)
and
\(y=nx^2+mx+k\)
. If the maximum amount of terms (1 greater than the degree) is reached and the polynomial is still not a good fit, then the degree can be increased.
Graphing technology can be a very helpful tool for finding models in this way, and students should be encouraged to use it. Graphing calculators can automatically find polynomials of specified forms that are the closest possible fit to a set of data. Desmos also has this feature. Calculators may report the value of
\(R^2\)
for the functions they produce.
\(R^2\)
measures how closely a function fits a set of data points. Tell students that the closer to 1
\(R^2\)
is, the better the function fits the data.
To conclude the task, invite students to share their models for the relationship between the orbital period and distance, and for the relationship between the orbital period and the square root of the distance. The square root should have a much simpler relationship to the orbital period: if we cube the square root of the distance and multiply it by some constant, we get the period. This is exactly what Kepler found. Share with students the statement of Kepler’s third law, which is: “The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit”. What this means is that the square of the orbital period is some constant times the cube of the distance. This can be expressed as
\(p^2 = n \boldcdot d^3\)
. This can be rearranged into something of the form
\(p= k (d^{1/2})^3\)
, which is the relationship that students have been investigating.
Once Kepler saw this relationship, he could divide the square of any planet’s period by the cube of its distance in order to find out what the proportionality constant is. Students can try this for themselves and verify that all planets have the same proportionality constant. They may also want to compare what happens with Jupiter’s moons—do they have the same proportionality constant as the planets? Newton’s laws of gravity later explained why this relationship between distance and period is true.
Student-Facing Statement
In the early 1600s, Johannes Kepler (1571–1630) studied the motions of the planets to find a good mathematical model for them. In 1619, he published his third law of planetary motion, which says how the orbital periods of the planets are related to their distances from the sun. In Kepler’s time, Uranus and Neptune had not been discovered, but here is the data for all 8 planets:
| planet  | distance (millions of km) | period (days) |
|---------|---------------------------|---------------|
| Mercury |           57.9            |     88.0      |
|  Venus  |           108.2           |     224.7     |
|  Earth  |           149.6           |     365.2     |
|  Mars   |           227.9           |     687.0     |
| Jupiter |           778.6           |     4,331     |
| Saturn  |          1,433.5          |    10,747     |
| Uranus  |          2,872.5          |    30,589     |
| Neptune |          4,495.1          |    59,800     |
Plot the distance (
\(x\)
) and period (
\(y\)
) of each planet, and find a polynomial model that fits the data as well as possible. You may have to experiment with both the degree of your polynomial function and the number of terms.
Make another plot that uses the square root of each distance instead, and find a polynomial model that fits those points as well as possible.
Which model do you think is best?
Jupiter has a lot of moons. Look up the periods and distances of some of them, and use that data to make a polynomial model of the relationship between period and distance for Jupiter’s moons.
Use your model to predict the period of another one of Jupiter’s moons using the radius of its orbit.
How good was the prediction? What are some possible sources of error?
Teacher Instructions
Although students saw how to find equations to fit data in an earlier course, this is the first time they fit polynomial equations to data. If students need a hint about how to find a polynomial equation that fits the data better, it may help them to know that polynomials of higher degree can fit data points more closely. One good strategy is to start with a polynomial of degree 1 or 2 and then add more terms. For example, students could start with an equation of the form
\(y=nx^2\)
and then move to
\(y=nx^2+mx\)
and
\(y=nx^2+mx+k\)
. If the maximum amount of terms (1 greater than the degree) is reached and the polynomial is still not a good fit, then the degree can be increased.
Graphing technology can be a very helpful tool for finding models in this way, and students should be encouraged to use it. Graphing calculators can automatically find polynomials of specified forms that are the closest possible fit to a set of data. Desmos also has this feature. Calculators may report the value of
\(R^2\)
for the functions they produce.
\(R^2\)
measures how closely a function fits a set of data points. Tell students that the closer to 1
\(R^2\)
is, the better the function fits the data.
To conclude the task, invite students to share their models for the relationship between the orbital period and distance, and for the relationship between the orbital period and the square root of the distance. The square root should have a much simpler relationship to the orbital period: if we cube the square root of the distance and multiply it by some constant, we get the period. This is exactly what Kepler found. Share with students the statement of Kepler’s third law, which is: “The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit”. What this means is that the square of the orbital period is some constant times the cube of the distance. This can be expressed as
\(p^2 = n \boldcdot d^3\)
. This can be rearranged into something of the form
\(p= k (d^{1/2})^3\)
, which is the relationship that students have been investigating.
Once Kepler saw this relationship, he could divide the square of any planet’s period by the cube of its distance in order to find out what the proportionality constant is. Students can try this for themselves and verify that all planets have the same proportionality constant. They may also want to compare what happens with Jupiter’s moons—do they have the same proportionality constant as the planets? Newton’s laws of gravity later explained why this relationship between distance and period is true.
Student-Facing Statement
In the early 1600s, Johannes Kepler (1571–1630) studied the motions of the planets to find a good mathematical model for them. In 1619, he published his third law of planetary motion, which says how the orbital periods of the planets are related to their distances from the sun. In Kepler’s time, Uranus and Neptune had not been discovered, but here is the data for all 8 planets:
| planet  | distance (millions of km) | period (days) |
|---------|---------------------------|---------------|
| Mercury |           57.9            |     88.0      |
|  Venus  |           108.2           |     224.7     |
|  Earth  |           149.6           |     365.2     |
|  Mars   |           227.9           |     687.0     |
| Jupiter |           778.6           |     4,331     |
| Saturn  |          1,433.5          |    10,747     |
| Uranus  |          2,872.5          |    30,589     |
| Neptune |          4,495.1          |    59,800     |
Plot the distance (
\(x\)
) and period (
\(y\)
) of each planet, and find a polynomial model that fits the data as well as possible. You may have to experiment with both the degree of your polynomial function and the number of terms.
Make another plot that uses the square root of each distance instead, and find a polynomial model that fits those points as well as possible.
Which model do you think is best?
Jupiter has a lot of moons. Here are the periods and distances of the Galilean moons, which were discovered in 1610:
|   moon   | distance (thousands of km) | period (days) |
|----------|----------------------------|---------------|
|    Io    |           421.8            |     1.77      |
|  Europa  |           671.1            |     3.55      |
| Ganymede |           1070.4           |     7.16      |
| Callisto |           1882.7           |     16.69     |
Use the data to make a polynomial model of the relationship between period and distance for Jupiter’s moons.
Another moon of Jupiter, number LXXI, was discovered in 2018. Its distance from Jupiter is 11,483 thousand km. According to your model, what should its period be?
The actual period of the moon is 252.0 days. How close was the prediction? What are some possible sources of error?


correct_option_index: 25