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: Prove and apply trigonometric identities.

Use functions to model relationships between quantities.

Understand the concept of a function and use function notation.

Model periodic phenomena with trigonometric functions.

Build new functions from existing functions.

Define, evaluate, and compare functions.

Construct and compare linear and exponential models and solve problems.

Analyze functions using different representations.

Interpret functions that arise in applications in terms of the context.

Extend the domain of trigonometric functions using the unit circle.

Interpret expressions for functions in terms of the situation they model.

Build a function that models a relationship between two quantities.
problem_activity: Optional activity
This activity is optional.  This activity is similar to the previous (optional) one from a function point of view, but now students investigate the volume of a cylinder instead of a rectangular prism. Students continue working with functions to investigate what happens to the volume of a cylinder when you halve the height. The exploration and representations resemble what was done in the previous activity, and students continue to identify the effect of the changing dimension on the graph and the equation of this function. As groups work on the task, identify those who make the connection between the graph and equation representations, and encourage groups to look for similarities and differences between what they see in this activity and what they saw in the previous activity. Invite these groups to share during the whole-class discussion.
Digital
Launch
Keep students in the same groups. Tell students that this activity is similar to the previous one, but they will work with a cylinder instead of a rectangular prism. Give students 2–3 minutes of quiet work time followed by time to discuss the last question with their partner. Follow with a whole-class discussion.
Students using the digital activity can graph their equations using the applet.
Student Facing
There are many cylinders with radius 5 units. Let
\(h\)
represent the height and
\(V\)
represent the volume of these cylinders.
Write an equation that represents the relationship between
\(V\)
and
\(h\)
. Use 3.14 as an approximation of
\(\pi\)
.
Graph this equation.
What happens to the volume if you halve the height,
\(h\)
? Where can you see this in the graph? How can you see it algebraically?
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
Keep students in the same groups. Tell students that this activity is similar to the previous one, but they will work with a cylinder instead of a rectangular prism. Give students 2–3 minutes of quiet work time followed by time to discuss the last question with their partner. Follow with a whole-class discussion.
Students using the digital activity can graph their equations using the applet.
Student Facing
There are many cylinders with radius 5 units. Let
\(h\)
represent the height and
\(V\)
represent the volume of these cylinders.
Write an equation that represents the relationship between
\(V\)
and
\(h\)
. Use 3.14 as an approximation of
\(\pi\)
.
Graph this equation and label the axes.
[IMG]
What happens to the volume if you halve the height,
\(h\)
? Where can you see this in the graph? How can you see it algebraically?
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?
Suppose we have a rectangular prism with dimensions 2 units by 3 units by 6 units, and we would like to make a rectangular prism of volume 216 cubic units by stretching
one
of the three dimensions.
What are the three ways of doing this? Of these, which gives the prism with the smallest surface area?
Repeat this process for a starting rectangular prism with dimensions 2 units by 6 units by 6 units.
Can you give some general tips to someone who wants to make a box with a certain volume, but wants to save cost on material by having as small a surface area as possible?
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
Ask previously identified groups to share their graphs and equations.
If students completed the optional activity, display student graphs from both activities and ask:
“Compare the graph in this activity to the graph in the last activity. How are they alike? How are they different?”
“Compare the equation in this activity to the equation in the last activity. How are they alike? How are they different?”
“How can you tell that this is a linear relationship?”
For the last question, make sure students understand that “looks like a line” is insufficient evidence for saying a relationship is linear since the scale of the axes can make non-linear graphs look linear. It is important students connect that the equation is linear (of the form y=mx+b) or relate back to the relationship between the height and volume: that the volume is
\(25\pi\)
, or about 78.5, times the height.
If students did not complete the optional activity, ask:
“Which of your variables is the independent? The dependent?” (The height,
\(h\)
, is the independent variable, and volume, V, is dependent variable.)
“Which variable is a function of which?” (Volume is a function of height.)
“How can you tell that this is a linear relationship?”
See notes above on the last question in this list. If it has not been brought up in students’ explanations, ask what the volume equation looks like when we double the height
\(h\)
. Display volume equation
\(V=78.5(2h)\)
for all to see. Ask how we can write this equation to show that the volume doubled when
\(h\)
doubled (using algebra, we can rewrite
\(V=78.5(2h)\)
as
\(V=2(78.5h)\)
. Since the volume for
\(h\)
was
\(78.5h\)
, this shows that the volume for
\(2h\)
is twice the volume for
\(h\)
).
Representing, Listening, Speaking: MLR7 Compare and Connect.
Invite students to create a visual representation that shows what is the same and what is different between this activity and the previous one. If students did not complete the optional activity, students can create multiple representations that illustrate why volume is a function of height. Give students time to do a gallery walk of the displays. Look for opportunities to highlight approaches that compare the multiple representations of the function of volume. Lead a whole-class discussion that connects features on the displays, such as comparing the equations and graphs. This will help students communicate about the multiple representations of volume functions while using mathematical language.
Design Principle(s): Optimize output; Maximize meta-awareness


correct_option_index: 1

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