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: Analyze, compare, create, and compose shapes.

Classify two-dimensional figures into categories based on their properties.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Experiment with transformations in the plane.

Define trigonometric ratios and solve problems involving right triangles.

Solve real-world and mathematical problems involving area, surface area, and volume.

Use coordinates to prove simple geometric theorems algebraically.

Apply geometric concepts in modeling situations.

Understand and apply theorems about circles.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Translate between the geometric description and the equation for a conic section.

Apply trigonometry to general triangles.

Prove geometric theorems.

Explain volume formulas and use them to solve problems.

Reason with shapes and their attributes.

Make geometric constructions.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Understand similarity in terms of similarity transformations.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Understand congruence in terms of rigid motions.

Understand and apply the Pythagorean Theorem.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Visualize relationships between two-dimensional and three-dimensional objects.

Prove theorems involving similarity.

Find arc lengths and areas of sectors of circles.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
problem_activity: Activity
Students generalize the process for finding the volume of a triangular pyramid, concluding it applies to all pyramids. As they compare cross sections across different solids, they are looking for and making use of structure (MP7).
Launch
Tell students that in this activity, they’ll decide whether their method to find the volume of triangular pyramids extends to other kinds of pyramids and cones.
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, remind students how to calculate the area of a dilated triangle given the area of the original triangle and a scale factor. Ask students how they can use this method to calculate the area of each of the three cross sections.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
Each solid in the image has height 6 units. The area of each solid’s base is 10 square units. A cross section has been created in each by dilating the base using the apex as a center with scale factor
\(k=0.5\)
.
[IMG]
Calculate the area of each of the 3 cross sections.
Suppose a new cross section was created in each solid, all at the same height, using some scale factor
\(k\)
. How would the areas of these 3 cross sections compare? Explain your reasoning.
What does this information about cross sections tell you about the volumes of the 3 solids?
Calculate the volume of each of the solids.
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?
An octahedron is a solid whose faces consist of 8 equilateral triangles. Find the volume of an octahedron with edge length
\(\ell\)
.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may believe the cross sections have area 5 square units rather than 2.5 square units. Remind them that if a two-dimensional figure is dilated by a factor of
\(k\)
, the area is multiplied by
\(k^2\)
. However, the exact value of the area isn’t as important as the concept that the areas are the same for all 3 cross sections.
Activity Synthesis
In the synthesis discussion, help students use the reasoning developed in the task to extend the formula
\(V=\frac13Bh\)
to all pyramids, not just triangular ones. Ask students:
“How did you calculate the volume of the triangular prism?” (Multiplied the area of the base, 10 square units, by the height, 6 units, then multiplied by
\(\frac13\)
to get 20 cubic units.)
“How did you find the volume of the remaining solids?” (The volumes of all the solids are the same.)
“Suppose we didn’t know the actual values of the height or the area of the base, but we knew all the bases had the same area
\(B\)
and the solids all had the same height
\(h\)
. Would all 3 solids still have the same volume?” (Yes. Each set of cross sections would still have the same areas. Instead of
\(10k^2\)
, each cross section would have area
\(Bk^2\)
.)
“Does the expression
\(\frac13 Bh\)
give the volume for
any
pyramid or cone? Why or why not?” (Yes, it does. For any pyramid or cone, we can draw a triangular pyramid with identical volume, the same height, and a base with equal area. We know that
\(\frac13 Bh\)
works for this triangular pyramid, so it also works for the particular pyramid or cone we’re looking at, too.)
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Before students share their solutions to the last question, present an incorrect answer and explanation. For example, “The volume of the pyramid is 60 cubic units because the volume of a pyramid is the area of the base times the height. The area of the base is 10 square units and the height is 6 units, so the volume is 10 times 6, which is 60.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who clarify that the volume of a pyramid is a third of the volume of a prism with a congruent base and equal height. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to justify why the volume of the pyramid is 20 cubic units. This will help students evaluate and improve on the written mathematical arguments of others.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness


correct_option_index: 14