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 geometric theorems.

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

Prove theorems involving similarity.

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

Understand and apply the Pythagorean Theorem.

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

Experiment with transformations in the plane.

Use coordinates to prove simple geometric theorems algebraically.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Understand and apply theorems about circles.

Understand congruence in terms of rigid motions.

Analyze, compare, create, and compose shapes.

Explain volume formulas and use them to solve problems.

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

Apply trigonometry to general triangles.

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

Define trigonometric ratios and solve problems involving right triangles.

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

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

Apply geometric concepts in modeling situations.

Understand similarity in terms of similarity transformations.

Reason with shapes and their attributes.

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

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

Make geometric constructions.

Find arc lengths and areas of sectors of circles.

Visualize relationships between two-dimensional and three-dimensional objects.
problem_activity: Activity
In this activity students continue to use coordinate transformation notation, but now without the scaffolding of an organizing structure like a table. Students apply transformation rules, then describe the results. One transformation is of a type students have not seen before (the horizontal stretch that is not a dilation).
In the synthesis, students will connect the geometric description of a reflection to the coordinate rule that produces it. They will also draw a conclusion about rules that produce a dilation versus those that produce a stretch in only 1 direction.
Monitor for students who erroneously call the horizontal stretch a dilation. Encourage students to refer to their reference charts for help in attending to precision when describing transformations they’ve learned about.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Conversing: MLR2 Collect and Display.
As students work on this activity, listen for and collect the language students use to describe each transformation of quadrilateral
\(ABCD\)
. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a statement such as, “The shape was flipped upside down” can be improved with the statement, “Quadrilateral
\(ABCD\)
was reflected across the
\(x\)
-axis.” This will help students use the mathematical language necessary to precisely describe transformations of figures on the coordinate plane.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
[IMG]
Here are some transformation rules. Apply each rule to quadrilateral
\(ABCD\)
and graph the resulting image. Then describe the transformation.
Label this transformation
\(Q\)
:
\((x,y) \rightarrow (2x,y)\)
Label this transformation
\(R\)
:
\((x,y) \rightarrow (x,\text-y)\)
Label this transformation
\(S\)
:
\((x,y) \rightarrow (y,\text-x)\)
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?
[IMG]
Plot the quadrilateral with vertices
\((4,\text-2),(8,4),(8,\text-6),\)
and
\((\text-6,\text-6)\)
. Label this quadrilateral
\(A\)
.
Plot the quadrilateral with vertices
\((\text-2,4),(4,8),(\text-6,8),\)
and
\((\text-6,\text-6)\)
. Label this quadrilateral
\(A'\)
.
How are the coordinates of quadrilateral
\(A\)
related to the coordinates of quadrilateral
\(A'\)
?
What single transformation takes quadrilateral
\(A\)
to quadrilateral
\(A'\)
?
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
If students struggle to keep their work organized, suggest they create a table of inputs and outputs, or create another organizational structure that works for them.
Some students may not be sure how to work with the rule
\((x,y)\rightarrow (y,\text-x)\)
. Ask these students, “What are the coordinates of point
\(A\)
?” (
\((3,\text-4)\)
.) “Which of those is the
\(x\)
-coordinate?” (3.) “In the transformation rule, where does the
\(x\)
land, and what happens to it?” (The
\(x\)
-coordinate lands in the
\(y\)
spot, and it takes on an opposite sign.) Another option is to suggest students write out
\(x=3\)
and
\(y=\text-4\)
, then substitute each value into the transformation rule.
Activity Synthesis
The goal of the discussion is to make connections between the coordinate rules and the geometric descriptions of transformations. Start with these discussion questions:
“In the previous activity, one rule was
\((x,y)\rightarrow (2x,2y)\)
. In this activity, one rule was
\((x,y)\rightarrow (2x,y)\)
. Compare and contrast these rules.” (In the first rule, both coordinates were multiplied by 2. It is stretched both vertically and horizontally by the same factor. The second rule just had the
\(x\)
-coordinate multiplied by 2. It was only stretched in one direction.)
“Can we call
\((x,y)\rightarrow (2x,y)\)
a dilation? Why or why not?” (No. In a dilation, the figure is stretched in all directions by the same factor, not just 1 direction. Also, in this example we can see that the angles of the dilated figure are not congruent to the angles in the original.)
“Can we call
\((x,y)\rightarrow (2x,y)\)
a translation? Why or why not?” (No. A translation is a rigid motion—it produces a result that does not change size or shape.)
Now ask students, “The rule
\((x,y) \rightarrow (x, \text-y)\)
is a reflection across the
\(x\)
-axis. Look at the definition of a reflection on your reference chart. Why does this rule produce a reflection?” Guide students through the different aspects of the geometric definition of a reflection. Consider displaying an image of the solution and drawing segments that connect pairs of original and image points. Important ideas that should surface are:
The
\(y\)
-coordinates of the original and image points have the same absolute value, so the distance from the
\(x\)
-axis is the same for each.
The sign of the image’s
\(y\)
-coordinate is opposite the original, so the image is on the opposite sign of the axis as the original.
Connecting the image and original points, we get a vertical line because the
\(x\)
-coordinates are the same. So this segment is perpendicular to the line of reflection, or the
\(x\)
-axis.


correct_option_index: 6