few_shots: Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Standard options: 
A. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
B. Solve quadratic equations in one variable.


Thought: This task shows one quadratic equation with one variable, x.
Answer: B

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 $P$.


Standard options: 
A. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
B. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.


Thought: This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.
Answer: A

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, Ö3) lies on the circle centered at the origin and containing the point (0, 2).

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
problem_activity: Activity
In this activity students use the structure of the coordinate plane to examine the observation that the altitudes of a triangle all intersect at a single point. Students practice writing equations for perpendicular lines as they represent altitudes algebraically. Then they solve the system of equations (a fairly simple system with
\(x=2\)
as one equation) to show that the altitudes all intersect at a single point.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Representation: Internalize Comprehension.
Activate or supply background knowledge about perpendicular bisectors. Display the following information: the perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it. Provide a visual example to accompany this definition.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Triangle
\(ABC\)
is graphed.
[IMG]
Find the slope of each side of the triangle.
Find the slope of each altitude of the triangle.
Sketch the altitudes. Label the point of intersection
\(H\)
.
Write equations for all 3 altitudes.
Use the equations to find the coordinates of
\(H\)
and verify algebraically that the altitudes all intersect at
\(H\)
.
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?
Any triangle
\(ABC\)
can be translated, rotated, and dilated so that the image
\(A’\)
lies on the origin,
\(B’\)
lies on the point
\((1,0)\)
, and
\(C’\)
has position
\((a,b)\)
. Use this as a starting point to prove that the altitudes of all triangles all meet at the same point.
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 aren’t sure how to find the slopes of the altitudes, ask them about the relationship between the slope of a side and the slope of the altitude through that side. (The product of the slopes is -1 since they are perpendicular.)
If students struggle to verify algebraically that
\(H\)
is the point of intersection, ask them what’s true if a particular point is on 3 different lines (the lines must intersect at that point, unless they coincide). How can students test if
\(H\)
is on each line?
As in the previous activity, an index card can be a useful too to help visualize the altitudes.
Activity Synthesis
Ask students what the relationship is between the slope of a side and the slope of the altitude through that side. (The product of the slopes is -1 since they are perpendicular.) Invite students to share strategies for verifying their coordinates of
\(H\)
. (Sample response: Start with
\(x=2\)
, find the
\(y\)
-value using one equation, and test that point in the other equation.)
Speaking: MLR8 Discussion Supports.
As students discuss the relationship between the slope of a side and the slope of the altitude through that side, press for details by asking students how they know that the product of the slopes is
\(\text-1\)
. Also ask how they determined the equations for all three altitudes. Show concepts multi-modally by drawing and labeling a slope triangle for segment
\(AC\)
or
\(BC\)
and the altitude through that side. This will help students justify the slopes and equations for all three altitudes of the triangle.
Design Principle(s): Support sense-making; Optimize output (for justification)


correct_option_index: 2