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: Write, read, and evaluate expressions in which letters stand for numbers.

Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

Write and evaluate numerical expressions involving whole-number exponents.
problem_activity: Optional activity
This activity allows students to practice identifying the base and height of triangles and using them to find areas.
Because there are no directions on which base or height to use, and because not all sides would enable them to calculate area easily, students need to think structurally and choose strategically. All triangles in the problems have either a vertical or a horizontal side. Choosing such a side as the base makes it easier to identify the corresponding height.
In some cases, students may opt to use a combination of area-reasoning strategies rather than finding the base and height of the shaded triangles and applying the formula. For instance, they may enclose a shaded triangle with a rectangle and subtract the areas of extra triangles (with or without using the formula on those extra triangles). Notice students who use such strategies so they could share later.
Launch
Keep students in groups of 2. Explain that they will now practice locating or drawing heights and using them to find area of triangles. Give students 8–10 minutes of quiet think time and time to share their responses with a partner afterwards. Provide access to their geometry toolkits (especially index cards).
If time is limited, consider asking students find the area of two or three triangles instead of all four.
Student Facing
For each triangle, identify and label a base and height. If needed, draw a line segment to show the height.
Then, find the area of the triangle. Show your reasoning. (The side length of each square on the grid is 1 unit.)
[IMG]
Student Response
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Student Facing
Are you ready for more?
Find the area of this triangle. Show your reasoning.
[IMG]
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may think that a vertical side of a triangle is the height regardless of the segment used as the base. If this happens, have them use an index card as a straightedge to check if the two segments they are using as base and height are perpendicular.
Some students may not immediately see that choosing a side that is either vertical or horizontal would enable them to find the corresponding height very easily. They may choose a non-vertical or non-horizontal side and not take advantage of the grid. Ask if a different side might make it easier to determine the base-height lengths without having to measure.
Activity Synthesis
Focus the whole-class discussion how students went about identifying bases and heights. Discuss:
“Which side did you choose as the base for triangle B? C? Why?”
“Aside from choosing a vertical or horizontal side as the base, is there another way to find the area of the shaded triangles without using their bases and heights?” (Invite a couple of students who use the enclose-and-subtract method to find the area of B, C, or D to share.)
“Which strategy do you prefer or do you think is more efficient?”
“Can you think of an example where it might be preferable to find the base and height of the triangle of interest?” (Students may point to any of the triangles in the task.)
“Can you think of an example where it might be preferable to enclose the triangle of interest and subtract other areas?” (Students may point to the triangle shown in "Are you ready for more?", where none of the shaded triangle's sides are horizontal or vertical.)
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. Provide sentence frames to help students produce statements that describe the strategies they use to identify bases and heights, and to find area. For example, “For triangle ___, I chose side ___ as the base because . . .” or “The next time I need to find the area of a triangle, the strategy I will use is _____, because . . . .”
Design Principle(s): Support sense-making


correct_option_index: 0