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: Analyze and solve pairs of simultaneous linear equations.

Solve linear equations in one variable.
problem_activity: Optional activity
In this activity, students solve a variety of systems of equations, some involving fractions, some involving substitution, and some involving inspection. This gives students a chance to practice using the methods they have learned in this section for solving systems of equations to solidify that learning. Some of the systems listed are ones students could have used in an earlier activity in this lesson, to describe the situations there. In the discussion, students compare the systems here to the ones they wrote in that activity and interpret the answer in that context.
Launch
Keep students in groups of 2. Allow students 5–7 minutes of partner work time followed by a whole-class discussion.
Representation: Internalize Comprehension
. Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Here are a lot of systems of equations:
\(\begin{cases} y=\text-2x+6 \\ y=x-3 \end{cases}\)
\(\begin{cases} y=5x-4 \\ y=4x+12 \end{cases}\)
\(\begin{cases} y=\frac23x-4 \\ y=\text-\frac43x+9 \end{cases}\)
\(\begin{cases} 4y + 7x = 6 \\ 4y+7x = \text-5 \end{cases}\)
\(\begin{cases} y=x - 6\\ x=6 + y \end{cases}\)
\(\begin{cases} y=0.24x\\ y=0.18x+0.9 \end{cases}\)
\(\begin{cases} y=4.5x+15 \\ y=5x+12.5 \end{cases}\)
\(\begin{cases} y=3x \\ x+y=52 \end{cases}\)
Without solving, identify 3 systems that you think would be the least difficult for you to solve and 3 systems you think would be the most difficult. Be prepared to explain your reasoning.
Choose 4 systems to solve. At least one should be from your "least difficult" list and one should be from your "most difficult" list.
Student Response
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Activity Synthesis
There are two key takeaways from this discussion. The first is to reinforce that some systems can be solved by reasoning whether its possible for a solution to one equation to also be a solution to another. The second takeaway is that there are some systems that students will only be able to solve after learning techniques in future grades.
For each problem, ask students to indicate if they identified the system as least or most difficult. Record the responses for all to see.
Bring students' attention to this system:
\(\begin{cases} 4y + 7x = 6 \\ 4y+7x = \text-5 \end{cases}\)
Ask students what the two equations in the system have in common to each other and to think about whether a solution to the first equation could also be a solution to the second. One can reason there is no solution because
\(4y+7x\)
cannot be equal to both 6 and -5.
Ask students to return to the earlier activity and see if they can find any of those systems in these problems. (Lin's family ride is the sixth system. Noah's kayaking trip is the seventh system. Diego's baking is the eighth system.) After students notice the connection, invite students who chose those systems to solve to interpret the numerical solutions in the contexts from the earlier activity.
Students may be tempted to develop the false impression that all systems where both equations are given as linear combinations can be solved by inspection. Conclude the discussion by displaying this system that students defined in the last activity about sugar and flour:
\(\begin{cases} 0.8x+0.5y = 9 \\ x+y = 15 \end{cases}\)
Tell students that this system has one solution and they will learn more sophisticated techniques for solving systems of equations like this in future grades.
Speaking, Listening, Conversing: MLR8 Discussion Supports.
Before the whole-class discussion, use this routine to give students an opportunity to discuss their reasons for labeling systems of equations as “least difficult” or “most difficult.” Provide the following sentence frame: “_____ is the least/most difficult to solve because _____.” Encourage listeners to press for detail by asking questions such as “What would make that problem easier to solve?”; “What do your least/most difficult problems have in common?” This will help students make connections between the structures of given systems of equations, and possible strategies they can use to solve them.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


correct_option_index: 0