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

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im_task_000870

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options: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
problem_activity: Activity
This activity gives students another opportunity to write and rearrange equations in two variables and to do so in context.
Unlike in previous activities, students no longer start by computing one quantity given numerical values of the other quantity and then generalizing the process. Instead, they are prompted to write one equation, solve for each variable, and interpret the solution. Students also articulate why one model might be more helpful than the other under a certain circumstance. Along the way, students reason quantitatively and abstractly (MP2) and engage in aspects of modeling (MP4).
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Familiarize students with the idea of organizational budgets, if needed. Explain that every governmental agency has a budget (not unlike a budget for a pizza party) and needs to decide how to allocate a limited amount of funds. Ask students to think of examples of expenses that a city's Department of Streets might have (road repairs, new street construction, additional workers, new equipment, and so on). Given a set budget, spending more on one thing means having less to spend on something else, so the agency would need to weigh their options and prioritize their needs.
Reading, Listening, Conversing: MLR6 Three Reads.
Ask students to keep their books or devices closed and display only the image and the task statement without revealing the questions that follow. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (a city department has money to resurface roads this year). Use the second read to identify quantities and relationships. Ask students “What can be counted or measured?” Listen for, and amplify, the important quantities that vary in relation to each other in this situation: the total amount of money available, the cost to resurface 1 mile of a 2-lane road, the average worker’s salary, and the number of miles of road to resurface. After the third read, reveal the first question and ask students to brainstorm possible strategies to write an equation. This routine helps students interpret the language within a given situation needed to create an equation.
Design Principle(s): Support sense-making
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
The Department of Streets of a city has a budget of \$1,962,800 for resurfacing roads and hiring additional workers this year.
The cost of resurfacing a mile of 2-lane road is estimated at \$84,000. The average starting salary of a worker in the department is \$36,000 a year.
[IMG]
Write an equation that represents the relationship between the miles of 2-lane roads the department could resurface,
\(m\)
, and the number of new workers it could hire,
\(p\)
, if it spends the entire budget.
Take the equation you wrote in the first question and:
Solve for
\(p\)
. Explain what the solution represents in this situation.
Solve for
\(m\)
. Explain what the solution represents in this situation.
The city is planning to hire 6 new workers and to use its entire budget.
Which equation should be used to find out how many miles of 2-lane roads it could resurface? Explain your reasoning.
Find the number of miles of 2-lane roads the city could resurface if it hires 6 new workers.
Student Response
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Anticipated Misconceptions
Some students may have trouble sorting out the given information because of unfamiliarity with the context or with certain terms. Ask them to explain the setup of the problem as best they understand it, and then point out any information that might be missing.
Activity Synthesis
Verify that the equations students wrote correctly isolate each variable. If students wrote expressions in different forms for a variable (for instance,
\(\dfrac {4,\!907 - 210m}{90}\)
and
\(\dfrac {1,\!962,\!800 -84,\!000m}{36,\!000}\)
for
\(p\)
), discuss how the expressions are equivalent.
If time permits, consider asking students to use their equations to answer these questions:
"How many people could be hired if the department is resurfacing 16 miles of roads?" (17 people) "3 miles?" (47 people)
"How many miles of roads could be resurfaced if the department is hiring 2 new workers?" (22.5 miles) "If no new workers are hired?" (About 23.37 miles)
Make sure students understand that solving for
\(p\)
makes it possible to quickly find the number of people that the department could hire given some miles of road to be resurfaced (and while sticking to the budget). Similarly, solving for
\(m\)
makes it possible to quickly find the miles that could be resurfaced given any number of new hires.


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