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: Solve linear equations in one variable.

Analyze and solve pairs of simultaneous linear equations.
problem_activity: Activity
In this activity, students reason about situations involving two different relationships between the same two quantities. Then they invent their own problem of the same type. While students are encouraged by the language of the activity to use a system of equations to solve the problems, they may elect to use a different representation to explain their thinking (MP1, MP3).
As students work through the first three problems, notice the ways students reason about the problems with and without systems of equations. Identify some groups with particularly compelling or clear reasoning to share later.
Launch
Arrange students in groups of 2. Provide tools for creating a visual display.
Once students have completed the first three problems, select previously identified groups to share their solutions. Bring out why these solutions are particularly good (i.e., clarity, efficiency), and discuss the connections between them, particularly the connections between groups that did and did not use systems of equations where possible. Next, have students begin the second part of the activity and write their own problem to trade with another group.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. After students have solved the first 2–3 problems, check-in with either select groups of students or the whole class. Invite students to share the strategies they have used so far, as well as any questions they have, before continuing.
Supports accessibility for: Organization; Attention
Representing, Conversing: MLR8 Discussion Supports.
Use this routine to help students consider audience when preparing a visual display of their work. When students create their displays for the problems they invent, invite them to consider how to display their strategies so that another student can interpret them. For example, students may wish to add notes or details to their equations to help communicate their thinking. When students receive the problem created by another group, provide 2–3 minutes of quiet think time for students to read and interpret the display before they begin their small-group discussion. During whole-group discussion, ask students to share the kinds of details that were most helpful when interpreting another group's display.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Student Facing
Solve each problem. Explain or show your reasoning.
Two friends live 7 miles apart. One Saturday, the two friends set out on their bikes at 8 am and started riding towards each other. One rides at 0.2 miles per minute, and the other rides at 0.15 miles per minute. At what time will the two friends meet?
Students are selling grapefruits and nuts for a fundraiser. The grapefruits cost
$
1 each and a bag of nuts cost
$
10 each. They sold 100 items and made
$
307. How many grapefruits did they sell?
Jada earns
$
7 per hour mowing her neighbors’ lawns. Andre gets paid
$
5 per hour for the first hour of babysitting and
$
8 per hour for any additional hours he babysits. What is the number of hours they both can work so that they get paid the same amount?
Pause here so your teacher can review your work. Then, invent another problem that is like one of these, but with different numbers. Solve your problem.
Create a visual display that includes:
The new problem you wrote, without the solution.
Enough work space for someone to show a solution.
Trade your display with another group, and solve each other's new problem. Make sure that you explain your solution carefully. Be prepared to share this solution with the class.
When the group that got the problem you invented shares their solution, check that their answer is correct.
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?
On a different Saturday, two friends set out on bikes at 8:00 am and met up at 8:30 am. (The same two friends who live 7 miles apart.) If one was riding at 10 miles per hour, how fast was the other riding?
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 write a system of equations, ask them to identify any unknown quantities in the problem and assign variables to them. Then ask them if there are ways to describe the relationships between the variables. If students still struggle to think about the relationships, ask them about some possible values for each of the variables including some that make sense (such as 20 grapefruits) and some that do not (such as 1,000 grapefruits). Have students explain why some values are not possible to help them understand the relationships between variables.
Activity Synthesis
Most of the discussions happen within and between groups, but the last question requires a whole-class discussion. Have each group share the peer-generated question they were assigned and the solution. Though the group that wrote the question will be responsible for confirming the answer, encourage all to listen to the reasoning each group used.
Alternatively, after groups have checked the work of the group that solved their problem, have students complete a gallery walk to see all the created problem. Ask students to look for situations similar to theirs and to identify the most common solution methods used. After the gallery, select a few groups to share a problem and how they solved it.


correct_option_index: 1