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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. 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.

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


Concept/skill: 
Solve quadratic equations in one variable.

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

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 $P$.


Concept/skill: 
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.

Answer: 
Yes. 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.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
problem_activity: Activity
The purpose of this activity is for students to practice generating equivalent equations. The structure of taking turns and justifying your thinking to your partner supports students to check their work as they go, and the class has a chance to check for equivalence when the total score is recorded.
Students also have to justify, whether formally or informally, why each move keeps the equation equivalent. This will be helpful for students in an associated Algebra 1 lesson when they must determine what moves are acceptable to make to an equation without changing its solution.
Monitor for students who solve the equation first and then work to create new equations with the same answer, and for students who simply create new equivalent equations from the given equation. Monitor for students who add or multiply by the same number on each side of an equation and students who use properties of operations (distributive, commutative, associative).
Launch
Arrange students in groups of 2. Explain to students that they are going to play a cooperative game in which the class tries to come up with as many different equations with the same solutions as they can. You give them an equation, and their job is to come up with other equations with the same solution as the original equation. The partner’s job is to check that the new equation is equivalent to the original by listening to their partner’s reasoning and making sure they agree. Each partner should create their own equations before moving to the next question.
At the end of each round, you will ask them to share the equations they came up with, and keep track of how many different equations the class came up with. Set a goal for the second round of coming up with 4 more equations than the first round, and continue to set meaningful challenge goals with each round.
Calling on previously identified students to share strategies may help the class be more successful.
Display these equations, one at a time:
\(x = 5\)
\(x + 1 = 0\)
\(2(x + 1) = 10\)
\(\frac12x + \frac32 = \frac72\)
Student Facing
Your teacher will display an equation. Take turns with your partner to generate an equivalent equation—an equation with the same solution. Generate as many different equations with the same solution as you can. Keep track of each one you find.
For each change that you make, explain to your partner how you know your new equation is equivalent. Ask your partner if they agree with your thinking.
For each change that your partner makes, listen carefully to their explanation about why their new equation is equivalent. If you disagree, discuss your thinking and work to reach an agreement.
Student Response
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Activity Synthesis
Remind students that today they did a lot of thinking about equivalent equations and moves they could do to equations that wouldn’t change the solution.
Make a semi-permanent display of “moves that won’t change the solutions to equations.” Sort the list by moves that are done to each side, and moves that are done to one side. Use students’ language, adding formal language if that is an emphasis in your school.
Possible list:
Add the same value to both sides.
Subtract the same value from both sides.
Multiply both sides by the same value (but not zero!). Divide both sides by the same value (but not zero!).
Change the order of terms (on one side) being added or multiplied (commutative property).
Change the grouping of terms (on one side) being added or multiplied (associative property).
Distributive property:
\(a(b + c) = ab + ac\)
.


aligns: no