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: 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
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
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:
\(a(b + c) = ab + ac\)
.


correct_option_index: 1