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: Explain why addition and subtraction strategies work, using place value and the properties of operations. (Explanations may be supported by drawings or objects.)

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Add up to four two-digit numbers using strategies based on place value and properties of operations.
problem_activity: Narrative
The purpose of this activity is for students to choose methods flexibly for finding the value of differences. Students might subtract by place, count on, or make an easier problem. There is no right answer to which method should be used for each problem. Students should choose a method that makes sense to them and justify their choice (MP3).
Engagement: Provide Access by Recruiting Interest.
Provide choice. Invite students to decide the order of the problems to complete the task. They can choose to work through in any order.
Supports accessibility for: Social-Emotional Functioning, Organization
Required Materials
Materials to Gather
Base-ten blocks
Launch
Groups of 2
Give students access to base-ten blocks.
“We’ve learned how to decompose units to subtract by place and different ways to represent that. We’ve also learned other methods for subtracting. We use them when they make sense to us or when they make sense for the numbers in an expression.”
Activity
“Find the value of each expression using a method that makes sense to you. You’ll have a chance to share your work with others.”
6 minutes: independent work time
Monitor for expressions that most students find the same way and expressions that many students find in different ways.
“Find a partner who found the value of _____ the same way as you.”
1–2 minutes: partner discussion
“Now find a partner that found the value of _____ in a different way than you. Share your thinking.”
2–3 minutes: partner discussion
Repeat for additional expressions as desired.
Student Facing
Find the value of each expression in a way that makes sense to you. Show your thinking. Organize it so it can be followed by others.
\(535 - 214\)
\(700 - 589\)
\(683 - 398\)
\(918 - 608\)
\(735 - 457\)
\(602 - 487\)
Student Response
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Advancing Student Thinking
If students use the same method for each problem, ask students to think about how they could use Jada’s way to find the value of
\(701 - 599\)
. Consider asking:
“What do you notice about the numbers in this expression that makes it easier to use Jada’s way?”
“What other expressions do you see where you might try Jada’s way?”
Activity Synthesis
Have 3–4 students share a method or representation that someone they talked to shared.
“What methods or representations do you want to try more?”


correct_option_index: 3