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: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
problem_activity: Activity
In general,
\(P\%\)
of something is
\(\frac{P}{100}\)
times that thing. The purpose of this activity is to make this explicit. In this activity, students are asked to find 80% of several different values, generalize their process, and express their generalization in different ways. As they make repeated calculations, students look for and express regularity in their work (MP8) and see more explicitly that
\(P\%\)
of a number is
\(\frac{P}{100}\)
times that number. Once they arrive at one or more generalizations, students practice articulating why each generalization always works (MP3).
Launch
Give students quiet think time to complete the activity and then time to share their explanation with a partner.
Representation: Internalize Comprehension.
Activate or supply background knowledge about equivalent fractions. Incorporate explicit opportunities for review and practice if necessary. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time.
Use this with successive pair shares to give students a structured opportunity to revise and refine their own strategies to determine 80% of any value to find a sale price. Make sure students include a table as well as an expression in their explanations. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., “Can you explain how…”, “Will this expression always work? How do you know?”, etc.). Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students refine their own strategies and learn about other ways to find the sale price of an item.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
During a sale, every item in a store is 80% of its regular price.
If the regular price of a T-shirt is
$
10, what is its sale price?
The regular prices of five items are shown here. Find the sale price of each item.
|               | item 1 | item 2 | item 3 | item 4 | item 5 |
|---------------|--------|--------|--------|--------|--------|
| regular price |        |        |        |        |        |
|  sale price   |        |        |        |        |        |
You found 80% of many values. Was there a process you repeated over and over to find the sale prices? If so, describe it.
[IMG]
Select
all
of the expressions that could be used to find 80% of
\(x\)
. Be prepared to explain your reasoning.
\(\frac{8}{100} \boldcdot x\)
\(\frac{80}{100} \boldcdot x\)
\(\frac{8}{10} \boldcdot x\)
\(\frac{4}{10} \boldcdot x\)
\(\frac85 \boldcdot x\)
\(\frac45 \boldcdot x\)
\(80 \boldcdot x\)
\(8 \boldcdot x\)
\((0.8) \boldcdot x\)
\((0.08) \boldcdot x\)
Student Response
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Activity Synthesis
Make sure students have the correct values in the table for the second question. Then, discuss the expressions, starting with
\(\displaystyle \frac{80}{100} \boldcdot x.\)
Guide students to see that if we know the value of 100%, dividing that value by 100 (or equivalently, multiplying it by
\(\frac{1}{100}\)
) tells us the corresponding value for 1%. We can then multiply that value by the desired percentage. Consider using a table to organize this argument, as shown below:
| percentage |         regular price          |
|------------|--------------------------------|
|    100     |             \(x\)              |
|     1      | \(\frac{1}{100} \boldcdot x\)  |
|     80     | \(\frac{80}{100} \boldcdot x\) |
After everyone understands where this expression comes from, ask students to discuss the remaining expressions with a partner. This is a good opportunity for just-in-time review on equivalent fractions, if needed.


correct_option_index: 1