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


Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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: 
Construct a tangent line from a point outside a given circle to the circle.

Answer: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_neg_0

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standard_description: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
problem_activity: Activity
In this activity, students are given a percent increase and use it to calculate the new value, rather than being given the original and new values to calculate the percent increase. Students can solve the problems using the double number lines but the discussion that follows will be the connection to what the words specifically mean (MP6).
As students work, monitor for students who complete the double number line diagram and any student that uses other methods to correctly reason about the problem.
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by partner then whole-class discussion.
Student Facing
​​
A cereal box says that now it contains 20% more. Originally, it came with 18.5 ounces of cereal. How much cereal does the box come with now?
[IMG]
​​
The price of a shirt is \$18.50, but you have a coupon that lowers the price by 20%. What is the price of the shirt after using the coupon?
[IMG]
Student Response
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Anticipated Misconceptions
Have students use the double number line diagram if they need help figuring out 20% more.
Activity Synthesis
Select students to share their reasoning for each problem. Start with a student that used the double number line diagram to solve the problem. Then, have students share other methods they used to solve the problem, such as multiplying by 1.2 and 0.8.
Ask students:
Did the number of ounces of cereal in the cereal box increase or decrease?
What percentage of the original amount of cereal is the new amount of cereal?
Did the price of the shirt increase or decrease?
What percentage of the original price of the shirt is the new price of the shirt?
Tell students that the change on the cereal box is an example of a
percent increase
. The discount on the shirt is an example of a
percent decrease
. You might want to mention that in both cases, 100% always corresponds to the original amount before the change; however, future activities will address this concept in depth.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of percent increase and percent decrease.
Supports accessibility for: Conceptual processing; Language
Writing: MLR3 Clarify, Critique, Correct.
Before students share their reasoning for each problem, present an incomplete strategy for finding the discount price for the shirt. For example, “The discount is 20%, so the price of the shirt is $3.70.” Ask students to critique the reasoning, and work with a partner to write an improved explanation. Listen for and amplify the language students use to make sense of what is displayed (e.g., “I think they ___ because ____.”), as well as the mathematical language that students use that strengthen their explanations. This will support student understanding of mathematical language related percent increase and percent decrease.
Design Principle(s): Maximize meta-awareness


aligns: yes