few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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:
Concept/skill: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

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


Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
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.

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 $P$.


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, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Use square root and cube root symbols to represent solutions to equations of the form x^(2) = p and x^(3) = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that Ö2 is irrational.
problem_activity: Optional activity
This activity is similar to the previous activity in that students are interpreting a graph and making sense of what situation the graph is representing. The difference here is that the specificity with numbers has been removed, so students need to think a bit more abstractly about what the changes in the graph represent and how they connect to the situation.
Monitor for students describing the graph with different levels of detail, particularly for any students who state that the car got up to speed faster than the car slowed down to 0.
Teacher Notes for IM 6–8 Math Accelerated
This activity is required.
Launch
Give students 1–2 minutes of quiet work time, followed by a whole-class discussion.
Engagement: Develop Effort and Persistence.
Break the class into small discussion groups and then invite a representative from each group to report back to the whole class. All group members should be prepared to share if invited.
Supports accessibility for: Attention; Social-emotional skills
Student Facing
[IMG]
The graph shows the speed of a car as a function of time. Describe what a person watching the car would see.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The graph models the speed of a car over a function of time during a
3-hour trip. How far did the car go over the course of the trip?
There is a nice way to visualize this quantity in terms of the graph. Can you find it?
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of this activity is for students to connect what is happening in a graph to a situation. Display the graph for all to see. Select previously identified students to share the situation they came up with. Sequence students from least descriptive to most descriptive. Have students point out the parts on the graph as they share their story about the situation.
Consider asking the following questions:
“Did the car speed up faster or slow down faster? How do you know?” (The car sped up faster because the first part of the model is steeper than the third part of the model.)
“How did you know that the car stayed that speed for a period of time?” (The graph stays at the same height for a while, so the speed was not changing during that time.)
Reading, Writing, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to provide students with a structured opportunity to revise and refine the explanation of their thinking about the shape and patterns in the graph. After students have written a response to the task statement, ask students to meet with 2–3 partners in a row for feedback. Each time, partners should explain their thinking without reading from their written work. Provide students with prompts for feedback that will help students strengthen their ideas and clarify their language (e.g., “How do you know that from the graph?”, “What in the graph makes you think that?”, “Can you give an example?”, etc.). Students can borrow ideas and language from each partner to strengthen their final version.
Design Principle(s): Optimize output (for explanation)


aligns: no