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: Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
problem_activity: Narrative
The purpose of this activity is to represent and solve a measurement problem with fractions. Students may approach this activity in multiple ways and are invited to apply what they know about operations with fractions, comparing fractions, and fraction equivalence to make sense of and solve the problems (MP2). Throughout the activity, listen for the ways students use what they know about comparing fractions and fraction equivalence as they make sense of the problem. Although students may consider ways to subtract fractions with unlike denominators, this is not a requirement for grade 4. Focus the conversation during the activity and the synthesis on how students can solve the problem by reasoning about equivalent fractions and the representation they use to make sense of the problem.
MLR8 Discussion Supports.
Synthesis: Create a visual display of the diagrams. As students share their strategies, annotate the display to illustrate connections. For example, next to each representation, write how the diagram relates to the situation.
Advances: Speaking, Representing
Launch
Groups of 2
Activity
5 minutes: independent work time
5 minutes: partner work
Monitor for different representations of the situation (for example, equations, drawings, and number lines).
Student Facing
In science class, Noah, Tyler, and Jada are building a model of a roller coaster out of 1-foot-long paper straws.
Noah needs a piece that is
\(\frac{7}{12}\)
foot long.
Tyler needs one that is
\(\frac{1}{4}\)
foot long.
Jada needs one that is shorter than the other two.
Jada says, “We can just use one straw for all these pieces.”
[IMG]
Draw a diagram to represent this situation and explain to your partner how it matches the situation. Then, find the length of the piece of straw that could be Jada’s piece.
Did Noah use more than
\(\frac{1}{2}\)
foot or less than
\(\frac{1}{2}\)
foot of straw? Explain or show your reasoning.
Tyler says, “If Jada uses a piece that is
\(\frac{1}{6}\)
foot long, there would be a piece of straw that is
\(\frac{1}{12}\)
foot left.”
Do you agree or disagree with Tyler? Explain your reasoning.
Student Response
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Advancing Student Thinking
If students create representations that do not match the quantities in the problem, consider asking:
“How does your representation show the length of the straw? How does it show Noah and Tyler’s pieces?”
“How does your representation show the action in the situation?”
Activity Synthesis
Invite previously selected students to share their arguments and allow peers to extend and add ideas to the conversation.
“How does each representation match this situation about the straw?” (They each show the total length of the straw and ways it could be broken into smaller parts. They each show the length of Noah and Tyler’s pieces. They show how long Jada’s piece could be.)


aligns: yes