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: Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
problem_activity: Narrative
The purpose of this activity is for students to use the location of a unit fraction to locate another fraction with a different denominator on the number line. Students can use their knowledge from the previous activity to place 1 on the number line and then use that to partition the interval from 0 to 1 to find other numbers. Because students have only located fractions with the same denominator on a single number line, they may want to use more than one number line in this activity. They may or may not label the points they find along the way to
\(\frac{3}{4}\)
. Encourage them to use whatever strategy makes sense to them.
Monitor for students who use a single number line to show both thirds and fourths and those who use separate number lines. Select them to share during activity synthesis.
This activity is optional because it goes beyond the depth of understanding required to address grade 3 standards.
MLR8 Discussion Supports.
Synthesis: As students share the similarities and differences between the strategies, use gestures to emphasize what is being described. For example, point to each fraction and show with your fingers the partitions such as thirds and fourths, that are being discussed.
Advances: Listening, Representing
Launch
Groups of 2
“Now we’re going to try something a little bit different. Let’s use the location of a unit fraction to find a fraction with a different denominator.”
Activity
“Take a few minutes to locate
\(\frac{3}{4}\)
on this number line. Use any strategy that makes sense to you.”
3–5 minutes: independent work time
As students work, consider asking:
“How did you decide whether to use one or two number lines?”
“Did you locate any numbers before locating
\(\frac{3}{4}\)
?”
“Where is 1 on your number line(s)?”
“Share your strategy with your partner.”
2–3 minutes: partner discussion
Student Facing
Locate and label
\(\frac{3}{4}\)
on the number line. Be prepared to explain your reasoning.
[IMG]
[IMG]
Student Response
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Activity Synthesis
Ask the two selected students to display their work side-by-side for all to see.
“What do these strategies have in common? How are these representations different?” (They both located 1 before locating
\(\frac{3}{4}\)
. In the first strategy, the thirds are one number line and used to find 1. Then, 1 is on the second number line and used to find
\(\frac{3}{4}\)
. In the second strategy, the thirds and fourths are marked on the same number line.)
Consider asking: “What questions do you have about these representations?”


aligns: no