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 subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.
problem_activity: Narrative
The purpose of this activity is for students to write the symbol or number that makes a comparison statement true. Students then read the comparison statement. This activity has two parts. In the first part, students are given two numbers with a blank space in which to write a comparison symbol that makes the statement true. After students write the symbol, they read the comparison statement. Reading the statement encourages students to relate the language of comparison to the symbols (MP6). In the second part of the activity, students are given a comparison symbol and either one number or neither number. Students determine a number or numbers that will make the comparison true.
MLR8 Discussion Supports.
Before pairs begin working together, remind students to use phrases such as greater than and less than. Model how to use these phrases when comparing two numbers. Invite students to chorally repeat the phrases in context.
Advances: Speaking, Conversing
Required Materials
Materials to Gather
Connecting cubes in towers of 10 and singles
Launch
Groups of 2
Give students access to connecting cubes in towers of 10 and singles.
Activity
Read the task statement.
8 minutes: partner work time
Student Facing
Compare the numbers.
Write <, >, or = in each blank.
Then read the comparison statement.
\(56 \underline{\hspace{1 cm}} 26\)
\(72 \underline{\hspace{1 cm}} 78\)
\(6 \underline{\hspace{1 cm}} 55\)
\(92 \underline{\hspace{1 cm}} 29\)
\(23 \underline{\hspace{1 cm}} 23\)
[IMG]
Fill in each box with a number to make each statement true.
\(\boxed{\phantom{\frac{aaai}{aaai}}} > 78\)
\(39 < \boxed{\phantom{\frac{aaai}{aaai}}}\)
\(13 = \boxed{\phantom{\frac{aaai}{aaai}}}\)
\(\boxed{\phantom{\frac{aaai}{aaai}}} < \boxed{\phantom{\frac{aaai}{aaai}}}\)
\(\boxed{\phantom{\frac{aaai}{aaai}}} > \boxed{\phantom{\frac{aaai}{aaai}}}\)
Student Response
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for free access to Student Response.
Advancing Student Thinking
If students create statements that are not true, consider asking:
“Read your statement. How could you prove that it is a true statement?”
“What other numbers could you use to make this a true statement? Explain how you know.”
Activity Synthesis
Display
\(\boxed{\phantom{3}} > 78\)
.
“How did you know what number would make the statement true?” (I knew it had to be greater than 78 because I read the statement ‘blank is greater than 78’. I put in a number and read the statement out loud to see if it was true. I chose a number with more than 7 tens so I knew it would be greater than 78.)
Display
\(39 < \boxed{\phantom{3}}\)
.
“How did you know what number would make the statement true?” (I knew it had to be greater than 39 because I read the statement ‘39 is less than blank’. I put in a number and read the statement out loud to see if it was true. I chose a number with more than 3 tens so I knew that it would be greater than 39.)
Invite students to share comparisons they made for
\(\boxed{\phantom{3}}< \boxed{\phantom{3}}\)
and
\(\boxed{\phantom{3}} >\boxed{\phantom{3}}\)
.
\(\)
For each comparison shared, have the class decide if it is true or not.


aligns: no