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: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with one variable, x.

Example 3:
Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
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, 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_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
problem_activity: Narrative
The purpose of this activity is for students to be introduced to the terms
halves
and
fourths
. In the launch, students explore examples and non-examples of shapes split into halves and fourths and develop a shared understanding of what these terms mean (MP6). Students then draw lines to partition shapes into halves or fourths. During the synthesis students learn that the term 'quarters' means the same as fourths. Fourths and quarters will be used interchangeably throughout this section.
Action and Expression: Develop Expression and Communication.
Give students access to a straight edge or ruler.
Supports accessibility for: Fine Motor Skills, Visual-Spatial Processing
Launch
Groups of 2
Display the categories of shapes from the student workbook.
“Here are some shapes split into pieces.”
“What do you notice?” (One table shows shapes split into two pieces and the other table shows shapes split into four pieces. The pieces are equal in the first group.)
1 minute quiet think time
2 minutes: partner discussion
Record responses.
“Based on what you observed, what do you think split into halves means? What do you think split into fourths means?”
1 minute quiet think time
Record responses.
“
Halves
are the pieces created when a shape is split into two pieces that are the same size.
Fourths
are the pieces created when a shape is split into four pieces that are the same size.”
Activity
Read the task statement.
4 minutes: independent work time
Monitor for students who accurately partition the shapes into fourths in different ways.
Student Facing
[IMG]
[IMG]
Split each shape into halves.
[IMG]
Split each shape into fourths.
[IMG]
Split each shape into fourths.
[IMG]
Split each shape into halves.
[IMG]
Split each shape into halves.
[IMG]
Split each shape into fourths.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
Students may split the shapes into halves accurately by drawing a vertical or horizontal line down the middle of each shape. These students may attempt to create fourths, for the rectangles and circles, using the same technique by guessing-and-checking. Ask these students to explain their method and what they hope to achieve. Consider asking:
“How many pieces are you trying to create?”
“How can you make sure they are the same size?”
“How would you fold this shape to make __ equal pieces?”
“How can you use the table of shapes we looked at earlier to help plan how to make __ equal pieces?”
Activity Synthesis
Invite previously identified students to share.
“Do all of these shapes show fourths? How do you know?” (Yes, each of the shapes have 4 equal pieces. Each of the pieces are the same size.)
“When we split a shape into four equal-sized pieces, the shape is split into fourths. We can also say the shape is split into quarters.”


aligns: no