few_shots: 
Example 1:
Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

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


Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Explain and use the relationship between the sine and cosine of complementary angles.
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