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: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
problem_activity: Narrative
The purpose of this activity is for students to compare shapes by covering them with pattern blocks. Students experience tiling as a way to see which shape covers the most space. There are several ways to tile the shapes, but it may prove most useful to use the same units, such as triangles. The rectangle can only be fully tiled with square pattern blocks. To compare shapes B and C, students will need to notice that the rectangle and parallelogram can be made the same length, but the square pattern blocks used to tile the rectangle are taller than the blocks used to tile the parallelogram.
[IMG]
[IMG]
The work here prepares students to tile figures with square tiles in the next lesson and to think of area in terms of square units.
MLR8 Discussion Supports.
Synthesis: Create a visual display of shapes A, B, and C. As students discuss their comparisons, annotate the display with their observations. For example, when comparing shape B to C, write “same length, but taller.”
Advances: Speaking, Listening
Action and Expression: Develop Expression and Communication.
Synthesis: Identify connections between strategies that result in the same outcomes but use differing approaches.
Supports accessibility for: Conceptual Processing
Required Materials
Materials to Gather
Pattern blocks
Scissors
Materials to Copy
Pattern Blocks to Compare Shapes
Required Preparation
Each group of 2 needs at least 2 hexagons and trapezoids,  4 squares and rhombuses, and 8 triangles.
Launch
Groups of 2
Give each student a copy of the blackline master.
Give each group pattern blocks.
Display the first two shapes (the hexagon and the heart-shaped octagon).
“What do you notice? What do you wonder?” (Students may notice: Both shapes could be made with pattern blocks. The second shape looks like the first shape, but it’s been bent. Students may wonder: Which shape covers more space? Which shape is larger?)
1 minute: quiet think time
Share and record responses.
“Which shape covers more space? How do you know?” (We can cover them with the same blocks, so they cover the same amount of space.)
2 minutes: partner discussion
Share and record responses.
Activity
“Let's continue comparing shapes. Work with your partner to decide which of these shapes covers the most space. Use pattern blocks if they are helpful.”
5 minutes: partner work time
Monitor for students who:
use pattern blocks to determine which shape covers the most space
notice that shape B and C would be the same length if one of the ends of B was moved over, but C covers more space because it is taller
Student Facing
Your teacher will give you handouts with some shapes on them.
Use pattern blocks to decide which shape covers the most space. Be ready to explain your reasoning.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Ask selected students to share strategies for tiling each shape with pattern blocks and how the tiling helped them decide which shape covers the most space.
“When we decide which shape covers the most space, we are talking about the area of a shape. We can think about the
area as the amount of space covered by a shape.”
“Which of these shapes has the greatest area?” (C takes up the most space, so it has the greatest area.)


aligns: no