few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


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.

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


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).

Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
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 $P$.


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.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

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 facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
problem_activity: Optional activity
The goal of this task is to introduce the notion of a tessellation and then carefully examine how to create a tessellation with each pattern block shape (square, rhombus, equilateral triangle, isosceles trapezoid). For hexagons, there is only one way to fit them together because there will be gaps unless three hexagons meet at each vertex. The other shapes offer much more flexibility, and students have an opportunity to use their artistic creativity.
Students look for and make use of structure (MP7), both when they try to put copies of the shape together to build a tessellation and when they examine whether or not it is possible to construct a different tessellation.
Digital
Launch
Introduce the definition of a
tessellation
of the plane by polygons: a tessellation covers the plane with copies of the shape with no gaps and no overlaps.
Show the students the image from the previous notice and wonder activity, and ask them if these are tessellations. (Yes.)
Demonstrate how to use tracing paper to create a tessellation.
Arrange students in groups of 2. Each pair uses the same shape for their tessellations.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Include the following term and maintain the display for reference throughout the lesson: tessellation. Consider providing step-by-step directions for students to use tracing paper to create a tessellation. Invite students to suggest language or diagrams to include that will support their understanding.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports.
Amplify students' uses of mathematical language to communicate about tessellations; edge to edge, vertex, symmetrical, plane, rigid motion, etc. Encourage students to use these words, revoicing students’ ideas as necessary. Ask students to chorally repeat the phrases that include these words in context.
Design Principle(s): Support sense-making; Optimize output (for explanation)
Student Facing
With your partner, choose one of the six shapes in the toolbar that you will both use.
Select the shape tool by clicking on it. Create copies of your shape by clicking in the work space.
When you have enough to work with, click on the Move tool (the arrow) to drag or turn them.
If you have trouble aligning the shapes, right click to turn on the grid.
When you are finished making your tessellation, you can hide the points.
Compare your tessellation to your partner’s. How are they similar? How are they different?
If possible, make a third tessellation of the plane with your shape (different from the ones you and your partner already created). If not possible, explain why it is not possible.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Introduce the definition of a
tessellation
of the plane by polygons: a tessellation covers the plane with copies of the shape with no gaps and no overlaps.
Show the students the image from the previous notice and wonder activity, and ask them if these are tessellations. (Yes.)
Demonstrate how to use tracing paper to create a tessellation.
Arrange students in groups of 2. Each pair uses the same shape for their tessellations.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Include the following term and maintain the display for reference throughout the lesson: tessellation. Consider providing step-by-step directions for students to use tracing paper to create a tessellation. Invite students to suggest language or diagrams to include that will support their understanding.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports.
Amplify students' uses of mathematical language to communicate about tessellations; edge to edge, vertex, symmetrical, plane, rigid motion, etc. Encourage students to use these words, revoicing students’ ideas as necessary. Ask students to chorally repeat the phrases that include these words in context.
Design Principle(s): Support sense-making; Optimize output (for explanation)
Student Facing
[IMG]
Pick one of the shapes. Create a
tessellation
by tracing copies of your shape. Make sure to use the same shape as your partner.
Compare your tessellation to your partner’s. How are they similar? How are they different?
If possible, make a third tessellation of the plane with your shape (different from the ones you and your partner already created). If not possible, explain why it is not possible.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Tessellations do not need to be edge to edge. That is, pieces do not need to fit together with edges of the same lengths matching exactly.
Tessellations do not need to have symmetric, repeating patterns, though sometimes the shape forces it (as with the regular hexagons).
Activity Synthesis
Important questions to address include:
“Were you able to make different tessellations with your shape?” (It depends on the shape.)
“If not, why not?” (Three hexagons have to come together at each vertex: once the first hexagon of the pattern is placed, everything else has no flexibility.)
“If so how?” (Triangles can be built into hexagons, or they can make rows that can be translated and stacked on top of one another. Parallelograms can also be made into hexagons or rows. Trapezoids can be made into hexagons or rows. Rhombuses can be made into rows and translated. Squares can also be made into rows and translated.)
“What does it look like to
not
define a tessellation?” (Two octagons can be put together sharing a vertex, but there is a gap that is not large enough for a third octagon.)
Make sure, as students share their ideas for tessellations, to use the language of rigid motions to describe the tessellations. For example, if a student has built a tessellation with parallelograms, choose two parallelograms and ask:
“How can I use rigid motions to move one of these parallelograms into the position of the other?”
“Are there other ways I could do this?”


aligns: no