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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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:
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_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Recognize and represent proportional relationships between quantities.
problem_activity: Activity
This activity serves two goals: to uncover the defining features of
prisms
and
pyramids
as well as to introduce
nets
as two-dimensional representations of polyhedra.
Students first analyze prisms and pyramids and try to define their characteristics. Next, they learn about nets and think about the polygons needed to compose the nets of given prisms and pyramids. They then use their experience with the nets of prisms and pyramids to sharpen and refine their definitions of these polyhedra.
Ask students discuss the features of prisms and pyramids, encourage them to use the terms face, edge, and vertex (vertices) in their descriptions.
Launch
Arrange students in groups of 3–4.
For the first question:
Tell students that Polyhedra A–F are all prisms and Polyhedra P–S are all pyramids. (Display and pass around the prisms and pyramids in the task statement, if available.)
Give students 2–3 minutes of quiet time for the first question and 2–3 minutes to discuss their observations in their groups. Ask them to pause for a class discussion before moving on.
Solicit students' ideas about features that distinguish prisms and pyramids. Record students' responses in a two columns—one for prisms and the other for pyramids. It is not important that the lists are complete at this point.
Next, tell students that we are going to use nets to better understand prisms and pyramids. Explain that a
net
is a two-dimensional representation of a polyhedron.
Display a cube assembled from the net provided in the blackline master for the warm-up, as well as a cutout of an unfolded net (consider removing the flaps). Demonstrate how the net with squares could be folded and assembled into a cube, or use this book of digital applets,
https://ggbm.at/rcu3Ka3j
, created in GeoGebra by the
GeoGebra DocuTeam
. Point out how the number and the shape of the faces on the cube correspond to the number and the shape of the polygons in the net.
For the second question:
Give groups a minute of quiet think time and a minute to discuss their response.
To verify their answer, give each group one of the three nets from the first page of the blackline master. Ask them to try to assemble a triangular pyramid from their net.
Invite groups to share with the class whether it can be done. Discuss why net 3 cannot be assembled into Pyramid P (two of the triangles would overlap).
For the last question, tell students that they will create a net of another prism or pyramid:
Assign each group a prism or a pyramid from the task statement (except for Prism B and Pyramid P).
Give each group a set of pre-cut polygons from the last two pages of the blackline master.
Tell students to choose the right kind and number of polygons that make up their polyhedron. Then, arrange the polygons so that, when taped and folded, the arrangement is a net and could be assembled into their prism or pyramid. Encourage them to think of more than one net, if possible.
Representation: Access for Perception.
Provide access to concrete manipulatives. Provide prisms and pyramids for students to view or manipulate. These hands-on models will help students identify characteristics or features, and support net building for each polyhedra.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Speaking, Conversing: MLR2 Collect and Display.
As students work on creating a net of their assigned polyhedron, circulate and listen to the language students use when talking about the polygons that make up their polyhedra, as well as the characteristics of their polyhedra (e.g., triangle, rectangle, square, hexagon, pentagon, vertex, edge, face). Collect this language, with corresponding drawings, and display it for all students to see. Remind students to borrow language from the display as they describe the features of prisms and pyramids. This will help students produce mathematical language to describe and define characteristics of polyhedra.
Design Principle(s): Support sense-making
Student Facing
Here are some polyhedra called
prisms
.
[IMG]
Here are some polyhedra called
pyramids
.
[IMG]
Look at the prisms. What are their characteristics or features?
Look at the pyramids. What are their characteristics or features?
Which of these
nets
can be folded into Pyramid P? Select all that apply.
[IMG]
Your teacher will give your group a set of polygons and assign a polyhedron.
Decide which polygons are needed to compose your assigned polyhedron. List the polygons and how many of each are needed.
Arrange the cut-outs into a net that, if taped and folded, can be assembled into the polyhedron. Sketch the net. If possible, find more than one way to arrange the polygons (show a different net for the same polyhedron).
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
What is the smallest number of faces a polyhedron can possibly have? Explain how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select groups to share their arrangements of polygons. If time permits and if possible, have students tape their polygons and fold the net to verify that it could be assembled into the intended polyhedron. Discuss:
“What do the nets of prisms have in common?” (They all have rectangles. They have a pair of polygons that may not be rectangles.)
“What do the nets of pyramids have in common?” (They all have triangles. They have one polygon that may not be a triangle.)
“Is there only one possible net for a prism or a pyramid?” (No, the polygons can be arranged in different ways and still be assembled into the same prism or pyramid.)
Explain the following points about prisms and pyramids:
A prism has two parallel, identical faces called
bases
and a set of rectangles connecting the bases.
Prisms are named for the shape of the bases. For example, if the base of a prism is a pentagon, then the prism is called a “pentagonal prism.”
A pyramid has one face called the base that can be any polygon and a set of faces that are all triangles. Each edge of the base is shared with an edge of a triangle. All of these triangles meet at a single vertex.
Pyramids are named for the shape of their base. For example, if the base of a pyramid is a square, then the pyramid is called a “square pyramid.”


aligns: no