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: 
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_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
problem_activity: Activity
The purpose of this activity is to begin to think of a dilation with a scale factor as a rule or operation on points in the plane. Students work on a circular grid with center of dilation at the center of the grid. They examine what happens to different points on a given circle when the dilation is applied and observe that these points all map to another circle whose radius is scaled by the scale factor of the dilation. For example, if the scale factor is 3 and the points lie on a circle whose radius is 2 grid units, then the dilated points will all lie on a circle whose radius is 6 units. Students need to explain their reasoning for finding the scale factor (MP3).
Students discover that the circular grid is a powerful tool for representing dilations and they will continue to use the circular grid as they study what happens when dilations are applied to shapes other than grid circles.
In the digital activity, students encounter some new tools and the circular grid.
Digital
Launch
Ask students if they have ever seen a pebble dropped in a still pond, and select students to describe what happens. (The pebble becomes the center of a sequence of circular ripples.) Display the image from the task statement, and ask students to think about how it is like a pebble dropped in a still pond. Demonstrate that distance on the circular grid is measured by counting units along one of the rays that start at the center,
\(P\)
. Use MLR 8 (Discussion Supports) to draw students’ attention to a few important words in the task:
“When we say ‘on the circle,’ we mean on the curve or on the edge. (We do not mean the circle’s interior.)”
“Remember that a ray starts at a point and goes forever in one direction. Their rays should start at
\(P\)
and be drawn to the edge of the grid.”
If using the digital activity, you may want to demonstrate dilating a point before having students begin the task.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of these terms. Include the following terms and maintain the display for reference throughout the unit: circular grid, “on the circle,” ray, scale factor, and distance. Include a circular grid on the display, give examples, and label the features.
Supports accessibility for: Conceptual processing; Language
Student Facing
The larger Circle d is a
dilation
of the smaller Circle c.
\(P\)
is the
center of dilation
.
Draw four points
on
the smaller circle using the Point on Object tool.
[IMG]
Draw the rays from
\(P\)
through each of those four points. Select the Ray tool, then point
\(P\)
, and then the second point.
[IMG]
Mark the intersection points of the rays and Circle d by selecting the Intersect tool and clicking on the point of intersection.
[IMG]
Complete the table. In the row labeled S, write the distance between
\(P\)
and the point on the smaller circle in grid units. In the row labeled L, write the distance between
\(P\)
and the corresponding point on the larger circle in grid units. Measure the distances between pairs of points by selecting the Distance tool, and then clicking on the two points.
[IMG]
|   | \(A\) | \(B\) | \(C\) | \(D\) |
|---|-------|-------|-------|-------|
| S |       |       |       |       |
| L |       |       |       |       |
The center of dilation is point
\(P\)
. What is the
scale factor
that takes the smaller circle to the larger circle? Explain your reasoning.
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
Ask students if they have ever seen a pebble dropped in a still pond, and select students to describe what happens. (The pebble becomes the center of a sequence of circular ripples.) Display the image from the task statement, and ask students to think about how it is like a pebble dropped in a still pond. Demonstrate that distance on the circular grid is measured by counting units along one of the rays that start at the center,
\(P\)
. Use MLR 8 (Discussion Supports) to draw students’ attention to a few important words in the task:
“When we say ‘on the circle,’ we mean on the curve or on the edge. (We do not mean the circle’s interior.)”
“Remember that a ray starts at a point and goes forever in one direction. Their rays should start at
\(P\)
and be drawn to the edge of the grid.”
If using the digital activity, you may want to demonstrate dilating a point before having students begin the task.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of these terms. Include the following terms and maintain the display for reference throughout the unit: circular grid, “on the circle,” ray, scale factor, and distance. Include a circular grid on the display, give examples, and label the features.
Supports accessibility for: Conceptual processing; Language
Student Facing
The larger Circle d is a
dilation
of the smaller Circle c.
\(P\)
is the
center of dilation
.
Draw four points
on
the smaller circle (not inside the circle!), and label them
\(E\)
,
\(F\)
,
\(G\)
, and
\(H\)
.
Draw the rays from
\(P\)
through each of those four points.
Label the points where the rays meet the larger circle
\(E’\)
,
\(F’\)
,
\(G’\)
, and
\(H’\)
.
[IMG]
4. Complete the table. In the row labeled c, write the distance between
\(P\)
and the point on the smaller circle in grid units. In the row labeled d, write the distance between
\(P\)
and the corresponding point on the larger circle in grid units.
|   | \(E\) | \(F\) | \(G\) | \(H\) |
|---|-------|-------|-------|-------|
| c |       |       |       |       |
| d |       |       |       |       |
5. The center of dilation is point
\(P\)
. What is the
scale factor
that takes the smaller circle to the larger circle? Explain your reasoning.
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
For question 5, students might think the scale factor is 4, because the distance between the smaller and larger circle for each point increases by 4. If this happens, ask students how many grid units circle
\(c\)
is from the center (2) and how many grid units circle
\(d\)
is from the center (6). Then remind them that
scale factor
means a number you multiply by.
Activity Synthesis
Ask students if they made a strategic choice of points, such as points that lie on the grid lines coming from the center point
\(P\)
. Why are these points good choices for dilating?
Ask students what they think would happen if a circle were dilated about its center with a scale factor of 2 or 4. (The result would be a circle with twice the radius and 4 times the radius, respectfully, all sharing the same center.)
Two important observations coming from the lesson are:
The scale factor for this dilation is 3 so distances from the center of the circles triple when the dilation is applied.
The large circle is the dilation of the small circle, that is each point on the circle with radius 6 units is the dilated image of a point on the circle of radius 2 units. (To find which one, draw the line from the point to the center and see where it intersects the circle of radius 2 units.)
Speaking, Listening: MLR8 Discussion Supports.
As students share the scale factor that takes the smaller circle to the larger circle, press for details in students’ reasoning by asking how they know the scale factor is 3. Listen for students’ explanations that reference the table with distances between the center of dilation and points on the circles. Amplify statements that use precise language such as, “The distance between point P and any point on the smaller circle is multiplied by a scale factor of 3 to get the distance between point P and the corresponding point on the larger circle.” This will a support rich and inclusive discussion about how the scale factor affects the distance between the center of dilation and points on the circle.
Design Principle(s): Support sense-making


aligns: yes