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: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_neg_0

HS_2
standard_description: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
problem_activity: Activity
This activity continues studying dilations on a circular grid, this time focusing on what happens to points lying on a polygon. Students first dilate the vertices of a polygon as in the previous activity. Then they examine what happens to points on the sides of the polygon. They discover that when these points are dilated, they all lie on a side of another polygon. Just as the image of a grid circle is another circle, so the dilation of a polygon is another polygon. Moreover the dilated polygon is a scaled copy of the original polygon. These important properties of dilations are not apparent in the definition.
Monitor for students who notice that the sides of the scaled polygon
\(A'B'C'D'\)
are parallel to the sides of
\(ABCD\)
and that
\(A'B'C'D'\)
is a scaled copy of
\(ABCD\)
with scale factor 2. Also monitor for students who notice the same structure for
\(EFGH\)
except this time the scale factor is
\(\frac{1}{2}\)
. Invite these students to share during the discussion.
Digital
Launch
Provide access to geometry toolkits. Tell students that they are going to dilate some points. Before they begin, demonstrate the mechanics of dilating a point using a center of dilation and a scale factor. Tell students, “In the previous activity, each point was dilated to its image using a scale factor of 3. The dilated point was three times as far from the center as the original point. When we dilate point
\(D\)
using
\(P\)
as the center of dilation and a scale factor of 2, that means we’re going to take the distance from
\(P\)
to
\(D\)
and place a new point on the ray
\(PD\)
twice as far away from
\(P\)
.” Display for all to see:
[IMG]
If using the digital activity, demonstrate the mechanics of dilating using the applet. You can also use the measurement tool to confirm.
Representation: Internalize Comprehension.
Begin with a physical demonstration of the process of dilating a point using a center of dilation and a scale factor to support connections between new situations and prior understandings. Consider using these prompts: “How does this build on the previous activity in which the main task was to find distances and scale factor?” or “How does the point D’ correspond to the points D and P?”
Supports accessibility for: Conceptual processing; Visual-spatial processing
Conversing, Reading: MLR2 Collect and Display.
Circulate and listen to students as they make observations about the polygon with a scale factor of 2 and the polygon with a scale factor of
\(\frac12\)
. Write down the words and phrases students use to compare features of the new polygons to the original polygon. As students review the language collected in the visual display, encourage students to clarify the meaning of a word or phrase. For example, a phrase such as “the new polygon is the same as the original polygon but bigger” can be clarified with the phrase “the new polygon is a scaled copy with scale factor 2 of the original polygon.” A phrase such as “the polygons have the same angles” can be clarified with the phrase “each angle in the original polygon is the same as the corresponding angle in the new polygon.” This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Here is a polygon
\(ABCD\)
Dilate each vertex of polygon
\(ABCD\)
using
\(P\)
as the center of dilation and a scale factor of 2.
Draw segments between the dilated points to create a new polygon.
What are some things you notice about the new polygon?
Choose a few more points on the sides of the original polygon and transform them using the same dilation. What do you notice?
Dilate each vertex of polygon
\(ABCD\)
using
\(P\)
as the center of dilation and a scale factor of
\(\frac{1}{2}\)
.
What do you notice about this new polygon?
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
Provide access to geometry toolkits. Tell students that they are going to dilate some points. Before they begin, demonstrate the mechanics of dilating a point using a center of dilation and a scale factor. Tell students, “In the previous activity, each point was dilated to its image using a scale factor of 3. The dilated point was three times as far from the center as the original point. When we dilate point
\(D\)
using
\(P\)
as the center of dilation and a scale factor of 2, that means we’re going to take the distance from
\(P\)
to
\(D\)
and place a new point on the ray
\(PD\)
twice as far away from
\(P\)
.” Display for all to see:
[IMG]
If using the digital activity, demonstrate the mechanics of dilating using the applet. You can also use the measurement tool to confirm.
Representation: Internalize Comprehension.
Begin with a physical demonstration of the process of dilating a point using a center of dilation and a scale factor to support connections between new situations and prior understandings. Consider using these prompts: “How does this build on the previous activity in which the main task was to find distances and scale factor?” or “How does the point D’ correspond to the points D and P?”
Supports accessibility for: Conceptual processing; Visual-spatial processing
Conversing, Reading: MLR2 Collect and Display.
Circulate and listen to students as they make observations about the polygon with a scale factor of 2 and the polygon with a scale factor of
\(\frac12\)
. Write down the words and phrases students use to compare features of the new polygons to the original polygon. As students review the language collected in the visual display, encourage students to clarify the meaning of a word or phrase. For example, a phrase such as “the new polygon is the same as the original polygon but bigger” can be clarified with the phrase “the new polygon is a scaled copy with scale factor 2 of the original polygon.” A phrase such as “the polygons have the same angles” can be clarified with the phrase “each angle in the original polygon is the same as the corresponding angle in the new polygon.” This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Here is a polygon
\(ABCD\)
.
Dilate each vertex of polygon
\(ABCD\)
using
\(P\)
as the center of dilation and a scale factor of 2. Label the image of
\(A\)
as
\(A’\)
, and label the images of the remaining three vertices as
\(B’\)
,
\(C’\)
, and
\(D’\)
.
Draw segments between the dilated points to create polygon
\(A’B’C’D’\)
.
What are some things you notice about the new polygon?
[IMG]
4. Choose a few more points on the sides of the original polygon and transform them using the same dilation. What do you notice?
5. Dilate each vertex of polygon
\(ABCD\)
using
\(P\)
as the center of dilation and a scale factor of
\(\frac{1}{2}\)
. Label the image of
\(A\)
as
\(E\)
, the image of
\(B\)
as
\(F\)
, the image of
\(C\)
as
\(G\)
and the image of
\(D\)
as
\(H\)
.
6. What do you notice about polygon
\(EFGH\)
?
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?
Suppose
\(P\)
is a point not on line segment
\(\overline{WX}\)
. Let
\(\overline{YZ}\)
be the dilation of line segment
\(\overline{WX}\)
using
\(P\)
as the center with scale factor 2. Experiment using a circular grid to make predictions about whether each of the following statements must be true, might be true, or must be false.
\(\overline{YZ}\)
is twice as long
\(\overline{WX}\)
.
\(\overline{YZ}\)
is five units longer than
\(\overline{WX}\)
.
The point
\(P\)
is on
\(\overline{YZ}\)
.
\(\overline{YZ}\)
and
\(\overline{WX}\)
intersect.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may think only grid points can be dilated. In fact, any point can, but they may have to measure or estimate the distances from the center. Grid points are convenient because you can measure by counting.
Activity Synthesis
Display the original figure and its image under dilation with scale factor 2 and center
\(P\)
.
[IMG]
Ask selected students to share what they notice about the new polygon. Ensure that the following observations are made. Encourage students to verify each assertion using geometry tools like tracing paper, a ruler, or a protractor.
The new figure is a scaled copy of the original figure.
The sides of the new figure are twice the length of the sides of the original figure.
The corresponding segments are parallel.
The corresponding angles are congruent.
Ask students what happened to the additional points they dilated on polygon
\(ABCD\)
. Note that a good
strategic
choice for these points are points where
\(ABCD\)
meets one of the circles: in these cases, it is possible to double the distance from that point to the center without measuring. The additional points should have landed on a side of the dilated polygon (because of measurement error, this might not always occur exactly). The important takeaway from this observation is that dilating the polygon’s vertices, and then connecting them, gives the image of the entire polygon under the dilation.


aligns: no