few_shots: Example:
Problem/activity:
Narrative
The purpose of this activity is for students to create composite shapes and compose new shapes from composite shapes. For example, students may find that a rectangular prism can be made from four triangular prisms or from eight smaller cubes. Students then discover that they can combine two of the composed rectangular prisms to make a new rectangular prism.
Required Materials
Materials to Gather
Geoblocks
Launch
Groups of 4
Give each group geoblocks.
Display a large cube.
“This cube can be built from other shapes. See how many ways you can build a cube using other shapes.”
3 minutes: partner work time
Share responses.
Activity
“Choose a different shape. Try to find smaller shapes you can use to build that shape.”
5 minutes: partner work time
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
Display a cube and a composite cube—for example, a cube composed of four small cubes and one rectangular prism.
Demonstrate putting the shapes together.
“What new shape did I make with the cube and the cube I made?”
“Put your shapes together to make a rectangular prism.”
Display a rectangular prism and two triangular prisms.
“Find these shapes. Put them together. What new shape can you make?” (a cube, a rectangular block)


Mathematical concept/skill options: 
A. Reason with shapes and their attributes.
B. Understand similarity in terms of similarity transformations.
C. Use coordinates to prove simple geometric theorems algebraically.
D. Translate between the geometric description and the equation for a conic section.
E. Understand congruence and similarity using physical models, transparencies, or geometry software.
F. Analyze, compare, create, and compose shapes.
G. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
H. Understand congruence in terms of rigid motions.
I. Classify two-dimensional figures into categories based on their properties.
J. Draw, construct, and describe geometrical figures and describe the relationships between them.
K. Explain volume formulas and use them to solve problems.
L. Prove theorems involving similarity.
M. Prove geometric theorems.
N. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
O. Define trigonometric ratios and solve problems involving right triangles.
P. Apply trigonometry to general triangles.
Q. Visualize relationships between two-dimensional and three-dimensional objects.
R. Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
S. Make geometric constructions.
T. Apply geometric concepts in modeling situations.
U. Solve real-world and mathematical problems involving area, surface area, and volume.
V. Graph points on the coordinate plane to solve real-world and mathematical problems.
W. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
X. Understand and apply theorems about circles.
Y. Understand and apply the Pythagorean Theorem.
Z. Experiment with transformations in the plane.
AA. Find arc lengths and areas of sectors of circles.


Your response:
A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_lesson_009431

1
options: Find arc lengths and areas of sectors of circles.

Analyze, compare, create, and compose shapes.

Translate between the geometric description and the equation for a conic section.

Understand similarity in terms of similarity transformations.

Prove geometric theorems.

Experiment with transformations in the plane.

Define trigonometric ratios and solve problems involving right triangles.

Reason with shapes and their attributes.

Use coordinates to prove simple geometric theorems algebraically.

Solve real-world and mathematical problems involving area, surface area, and volume.

Visualize relationships between two-dimensional and three-dimensional objects.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Understand congruence in terms of rigid motions.

Make geometric constructions.

Classify two-dimensional figures into categories based on their properties.

Prove theorems involving similarity.

Apply trigonometry to general triangles.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Understand and apply theorems about circles.

Understand and apply the Pythagorean Theorem.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Explain volume formulas and use them to solve problems.

Apply geometric concepts in modeling situations.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
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.


correct_option_index: 11