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

Experiment with transformations in the plane.

Apply trigonometry to general triangles.

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

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

Prove geometric theorems.

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

Understand similarity in terms of similarity transformations.

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

Understand congruence in terms of rigid motions.

Explain volume formulas and use them to solve problems.

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

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

Find arc lengths and areas of sectors of circles.

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

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

Analyze, compare, create, and compose shapes.

Apply geometric concepts in modeling situations.

Prove theorems involving similarity.

Understand and apply the Pythagorean Theorem.

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

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

Make geometric constructions.

Understand and apply theorems about circles.

Reason with shapes and their attributes.

Define trigonometric ratios and solve problems involving right triangles.

Use coordinates to prove simple geometric theorems algebraically.
problem_activity: Activity
In this activity students continue to use coordinate transformation notation, but now without the scaffolding of an organizing structure like a table. Students apply transformation rules, then describe the results. One transformation is of a type students have not seen before (the horizontal stretch that is not a dilation).
In the synthesis, students will connect the geometric description of a reflection to the coordinate rule that produces it. They will also draw a conclusion about rules that produce a dilation versus those that produce a stretch in only 1 direction.
Monitor for students who erroneously call the horizontal stretch a dilation. Encourage students to refer to their reference charts for help in attending to precision when describing transformations they’ve learned about.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Conversing: MLR2 Collect and Display.
As students work on this activity, listen for and collect the language students use to describe each transformation of quadrilateral
\(ABCD\)
. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a statement such as, “The shape was flipped upside down” can be improved with the statement, “Quadrilateral
\(ABCD\)
was reflected across the
\(x\)
-axis.” This will help students use the mathematical language necessary to precisely describe transformations of figures on the coordinate plane.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
[IMG]
Here are some transformation rules. Apply each rule to quadrilateral
\(ABCD\)
and graph the resulting image. Then describe the transformation.
Label this transformation
\(Q\)
:
\((x,y) \rightarrow (2x,y)\)
Label this transformation
\(R\)
:
\((x,y) \rightarrow (x,\text-y)\)
Label this transformation
\(S\)
:
\((x,y) \rightarrow (y,\text-x)\)
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?
[IMG]
Plot the quadrilateral with vertices
\((4,\text-2),(8,4),(8,\text-6),\)
and
\((\text-6,\text-6)\)
. Label this quadrilateral
\(A\)
.
Plot the quadrilateral with vertices
\((\text-2,4),(4,8),(\text-6,8),\)
and
\((\text-6,\text-6)\)
. Label this quadrilateral
\(A'\)
.
How are the coordinates of quadrilateral
\(A\)
related to the coordinates of quadrilateral
\(A'\)
?
What single transformation takes quadrilateral
\(A\)
to quadrilateral
\(A'\)
?
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
If students struggle to keep their work organized, suggest they create a table of inputs and outputs, or create another organizational structure that works for them.
Some students may not be sure how to work with the rule
\((x,y)\rightarrow (y,\text-x)\)
. Ask these students, “What are the coordinates of point
\(A\)
?” (
\((3,\text-4)\)
.) “Which of those is the
\(x\)
-coordinate?” (3.) “In the transformation rule, where does the
\(x\)
land, and what happens to it?” (The
\(x\)
-coordinate lands in the
\(y\)
spot, and it takes on an opposite sign.) Another option is to suggest students write out
\(x=3\)
and
\(y=\text-4\)
, then substitute each value into the transformation rule.
Activity Synthesis
The goal of the discussion is to make connections between the coordinate rules and the geometric descriptions of transformations. Start with these discussion questions:
“In the previous activity, one rule was
\((x,y)\rightarrow (2x,2y)\)
. In this activity, one rule was
\((x,y)\rightarrow (2x,y)\)
. Compare and contrast these rules.” (In the first rule, both coordinates were multiplied by 2. It is stretched both vertically and horizontally by the same factor. The second rule just had the
\(x\)
-coordinate multiplied by 2. It was only stretched in one direction.)
“Can we call
\((x,y)\rightarrow (2x,y)\)
a dilation? Why or why not?” (No. In a dilation, the figure is stretched in all directions by the same factor, not just 1 direction. Also, in this example we can see that the angles of the dilated figure are not congruent to the angles in the original.)
“Can we call
\((x,y)\rightarrow (2x,y)\)
a translation? Why or why not?” (No. A translation is a rigid motion—it produces a result that does not change size or shape.)
Now ask students, “The rule
\((x,y) \rightarrow (x, \text-y)\)
is a reflection across the
\(x\)
-axis. Look at the definition of a reflection on your reference chart. Why does this rule produce a reflection?” Guide students through the different aspects of the geometric definition of a reflection. Consider displaying an image of the solution and drawing segments that connect pairs of original and image points. Important ideas that should surface are:
The
\(y\)
-coordinates of the original and image points have the same absolute value, so the distance from the
\(x\)
-axis is the same for each.
The sign of the image’s
\(y\)
-coordinate is opposite the original, so the image is on the opposite sign of the axis as the original.
Connecting the image and original points, we get a vertical line because the
\(x\)
-coordinates are the same. So this segment is perpendicular to the line of reflection, or the
\(x\)
-axis.


correct_option_index: 1