options: Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
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
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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
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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: 7