13.1: Transformations with Lists
This activity is intended to supplement Geometry, Chapter 12, Lessons 2 and 3.
ID: 10277
Time required: 40 minutes
Activity Overview
Students will graph a figure in the coordinate plane. They will use list operations to perform reflections, rotations, translations and dilations on the figure, and graph the resulting image using a scatter plot.
Topic: Transformational Geometry
- Perform reflections, rotations, translations and dilations using lists and scatter plots to represent figures on a coordinate plane.
Teacher Preparation and Notes
- This activity is designed to be used in a high school geometry or algebra classroom.
- If an original point on the coordinate plane is denoted by \begin{align*}(x, y)\end{align*}, then each of the following ordered pairs denotes a transformation:
\begin{align*}& (x, -y)\ reflect\ over\ x-axis && (-y, x)\ \ \ rotate\ 90^\circ \ around\ origin \\ & (-x, y)\ reflect\ over\ y-axis && (-x, -y)\ rotate\ 180^\circ \ around\ origin \\ & (y, x)\ \ \ reflect\ over\ y = x && (y, -x)\ \ \ rotate\ 90^\circ \ around\ origin\end{align*}
- To perform a translation, add or subtract a constant from the list with the \begin{align*}x-\end{align*}values or the \begin{align*}y-\end{align*}values of the figure.
- To perform a dilation, multiply a constant scale factor by the list with the \begin{align*}x-\end{align*}values or the \begin{align*}y-\end{align*}values of the figure.
- This activity is designed to be student-centered with the teacher acting as a facilitator while students work cooperatively. If desired, have students work in groups of 3. Each person in the group should enter a different combination of lists for Problem 2 and the group should discuss the results.
Associated Materials
- Student Worksheet: Translations with Lists http://www.ck12.org/flexr/chapter/9697
Problem 1 – Creating a Scatter Plot
Before beginning the activity, students need to clear all entries from the \begin{align*}Y=\end{align*} screen and all lists.
First, students will enter the data on the worksheet into lists \begin{align*}L_1\end{align*} and \begin{align*}L_2\end{align*}.
After setting up Plot1 for a scatter plot of \begin{align*}L_1\end{align*} vs \begin{align*}L_2\end{align*} and changing the window settings, students view the graph and sketch the figure on the worksheet. The shape should be an arrow, in the first quadrant, pointing to the right.
Problem 2 – Reflections and Rotations
In the list editor students are to enter the formulas \begin{align*}= -L_1\end{align*} and \begin{align*}= -L_2\end{align*} for \begin{align*}L_3\end{align*} and \begin{align*}L_4\end{align*} respectively. This will allow them to create several different reflections and rotations of the original figure.
To type \begin{align*}L_1\end{align*}, students need to press \begin{align*}2^{nd}\ [1]\end{align*}.
To type \begin{align*}L_2\end{align*}, students need to press \begin{align*}2^{nd}\ [2]\end{align*}.
For each combination of lists, students are to determine what type of reflection occurred.
A: \begin{align*}x \leftarrow L_3\end{align*} and \begin{align*}y \leftarrow L_2\end{align*}
\begin{align*}(-x, y)\end{align*} over \begin{align*}y-\end{align*}axis
B: \begin{align*}x \leftarrow L_1\end{align*} and \begin{align*}y \leftarrow L_4\end{align*}
\begin{align*}(x, -y)\end{align*} over \begin{align*}x-\end{align*}axis
C: \begin{align*}x \leftarrow L_2\end{align*} and \begin{align*}y \leftarrow L_1\end{align*}
\begin{align*}(y, x)\end{align*} over the line \begin{align*}y = x\end{align*}
Use Plot2 to create the following scatter plots. For each combination, determine what type of rotation occurred.
D: \begin{align*}x \leftarrow L_4\end{align*} and \begin{align*}y \leftarrow L_1\end{align*}
\begin{align*}(-y, x)\ 90^\circ\end{align*} around origin
E: \begin{align*}x \leftarrow L_2\end{align*} and \begin{align*}y \leftarrow L_3\end{align*}
\begin{align*}(-x, -y)\ 90^\circ\end{align*} around origin
F: \begin{align*}x \leftarrow L_3\end{align*} and \begin{align*}y \leftarrow L_4\end{align*}
\begin{align*}(y, -x)\ 180^\circ\end{align*} around origin
Problem 3 – Translations
In the list editor, students are to enter the formulas \begin{align*}=L_1-5\end{align*} and \begin{align*}=L_2+3\end{align*} for \begin{align*}L_3\end{align*} and \begin{align*}L_4\end{align*} respectively. This will allow them to translate the original figure.
Students should see that the image shifted to the left 5 units and up 3 units. Remind students that the tick marks on the graph are every 2 units.
Now students are to translate the scatter plot into Quadrant 3 by editing the formula bars for \begin{align*}L_3\end{align*} and \begin{align*}L_4\end{align*}. Possible formulas are below.
\begin{align*}L_3 \ \text{formula:} = L_1-15 && L_4 \ \text{formula:} = L_2-10\end{align*}
The image shifted 15 units to the left and 10 units down.
Extension – Dilations
Remind students that they have seen dilations in Chapter 7, Lesson 6.
In the list editor, students are to enter the formulas \begin{align*}=0.5^*L_1\end{align*} and \begin{align*}=0.5^*L_2\end{align*} for \begin{align*}L_3\end{align*} and \begin{align*}L_4\end{align*} respectively. This will allow them to dilate the original figure.
Students should see that the image decreased in size. If they have trouble seeing the image, they can the mark of the plot to the small dot.
Then students are to dilate the scatter plot into Quadrant 3 by editing the formula bars for \begin{align*}L_3\end{align*} and \begin{align*}L_4\end{align*}. Remind students that the scale factor needs to be the same for both lists. Possible formulas are below.
\begin{align*}L_3\ \text{formula:} & = -0.5^*L_1\\ L_4\ \text{formula:} & = -0.5^*L_2\end{align*}