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# 13.1: Transformations with Lists

Difficulty Level: At Grade Created by: CK-12

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 (x,y)\begin{align*}(x, y)\end{align*}, then each of the following ordered pairs denotes a transformation:

(x,y) reflect over xaxis(x,y) reflect over yaxis(y,x)   reflect over y=x(y,x)   rotate 90 around origin(x,y) rotate 180 around origin(y,x)   rotate 90 around origin\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 x\begin{align*}x-\end{align*}values or the y\begin{align*}y-\end{align*}values of the figure.
• To perform a dilation, multiply a constant scale factor by the list with the x\begin{align*}x-\end{align*}values or the y\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

## Problem 1 – Creating a Scatter Plot

Before beginning the activity, students need to clear all entries from the Y=\begin{align*}Y=\end{align*} screen and all lists.

First, students will enter the data on the worksheet into lists L1\begin{align*}L_1\end{align*} and L2\begin{align*}L_2\end{align*}.

After setting up Plot1 for a scatter plot of L1\begin{align*}L_1\end{align*} vs L2\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 =L1\begin{align*}= -L_1\end{align*} and =L2\begin{align*}= -L_2\end{align*} for L3\begin{align*}L_3\end{align*} and L4\begin{align*}L_4\end{align*} respectively. This will allow them to create several different reflections and rotations of the original figure.

To type L1\begin{align*}L_1\end{align*}, students need to press 2nd [1]\begin{align*}2^{nd}\ [1]\end{align*}.

To type L2\begin{align*}L_2\end{align*}, students need to press 2nd [2]\begin{align*}2^{nd}\ [2]\end{align*}.

For each combination of lists, students are to determine what type of reflection occurred.

A: xL3\begin{align*}x \leftarrow L_3\end{align*} and yL2\begin{align*}y \leftarrow L_2\end{align*}

(x,y)\begin{align*}(-x, y)\end{align*} over y\begin{align*}y-\end{align*}axis

B: xL1\begin{align*}x \leftarrow L_1\end{align*} and yL4\begin{align*}y \leftarrow L_4\end{align*}

(x,y)\begin{align*}(x, -y)\end{align*} over x\begin{align*}x-\end{align*}axis

C: xL2\begin{align*}x \leftarrow L_2\end{align*} and yL1\begin{align*}y \leftarrow L_1\end{align*}

(y,x)\begin{align*}(y, x)\end{align*} over the line y=x\begin{align*}y = x\end{align*}

Use Plot2 to create the following scatter plots. For each combination, determine what type of rotation occurred.

D: xL4\begin{align*}x \leftarrow L_4\end{align*} and yL1\begin{align*}y \leftarrow L_1\end{align*}

(y,x) 90\begin{align*}(-y, x)\ 90^\circ\end{align*} around origin

E: xL2\begin{align*}x \leftarrow L_2\end{align*} and yL3\begin{align*}y \leftarrow L_3\end{align*}

Undefined control sequence \ \begin{align*}(-x, -y)\ 90^\circ\end{align*} around origin

F: xL3\begin{align*}x \leftarrow L_3\end{align*} and yL4\begin{align*}y \leftarrow L_4\end{align*}

(y,x) 180\begin{align*}(y, -x)\ 180^\circ\end{align*} around origin

## Problem 3 – Translations

In the list editor, students are to enter the formulas =L15\begin{align*}=L_1-5\end{align*} and =L2+3\begin{align*}=L_2+3\end{align*} for L3\begin{align*}L_3\end{align*} and L4\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 L3\begin{align*}L_3\end{align*} and L4\begin{align*}L_4\end{align*}. Possible formulas are below.

L3 formula:=L115L4 formula:=L210\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 =0.5L1\begin{align*}=0.5^*L_1\end{align*} and =0.5L2\begin{align*}=0.5^*L_2\end{align*} for L3\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*}

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