Lesson 1 Garden Transformations Solidify Understanding

Learning Focus

Identify the defining features of the translation, rotation, reflection, and dilation transformations.

Use function notation to describe transformations.

What are the defining features of each of the following geometric transformations: translation, rotation, reflection, and dilation?

Why can transformations be treated as functions?

Open Up the Math: Launch, Explore, Discuss

Kiwan, Brielle, and Juan Carlos are preparing an interdisciplinary report and presentation for a school project. Their report will focus on the biology and mathematics of garden insects using the theme “transformations.” The biology portion of their report will focus on how useful insects help transform and maintain a garden. They have found the following statement in their research:

Insects help provide food that we eat, through pollination – Bees, beetles, moths, flies, and other insects carry the pollen that plants need to reproduce and to form fruits and vegetables. In the woods, ants spread seeds to new areas.
Some insects help keep garden “pests” under control – Many “beneficial” insects (and spiders) are predators. They live by hunting and eating the bugs that damage our garden plants.
Many insects are experts at recycling and cleanup – Insects break down rotting logs, dead animals, and wilted plants, recycling them into the soil.
Some help to keep weeds from taking over – Some insects live by eating seeds in meadows and weedy places.

Adapted from Beneficial of the Week, TPM/IPM Weekly Report for Arborists, Landscape Managers, and Nursery Managers, March 20, 2015, by the University of Maryland Extension Program https://openup.org/yhmWZ7

While Juan Carlos has been writing the biology portion of their report, Kiwan and Brielle have been taking photographs of their gardens, which they can use to illustrate their presentation. As they review the images, they realize they have been using the Live Photo® feature of their cameras. As smartphone technology has improved, some cameras have included a “live” feature that takes multiple images each time you take a photo. Instead of freezing an image in time with a still photo, a Live Photo® captures a $1 -second$ moving image. Kiwan and Brielle find it interesting to compare the first and last image in this sequence and to imagine what happened during the elapsed time in-between. This type of activity will form the mathematical focus of their presentation.

Kiwan and Brielle have produced a composite image of several garden insects in their “before and after” positions, based on a $1 -second$ interval of time. Examine this image and describe and illustrate the transformation that occurred between the before and after position of each insect. Your annotations on the diagram should show the path followed by the insect during this $1 -second$ interval of time. Your description should include precise language for describing the key defining features of the transformation, using words such as: concentric circles, parallel line segments, perpendicular bisector, center, congruent angles, vector, scale factor, ratio, line of reflection, etc.

1.

Ant transformation:

2.

Ladybug transformation:

3.

Butterfly transformation:

4.

Fly transformation:

5.

Grasshopper transformation:

Pause and Reflect

Transformation Notation:

Your descriptions of the above transformations used lots of words in the descriptions. Since transformations are functions with an input and output, we can name transformations using function notation. We will use the following notation:

Transformation	Notation	Description
Translation	$T_{⟨ a, b ⟩} (i m a g e)$	Translate the given pre-image $a$ units horizontally and $b$ unit vertically.
Rotation	$R_{(h, k), N °} (i m a g e)$	Rotate the given pre-image $N °$ about the point $(h, k)$ ; by convention, positive values for $N$ represent counterclockwise rotations, and negative values represent clockwise rotations.
Reflection	$r_{y = m x + b} (i m a g e)$	Reflect the given pre-image over the line $y = m x + b$ .
Dilation	$D_{(p, q), k} (i m a g e)$	Dilate the given pre-image relative to the point $(p, q)$ by a factor of $k$ .

6.

Use this transformation notation to symbolize each of the insect transformations you described in words above, (Note: the diagram shows the choices for the pre-image, figure 1, and the image, figure 2, in these descriptions.)

a.

Ant transformation:

b.

Ladybug transformation:

c.

Butterfly transformation:

d.

Fly transformation:

e.

Grasshopper transformation:

The following special notation is used when points are:

reflected over either of the coordinate axes or over the line $y = x,$
rotated $90 °$ , $180 °$ , $270 °$ or $360 °$ counterclockwise about the origin,
dilated relative to the origin or,
as alternative notation for translation of points.

7.

Complete the description or notation of each of the following transformations after testing out the notation or description on a couple of the points indicated on the coordinate grid. The notation and description of the translation has been given for you.

Transformation	Notation	Description
Translation	$(p, q) \to (p + a, q + b)$	Translate the point $(p, q)$ $a$ units horizontally and $b$ units vertically.
Reflection	$(p, q) \to (p, - q)$	Reflect the point $(p, q)$ .
Reflection	$(p, q) \to (- p, q)$	Reflect the point $(p, q)$ .
Reflection	$(p, q) \to (q, p)$	Reflect the point $(p, q)$ .
Rotation	$(p, q) \to \underset{―}{}$	Rotate the point $(p, q)$ $90 °$ counterclockwise about the origin.
Rotation	$(p, q) \to \underset{―}{}$	Rotate the point $(p, q)$ $180 °$ counterclockwise about the origin.
Rotation	$(p, q) \to \underset{―}{}$	Rotate the point $(p, q)$ $270 °$ counterclockwise about the origin.
Rotation	$(p, q) \to \underset{―}{}$	Rotate the point $(p, q)$ $360 °$ counterclockwise about the origin.
Dilation	$(p, q) \to \underset{―}{}$	Dilate the point $(p, q)$ by a factor of $k$ relative to the origin.

Ready for More?

It is possible to complete all rigid transformations using only a sequence of reflections. Show how you can transform each of the pre-images of the insects for problems 1 and 2 onto their corresponding images using only a sequence of reflections.

Takeaways

Translation:
Rotation:
Reflection:
Dilation:

Lesson Summary

In this lesson, we reviewed the defining characteristics of each of the three rigid transformations, which preserve angle and distance measurements from pre-image to image. We also reviewed the characteristics of the dilation transformation, which produces similar figures. We also examined notation for describing these transformations symbolically. The symbolic notation illustrates that geometric transformations are functions, with each set of points in the input image being mapped to a unique set of points in the output image.

Retrieval

1.

Use the congruence marks to complete the congruence statements.

$∠ A B C ≅$

$\overset{―}{F E} ≅$

$\overset{―}{D A} ≅$

Rewrite the phrases below using correct mathematical symbols.

2.

Angle $2$ is congruent to angle $7$ .

3.

The length of segment $P R$ is equal to the length of segment $M K$ .