Lesson 1 Leaping Lizards! Develop Understanding

Jump Start

Given the pre-image point $A$ located at $(5, 2)$ as shown in the diagram, find the location of the image point if point $A$ is:

Reflected over the $x$ -axis.
Reflected over the $y$ -axis.
Rotated $180 °$ counterclockwise around the origin.
Rotated $90 °$ clockwise around the origin.
Translated left $2 units$ and up $3 units$ .

Learning Focus

Identify features of translations, rotations, and reflections.

What tools and strategies do I use when I translate a figure? Rotate a figure? Reflect a figure?

What do these tools and strategies reveal about the transformation?

Although each of these transformations move figures in different ways, what do they all have in common?

Open Up the Math: Launch, Explore, Discuss

Animated films and cartoons are now usually produced using computer technology, rather than the hand-drawn images used in the past. Computer animation requires both artistic talent and mathematical knowledge.

Sometimes animators want to move an image around the computer screen without distorting the size and shape of the image in any way. This is done using geometric transformations such as translations (slides), reflections (flips), and rotations (turns), or perhaps some combination of the three. These transformations need to be precisely defined, so there is no doubt about where the final image will end up on the screen.

Today you will be animating the lizard shown on your handout by transforming it in various ways. The original lizard was created by plotting the following anchor points on the coordinate grid and then letting a computer program draw the lizard. The anchor points are always listed in this order: tip of nose, center of left front foot, belly, center of left rear foot, point of tail, center of rear right foot, back, center of front right foot.

Original lizard anchor points:

${(12, 12), (15, 12), (17, 12), (19, 10), (19, 14), (20, 13), (17, 15), (14, 16)}$

Each statement below describes a transformation of the original lizard. Do the following for each of the transformations:

Plot the anchor points for the lizard in its new location.
Connect the pre-image and image anchor points with line segments or circular arcs, whichever best illustrates the relationship between them.

1.

Lazy Lizard

Translate the original lizard so the point at the tip of its nose is located at $(24, 20)$ , making the lizard appear to be sunbathing on the rock.

2.

Lunging Lizard

Rotate the lizard $90 °$ about the point $A (12, 7)$ so it looks like the lizard is diving into the puddle of mud.

3.

Leaping Lizard

Reflect the lizard about the given line $y = \frac{1}{2} x + 16$ so it looks like the lizard is doing a back flip over the cactus.

Ready for More?

Refer to Lazy Lizard, the lizard lounging on the rock. Lazy Lizard is rotated $90 °$ counterclockwise about the point $(12, 7)$ . Find the coordinates of the points at the tip of the nose and the end of the tail for the rotated image of the lizard.

Describe how you found these points since they do not lie on the given portion of the coordinate grid.

Takeaways

What we noticed about all three transformations:

What we noticed about each transformation:

Translations:
Rotations:
Reflections:

Adding Notation, Vocabulary, and Conventions

Use this notation to describe a translation:

Vocabulary

Lesson Summary

In this lesson, we explored how to perform rigid transformations using a variety of tools, such as tracing paper, rulers, protractors, and compasses; and using a variety of methods, such as counting the squares on the coordinate grid, drawing parallel lines, or folding an image over a line. We used these tools and strategies to identify key features of each of the transformations.

Retrieval

1.

Determine the length of the missing side of the right triangle.

2.

Graph the function on the coordinate grid.

$f (x) = \frac{2}{3} x - 1$