Lesson 2 Leap Frog Solidify Understanding

Jump Start

Given the pre-image point $A$ located at $(2, 6)$ and the image point $A^{'}$ located at $(6, 2)$ as shown in the diagram, describe the following transformations:

A translation that will carry $A$ onto $A^{'}$

A reflection that will carry $A$ onto $A^{'}$

A rotation that will carry $A$ onto $A^{'}$

A sequence of two transformations that will carry $A$ onto $A^{'}$

Learning Focus

Determine the rigid transformation that carries one image onto another.

When given an image and its pre-image, what do I look for to identify a rigid transformation that has already occurred?

How do I recognize when a sequence of rigid transformations may be needed to carry one figure onto another, and how do I identify what transformations that sequence might include?

How do I justify that my proposed rigid transformation, or sequence of transformations, works?

Open Up the Math: Launch, Explore, Discuss

Josh is animating a scene in which a troupe of frogs is auditioning for the Animal Channel reality show, “The Bayou’s Got Talent.” In this scene the frogs are demonstrating their “leap frog” acrobatics act. Josh has completed a few key images in this segment, and now needs to describe the transformations that connect various images in the scene.

For each pre-image/image combination listed below, describe the transformation that moves the pre-image to the final image.

If you decide the transformation is a rotation, you will need to give the center of rotation, the direction of the rotation (clockwise or counterclockwise), and the measure of the angle of rotation.
If you decide the transformation is a reflection, you will need to give the equation of the line of reflection.
If you decide the transformation is a translation, you will need to describe the “rise” and “run” between pre-image points and their corresponding image points.
If you decide it takes a combination of transformations to get from the pre-image to the final image, describe each transformation in the order they would be completed.

1.

Pre-image	Final Image	Description
image $1$	image $2$
image $2$	image $3$
image $3$	image $4$
image $1$	image $5$
image $2$	image $4$

Ready for More?

How would you respond to these questions: Is it possible to find a sequence of transformations that will carry every image to every other image in the diagram if the first transformation in the sequence is always to translate the tip of the middle finger of the left hand of the first image to the corresponding point on the second image? Why or why not?

Takeaways

While working today on the rigid transformations

I noticed

I also noticed

and I am still wondering

Vocabulary

coincides (superimposed or carried onto)
corresponding points / sides
orientation
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we identified the transformation, or sequence of transformations, that would carry one image onto another. We justified our claims by describing or showing the essential features of each transformation, such as the center of rotation or the line of reflection.

Retrieval

1.

For each linear equation, write the slope of a line parallel to the given line and the slope of a line perpendicular to the given line.

a.

$y = \frac{2}{3} x + \frac{5}{3}$

Parallel slope:

Perpendicular slope:

b.

$3 x - 4 y = 6$

Parallel slope:

Perpendicular slope:

2.

Use the two points $(- 2, 3)$ and $(4, 5)$ to do the following:

Plot the points on the coordinate grid.
Find the slope of the line segment between these two points.
Use the Pythagorean theorem to find the distance between the two points.