Lesson 3 Leap Year Practice Understanding

Jump Start

Today’s task contains this calendar page for February 28. For each definition, draw and label a diagram to show the meaning of the words.

February 28

A circle is the set of all points in a plane that are equidistant from a fixed point called the center of the circle.

An angle is the union of two rays that share a common endpoint.

An angle of rotation is formed when a ray is rotated about its endpoint. The ray that marks the pre-image of the rotation is referred to as the “initial ray” and the ray that marks the image of rotation is referred to as the “terminal ray.”

Angle of rotation can also refer to the number of degrees a figure has been rotated around a fixed point, with a counterclockwise rotation being considered a positive direction of rotation.

1.

A circle...

2.

An angle...

3.

An angle of rotation...

4.

The calendar page for March 1 discusses some interesting facts about degree measurement, but doesn’t define it. What is a degree?

Learning Focus

Write precise definitions of the rigid transformations.

How do I use my intuition, and the insights gained during the past few tasks, to identify or produce a rigid transformation?

How can I make my intuitive and insightful thinking explicit in words and diagrams?

What can I add to the words slide, flip, and turn to more precisely define the rigid transformations—translation, reflection, and rotation?

Open Up the Math: Launch, Explore, Discuss

What is wrong with slide, flip, and turn as words for defining the rigid transformations?

Slide, flip, and turn are verbs; they describe an action.

Here are ways these words might be used in a sentence:

Slide: “Slide over here for a minute.”

Flip: “Flip the card over, so we can see which card it is.”

Turn: “Turn the lid counterclockwise to open the jar.”

1.

Think about the work you did in class with Leaping Lizards and Leap Frog. What is missing in each of these sentences that would need to be included to make these actions a translation, a reflection, or a rotation?

The word “flip” can create a misconception—a wrong way of thinking.

2.

Draw a picture to illustrate the sentence, “Flip the card over, so we can see which card it is.” (Be as accurate as possible when drawing the two pictures of the card.)

3.

Does your picture show a reflection? Why or why not?

Carlos and Clarita are discussing their latest business venture with their friend Juanita. They have created a daily planner that is both educational and entertaining. The planner consists of a pad of $365$ pages bound together, one page for each day of the year. The planner is entertaining because images along the bottom of the pages form a flip-book animation when thumbed through rapidly. The planner is educational because each page contains some interesting facts. Each month has a different theme, and the facts for the month have been written to fit the theme. For example, the theme for January is astronomy, the theme for February is mathematics, and the theme for March is ancient civilizations. Carlos and Clarita have learned a lot from researching the facts they have included, and they have enjoyed creating the flip-book animation.

The twins are excited to share the prototype of their planner with Juanita before sending it to printing. Juanita, however, has a major concern. “Next year is leap year,” Juanita explains. “You need $366$ pages.”

So now Carlos and Clarita have the dilemma of needing to create an extra page to insert between February 28 and March 1.

Here are the planner pages they have already designed.

February 28

A circle is the set of all points in a plane that are equidistant from a fixed point called the center of the circle.

An angle is the union of two rays that share a common endpoint.

Angle of rotation can also refer to the number of degrees a figure has been rotated around a fixed point, with a counterclockwise rotation being considered a positive direction of rotation.

March 1

Why are there $360 °$ in a circle?

One theory is that ancient astronomers established that a year was approximately $360$ days, so the sun would advance its path, relative to the earth, approximately $1 / 360$ of a turn, or one degree, each day. (The $5$ extra days in a year were considered unlucky days.)

Another theory is that Babylonians first divided a circle into parts by inscribing a hexagon consisting of six equilateral triangles inside a circle. The angles of the equilateral triangles, located at the center of the circle, were further divided into $60$ equal parts, since the Babylonian number system was base- $60$ (instead of base- $10$ , like our number system).

Another reason for $360 °$ in a circle may be the fact that $360$ has $24$ divisors, so a circle can easily be divided into many smaller, equal-sized parts.

Since February’s theme is mathematics, Clarita suggests that they write formal definitions of the three rigid-motion transformations they have been using to create the images for the flip-book animation.

How would you complete each of the following definitions?

4.

A translation of a set of points in a plane…

5.

A rotation of a set of points in a plane…

6.

A reflection of a set of points in a plane…

7.

Translations, rotations, and reflections are rigid transformations because...

Note: Carlos and Clarita used these words and phrases in their definitions: perpendicular bisector, center of rotation, equidistant, angle of rotation, concentric circles, parallel, image, pre-image, preserves distance and angle measures within the shape. Revise your definitions to include these words or phrases.

Ready for More?

In addition to writing new facts for February 29, the twins also need to add another image in the middle of their flip-book animation. The animation sequence is of Dorothy’s house from The Wizard of Oz as it is being carried over the rainbow by a tornado. The house in the February 28 drawing has been rotated to create the house in the March 1 drawing. Carlos believes that they can get from the February 28 drawing to the March 1 drawing by reflecting the February 28 drawing, and then reflecting it again.

Verify that the image Carlos inserted between the two images that appeared on February 28 and March 1 works as they intended. For example:

What convinces you that the February 29 image is a reflection of the February 28 image about the given line of reflection?
What convinces you that the March 1 image is a reflection of the February 29 image about the given line of reflection?
What convinces you that the two reflections together complete a rotation between the February 28 and March 1 images?

Takeaways

The essential elements of each definition are as follows:

Translation:
Rotation:
Reflection:

Vocabulary

Lesson Summary

In this lesson, we wrote precise definitions of the three rigid-motion transformations: translation, rotation, and reflection, and explored why the words slide, turn, and flip were not adequate to use as definitions.

Retrieval

Give the name of a geometric figure that fits the following characteristics:

1.

A quadrilateral with four congruent sides

2.

A quadrilateral with four congruent angles

3.

A quadrilateral with opposite sides parallel