Lesson 2

Naming the Moves

Let’s be more precise about describing moves of figures in the plane.

2.1: A Pair of Quadrilaterals

Quadrilateral A can be rotated into the position of Quadrilateral B. 

Quadrilateral A and rotated image quadrilateral B.

Estimate the angle of rotation.

2.2: How Did You Make That Move?

Here is another set of dance moves.

6 panels showing the same figure in different positions and orientations.

  1. Describe each move or say if it is a new move.

    1. Frame 1 to Frame 2.

    2. Frame 2 to Frame 3.

    3. Frame 3 to Frame 4.

    4. Frame 4 to Frame 5.

    5. Frame 5 to Frame 6.

  2. How would you describe the new move?

2.3: Card Sort: Move

Your teacher will give you a set of cards. Sort the cards into categories according to the type of move they show. Be prepared to describe each category and why it is different from the others. You can explore the applets below to see the ways the images move.

Drag the red point. Explore how the image changes.

Click on the box to show the transformed image. Move the yellow points and the red segment to see how the image changes.

Summary

Here are the moves we have learned about so far:

  • A translation slides a figure without turning it. Every point in the figure goes the same distance in the same direction. For example, Figure A was translated down and to the left, as shown by the arrows. Figure B is a translation of Figure A.

    Two figures, one labelled A, and its translation, labelled B.

  • A rotation turns a figure about a point, called the center of the rotation. Every point on the figure goes in a circle around the center and makes the same angle. The rotation can be clockwise, going in the same direction as the hands of a clock, or counterclockwise, going in the other direction. For example, Figure A was rotated \(45^\circ\) clockwise around its bottom vertex. Figure C is a rotation of Figure A.

    Two figures, one labelled A, and its rotation, labelled C.

  • A reflection places points on the opposite side of a reflection line. The mirror image is a backwards copy of the original figure. The reflection line shows where the mirror should stand. For example, Figure A was reflected across the dotted line. Figure D is a reflection of Figure A.

    Two figures, one labelled A, and its reflection, labelled D

We use the word image to describe the new figure created by moving the original figure. If one point on the original figure moves to another point on the new figure, we call them corresponding points.

Glossary Entries

  • clockwise

    Clockwise means to turn in the same direction as the hands of a clock. The top turns to the right. This diagram shows Figure A turned clockwise to make Figure B.

  • counterclockwise

    Counterclockwise means to turn opposite of the way the hands of a clock turn. The top turns to the left.

    This diagram shows Figure A turned counterclockwise to make Figure B.

    Two figures. Figure A turned counterclockwise makes Figure B.

     

  • reflection

    A reflection across a line moves every point on a figure to a point directly on the opposite side of the line. The new point is the same distance from the line as it was in the original figure.

    This diagram shows a reflection of A over line \(\ell\) that makes the mirror image B.

    Two triangles, A and B. A reflection of A over a line makes the mirror image B.
  • rotation

    A rotation moves every point on a figure around a center by a given angle in a specific direction.

    This diagram shows Triangle A rotated around center \(O\) by 55 degrees clockwise to get Triangle B.

    Two trianges, A and B. Triangle A rotated by 55 degrees clockwise gets Triangle B.
  • translation

    A translation moves every point in a figure a given distance in a given direction.

    This diagram shows a translation of Figure A to Figure B using the direction and distance given by the arrow.

    A translation of triangle A to triangle B
  • vertex

    A vertex is a point where two or more edges meet. When we have more than one vertex, we call them vertices.

    The vertices in this polygon are labeled \(A\), \(B\), \(C\), \(D\), and \(E\).

    Polygon with 5 sides