Students continue to investigate the effects of transformations. The new feature of this lesson is the coordinate plane. In this lesson, students use coordinates to describe figures and their images under transformations in the coordinate plane. Reflections over the \(x\)-axis and \(y\)-axis have a very nice structure captured by coordinates. When we reflect a point like \((2,5)\) over the \(x\)-axis, the distance from the \(x\)-axis stays the same but instead of lying 5 units above the \(x\)-axis the image lies 5 units below the \(x\)-axis. That means the image of \((2,5)\) when reflected over the \(x\)-axis is \((2,\text-5)\). Similarly, when reflected over the \(y\)-axis, \((2,5)\) goes to \((\text-2,5)\), the point 2 units to the left of the \(y\)-axis.
Using the coordinates to help understand transformations involves MP7 (discovering the patterns coordinates obey when transformations are applied).
- Draw and label a diagram of a line segment rotated 90 degrees clockwise or counterclockwise about a given center.
- Generalize (orally and in writing) the process to reflect any point in the coordinate plane.
- Identify (orally and in writing) coordinates that represent a transformation of one figure to another.
Let’s transform some figures and see what happens to the coordinates of points.
- I can apply transformations to points on a grid if I know their coordinates.
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.
The coordinate plane is a system for telling where points are. For example. point \(R\) is located at \((3, 2)\) on the coordinate plane, because it is three units to the right and two units up.
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.
An image is the result of translations, rotations, and reflections on an object. Every part of the original object moves in the same way to match up with a part of the image.
In this diagram, triangle \(ABC\) has been translated up and to the right to make triangle \(DEF\). Triangle \(DEF\) is the image of the original triangle \(ABC\).
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.
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.
sequence of transformations
A sequence of transformations is a set of translations, rotations, reflections, and dilations on a figure. The transformations are performed in a given order.
This diagram shows a sequence of transformations to move Figure A to Figure C.
First, A is translated to the right to make B. Next, B is reflected across line \(\ell\) to make C.
A transformation is a translation, rotation, reflection, or dilation, or a combination of these.
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 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 \(E\).