# Common Core Standards: Math

### Geometry 8.G.A.1

1. Verify experimentally the properties of rotations, reflections, and translations.

Have you ever seen a baby looking at herself in a mirror? For starters, she has no idea that the little person staring at her is her! This other person thinks almost exactly the same way she does—she raises her hand, and so does the other kid! She jumps, and the other kid does, too.

Of course, what she doesn't realize is that when she raises her left hand, the reflection is raising her right hand. (Honest! If you don't believe us, try it yourself.)

Reflections are just one of the three main types of transformationsthat shapes (and babies) can undergo, and we're not talking about an extreme makeover. A transformation is a fancy term for a movement on the coordinate plane.

Even if your students aren't doing too well in Spanish, they can still translate, or shift, shapes to their hearts' content. A reflection can be thought of as "flipping" a shape across an axis of symmetry, while rotation is the same as turning a shape around a central point.

Students should be aware of these different transformations and understand how to perform them. In order to explore the different properties (ahem, congruence) of these different transformations, students can experiment with them on the coordinate plane. With these concepts, hands-on activities are the way to go.

#### Drills

1. Which of the following rotations will yield a line segment that is in the same position as the original?

360°

Rotating a line segment 360 degrees brings it back to its original position. Actually, this is true for any figure and any center of rotation. If it helps, think of Tony Hawk skateboarding tricks.

2. Line symmetry exists as the result of which kind of a transformation?

Reflection

When you reflect a figure over a line, you end up with line symmetry. If you see your reflection across a mirror, the mirror is the "line" of symmetry. Plus, rotation occurs relative to a point and translations don't require axes or points. Transformation is too broad to be the right answer.

3. A horizontal line is translated seven units to the right. Which of the following is true?

The new line and the original line are the same

Since lines have infinite length, moving a horizontal line from side to side won't change the slope or y-intercept of the line. Since those two parameters define a line, the new line and the original line are the same exact line. This might be difficult to model because infinite length might be a bit difficult to reproduce, but a really long string might do the trick.

4. A line segment is translated. Which of the following is true?

The location of the segment changes

Translations consist of moving the shape (line, figure, image, object, whatever) as is. There is no spinning, turning, or flipping involved. Applying that to a line segment, that won't change its length or slope, but it will change its location. Also, (A) is impossible because lines only have length, not width or thickness.

5. A line is rotated 90° clockwise. Which of the following is false?

The original line and the rotated line will have the same slope

It's important to remember that lines are infinitely long, so if they aren't parallel then they will intersect at some point. Since only a rotation of 180° can make a parallel line, we know that (C) and (D) are true. Since two lines with the same slope will be parallel, (A) must be the right answer. By process of elimination (or knowledge that a right angle has a measure of 90°), we know that (B) must also be true.

6. When an angle is reflected across a line, what happens to its measure?

It remains the same

If it helps, we can imagine an angle as two line segments with a common endpoint. Whatever we do to one must do to the other. If we reflect one across a line, we must reflect the other in the exact same way. Even when we shift, turn, or flip the angle, the number of degrees that defines the angle doesn't change because the two segments stay the same relative to each other.

7. A figure has rotational symmetry if it can be rotated some number of degrees less than 360 and still look like the original figure. How many degrees would a normal five-pointed star have to be rotated to look like it hasn't been moved?

72°

Since the star has five points, divide 360° (one complete revolution) by 5. Since 360 ÷ 5 = 72, a rotation of 72° will make a star that's identical to the original image. All other answers are incorrect because they won't form a star identical to the original one.

8. Two parallel lines are separated by a distance of 5 units. If one is translated 3 units to the right, will the lines remain parallel?

Yes, they will remain parallel

They may change position, but they'll remain lines in the same plane with the same slope, so they'll stay parallel. We can also be sure that the lines won't overlap because they are separated by 5 units and moved only 3. Regardless of their slopes or y-intercepts, translation keeps the lines parallel to each other.

9. Two parallel lines are both rotated 90° clockwise. What happens?

They remain parallel but change direction

Since both lines rotate the same amount together, they stay parallel. Even though they are no longer in the same position. Imagine a game of spin the bottle. Rotating the bottle 90° doesn't make the sides of the bottle intersect. They're both rotated in the same way, so even though the two lines do change direction, they're still parallel to one another.

10. Two parallel lines are rotated 180° counterclockwise. What happens?