Why Move Things Around?

Examples, videos, and solutions to help Grade 8 students learn the vocabulary and notation related to rigid motions and transformations.

New York State Common Core Math Module 2, Grade 8, Lesson 1

Worksheets for Grade 8

Lesson 1 Student Outcomes

• Students are introduced to vocabulary and notation related to rigid motions (e.g., transformation, image, and map).
• Students are introduced to transformations of the plane and learn that a rigid motion is a transformation that is distance preserving.
• Students use transparencies to imitate a rigid motion that moves or maps one figure to another figure in the plane.

Lesson 1 Summary
A transformation of the plane, to be denoted by F, is a rule that assigns to each point of the plane, one and only one (unique) point which will be denoted by F(P).
• So, by definition, the symbol F(P) denotes a specific single point.
• The symbol F(P) shows clearly that F moves to F(P)
• The point F(P) will be called the image of P by F
• We also say F maps P to F(P)
If given any two points P and Q, the distance between the images F(P) and F(Q) is the same as the distance between the original points P and Q, then the transformation F preserves distance, or is distance-preserving.
• A distance-preserving transformation is called a rigid motion (or an isometry), and the name suggests that it “moves” the points of the plane around in a “rigid” fashion.

Classwork
Concept Development
• Given two segments AB and CD which could be very far apart, how can we find out if they have the same length without measuring them individually? Do you think they have the same length? How do you check?
• For example, given a quadrilateral ABCD where all four angles at A, B, C, D are right angles, are the opposite sides AB, CD of equal length?
• Similarly, given angles ∠ AOB and ∠ A'O'B' how can we tell whether they have the same degree without having to measure each angle individually?
• For example, if two lines L and L' are parallel and they are intersected by another line, how can we tell if the angles and (as shown) have the same degree when measured?

Exploratory Challenge
1. Describe, intuitively, what kind of transformation will be required to move the figure on the left to each of the figures (1–3) on the right. To help with this exercise, use a transparency to copy the figure on the left. Note that you are supposed to begin by moving the left figure to each of the locations in (1), (2), and (3).
2. Given two segments and , which could be very far apart, how can we find out if they have the same length without measuring them individually? Do you think they have the same length? How do you check? In other words, why do you think we need to move things around on the plane?

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