Lesson 1 Photocopy Faux Pas Develop Understanding
Learning Focus
Describe the essential features of a dilation transformation.
How do I use a dilation to enlarge or shrink a figure?
How are distance and area in a dilated image related to corresponding distances and area in the original image?
Open Up the Math: Launch, Explore, Discuss
Burnell has a new job at a copy center helping people use the photocopy machines. Burnell thinks he knows everything about making photocopies, and so he didn’t complete his assignment to read the training manual.
Mr. and Mrs. Donahue are making a scrapbook for Mr. Donahue’s grandfather’s 75th birthday party, and they want to enlarge a sketch of their grandfather which was drawn when he was in World War II. They have purchased some very expensive scrapbook paper, and they would like this image to be centered on the page. Because they are unfamiliar with the process of enlarging an image, they have come to Burnell for help.
“We would like to make a copy of this image that is twice as big, and centered in the middle of this very expensive scrapbook paper,” Mrs. Donahue says. “Can you help us with that?”
“Certainly,” says Burnell. “Glad to be of service.”
Burnell taped the original image in the middle of a white piece of paper, placed it on the glass of the photocopy machine, inserted the expensive scrapbook paper into the paper tray, and set the enlargement feature at 200%.
In a moment, this image was produced.
“You’ve ruined our expensive paper,” cried Mrs. Donahue. “Much of the image is off the paper instead of being centered.”
“And this image is more than twice as big,” Mr. Donahue complained. “One-fourth of Grandpa’s picture is taking up as much space as the original.”
In the diagram provided in problem 2, both the original image—which Burnell taped in the middle of a sheet of paper—and the copy of the image have been included on the same figure.
1.
Explain how the photocopy machine produced the partial copy of the original image.
2.
Using a “rubber band stretcher” finish the rest of the enlarged sketch.
Pause and Reflect
3.
Where should Burnell have placed the original image if he wanted the final image to be centered on the paper?
4.
Mr. Donahue complained that the copy was four times bigger than the original. What do you think? Did Burnell double the image or quadruple it? What evidence would you use to support your claim?
5.
Transforming a figure by shrinking or enlarging it in this way is a called dilation. Based on your thinking about how the photocopy was produced, list all of the things you need to pay attention to when dilating an image.
Ready for More?
Consider the following question: Is it possible for Burnell to locate a position to place the original photo on the paper so the enlarged image will be located anywhere on the paper that Mr. and Mrs. Donahue may select?
Takeaways
Transforming a figure by shrinking or enlarging it is called a dilation. Based on your thinking about how the photocopy was produced, list all of the things you need to pay attention to when dilating an image.
We observed the following relationships between the pre-image and image figures of a dilation:
Adding Notation, Vocabulary, and Conventions
Words we use when describing a dilation, and what they tell us about the dilation:
Center of dilation:
Scale factor of dilation:
Vocabulary
- center of dilation
- collinear, collinearity
- dilation
- scale factor
- Bold terms are new in this lesson.
Lesson Summary
In this lesson, we observed the key features of a dilation transformation while figuring out how a photocopy machine enlarges an image. We learned how to locate points on a dilated image by using the center and scale factor that define a specific dilation. We observed that “the same shape, different size” relationship between the pre-image and image figures are a consequence of the way dilations are defined.
The pairs of figures below are similar to one another. Find the scale factor between the figures, and find the unknown measure indicated by the letter.
1.
2.
3.
When figures are similar, what will you know besides the fact that they have a scale factor?
Consider each of the following functions represented and determine whether they are linear, exponential, or quadratic.
4.
A.
linear
B.
exponential
C.
quadratic
5.
A.
linear
B.
exponential
C.
quadratic
6.
A.
linear
B.
exponential
C.
quadratic