Verifying Properties of Dilations

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Objective

SWBAT verify the effects of dilations on lines and line segments.

Big Idea

Conceptions vs. Reality....Let's see how student predictions stack up against the properties of dilations.

Activating Prior Knowledge

15 minutes

The activity in this lesson requires students to be familiar with notation for lines, segments and their lengths. It also requires students to know the difference between parallel lines and coinciding lines. APK_Verifying Properties of Dilations is a quick warm-up activity to make sure that students are in the right frame of mind to start the lesson.

I plan to have my students work on this warm-up independently for 5 minutes. Then, for three minutes or so, students will compare their answers with those of a neighbor. If necessary, we will go over the correct answers together to clarify any misudnerstandings.

Making Predictions

15 minutes

In the first part of this activity I describe a dilation for my students and they must predict its effect on the relationship between corresponding pairs of lines and segment lengths on the preimage and image (see Verifying Properties of Dilations). As they begin I remind students to pay close attention to the notation used in the prompts and to be as precise as possible when making their predictions. I have them work independently on the front side of the resource for about 5 minutes. When the majority of students are finished with the front side, I will call on a few students to share their predictions. As I hear the responses, I will elaborate and clarify as needed, based on my observations of the students' work during the opening minutes.

Next, I will have students turn to Page 2 of Verifying Properties of Dilations to read about the properties of dilations. A-B partners will work together to make sense of the properties after they read the worksheet. I will direct one partner to leave their packet showing the diagram on the first page and the other partner to turn to the two properties on the second page. Using the diagrams as a visual aid, they then follow this protocol:

  1. Referencing the diagram, A explains to B what the property in 6(a) means
  2. B echoes/mirrors to A what (s)he has heard/seen
  3. Reverse roles and repeat for the property in 6(b).

This process should prepare students to complete #6 (i), (ii) and (iii) on the worksheet, which I have them do at this time.

As students are finishing, I select an exemplar or two to show model responses for #6 (i), (ii) and (iii). I plan to select model responses that are precise, concise and thorough. For example, a model response for (i) would say something like:

Line BD does not pass through the center of dilation, therefore it should be taken to a parallel line. I predicted that line B'D' would be parallel to line BD, so my prediction is consistent with the property. 

At this point, students are ready to work on the remainder of the activity independently or cooperatively. I continue walking the room making sure students are on track and moving at a reasonable pace. If students are stuck, I try referring them to what they have done previously in the activity in order to get them to see connections to what they are currently attempting.

Rehearsing the Properties

15 minutes

This is a good-old-fashioned memorization activity. We've already fleshed out the meaning of the properties of dilations and applied them in an example. Now it's time to help ensure that my students remember them over the long term.

Specifically, I want my students to be able to internalize the following script:

 

1. A dilation takes a line not passing through the center of dilation to a parallel line and leaves a line passing through the center unchanged. 

2. The dilation of a line segment is longer or shorter in the ratio of the scale factor.

 

As we know, when students just memorize words without really thinking about what they are memorizing there is little value and lots of potential for misconception. So my goal here is to help students develop schema that will lock the meaning of these statements into meaningful long-term memory.

There are two parts to this.

 

First I want my students to understand the organization of the statements. The organization is basically:

1. Talk about the effect on lines (Those not passing through center vs. those passing through the center)

2. Talk about the effect on line segments

To promote awareness of this organization, I write on the board:

 

"Organization of the Properties of Dilations Statements:

 I. Effect on Lines

A. Lines NOT passing through the center (Taken to parallel lines)

B. Lines passing through the center (left unchanged)

 

II. Effect on line segments (longer or shorter in ratio of scale factor)"

 

Alternately, I might project schema_dilation properties on the projector instead of writing on the board.

 

 

Second, I want my students to visualize what they are saying so that it becomes real to them. To do this I give the students hand gestures to represent the words they are saying. See the following video for a brief demonstration of this.

 

Next, I have the students get together with their A-B partner and rehearse reciting the properties and using the hand gestures. I give them the Memorizing and Understanding Properties of Dilations Partner Rehearsal Protocol to structure their rehearsal.

To make sure they are putting maximum effort into it, I tell my students that I will be calling on a large number of them to recite the properties from memory in front of the class. After 5 - 7 minutes, I begin selecting students to recite the properties. Usually for the first few students, I only ask them to recite one of the properties. By the time I get to the fourth student, they will be reciting both properties. I try to get as many students to recite as time will allow. I give targeted praise to students whose recitations and hand signaling suggest that they have internalized the properties and are not merely regurgitating the words.

 

Checking for Understanding

10 minutes

This Check for Understanding will help me to assess how well my students have assimilated the main definitions and properties from the first two lessons of this unit. I explain to students that this is a formative assessment that will help me to fine tune instruction. I tell them that, although it is not graded, it works best when every student does his or her own personal best, no more, no less. With that introduction, I will pass out the Check for Understanding_Properties of Dilations resource and give students 5-7 minutes to complete it.

When I collect the formative assessment I will remind students that they will be completing #8 from Verifying Properties of Dilations for homework. This task requires students to use Geometer's Sketchpad or Geogebra at home to explore and verify the dilation properties that were the topic of this lesson.