We started a unit on dilations last week.
Our I can statements:
Unit 6 – Similarity
Level 1: I can identify, define, and perform dilations. G-SRT 1
Level 2: I can determine the similarity of two figures using similarity transformations. G-SRT 2, G-SRT 3
Level 3: I can prove theorems about triangles. G-SRT 4, G-C 1
Level 4: I can solve for and prove relationships in geometric figures using similarity criteria. G-SRT 5
And the standards:
Understand similarity in terms of similarity transformations
G-SRT 1. Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G-SRT 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G-SRT 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity
G-SRT 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G-SRT 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Understand and apply theorems about circles
G-C 1. Prove that all circles are similar.
This lesson needed rewriting from last year. We used part of the Geometry Nspired activity Corresponding Parts of Similar Triangles, but it just wasn’t all sequenced as it should have been.
So we tried again. We started with a task that I learned about in an NCSM session – Hannah’s Rectangle Problem.
I gave the students a piece of wax paper and a straightedge to determine which rectangles were similar to rectangle a. I sent a poll to find out what students thought. One student asked whether I was grading the poll. Everyone stared at him. I’m planning to have a conversation with him about what formative assessment really is sometime this week. I told them I wasn’t going to show them the correct answers yet (even though I have them marked in the screen capture below).
Our unit is on dilations. What is your previous experience with the term dilation? Students paired to talk with each other. I heard “eyes” and “enlargement”.
Next we thought about what we need for a dilation. If we were going to dilate the given hexagon what do we need to know?
Someone suggested that we need a number. He called it a factor. Okay, so we need a number to know how big or small to make the image. We’re going to call that number a scale factor. What else do we need for a dilation?
I have some cartoon dilations on the wall of my classroom. We looked at those. What do they all have in common? Lines. (Have you seen a cartoon dilation before? We used to do them in class. Not anymore, although I do think they give students a good understanding of dilations. We did these cartoon dilations with compasses and straightedges – not grids.) What do the lines have in common? A point. They all intersect at a point. Okay, so we are going to call that point the center of dilation.
Using our dynamic geometry software, we performed a dilation of the hexagon. We clicked on the hexagon, we clicked on a point outside the hexagon for the center of dilation, and we typed in a number for the scale factor. I asked students to explore a few ideas. What happens when the center of dilation is on, inside, or outside of the figure? What happens when the scale factor changes? What if the scale factor is negative? I am not always so specific when I give students time to explore, but I tried it this time to see what happened. I set the timer for 4 minutes. I used the Class Capture feature of TI-Nspire Navigator to see what they figured out.
We had a conversation about what happens when the center of dilation is on the figure compared to outside. We had a conversation about what happens when the center of dilation is inside the figure.
When is there a reduction? When the scale factor is a fraction.
Does everyone agree?
What if the scale factor is 5/2? Oh. Not just any fraction – a fraction less than 1.
Does everyone agree? –3 is less than one, but it isn’t a reduction. Fractions between –1 and 0 and between 0 and 1.
What happens when the scale factor is negative? The figure is reflected. We didn’t describe what is the line of reflection.
Next up: some measurements. On this page, I want you to play with the slider. If I give you two of the measurements on the page, what can you do to determine the 3rd?
Which figure is the image and which is the pre-image? How do you know?
If the scale factor on the first picture were 0.5, which figure is the image and which is the pre-image?
On the next page, we began to think about how many copies of segment AY it takes to make segment DE. What if we change the scale factor to 3? What if we change it to 0.5? We also began to think about how many copies of ∆AYM it takes to make ∆EDT. What if we change the scale factor to 3?
We are going to define two figures as similar if there is a dilation (and if needed, a set of rigid motions) that would map one figure onto another. Can you show that AYM~DET? After students explore on their own, I made CA the Live Presenter so that she could share what she did. She rotated ∆DET using S to control the angle of rotation. Then she translated ∆DET by vector YC. Then she undid the dilation by changing the scale factor to 1. Did everyone do the same? No – some undid the dilation first; others rotated and undid the dilation without a translation.
How can you show the two figures are similar? With a reflection and a dilation.
We moved to paper.
After a few minutes, a student shared how they dilated the triangle.
Did everyone work it the same way?
Students saw all kinds of proportional relationships, and many thought about slope.
What if we want to dilate points A and B about O using a scale factor of 3?
We started on paper.
And moved to the dynamic software.
That was it for the first day.
We came back to a few problems the next day to continue our work on dilations.
One figure is a dilation of the other. Where is the center of dilation?
Some students emphasized that we don’t only consider the vertices, but every pair of corresponding points on the pre-image and the image.
How can we show the two figures are similar?
What Standards for Mathematical Practice did the students have the opportunity to employ during our lesson on dilations? The dynamic software definitely provided them the opportunity to look for regularity in repeated reasoning.
Does the lesson still need some work? Absolutely. Is it better than last year? Absolutely. And so the journey continues ….