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.
I can statements:
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
6B Similarity Theorems
6C-6D The Similarity Ratio
6E Pythagorean Relationships
6F Altitude to the Hypotenuse
6G Dilations Performance Assessment – Dilating a line and a circle
6H Dilations Performance Assessment – Bank Shot
In which Standard for Mathematical Practice did you engage most often during the unit?
Most students chose MP1, make sense of problems and persevere in solving them, and MP 6, attend to precision.
I asked students whether any of the content seemed like repeats of previously learned material.
- Aside from problems that were solved by utilizing the Pythagorean Theorem, all of the content from this unit was new to me.
- I’ve gone over pythagorean theorem multiple times in the past year, however the rest of this is new to me, and frankly quite difficult.
- I can honestly say I learned everything as new in this unit. This was entirely new for me. I feel I had an easier time learning this material than the other material in other units. I caught on very quickly in this unit compared to others
- the only thing I already knew was that for a figure to be similar all the side lengths had to be proportional.
- The only thing i knew about unit 6 before we learned it was that dilations have to do with making an object larger or smaller.
- There was nothing that was repeated in this lesson. I knew what a dilation was, but I’d never done anything with it, so basically I knew the first 15 minutes of this unit, and the rest was completely new to me.
Which lesson helped you the most in this unit?
- There is not one lesson or activity in particular that stands out to me, but I do think the extra homework helped, even though the thought of extra homework is not a very appealing thought.
- 6C-6D was very helpful. If I hadn’t understood similarity ratio, I would have not understood any of the material we learned this unit. I have never gone into so much detail about similarity and ratios. I’m glad I understand them more.
- The day when we used wax paper to decide if objects were similar or not helped me because it let me visually understand it.
- The homework was extremely helpful. It provided practice for me, and showed me what I needed to work on. It deffinitely made the lesson more clear.
- I think that lessons 6A and 6B, dilations and similarity theorems, really prepared me for all the material that was taught after that. Learning these two skills taught me the basics for learning the rest of the unit.
What did you learn during this unit?
- you have to be very precise and make sure you evaluate your answer
- I have learned about the similarity ratio and how to dilate a line and a circle.
- I have learned my favorite transformation: Dilations. I love dilations now and they are my favorite. I figured out how to tell if figures are similar through a rigid motion and dilation or just through a dilation. I learned that the hypotenuse altitude is the geometric mean between the two sides it divides, which helped a ton in figuring out measurements of the sides of a right triangle. I think this was my favorite unit and I learned so many new things.
- I have learned how to use, apply, and perform dilations to geometric figures. I have also learned that listening to the opinions and ideas of others can lead you to find the answer or the reasoning behind certain answers.
And so the journey continues …