Proof is central to the development of math.
When a theorem is an idea without a proof, it is referred to as a hypothesis. Once a proof has been written and peer-reviewed, the hypothesis becomes a theorem. That theorem is then used to develop and prove other theorems.
When Euclid (365 – 300 BCE) wrote his book ‘The Elements’ he began this tradition of building new theorems on old theorems. He organised the geometry theorems known at the time into a hierarchy. No theorem was included if it couldn’t be proved from theorems already proved.
And for the last 2000+ years, math has continued to develop in this way!
Proofs are why we know the math works.
Proofs can be compared to poetry: it can be satisfying to learn a proof that someone else has written and it can be exceedingly satisfying to come up with your own proof to a theorem. Proof writing requires creativity and there can be many different proofs for the same theorem.
In precalculus 11, we have used several theorems. In this project, you are asked to research two of these. If you work with a partner, you need to choose three theorems from the list at the end of this page.
Prepare a google slides file/powerpoint file to showcase proofs for 2 (or 3) of the theorems that we rely on in precalculus 11.
- Find a video or website online that offers a proof for each of the theorems listed at the end of this page.
- Watch each of the videos a few times, as often as you need to begin to make sense of the argument. If your video is incomprehensible, look for another video.
- Choose 2 (or 3 if working with a partner) theorems to study in greater depth.
- Learn these 2 proofs with sufficient fluency that you can explain them (eg to another class member) without referring to the video you learned them from.
- Create a paper slide video for each proof, explaining them on video as you would explain them to a fellow class member.
When you have completed your research on the proofs of your chosen theorems and made your own proof videos, it is time to put together a google slides document.
Slide 1: Title Page. Your name, the name of your chosen theorems
Slide 2: Theorem 1. State the theorem as an ‘if’ ‘then’ statement. Create a diagram or example to demonstrate that the theorem works (use paper, pencil, ruler/protractor for geometry, or use GeoGebra). Demonstrate a numerical example showing how the theorem is used to calculate an unknown value.
Slide 3: Embed the best video you have found from youtube that explains a proof of your first theorem. Be sure to copy and paste the URL.
Slide 4: Embed your own video where you present the proof.
Slide 5, 6,7 As for 2,3,4 but with the second theorem you choose.
Slide 8: A list of all the sources you used in your research.
This is a suggested structure. As long as all the elements are included and your content is organised, you are welcome to use your own structure.
Please read the assessment criteria so that you know how the project will be assessed. Grade 11 PC Proof Project Assessment Criteria
Theorems to Choose From
- Pythagoras’ Theorem
- The Quadratic Formula
- The Central Angle Theorem
- The Sine rule
- The Cosine rule
- Angles in a triangle add to 180 degrees