Tuesday, April 22, 2014

Geometry Tech Tool: Proofs involving Congruent Triangles

The concept of congruence and proofs is a difficult one to cover exclusively through Projects and Problems. I've mentioned another tech tool before, Illuminations-Congruence Theorems, that I have used to help me help students develop a conceptual understanding. This tech tool comes from another colleague of mine and is helpful with completing the proof-writing process involved with congruent triangles.
Original CEP Post

Curator: Hilary P.

Name & Link to Tech Tool or Tool homepage: Proofs Involving Congruent Triangles

Brief Description of Tech Tool: This tech tool provides a way for students to practice writing two-column triangle congruence proofs and get feedback about whether or not they are writing proofs correctly (students simply click "proof" to reveal the answer). I find it is sometimes difficult to find resources for writing proofs and it seems different textbooks provide very similar if not the same proofs, so it is great to see some different proofs on this website. The proofs get more difficult as students progress through the practice so it could easily be used as a type of leveled practice. The higher number proofs also use CPCTC to prove corresponding segments/angles are congruent. Finally, this tool should be used after all congruence properties are covered as it uses ASA, SSS, AAS, and SAS.

Technical & Cost considerations: There is no cost to use Regents Prep. Students can easily access the website and use the "proof" button to see the correct answers.


Description of Learning Activity

I would use this activity as an independent practice. This practice must be implemented after students have an understanding of each of the the triangle congruence postulates as well as how to use CPCTC in a proof. This is a great way to change up the flow of a lesson (instead of simply practicing paper/pencil proofs). Students can visit website, and write their proofs on paper and easily check their answers. As we know as teachers, it is very time consuming for us to write answer keys for proof practice, so I like that this comes with a "built-in" answer key.

1. Learning Activity Types

  • LA-Practice - practicing for fluency
    • This activity is a great way to have students practice writing two column proofs. Students can easily click the "proof" button to check their answers, and monitor how well their proof writing is going. The practice proofs get more difficult and more involved as you progress through the problems, so it can also be used as a leveled practice.

2. What mathematics is being learned

NCTM Standards

NCTM-Geo-analyze - analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;

Proficiency Strands

  • PS-procedural fluency
    • This website provides a total of 10 two-column triangle congruence proofs. Students can work at their own pace through these problems, and can check their answers against the key as they go.
  • PS-adaptive reasoning
    • Students are constructing mathematical arguments as they write two-column proofs. Students make statements and must be able to justify their statements in order to write a cohesive proof. Students are not told which triangle congruence postulate to use, but must be able to analyze the diagrams and givens to determine how to prove the two triangles are congruent.

Additional comments on what is being learned

This website is designed to help students practice writing two-column triangle congruence proofs. Students must use statements and the correct mathematical reasons to write a cohesive proof. Students must use a variety of triangle congruence postulates such as SSS, SAS, AAS, ASA. Additionally, students may also need to use CPCTC in their mathematical proof to prove that corresponding sides and angles are congruent.

3. How is the mathematics represented?

The students are given both a diagram of the two triangles as well as mathematical statements that list what information was given. When students click on "proof" a two-column proof answer key opens in a additional window, and students can use this to check their answers.

4. What role does technology play?

Advantages: Students are provided with different proofs. As mentioned earlier, there are often several different proof problems that repeat from textbook to textbook, so it is nice to find a website with so many practice problems. Students are interacting more with this website much like a textbook, but at a click of a button are told the answers to the two-column proof.

Disadvantages: As the website notes, there is sometimes more than one way to write a mathematical proof and the answers only give one answer. Additionally, sometimes the statements or reasons might be written differently than students were taught from the textbook and may cause some confusion. Also, students are asked to use integrity when interacting with this website. Students should be writing two column proofs on paper and then checking their answer against the key. The teacher will want to closely monitor this interaction. Finally, I wish the website would offer hints for each step before just giving students the answer - this would be a great addition to this tool.

Affordances of Technology for Supporting Learning

  • Computing & Automating -
  • Representing Ideas & Thinking - This website provides students with proofs of varying levels of difficulty and students have to visualize which triangle congruence postulates can be used to prove congruent triangles.
  • Accessing Information - This website provides students with an answer key for each of the proofs they write. This is immediate access to information and can be very helpful formative data for students.
  • Communicating & Collaborating -
  • Capturing & Creating -

5. How does the technology fit or interact with the social context of learning?

This tech tool would best be used by individuals and does not foster tons of interaction between peers. However, I hope that providing the answer key will spark questions within students' minds about the way they wrote something in their proof, or why the textbook writes it differently than the website. Hopefully, the teacher could answer these questions as he or she circulated the room.

6. What do teachers and learners need to know?

As mentioned, teachers must monitor student use of the website. The answer keys should not be revealed until students have first written the proofs. Again, students must have the background knowledge of all triangle congruence theorems as well as CPCTC in order to fully engage with this website. The proofs get more difficult, and number 8 and 9 in particular, require some properties that may or may not be covered in you course, so you may choose to skip those. Overall, this is a good website, and a nice way to have students use computers to practice proofs which is traditional simply a pencil and paper activity.

How this Supports & Supplements PBL/PrBL
As I mentioned at the beginning of this post, proofs are a difficult concept to address directly through projects and problems. Students can most certainly practice attending to precision and justifying their statements and claims in projects and problems, but create a direct proof using geometry theorems is a tough subject to master through problems alone. (I say just problems, because I have yet to do a project with proofs, though I frequently run activities and seminars that are problem-based).