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

Overview

Curator: Hilary P.

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

http://www.regentsprep.org/Regents/math/geometry/GP4/PracCongTri.htm

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.

Evaluation

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).

Geometry Tech Tool: Geometer's Sketchpad Tessellations that use Rotations

You'll see in another post that the Geometer's Sketchpad and it's iPad version: Sketchpad Explorer are wonderful tools to use to complement a PBL/PrBL classroom. Below is a review of a specific tech tool that uses the software, written by a colleague of mine: Terry C.

Original CEP Posting

Overview

Curator: Terry C.

Name & Link to Tech Tool or Resource homepage: Tessellations That Use Rotations

Link to parent Wiki page: Geometer's Sketchpad

Link to Other GSP Activities: Geometer's Sketchpad Activities

Brief Description of Specific Math Activity: Students construct an irregularly shaped tile based on an equilateral triangle, and then use rotation to tessellate the plane with it (Key Curriculum Press, 2009).

Evaluation

Description of Learning Activity

By creating the tessellation and then dynamically changing the original tile to see the effects, students get a deeper understanding of what makes the tessellation work (Key Curriculum Press, 2009).

1. Learning Activity Types

• LA-Present - (read or attend to) presentation of new content/ideas
• LA-Present-Demo - demonstration
• LA-Present-Explain - explanation
• LA-Explore - exploring/investigating mathematical ideas
• LA-Apply - applying mathematics to problems and situations

2. What mathematics is being learned?

The objective of this activity is to use rotation to tessellate and to explore rotational symmetry (Key Curriculum Press, 2009).

NCTM Standards

• NCTM-Geo-analyze - analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;
• NCTM-Geo-specify locations - specify locations and describe spatial relationships using coordinate geometry and other representational systems;
• NCTM-Geo-visualization - use visualization, spatial reasoning, and geometric modeling to solve problems.

Proficiency Strands

• PS-conceptual understanding - Students work with a specific example but the activity tries to promote the idea that students can wander down a path of "What if I did this?" and still reach similar results.
• PS-adaptive reasoning - Throughout the construction and manipulation, students have to understand what is happening, how its happening, and why its happening.
• PS-productive disposition - Seeing the tessellation animated at the end of the activity is fun and rewarding.

Prerequisite Knowledge Required: Experience with equilateral triangles, translation, tessellation, and rotation.

3. How is the mathematics represented?

Mathematics can be represented symbolically and graphically in this activity. Geometer's Sketchpad allows for both dynamic and static representations of mathematics, and it depends on the inputs made by the creator how the mathematics is represented in the tool. Through the instructions, this activity guides students to creating graphical and symbolic representations of mathematics that act as virtual manipulatives.

4. What role does technology play?

Geometer's Sketchpad makes a unique contribution in that it automates tessellation of a plane through tile rotation, so that students can see the effects more easily and quickly.

Affordances of Technology for Supporting Learning

• Computing & Automating - Sketchpad allows students to construct shapes on a computer more easily than if they were drawing them with pencil and paper. Sketchpad also allows them to manipulate and change the shapes more easily than with pencil, paper, and eraser. The effect of rotating the tile to tessellate the plane is achieved through technology; this cannot be done on paper.
• Representing Ideas & Thinking - Sketchpad allows students to visualize and explore mathematics problems.
• Accessing Information -
• Communicating & Collaborating -
• Capturing & Creating -

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

According to the instructions, this activity is meant to be performed in student pairs. It can also be modified for a whole class. The technology facilitates the ease in which students can construct tessellations together (or side-by-side on separate computers) and then share the results with each other. Using technology allows students to rotate the original tile dynamically, which certainly is not easily accomplished with pencil and paper.

6. What do teachers and learners need to know?

This activity requires that students and teachers have access to Geometer's Sketchpad, and intermediate knowledge of Sketchpad is required to do this activity. The suggested grade level is 9 to 10; however, advanced seventh or eighth grade students could work on this activity. The activity includes an introduction on the activity for the teacher, notes, instructions, and discussion points for the teacher to guide the class through the activity, and a worksheet for the students that will guide them through the activity in Sketchpad and ask them questions at various intervals. The recommended duration for this activity is 45 minutes.

How this Supports and Supplements PBL/PrBL

 This activity is a great way for students to work on developing a conceptual understanding of tessellations as well as the procedural process of making tessellations, both of which can be daunting and time consuming for students. A lot of great projects and problems can focus on tessellations, or can take a tessellation turn (try turning ANY tiling problem into a tessellation problem!). The exposure students get by using this tool and doing this activity helps familiarize them with the concepts well enough to prepare them for thinking this way in projects and problems.

Geometry Tech Tool: Illuminations-Geometric Solids

Overview

Name & Link to Tech Tool or Tool homepage: Illuminations - Geometric Solids

Brief Description of Tech Tool: From their website:

This tool allows you to learn about various geometric solids and their properties. You can manipulate and color each shape to explore the number of faces, edges, and vertices, and you can also use this tool to investigate the following question:

• For any polyhedron, what is the relationship between the number of faces, vertices, and edges?

This tool could also be used to explore other relationships among 2D and 3D figures while focusing on nets of solids and can be adapted to fit a variety of needs from pre-k through high school. This tool provides a virtual manipulative to help students develop spatial reasoning regarding these figures.

Technical & Cost considerations: As with other Illuminations tools, lessons and activities, the resources are free. This applet is iPad and tablet compatible. It also runs in a variety of web browsers on computers and Macs.

Evaluation

Description of Learning Activity

For lower elementary students, this applet could be used to explore and identify the various 3D figures and the 2D figures that make them up. They could also practice counting the number of 2D shapes and relating that to the name for the 3D figure.

Upper elementary students could practice exploring the nets that are created as they "unfold" each of the 3D shape through the virtual manipulative. Students could then practice designing their own nets to be printed out to see if their net creates a 3D figure. Students could compare and contrast their findings with one another to see if they can come up with any generalizations for nets.

Students in the middle grades could expand on the activities for the upper elementary students and could complete the Geometric Solids Exploration worksheet to try to come up with Euler's formula on their own. They can then try to develop an informal proof for it.

Students in high school could expand upon the middle grades activities to develop Euler's formula and a formalized proof for it. They could also explore and establish their own criteria for creating "working" nets and "non-working" nets and generate their own examples for this using the "My Own Net" feature and printing it out.

1. Learning Activity Types

• LA-Present - (read or attend to) presentation of new content/ideas
• LA-Present-Demo - This tool could be used by a teacher to demonstrate what a net is in relation to a 3D figure.
• LA-Explore - This tool is particularly useful for helping students to explore the various relationships between 2D and 3D figures at any level.

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;

NCTM-Geo-visualization - use visualization, spatial reasoning, and geometric modeling to solve problems.

Proficiency Strands

• 
  PS-conceptual understanding Students develop a conceptual understanding of the 2D relationships to 3D models through experimenting and visualization with this tool
• PS-strategic competence Students build strategic competence through their continual trial and error with the creation of nets that create 3D figures.
• PS-adaptive reasoning Students' reasoning is likely to adapt as they continue to explore various combinations of polygons that create nets of 3D figures.
• PS-productive disposition Students will likely struggle with net creation at first, but as they print out and examine more nets from the program, they will become better at creating nets for 3D figures.

Additional comments on what is being learned

This tool is accessible to students at all levels of learning. Younger students made need assistance in using the tool, but once the students have learned basic mouse control, they should be able to use the manipulative to analyze the characteristics of 3D figures in terms of 2D figures to give them a solid foundation for their conceptual understanding of the relationship between the two.

Older students will have to work strategically and diligently on their ideas as they develop Euler's formula as well as when creating their nets to ensure that they create a solid figure. Both of these ideas may take several attempts for the student, but each time, they should be able to adapt their reasoning and work through until they find success. This may require more teacher encouragement and feedback depending on the students' mindset about mathematics.

These features help students to visualize the relationship between the 2D and 3D figures that is outlined in both the CCSS and the NCTM standards, with the upper grades being able to reach the level of developing argumentation about the relationship and proving their conjectures.

3. How is the mathematics represented?

This tool is a virtual manipulative that can take the place of traditional cut and tape nets in the math classroom. This tool could be particularly beneficial for visual learners in helping them to deepen their conceptual understanding of the relationship between 2D and 3D figures and to improve their visual-spatial reasoning. However, for more tactile and kinesthetic learners, this tool may be more of a starting point for a student to see the process of unfolding a solid to create a net, and then the student can use the "My Own Net" feature to create their own nets to print out and try working with.

4. What role does technology play?

This tool provides a great advantage in initially learning and exploring the relationship between 2D and 3D figures as students can continually fold and unfold their 3D figure, as well as color the various aspects of the figure to see just how the net is a mapping of the 3D figure.The ability to look at the figure from multiple perspectives and as a filled solid or a transparent solid is really unique and provides perspectives you don't get with a hands-on manipulative.

Affordances of Technology for Supporting Learning

• Representing Ideas & Thinking - This tool enables the user to manipulate the way the 3D figure is being represented as they explore the relationships between 2D & 3D figures.
• Capturing & Creating - This tool allows the student to create their own nets in an attempt to create a net of a solid figure.

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

This tool can be used by individuals or partners to support the students' exploration as they work with the applet. Students can play around with the tool individually and form their own conjectures about the various relationships between the 2D and 3D figures that they are able to create and can then share out these ideas with partners and continue their individual exploration from there. Alternatively, students can begin their explorations in a partnership and continually discuss the math that is unfolding before their eyes as they work with this applet. Partner work may prove beneficial for promoting good discussion when using the "My Own Net" explorations.

6. What do teachers and learners need to know?

Users should be comfortable with basic mouse control and should be familiar with the terms: face, edge and vertices when using this applet. Users should also be provided access to a printer to get the most out of their experience and explorations with the applet.

How it Supports & Supplements PBL/PrBL

The various representations of 2D/3D figures comes up quite often in my projects in the geometry classroom, as students are often sketching buildings, bridges, sculptures and other designs and trying to determine how to best construct them. This applet is useful in the classroom because it allows students to get a good visual representation of the way in which the two models are related to one another. I've given students what I consider simple nets before, and they've had no idea what it would become. This applet helps those students to see the changes that happen in a net to see how it maps onto a 3D figure. After getting a conceptual understanding here, students can practice and implement the relationship in their projects.

Geometry Tech Tool: Illuminations-Congruence Theorems

Original Posting for CEP 805

Overview

Name & Link to Tech Tool or Tool homepage: Illuminations - Congruence Theorems

Brief Description of Tech Tool: This tool allows for students to explore the idea of triangle congruence through combining any three attributes of a triangle. Students create their first triangle and then try to create exact copies of the triangle to determine congruence.

Technical & Cost considerations: This is a free tool provided from Illuminations. The tool works in multiple web browsers on computers.




Evaluation

Description of Learning Activity

This tool can be used for an in-class or at-home exploration of the triangle congruence criteria. Students can work through any combination of three triangle attributes to determine if triangle congruence is possible. This could be used as a precursor for students studying triangle congruence to help them explore the idea and see for themselves why certain combinations work and others don't. This would help students to gain a basic conceptual understanding of congruent figures, and triangle congruence in particular.


1. Learning Activity Types
LA-Explore -Students should use this tool primarily as an exploration tool where they are seeking out combinations to create triangle congruence. Students can work on any combination of elements that they choose and will have the ability to reason through each situation to help see why triangle congruence is or is not possible with their chosen combination.


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-conceptual understanding Students work on developing a conceptual understanding of triangle congruence by developing and testing their own hypothesis'.

PS-strategic competence Students work on strategic competency through experimentation with each of the various combinations of triangle components for congruence.

PS-adaptive reasoning Students work on developing adaptive reasoning through continual trial and error with various combinations. Students should improve in time.

Additional comments on what is being learned

Students will be exploring triangle congruence through their own means. As such, students will likely have their own initial thoughts on which scenarios will work for triangle congruence and which won't work. As students work through their explorations, they will likely get better at analyzing the attributes needed and may become better "guess"-ers in the exploration. This tool alone may not be enough for the students to work on all of the proficiency standards, but through strategic lesson planning and guidance it could be. As a standalone, however, students should develop a basic conceptual understanding of congruence.

3. How is the mathematics represented?

This tool is a virtual manipulative that allows users to change triangle attributes and orientation to attempt to come up with pairs of triangles that are(n't) congruent to one another. While students can choose their attributes, they cannot change the features about those attributes (segment length or angle degree) this could cause some confusion among students about whether or not certain triangles, like obtuse and right, can be congruent. Also, the CCSS calls for the understanding of congruence through rigid motions which is not inherently a part of this tool. This tool could be used as a jumping off point to understand what congruence means in terms of triangle attributes, but then another tool would be better suited for students to practice the rigid motion transformations of congruent figures.


4. What role does technology play?

Technology is incredibly helpful in exploring this difficult topic for the students because it allows for them to create and play with their own combination of triangle attributes and do this many times over. Students can work on manipulating their triangles in any manner they choose, something that would be difficult with a hands-on manipulative. However, as noted above, the tool does not allow the user to change the degree or the length of the initial segments/angles chosen, which leaves some types of triangles under-represented in the exploration.


Affordances of Technology for Supporting Learning

Computing & Automating - This tool helps to automate the process of creating multiple triangles with given criteria.

Representing Ideas & Thinking - This tool is particularly useful for creating multiple representations of triangles with given criteria, allowing for students to change the orientation of triangles in an attempt to create triangles that are(n't) congruent to one another.



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


I could see this activity and exploration being used in the classroom as an individual or partner activity. Students may benefit from having a partner to discuss their explorations with to come up with better "guesses" as to creating congruent criteria, however students could also write down their thinking if working independently. If working in partners, students could make this into a mildly competitive game where they try to come up with combinations that they think will or won't work to try to trick their partners. If working alone, students could try to find all the possible triangle congruence combinations in the fewest attempts possible and then compare their answers with others in the class.



6. What do teachers and learners need to know?


Users need only basic technology knowledge, such as mouse or clicker control, in order to successfully use this tool. Teachers preparing for this lesson should know the criteria that create congruent triangles and which criteria don't create congruent triangles and why so that they are prepared to guide students through this exploration.


How it Supports & Supplements PBL/PrBL

Congruence can be a difficult subject for students to understand initially, especially as they are posing and testing conjectures. This tool helps students to gain a conceptual understanding of congruence in polygons that they can carry forward with them when they are exploring the properties of polygons in problems and projects.