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.