Congruence Theorem Applet
The Congruence Theorem applet is an interactive tool that is designed primarily for high school level math, grades 9-12, with an emphasis coming in either 9th or 10th grade. The purpose of this applet is to get students to explore conditions that result in congruent triangles. Students start by selecting any three elements in the top right corner of the screen the applet is displayed in. The elements will then appear in the blue area of the screen. Students are then able to click and drag the elements to form a triangle. After the first triangle is formed, a new set of congruent elements will appear in the blue area of the screen. Students are expected to form another triangle with the new elements. Then, the students will be asked if the second triangle is congruent to the first one. They will be able to click and drag the second triangle over the first triangle to compare. If the two triangles are congruent, then the student will be asked to attempt make a third triangle that is not congruent to the original triangle.
Below is a screen shot of the applet. Click here to access the applet.
Mathematical Standards
Georgia Performance Standards:
Content
MM1G3. Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons. c. Understand and use congruence postulates and theorems for triangles (SSS, SAS, ASA, AAS, HL).
Process
MM1P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving.
MM1P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjecture. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof.
In relating to the process standards, this applet clearly addresses the fact that students will solve problems using appropriate technology. Students should hopefully build upon their knowledge of congruence theorems through use of this technology. While exploring and using the applet, students are actively investigating mathematical arguments and proofs. Also, students should recognize reasoning and proof about congruence between triangles.
Common Core
Content
MCC9‐12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
MCC9‐12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Process
3 Construct viable arguments and critique the reasoning of others. 5 Use appropriate tools strategically.
While interacting with the technology, students should be able to make their own conjectures about which sets of conditions result in congruent triangles. In turn, they should be able to construct an argument to support their reasoning and also may critique the work of other students. This technology can be thought of as a tool itself. Students will use this tool to gain a better understanding of congruence theorems between triangles.
Critique of Technology
How well does it work?
The Congruence Theorem applet is a great interactive tool for students to explore the many different theorems dealing with congruent triangles. The options within the applet are very easy to understand and use. There are a few negatives however. One being that you must line up the elements exactly in order for them to "snap together." Also, it is unclear as to what elements work because it never says one does not work. You just have to realize it for yourself. This may be confusing to some students. They may believe they are doing it wrong when in fact they are doing it correctly.
Are the written materials well organized and useful?
The instructions are very helpful. Before reading them, it is hard to grasp exactly how to arrange the elements to form a triangle. However, the instructions clearly outline how to operate the applet and what the objectives are.
What are the purposes and goals for using this technology? Does the technology reach this goal?
The purpose of this applet is to get students to explore different conditions that result in congruent triangles. The goal is to broaden each student's understanding of congruence axioms such as SAS, ASA, and SSS. Students are given the opportunity to explore the different congruence axioms and are expected to make conjectures about which sets of elements result or do not result in congruent triangles. This applet accomplishes these goals. Students are able to explore different conditions and each set of conditions results in a new congruence theorem.
Is the technology relatively easy to learn how to use?
After reading the directions and exactly how to fit the different elements together, the technology is pretty easy to understand and use. It does not take long to get the hang of the different tools and options. Also, there is a very limited amount of things you can do within the applet so it is not hard to master everything.
Does this technology enhance or extend the teaching and learning process for the intended mathematics concepts? How and why?
With this technology, students will broaden their knowledge and understanding of the different congruence axioms involving triangles. They will gain a better understanding of why each axiom holds while using a visual representation of each congruence axiom. Also, this applet can be used as an extension of teaching why some sets of elements do not result in congruence axioms, such as AAA. This tool may be used as a distinction between congruency and similarity between figures. However, any sort of discussions will have to be teacher led since the applet itself does not address these things. It only provides a visual representation.
Would you recommend this product for purchase to a school? Why or why not?
The Congruence Theorem applet does not require purchase of anything to use it. With that said, I would still recommend it for use in a classroom. However, if there was a fee to use the applet, then I would suggest finding a different app. Although it is useful, I do not believe it would be worth purchasing. That aside, there are many other applets available on the NCTM Illuminations website also. This applet, in particular, could benefit students when learning about congruence between triangles. It gives a visual representation of the different congruence axioms and students can also see which combination of elements does not result in congruence. However, I would recommend another applet for students to get a better grasp of the congruence axioms if there exists such an applet. This applet may distract from the concepts and ideas that the applet is designed to explore.
Activity
The following is a handout designed to explore the applet. Click here.
The Congruence Theorem applet is an interactive tool that is designed primarily for high school level math, grades 9-12, with an emphasis coming in either 9th or 10th grade. The purpose of this applet is to get students to explore conditions that result in congruent triangles. Students start by selecting any three elements in the top right corner of the screen the applet is displayed in. The elements will then appear in the blue area of the screen. Students are then able to click and drag the elements to form a triangle. After the first triangle is formed, a new set of congruent elements will appear in the blue area of the screen. Students are expected to form another triangle with the new elements. Then, the students will be asked if the second triangle is congruent to the first one. They will be able to click and drag the second triangle over the first triangle to compare. If the two triangles are congruent, then the student will be asked to attempt make a third triangle that is not congruent to the original triangle.
Below is a screen shot of the applet. Click here to access the applet.
Mathematical Standards
Georgia Performance Standards:
Content
MM1G3. Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons.c. Understand and use congruence postulates and theorems for triangles (SSS, SAS, ASA, AAS, HL).
Process
MM1P1. Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
MM1P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjecture.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
In relating to the process standards, this applet clearly addresses the fact that students will solve problems using appropriate technology. Students should hopefully build upon their knowledge of congruence theorems through use of this technology. While exploring and using the applet, students are actively investigating mathematical arguments and proofs. Also, students should recognize reasoning and proof about congruence between triangles.
Common Core
Content
MCC9‐12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.MCC9‐12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Process
3 Construct viable arguments and critique the reasoning of others.5 Use appropriate tools strategically.
While interacting with the technology, students should be able to make their own conjectures about which sets of conditions result in congruent triangles. In turn, they should be able to construct an argument to support their reasoning and also may critique the work of other students. This technology can be thought of as a tool itself. Students will use this tool to gain a better understanding of congruence theorems between triangles.
Critique of Technology
How well does it work?
The Congruence Theorem applet is a great interactive tool for students to explore the many different theorems dealing with congruent triangles. The options within the applet are very easy to understand and use. There are a few negatives however. One being that you must line up the elements exactly in order for them to "snap together." Also, it is unclear as to what elements work because it never says one does not work. You just have to realize it for yourself. This may be confusing to some students. They may believe they are doing it wrong when in fact they are doing it correctly.
Are the written materials well organized and useful?
The instructions are very helpful. Before reading them, it is hard to grasp exactly how to arrange the elements to form a triangle. However, the instructions clearly outline how to operate the applet and what the objectives are.
What are the purposes and goals for using this technology? Does the technology reach this goal?
The purpose of this applet is to get students to explore different conditions that result in congruent triangles. The goal is to broaden each student's understanding of congruence axioms such as SAS, ASA, and SSS. Students are given the opportunity to explore the different congruence axioms and are expected to make conjectures about which sets of elements result or do not result in congruent triangles. This applet accomplishes these goals. Students are able to explore different conditions and each set of conditions results in a new congruence theorem.
Is the technology relatively easy to learn how to use?
After reading the directions and exactly how to fit the different elements together, the technology is pretty easy to understand and use. It does not take long to get the hang of the different tools and options. Also, there is a very limited amount of things you can do within the applet so it is not hard to master everything.
Does this technology enhance or extend the teaching and learning process for the intended mathematics concepts? How and why?
With this technology, students will broaden their knowledge and understanding of the different congruence axioms involving triangles. They will gain a better understanding of why each axiom holds while using a visual representation of each congruence axiom. Also, this applet can be used as an extension of teaching why some sets of elements do not result in congruence axioms, such as AAA. This tool may be used as a distinction between congruency and similarity between figures. However, any sort of discussions will have to be teacher led since the applet itself does not address these things. It only provides a visual representation.
Would you recommend this product for purchase to a school? Why or why not?
The Congruence Theorem applet does not require purchase of anything to use it. With that said, I would still recommend it for use in a classroom. However, if there was a fee to use the applet, then I would suggest finding a different app. Although it is useful, I do not believe it would be worth purchasing. That aside, there are many other applets available on the NCTM Illuminations website also. This applet, in particular, could benefit students when learning about congruence between triangles. It gives a visual representation of the different congruence axioms and students can also see which combination of elements does not result in congruence. However, I would recommend another applet for students to get a better grasp of the congruence axioms if there exists such an applet. This applet may distract from the concepts and ideas that the applet is designed to explore.
Activity
The following is a handout designed to explore the applet. Click here.References
Common Core Georgia Performance Standards CCGPS (2011). Common Core State Standards For Mathematics. Retrieved from http://www.corestandards.org/assets/CCSS_Math%20Standards.pdf
Georgia Department of Education (2006). Mathematics Georgia Performance Standards. Retrieved from https://www.georgiastandards.org/standards/Georgia%20Performance%20Standards/Math-I-Stds.pdf
Georgia Department of Education (2011). Common Core Georgia Performance Standards CCGPS. Retrieved from https://www.georgiastandards.org/Common-Core/Documents/CCGPS-Gr9-12-Math-Standards.pdf
NCTM Illuminations. (n.d.). Retrieved from http://illuminations.nctm.org/ActivityDetail.aspx?ID=4%20