Text Set #1: Triangle Congruence

Books/Print Resources

·      Geometry Workbook For Dummies

Mark Ryan

Grade level: 12 (for teachers to use in instruction)

This book has a good ‘plain language’ approach to geometry, though the reading level is a bit high.  I would certainly use it sparingly, and mostly on concepts that don’t have much in the way of an access point for students.  Some of the exercises are also presented well and broken down into small steps and would be appropriate for students who get overwhelmed by multi-step problems. 

·      “The History and Concept of Mathematical Proof”

Steven G. Krantz

Accessed from http://www.math.wustl.edu/~sk/eolss.pdf

Grade level: 8.6

This text might be nice for advancing students who perform highly and need activities that deepen their understanding of geometry, but section 4 on the history of geometry would be a good assignment for the whole class. It could even be a read-along for those classes that struggle with reading. 

Activities/Websites

·      Triangle Congruency By SSS and Properties of Isosceles Triangles Activity

http://education.ti.com/en/us/activity/detail?id=486E4EB82B77484BBC4C0097DF3FE3E1

Grade level: 6.6

This is a great in-depth application lesson on using triangle congruency and the properties of isosceles triangles.  The lesson includes an authentic problem about building a doghouse, and set ups for analysis based on symbolic, graphic, and verbal representations.  I find this activity useful because of the depth and variety of representations in analysis. 

·      Conditions That Prove Congruency

http://education.ti.com/en/us/activity/detail?id=359A3010A7CE45C89995C27ABEC063CB

Grade level: 7.7 and above

 This activity combines conceptual and abstract verbal representations with interactive portions that let students determine empirically the conditions necessary to prove congruence in triangles.  It lets students see first-hand how changing various parts of a triangle affects congruence. 

·      Math for Morons Like Us

http://library.thinkquest.org/20991/geo/ctri.html

Grade level: 10.1

This page is a good reference for the five triangle congruence theorems, and provides great examples of small, manageable proofs for each of the cases.  It also has a little quiz at the end to test student knowledge.

·      Regents Prep: Practice with Proofs Involving Congruent Triangles

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

Grade level:  8.3

This activity gives good triangle congruence proofs for increasingly complex figures.  It uses the ‘traditional’ two-column form used in Geometry classes, and challenges students to try the proof before clicking to reveal the answer. 

·      Khan Academy: Congruent Triangles

https://www.khanacademy.org/math/geometry/congruent-triangles

Grade level: 11.9

Khan Academy’s collections on triangle congruency combine visual/symbolic representations with aural presentations.  It is good for those students who favor auditory methods and who may get ahead of themselves with verbal representations, thus losing the sequential logic of the proof.  Of course, after the videos, Khan Academy has some stellar interactive quiz questions.

·      Khan Academy: Congruence, Isosceles and Equilateral Triangles

https://www.khanacademy.org/math/geometry/congruent-triangles/isoscleles_equil/e/geometry_proofs_intro

Grade level: 8.8

See previous entry on rationale for videos; proof questions here are constructed as a fill in the blank for parts of the triangle with a drop-down box supplying the properties to be used as justification. 

·      Wikipedia: Congruence (geometry)

http://en.wikipedia.org/wiki/Congruence_(geometry)

Grade level: 12

Honestly, Wikipedia is actually usually a pretty good place for establishing a baseline for a mathematical concept.  Now, I wouldn’t have have the students read the entire page because it gets technical, but the intro and selected sections are good, as are the .gifs in the sidebar. 

·      WyzAnt Resources: Congruent Triangles

http://www.wyzant.com/resources/lessons/math/geometry/triangles/congruent_triangles

Grade: 10.7

This webpage uses a nice, approachable conversational tone to discuss congruence, primarily in a verbal mode.   Also, the figures here are understandable and clearly marked with different colors to indicate congruent parts. 

·      Regents Prep: Lesson on Theorems for Congruent Triangles

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

Grade level:  9.3

The first part of this page is nothing special and is a less stellar version of some of the other resources I’ve included in this set, but the last third of the page is dedicated to a discussion on why AAA and ASS are not valid as congruency postulates.

·      Math is Fun: Congruent Triangles

http://www.mathsisfun.com/geometry/triangles-congruent.html

Grade level: 5.8

This would be a good resource for students who do not read at a high school level.  It is simple introduction to congruence with simple figures – nothing overly complicated. 

·      Math Warehouse: Isosceles Triangle Theorems and Proofs

http://www.mathwarehouse.com/geometry/congruent_triangles/isosceles-triangle-theorems-proofs.php

Grade level: 6.7

This site is good to look at proofs in an incremental fashion, if it is a bit garish. 

·      Geometry History

http://www.kidsmathgamesonline.com/facts/geometry/history.html

Grade level: 12.0

Despite having a Flech-Kinkaid level of 12, this page provides a few basic historical facts about Geometry and geometric constructions.  I’m not entirely convinced that the reading level is as listed – I think the Greek root words discussed skew the measure.

·      Math Open Reference: Introduction to Constructions

http://www.mathopenref.com/constructions.html

Grade level: 7.9

This site gives a little bit of background on construction proofs and provides a nice answer for that student who always says “Well why didn’t they just measure it?”  It also has a collection of links to pages on constructions and proofs.

Reflection 8: Wait, I Thought We Were Finished With Vocabulary?

Nope.  It never stops.  Which was one of the things that I found interesting about the Bromely article -- she acknowledged the fact that language is a living, changing thing and that new words are created every day in English, especially in STEM fields.  I also think it's significant that she took the time and space to talk about how our attitudes as teachers towards new vocabulary affect how students perceive the experience of encountering new vocabulary.  Sheer enthusiasm for language is the bedrock that all of these other strategies are built on.  I don't know about you, but I love a good $10 word; and that makes the experience of guiding students through vocabulary instruction or discovery much less like a chore.

In general, I found the Bromely article very interesting, even if some of the statistics made me raise my eyebrows a bit.  Seventy percent of the most common words having multiple meanings seems a bit iffy to me; that just seems high.  As well, I thought that saying that English is a simple and consistent language relative to other languages was a bit of an overstatement.  While I can concede that English would be simpler than tonal languages just on a literary level, that doesn't necessarily mean that we're consistent by any means.  English pulls grammatically and linguistically from so many places it would be difficult to remain internally consistent.  In addition, I really wish the Bromely article had gone more in depth about using connections and associations to learn new words.  I know this is the primary method I use to learn new words, and by its very nature this strategy is accessible to all.  Seeing how to implement this in a classroom would be a great resource for me.

In the Baumann and Graves article, I feel like it's worth mentioning that it's really important to me that they classified symbolic representations as their own category of vocabulary.  Understanding these and knowing how to read them are crucial for building comprehension in math and science.  I often liken math to a foreign language -- it's not English, but it is a language with vocabulary and a grammatical structure, and can be understood and used to communicate.  It is internally consistent and logical.  It's not magic or gibberish.

Text Set Topic

Triangle Congruency

Reflection 7: Oops, Vocabulary Again

The BBR Chapter 5 had some strategies that transfer really well to the content area of math, finally!  Math is all about content-specific vocabulary and 'common' vocabulary that has a specific meaning in the context of a mathematics classroom.  

I really like the Contextual Redefinition strategy because it explicitly acknowledges the difference between vocabulary as seen in 'common' circumstances and as seen in academic mathematics.  In addition (zing!), I find that it is the strategy that employs the most 'think-aloud' elements.  I think that think-aloud strategies are really important in mathematics because oftentimes people view a math problem as one giant insurmountable whole written in what might as well be a foreign language.  They need to see that good mathematicians read things in chunks and frequently call up definitions and characteristics of elements of a problem as they read it.  

I also really like the Etymologia and Morphologia strategies because they add a narrative element to understanding new vocabulary and they draw in the human element through history.  However, I think I might combine the two into one assignment/strategy in my classroom.  Why did the person who named this particular thing choose the words or morphemes that they did? 

Feature Analysis also struck me as being particularly useful in mathematics because we are constantly comparing concepts or things that share many characteristics but differ in a few minute but crucial ways.  Feature Analysis charts are a nice way of keeping track of the nitty gritty that sometimes gets lost in the concise mathematical definitions.  

Reflection 6: Help for Struggling Readers

The Daniels and Zemelman chapter on strategies for helping struggling readers helped me realize where some of the bumps in one of my classes may be coming from.  The idea of forming mental pictures and scouring word problems for pertinent data is something I should take into account when I assign word problems; I've noticed my students have trouble going from a purely verbal description to a diagram, a crucial skill in geometry.  I also liked that one of the main strategies that they focused on was that of having readers make note(s) of where they become confused.  This helps students develop self-monitoring skills as well as be aware of content area knowledge they may need extra help with.  I like that they mentioned that the teachers in the vignettes had set up a chain of actions that were to be taken at the points where students become confused, but I wish that they had explored those actions a little more deeply.  Where should students be going and what should they be doing after they realize they've lost the thread of a reading?

Reflection 5: Vocabulary Development

I loved the Bean, Baldwin and Readence chapter on developing vocab.  It has lots of awesome strategies for getting students physically and mentally engaged with the vocabulary that's essential for understanding.  (You don't learn new words by osmosis!)

We actually did a semantic mapping activity the other day to start off Module 2 in my Geometry class to talk about what it means to 'study harder' -- things you can do by yourself, things you can do with others, and things you can do in class. (This discussion was prompted by a reflection ticket I had them do after the Module 1 test; they were to list 3 good things they're doing, 2 bad things they're doing, and 1 thing they're willing to change in the next unit to do better.)  I used a semantic mapping software for the iPad called MindMeister, found here.  It's pretty lightweight and easy to use on the fly, after you get a feel for where to tap to add new bubbles.

I also really like the idea of the feature analysis chart in general to compare and contrast things, but I think that I might use it later this module to wrap up quadrilaterals.  It'll be a nice way to recap all of the different properties, and to look at which quadrilaterals are actually more than one type (a rectangle is also a parallelogram).  I'm also thinking about having students do something like the verbal/visual representations strategy for each shape, as we learn them.

I'm stoked that this reading had so many strategies that are easily transferred to my content area!