Text Set 2: Trigonometry

Text Set 2: Trigonometry

·      S.O.S. Mathematics: Trigonometry

http://www.sosmath.com/trig/trig.html

Grade level: 12.0

While the reading level of this collection of pages may be a bit high, it has excellent diagrams that present trigonometry in a variety of ways.  Here, the links between the geometric and the algebraic conceptions of trigonometry are made clear.  This is a good extension for a high performing student, or a student who has been accelerated and is studying independently.

·      Math is Fun: Introduction to Trigonometry

http://www.mathsisfun.com/algebra/trigonometry.html

Grade level:  6.8

This page from Math is Fun is written in a simple, conversational style and is accessible to both those reading below grade level and those whose math vocabulary may be weaker.  However, I find that the greatest value is in the interactive flash elements halfway down the page – students can move a point around the unit circle to create a right triangle and examine its trigonometric ratios.

·      Khan Academy:  Basic Trigonometry

https://www.khanacademy.org/math/trigonometry/basic-trigonometry

Grade level:  12.0

Khan Academy is a great resource for those students that prefer auditory methods of taking in information (most of the information is conveyed graphically or verbally).  The videos show trigonometric concepts and how to use them, in the form of notes that the author puts on a slide.  The author talks the students through what he is doing, and is clear and direct; as well, his mouse/pointer is always visible and follows which element of the notes at which he happens to be looking at any given point.  And of course, this lesson includes the interactive quiz questions that Khan Academy is famous for. 

·      Trig Cheat Sheet

http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf

Grade level: 2.4

This is a sheet of trig formulas and trig identities, conveniently condensed, with figures illustrating the general case.  There isn’t much reading (verbal) in this resource, but math-specific literacy is necessary for comprehension.

·      Cliff’s Quick Review Trigonometry

https://play.google.com/store/books/details?id=qo8GBrUfGIcC&source=productsearch&utm_source=HA_Desktop_US&utm_medium=SEM&utm_campaign=PLA&pcampaignid=MKTAD0930BO1

Grade level:  9.4

Cliff’s Quick Review is exactly what is says – a quick crash-course review of trigonometry concepts.  It is heavily verbal, though the reading level appears to be accessible for the majority of high school sophomores.  It is available as an e-book download from this link in the Google Play store (which is great for tech schools, though Apple users may run into some compatibility issues), as well as a hard copy from your local bookstore. 

·      Trigonometry (Corral, M.)

http://www.mecmath.net/trig/trigbook.pdf

Grade level:  11.0

Corral’s text on trigonometry is a good extension for students who would like to read trigonometry in the context of how mathematics looks when it is published within the field.  It connects trigonometry to history and other branches of mathematics.  

·      Wikibooks: Trigonometry

http://en.wikibooks.org/wiki/Trigonometry

Grade level:  7.8

Wikibooks has a good collection of information, free and available on the web to anyone, on trigonometry.  It gives us three books, organized by increasing difficulty and frequency of use of concepts present in other branches of mathematics (matrices, calculus, computing).  Topics are arranged by page, and include many diagrams.  This resource is also useful for students and teachers to think and talk about what makes a good figure – there are editors’ notes on how figures will be updated to be clearer and more useful.

·      Applications of Trigonometry (Clark University)

http://www.clarku.edu/~djoyce/trig/apps.html

Grade level: 10.9

This page gives a quick run-down of some trig applications, and the historical context in which they developed. 

·      Math Central: Applications of Trigonometry

http://mathcentral.uregina.ca/RR/database/RR.09.01/trigexamples1.html

Grade level: 8.2

This page gives a couple of unconventional examples of applying trig.  In addition, they’re examples that the average layperson can encounter, not just examples found in physics or engineering contexts. 

·      Fascinating Facts of Mathematics

http://malini-math.blogspot.com/2011/08/applications-of-trigonometry-in-real.html

Grade level: 8.6

A quick succinct list of traditional applications of trigonometry, complete with example pictures.  The pictures have figures overlaid that connect the trigonometric model with the real world context.  

·      Slideshare: Real World Uses of Trigonometry

http://www.slideshare.net/ihatetheses/real-world-application-of-trigonometry

Grade level: 12.0

The reading level on this is a bit high, but this slideshow is an excellent resource for connecting trigonometry to the real world.  It showcases how certain careers make frequent use of trigonometry, and then give details about those careers such as salary and areas of study needed. 

·      WISC-Online: Practical Trigonometry

http://www.wisc-online.com/Objects/ViewObject.aspx?ID=MSR3603

Grade level: 4.6

This is an interactive slideshow that poses some really great application problems within the context of a machine/prototyping shop.

·      Clark University: History of Trigonometry Online

http://www.clarku.edu/~djoyce/ma105/trighist.html

Grade level:  12.0

This is a brief online outline of the beginnings of trigonometry, complete with figures and historical context.

·      Applications; Web-Based Calculus

http://ualr.edu/lasmoller/trig.html

Grade-level: 12.0

This page gives a brief history of trigonometry, with a focus on its journey through various cultures and its (mis-)translations.

·      TI-Nspire: Sinusoidal Modeling

http://education.ti.com/en/timathnspired/us/detail?id=469A5FBBB74F4031882FEC9806D32867&t=3270A55B10BB4BF29C7D8883E3990203

Grade level: 6.9

This exploratory activity extends knowledge of trigonometry from geometry to algebra and statistics, and models the use of trig functions to model oscillating phenomena. 

Text Set Topic Redux

Trigonometry

Reflection 10: Technology

I think the strategies in BBR Chapter 10 were useful as a macro-structure for web-based searches, but I feel like some of the other strategies that are nested in these structures bear some investigation as well. How do students decide that a website is credible?  The -AND (Analyze and Note Details) and -SD- (Slow Down) steps from the SAND and ISSDaT strategies require students to evaluate the credibility of web sources, but do't mention more than making sure to stay away from .com sites and sites clearly trying to sell you something.  I can think of plenty of examples of .org websites that may be of dubious credibility -- I'm sure the KKK and crazy Doomsday/Rapture Preppers have websites ending in things other than .com, but they're hardly credible sources (except as primary sources to examine how bleeding insane they are).  I'm at a one-to-one school and the students almost exclusively use the web for resources when doing projects, but few of them know how to differentiate an academically credible source (e.g. a scholarly one) from one that appeals to sensationalism, trends, and popular culture -- beyond "Don't use Wikipedia."  I think a more useful strategy to analyze would be how to determine credibility, and how credibility requirements change depending on the project.

Web Resource 2: Khan Academy's Geometry section

https://www.khanacademy.org/math/geometry

Khan Academy's geometry section has been a fantastic resource for me because of the interactivity of the 'quiz' sections.  I feel like the quiz sections are almost more valuable than the videos; the videos are nice because they offer a clear verbal walkthrough of problems, but the interactive quiz sections let students experiment with figures to see if constructing counterexamples (per say) is even physically possible with the given conditions.  I feel like nothing is quite as instructive as a hands-on session of trial and error.

As well, students can sign up for an account through the website and earn points through a cute little game system and earn rewards.  In addition to the points and badges that accounts keep track of for students, student accounts can also be linked to a teacher account.  The teacher/coach account that is associated with it can view statistics for each student, including frequency of use and success rates.

One drawback to Khan Academy is that it requires student access to one-to-one technology to use successfully (in my opinion).  Even if you don't have one-to-one access, you might be able to make a couple of computers with a Khan Academy activity set up one center in a set of stations that the whole class participates in.

Reflection 9: CCSSI... everyone's favorite!

I liked reading the article that showed us that there is some difference of opinion on the CCSSI, or, more accurately, that there is some difference of opinion on how much preparation the transition will take.  I thought it was interesting to note that the article says there is some push back against CCSS adoptions, but that usually this isn't coming from teachers and parents, i.e. the educational community.  Teachers and parents seem to be largely in support of Common Core, though many think they may need some more time to develop their new curriculums and resources.  Detractors seem to be from camps that are concerned about 'looking bad' -- since CCSS will (ideally) reduce ratings inflation among schools (not every school can be 'above average'!).

One of the most interesting things mentioned in the article is some speculation on how students will react to this transition and how they will initially perform.  There is some fear that students will have difficulty with the new, rigorous, in-depth approaches to material presented under CCSS after having been acclimated to the drill-and-kill approach to material under NCLB.  I  can understand this fear, as I see it in my students every day.  The transition has not been kind to these children; being constantly asked 'why?' and 'explain' is totally foreign to these kids.  They're used to being able to pop out a numerical answer and then move on with their lives.  It's a shame that a there will be whole grade levels of students who are thrown under the bus b y this transition, because I think that in the end, after we're fully ensconced in CC, it'll be work it.

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.