Text Set #2: Trigonometric Functions and the Unit Circle

Books

Trigonometry Smarts!:

by Rebecca Wingard-Nelson

Age: 14 and up

This self-help, step-by-step book helps students learn and practice the trigonometric aspects of triangles in the unit circle in order to understand how to use sine and cosine. This book breaks down these concepts and many more into easy to follow steps, lessons, and examples for students in order for them to really learn how trigonometry works and how to feel confident when using these concepts.

Websites

Mathisfun.com: Introduction to Trigonometry:

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

Age: 14 and up

This helpful website is very user-friendly which allows students to get a basic understanding of how trigonometric functions work, what they represent, and how to compute with them. There are many easy to read examples that guide students through the steps, and there is even an interactive unit circle so students can get a visual representation of the circle and how the trigonometric functions work.This is a great site for students to use outside of the classroom, as well.

The Inverse Trigonometric Functions:

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

Age: 15-16 and up

Where there are trigonometric functions, there are their inverses. The website above shows the relationship between the trig function and its inverse, and it walks the reader through the background information, how to find them using a calculator, and how to solve with them in a written problem. It is a very supportive site that the reader can follow along with, perform problems, and read helpful hints and notes when it comes to using inverses trigonometric functions in triangles.

Trigonometric Functions:

http://www.intmath.com/trigonometric-functions/trig-functions-intro.php

Ages: 14 and up

In the opening page of this site, the title is what catches the reader. The title states, "Why learn trigonometric functions..?" I think this is a great header because many students see themselves asking this question many times, too. With this site, students read the opening page, and from there, they can explore the different aspects of trigonometric functions within this chapter of text. They have the freedom to chose where they want to go and what they want to discover, and the articles are links posted below the first page of text. These sub-articles also have great hooks to catch the reader's eye! 

The Six Trigonometric Functions and Reciprocal Functions:

http://www.regentsprep.org/regents/math/algtrig/ATT1/trigsix.htm

Ages: 14 and up 

The trig functions,along with their reciprocals, are laid out in an opening table that is great for students to read and make flashcards of if they wish. It is very spelled out for them to understand each function's parts and the parts of their reciprocal. Between the two sets, they can read and discover the relationship that they have. There are many charts that might seem overwhelming, but there are many examples for the students to work on and see how to solve problems with the functions and their reciprocals. The examples are very detailed and help students see the set up and the solution step-by-step. At the bottom, students can see how to use their calculator in order to solve the problems.  

Trigonometry Facts: Exact values of the Trigonometric Functions:

http://mathmistakes.info/facts/TrigFacts/learn/vals/sum.html

Ages: 16 and up (Students in Pre-Calculus)

With this site, students can test themselves on memorizing the values of the trigonometric functions on the unit circle where the angle and radian is presented.  It is very similar to flash cards which can be very helpful for students! The user can click the link in the box to show or hide the answer to the trig function, and they can memorize all of them! This is really useful for a pre-calculus student because they are working with the unit circle, the functions, and the values on a heavy and regular basis.

Basic Trigonometric Ratios:

http://www.purplemath.com/modules/basirati.htm

Ages: 13 and up

This is an interesting website because at the top of the page, the user is presented with two options. You can start reading the article below, or you can watch a tutorial on how to solve trigonometric ratios. I think this is neat because some students would rather read first than watch a short clip, and viseversa. The reading is very straight forward, and it is very thorough with the material. The reader discovers how trig functions came about, what they mean, how to memorize, and what the theta stands for in the function. The tables are helpful in keeping the information straight, but students might not use it. The video is also a great interactive that helps explain these same concepts.

Articles

The Tale of Unit Circle City

http://www.pixton.com/comic/11g197oo



Ages: 15 and up

Nothing beats a classic comic strip! This story of Sine and Cosine man making sure that the city of Unit Circle remains out of harms way is a great read! This comic presents trigonometric math in a very fun and exciting way that appeals to non-math students. It does include a lot of other mathematical functions such as logs, but it is still a fun read that gets students excited about reading math while giving them mathematical information.

Sine- Trigonometry for Kids!:

http://scienceforkids.kidipede.com/math/trigonometry/sine.htm

Ages: 12 and up

This article presents the use of sine, sine graph, and solving for sine in triangles in a very up-beat and engaging fashion. This site presents many topics and concepts that shows their history and how it is relevant today. There are many visuals for students to read and analyze, and the article makes excellent connections to angles and students daily lives. There is also a page for cosine and tangent for students to explore!

Trigonometry Basics Study Guide:

http://www.education.com/study-help/article/trigonometry-basics/

Ages: 14 and up

A quick and simple study guide is a great resource for students to access in any subject, but especially in math. This article gives the reader an overall lesson and refresher of trigonometric functions while helping students understand the basics of them, as well. Example problems are given and students are able to follow along or complete on their own. It is a lot to practice for students to complete, but it does cover all types of questions that they could possibly see. 

The Trig Functions- Overview: 


http://math2.org/math/algebra/functions/trig/

Ages: 14 and up

The article in this link is very straight-forward, and it is in its simplest form an overview of trigonometric functions. It also breaks down what trigonometric functions mean, how to set them up correctly, the relationships, and a written out way of how to piece together these complex functions. There is a quite a bit of reading in this article, but if there are some students who like reading, this should be a great source for them!

The Unit Circle: 

http://suite101.com/a/the-unit-circle-a64437

Ages: 15 and up

This article talks about the unit circle in its entirety, and it highlights the main points, no pun intended, of the unit circle. It also talks about how the unit circle helps us with trigonometry and when we are dealing with any right triangle. Graphing the unit circle is also a key point that this article highlights, and I think this can be useful! We don't always have time to talk about graphing the unit circle necessarily. Students are just given the unit circle and not given the time to explore with the graph. This article is easy to read, and it is enlightening, as well.

Interactives

Earth Math: 

http://earthmath.kennesaw.edu/main_site/review_topics/trig_functions.htm

Ages: 14 and up

This website is an interactive site that allows the audience to use and make connections with the unit circle and trigonometric functions. It first shows how angles in standard positions are formed in the x-y coordinate plane. Once understanding this concept, the article shows how the unit circle is derived with a simple equation. Finally, the site talks about terminal points on the unit circle, and students can click a link to participate in the interactive for terminal points on the unit circle. This interactive does have a restriction. In order to use the interactive, the user must have the updated Java software.

Inverse Sine, Cosine, and Tangent: 

http://www.mathwarehouse.com/trigonometry/inverse-sine-cosine-tangent/

Ages:  14 and up/ Teachers

Trigonometric functions are very useful, and so are their inverses. This website shows how finding the inverse compares to using the interior angle sum theorem, and it demonstrates how to use both methods. This relationship can be very helpful to see, and they do a great job at showing it. It also shows videos on how to input inverses into a calculator. There are interactive example problems for students to compute, and the site provides links to an online calculator and worksheet to practice more problems.

Degrees and Radians: 

http://www.mathinary.com/degrees_radians.jsp

Ages: 16 and up: Pre-Calculus students:

One important concept with trigonometric functions and the unit circle is converting between degrees and radians. Both of these measures are found on the unit circle, and there are ways to find them! With this site, the reader can briefly read  what each, degrees and radians, mean and how to convert either way. At the bottom, the user can use the interactive calculator to determine how to convert from degrees to radians and radians to degrees. This site is very straight forward with the reading and concepts, but it is a very good sight for those who struggle with heavy reading.

Reflection #10 Using Technology and Creditable Resources

At some point in all of our educational experiences, we all have had to analyze online resources. There are many resources out there on the world wide web that are very useful and professional in order to help people support any idea or topic that the need data or information for. However, there are also many websites that have reading material on them, but they are in no way, shape, or form useful for research purposes. This is why students are strongly discouraged from using Wikipedia. Obviously, this site is considered online reading material that is publicly well-known, but would students use this site as a creditable resource when it comes to research, writing a paper, or giving a presentation? Absolutely not. This site is not creditable enough for students to use this site, and it is important that students know how to decide which site is acceptable to use for research and what is not. In Bean, Baldwin, and Readence's Chapter 10, they offer two strategies for enhancing online reading comprehensions: SAND and ISSDAT. SAND, "helps students avoid retrieving misinformation from the Internet when they are searching for information." They must search, analyze, and note details of the sites they look up. The second strategy, ISSDAT, " helps students make intertextual connections across diverse websites when searching for a topic to be developed into a presentation, paper, or project." Here, the students must internet search, slow down, and think. After reading both of these strategies in this chapter, I think that they could come in handy when teaching the students how to conduct useful research no matter what field you are using research. At the college level, students are required to know how to research topics, and I think it is valuable to teach students these concepts in all aspects and domains during their high school career. I wouldn't particularly use this on a regular basis in my class, but I think that I could definitely mention it often.

Reflection #9: Common Core Flashpoint

Make way! Common Core State Standards are coming through! Yes, they are here to stay. Well, at least for right now. With Common Core making a stance into our nation, the change from NCLB (No Child Left Behind) to the CCSS is a shift that is not necessarily adopted by all in a loving fashion. In the article, CCSS Implementation Becomes a Flashpoint for State and Federal Action, Richard Long talks about how CCSS are viewed through the public’s eye, the educators’ eyes, and the policy-holder or political leaders’ eyes. The views are torn, and I can see why. Many people see that CCSS will help improve the education of our students, make them more successful for after high school in either college or the work force, and parents are in support of having more successful and educated students. However, I see the other perspective of this double-sided coin. As a new teacher myself, I am having a harder time finding Common Core tasks that I can implement on a regular basis (twice a week, every other day, etc.) One part of the article that I noted was that CCSS might have had some bad timing. Long states, “This all comes at a time when class sizes are expected to increase, the number of child living in poverty are rising, and there are fewer dollars for professional development, curriculum development, new materials and technology.” I could not agree with this more. We are expected to make these educational changes when our environmental conditions couldn’t be worse. Another excellent point that Long makes is about how students under the NCLB system will not be able to adjust accordingly to the challenges of CCSS. At the high school level, we see this all the time! Students who are already into their secondary education are having trouble converting to the demanding work of the CCSS, and the students are uncomfortable with this last minute transition. After reading this article and listening to both sides, I still see both sides as a high school teacher. I think Common Core is going to be highly beneficial for our students, but we should be implementing CCSS at a lower elementary level and continue with them as they go through the educational system. 

Web Resource #2: Quizlet.com

 This web resource that I am writing about today is for those who like the idea of flashcards and drill of certain skills, but hate taking the time to make them! Quizlet.com is a site that many teachers know fondly, and it is a great resource for students inside and outside the classroom. This site is known for many different uses and content areas, but I’m going to focus on using this site for a mathematical purpose.  For the Quizlet flashcards, children at many ages can work and use the flashcards. However, these flashcards are probably most functional for students who are age 7 and up. For learning formulas and practicing memorization of the mathematical formulas and facts, this site is very appropriate for in-school use. There are not any bad advertisements displayed, and all of the flashcards that you can search for are all school-appropriate topics. For the specific Quizlet flashcards for mathematics, there are many content area connections. There are flashcards for almost every skill in mathematics that students need to memorize or drill quickly. If the user clicks on “start flashcards,” the flashcards appear one at a time, and the student can practice while sitting at the computer. The action of flipping the flashcards and reviewing the mathematical connections is a great tool for students to use. One con to this resource is that the only interactive aspect of this site is that the students get to click on the flashcard to flip it. Thus, there is some small interactive aspect, but there is not a large interactive aspect to use in the classroom that gets everyone involved for a long period of time. Luckily, to use the flashcards on Quizlet.com, you do not have to have a log in and password. You can use the search bar to find flashcards about a certain mathematical or non-mathematical topic to help review facts for students of any age!

Text Set #1: Surface Area and Volume of Three-Dimensional Figures

General Resources for Surface Area and Volume of Shapes

Books:

• Pack It Up: Surface Area and Volume

Chloe Lane

Ages: 12 and up

While the student is reading this book, he/she follows a student through the process of moving, from packing boxes to decorating a new bedroom and everywhere in between. Students also calculate the surface area and volume of rectangular prisms and cylinders using nets during this book. I like this scenario because it is something that people do almost once in their life, and it puts it into a great perspective of answering the question, “When will I ever use this?” I’m not sure if the students will do the calculations during the reading, but they have that option. 

• Ancient Mathematics

Serafina Cuomo

Ages: 16 and up

This book consists of the up-most important mathematics theorems and discoveries we know of today: The theorem of Pythagoras, Euclid's "Elements", Archimedes' method to find the volume of a sphere. This book talks about the ancient mathematics and how they were thought about and derived during  the fifth century BC to the sixth century AD, with the focus on the Mediterranean region. I think this is great foundation for students to learn about with mathematics, and it appeals to the history buffs in our classes. I do not particularly like how it reaches to other ancient mathematics that I might not cover in my math classes, but this could be a blessing as well while seeing how math played into the daily lives and issues during this time.

• Finding 3-D Shapes in New York City

Julia Wall

Ages: 8 and up

Three dimensional shapes are everywhere, and this book helps students realize that we see and use them every day. The pictures and situations of skyscrapers, statues, and trees in Central Park are kid-friendly, real-life situations that show readers how they can put math to work in their day-to-day lives. A variety of problem-solving activities and graphic organizers make these books ideal for learners and readers of all ages and levels. I wish it would have done a non-specific city so students would not be unfamiliar with New York City, but it is still a fun resource to relate to lower-level readers. 

Websites

•  Surface Area Formulas:

http://www.math.com/tables/geometry/surfareas.htm

Ages: 13 and up

Every surface are formula for all three dimensional shapes are given in individual boxes as the students read about what surface area is, what the variables stand for, and gives the reader helpful hints throughout the passage. It isn't just the formulas on a page. This text takes the reader through the material and how each formula is derived. This provides further understanding and helps students outside the classroom lesson.

• How to find Surface Area:

http://www.ducksters.com/kidsmath/surface_area.php
Ages: 12 and up

This site seems targeted toward younger students, but it actually does a very detailed and thorough job at describing what exactly surface area is and gives students various examples on how to compute surface area. It is easy to read, and it does an excellent job at communicating how to solve these problems step-by-step.

• Maths Dictionary for Kids:

http://www.amathsdictionaryforkids.com/dictionary.html

Ages: 7 and up

 

Dictionaries can be overwhelming, but this online dictionary is very kid-friendly and easy to read. On the left hand side, they can click which letter the word they are looking for starts with, and they can find their word in a word bank from there. Clicking on the word brings up the definition, and the student has the option of being more specific with the term, see examples, or use it in a problem.  With all of the three-dimensional shapes, students can see how the shape's surface area and volume are computed. This teaches them how to use and be comfortable when it comes to using references and resources, looking up words, and having confidence utilizing them. 

 

• Surface Area and Volume Flashcards:

http://quizlet.com/21623119/new

Ages: 13 and up

 

Students can go to this website and view the flashcards that are provided. The cards also have lateral area which is great. The cards are for students to practice memorizing and relating which formula goes with what shape. Students can also use this site to create their own flashcards if they need a quick resource. However, students will need to be familiar with the variables and what they represent in the equation.  

 

• Intro to Cell Biology: Surface Area and Volume Relationships:

http://suite101.com/a/intro-to-cell-biology-surface-area-and-volume-relationships-a359072

Ages: 14 and up; Teachers

 

This article relates the mathematical function of surface area and volume to cells of all living organisms and their overall function. Students want/need to see how this relates outside of shapes, and this is a great example of that. It also gives those students who are interested in medical field, the science of the body, biology, or any other science functions something to enjoy. 

 

• Volume and Surface Area| Geometrical Relationship?

http://mathsisinteresting.blogspot.com/2008/08/volume-and-surface-area-geometrical.html

Ages: 12 and up

Teachers usually get this question when they teach surface area and volume. As a teacher, I don't want to crush their interest or thinking by giving them the answer. Thus, this blog post presents the answer to this question in a fun way while breaking down how the different diagrams can show that the relationship between surface area and volume can be mathematically displayed instead of just students saying, "I think they are...?" It isn't a long post, but it does justice. 

 

• How Much Water is on Earth? 

http://ga.water.usgs.gov/edu/earthhowmuch.html

Ages: 14 and up

 

After learning about surface area and volume of a sphere, lets put this into a real-world perspective. This article  informs about the amount of water the Earth holds. It refers this number to volume and the amount of area on Earth the water takes up. Students can learn a lot from this article about our environment, and it puts these mathematical computations into an interesting point of view.

• 3D Objects (Prisms, Cylinders, Cones, Spheres):

http://www.shmoop.com/basic-geometry/three-d-prisms-cylinders-cones-spheres.html

Ages: 12 and up 

 

Without a good understanding of the 3-Dimensional shapes, students have a hard time knowing how to compute surface area and volume. This site gives students clear definitions of the shapes, how they got the name,the sides, faces, etc. After reviewing the definitions, they can go on and look at surface area and volume definitions, as well. It is laid out very well, and students will be more confident during classroom conversations.

• Suface Area Zine: 

http://surfaceareazine.com/main/?page_id=261

Ages: 14 and up

This is an art magazine that features graphics artists, and it displays their pieces of self-expression. In some of the articles, it encourages students to draw their own piece of art that meets the surface area dimensions given.

Applications/Interactive Texts 

• Interactives: Geometry 3-D Shapes 

http://www.learner.org/interactives/geometry/area.html

Ages: 12 and up

From this home page, students can read more information about three-dimensional shapes. Once they feel comfortable with the basic shapes, they have many options to click on that are links in the tool bar. They can look at surface area and volume of the individual shapes, look more at 3-D shapes, visit and learn about Euler's Theorem, and then test their skills. This site gives great text for the students to read, but they might not need to click on all of this information. They do not need to get more confused, but it can also be looked for those students who look for enrichment. This site really breaks down the information on a level that relates to the middle or high school students, and the test your skills is a great interactive to have at the end.

• Measuring a Solid

http://msms.ehe.osu.edu/tag/volume-and-surface-area/

Ages: Teachers

This article from the Ohio State University's College of Education and Human Ecology shares how teachers can use and present how to use surface area and volume in various ways and hands on activities. They are very appealing to students, and they can help students see how these formulas work instead of memorizing them. Some activities are applets, and those might be a little foreign for some teachers if they do not understand what an applet is and how it works. 

• Geometry Nets

http://www.kidzone.ws/math/geometry/nets/

Ages: 12 and up; Teacher Resource

Nets can be very abstract to understand how they work, and it is the foundation to understanding three-dimensional objects, their surface area, and their volumes. There are many different three dimensional shapes, and this site offers printable nets for students to build, and it gives a great introduction of what nets are and why we use them. The text is student friendly, but it also is talking to the teacher audience which can be confusing if the reader is a student.