**Math Topics**- Common Core
- Initiatives
- Methodology
- Resources
- Projects
- Manipulatives
- Software
**Learning Support**- Standardized Test Prep
- Technology Integration
- Assisting Readers
**Professionalism**- Associations
- Journals
- News
- Professional Development
- Education Standards
- Education Research
- CT4ME Publications

Middle, Secondary, Post-Secondary School:

Subject Specific Resources

Plus Apps for Mobile Devices

Our collection of Math Resources has multiple pages designed for teaching within specific grade bands. You'll also find valuable collections of support and enrichment resources that will benefit all learners.

Math Resources (Page 1): Elementary and Middle School Levels: Basic Mathematics and Skills Development, plus apps for mobile devices

**Math
Resources (Page 2)**:
Middle,
Secondary, Post-Secondary: Subject Specific Resources also
include apps for your mobile devices:

- Algebra and Pre-Algebra
- Geometry
- Trigonometry and Precalculus
- Calculus
- Statistics and Probability
- Multiple Subject Sites
- Resources to Spice Up Your Math Class

Math Resources (Page 3): K-12 Supplementary Collections: Miscellaneous Math Collections; Practical Applications--Math in Careers, Daily Life, and Across the Curriculum; and Problem Solving

Math Resources (Page 4): Enrichment and Extra Help: Math Contests, Competitions, Challenges, and Camps; Study Skills and Homework Help, plus tips for parent involvement; and Dictionaries, Glossaries, Reference Sheets, and Math Encyclopedias

Math Resources (Page 5): Lesson Plans and Worksheets, Design and Manage Your Own Lesson Plans, and Publishers' Textbook Support Sites

Math Resources (Page 6): Resources for Podcasts and Vodcasts and for Whiteboards, including tips on using whiteboards

**Handbook of Essential Mathematics**

Teachers: You and your students might come to value the Handbook of Essential Mathematics from the Air Force Research Laboratory (2006) at Wright Patterson Air Force Base in Dayton, Ohio. It's particularly relevant for subjects addressed on this page of resources. This free online handbook is open source and has three sections "that will well serve both students and teachers alike from early grades through early college" (p. 3). Those are described as follows:

"Section I, “Formulas”, contains most of the mathematical formulas that a person would expect to encounter through the second year of college regardless of major. In addition, there are formulas rarely seen in such compilations, included as a mathematical treat for the inquisitive. Section I also includes select mathematical processes, such as the process for solving a linear equation in one unknown, with a supporting examples. Section II, “Tables”, includes both ‘pure math’ tables and physical-science tables," including some "just to nurture curiosity in a spirit of fun." ... Section III,“Applications in Personal Finance”, applies the language of algebra to everyday finances. (p. 4)

** Are your students ready for algebra?** Your students can find out by taking
(The College of the Redwoods)
Algebra Readiness Test. While most schools provide such a test, this
is a good online test with 50 multiple choice questions, answers, and detailed
solutions. Don't be concerned that it is a 2003 version.

Watch the overviews and get materials related to the National Math Panel: Critical Foundations for Algebra, a great resource for improving mathematics instruction, especially for elementary educators and elementary mathematics specialists.

In *Foundations for Success*, the National Mathematics Advisory Panel (2008) set
an expectation for school algebra that "students to be able to proceed
successfully at least through the content of Algebra II" (p. xvii). Major
topics recommended for school algebra include:

- Symbols and Expressions: polynomial and rational expressions, arithmetic and finite geometric series.
- Linear Equations, including linear inequalities
- Quadratic Equations
- Functions: linear, quadratic, polynomial, simple non-linear, exponential, logarithmic, trigonometric, and fitting simple mathematical models to data
- Algebra of Polynomials. This study would also extend to include complex numbers and operations, and mathematical induction.
- Combinations and permutations, as applications of the binomial theorem and Pascal’s Triangle, and Finite Probability. (p. 16).

**The following algebra resources should help to achieve those recommendations.**

**HOT**: Algebra 1 is a
online course developed by SAS Curriculum Pathways, which is designed to
address Common Core Mathematics Standards for algebra, including conceptual
understanding, procedural skills, and application. Best of all, it is "free
to students and educators in traditional, virtual, and home schools."
Learners will find real-world examples, images, animations, videos, and
feedback. Lessons have four sections: the introduction, instructional
content with interactivity and guided practice, additional problems, and a
summary.

Algebra 1A (units 1-4) and Algebra 1B (units 5-7) combine for a complete course in Introductory Algebra. The multimedia lessons, interactivity, immediate feedback, and practice lessons are excellent. Materials include activities and labs, assessments, homework, simulations, and video lectures. Brought to you by the National Repository of Online Courses at Monterey Institute for Technology and Education. A single course designed similarly in Elementary Algebra (7 chapters) is also available, which begins with a review of pre-algebra concepts in chapter 1.

Algebra 1 Online from Henrico County Public Schools in Virginia is a full course described as follows: "This course contains both content that reviews or extends concepts and skills learned in previous grades and new, more abstract concepts in algebra. Students will gain proficiency in computation with rational numbers (positive and negative fractions, positive and negative decimals, whole numbers, and integers) and algebraic properties. New concepts include solving two-step equations and inequalities, graphing linear equations, simplifying algebraic expressions with exponents, i.e. monomials and polynomials, factoring, solving systems of equations, and using matrices to organize and interpret data."

Algebra 2 Online from Henrico County Public Schools in Virginia is a full course described as follows: "This course contains both content that reviews or extends concepts and skills learned in previous grades and new, more abstract concepts in algebra. Students will gain proficiency in solving linear equations, inequalities, and systems of linear equations. New concepts include solving quadratic equations and inequalities, exploring conics, investigating polynomials, and applying and using matrices to organize and interpret data. Students will also investigate exponential and logarithmic functions."

Algebra: The Supplement is by Dan Meyer, a high school math teacher who also blogs about math. This site is an entire year's worth of supplementary lessons in Algebra complete with handouts, PowerPoint demonstrations, pdf files, and keynotes.

Algebasics.com contains 16 sections of step-by-step worked out problems on algebraic concepts. Use this site with middle school students and above to introduce concepts or for test review, including SAT review. Macromedia Flash is used accompanied with audio explanations from a math teacher. Select from hundreds of problems beginning with order of operations, rules for working with signed numbers, equations, ratio/proportion, inequalities, absolute value, exponents, polynomials, factoring, rational expressions/equations, linear equations and their systems, functions, radicals, quadratics, and applications.

AlgebraLab.org focuses on topics and skills from high school mathematics that students need for introductory science courses. You'll find topics in algebra I and II, geometry, trigonometry, statistics. There are lessons, hands-on activities, an interactive glossary, study aids with interactive practice problems, technical reading passages, word problems, connections between math and science to various careers, and more. The project was developed in conjunction with a Florida Department of Education Enhancing Education Through Technology Competitive Grant, a National Council of Teachers of Mathematics' Edward G. Begle Grant for classroom-based research in precollege mathematics, and Florida's Volusia County Schools.

Advanced Algebra II: Conceptual Explanations by Kenny Felder is an entire book online. Per the summary: "It is intended for students to read on their own to refresh or clarify what they learned in class. This text is designed for use with the "Advanced Algebra II: Homework and Activities" (http://cnx.org/content/col10686/latest/) and the "Advanced Algebra II: Teacher's Guide" (http://cnx.org/content/col10687/latest/) collections to make up the entire course."

Beginning Algebra is a full online course posted at Saylor.org, which is free to use under a Creative Commons license. The course was "developed through a partnership with the Washington State Board for Community and Technical Colleges." It includes the syllabus, readings, assignments, media, assessments, and final exam and begins with a review of some pre-algebra concepts. Whether or not you use the full course for learning, the You Tube videos in the Web Media section are a huge plus for visual learners for demonstrations on how to solve beginning algebra problems. Units have application problems, depending on concepts learners are exposed to.

Complex Numbers--*An Interactive Introduction to Complex
Numbers* includes two Java applets, explanatory text, and
exercises for exploring basic principles of complex numbers (e.g., basic
arithmetic, conjugation, powers, roots). Both applets allow users to
"input complex numbers in either cartesian or exponential form and display them
in vector form." Content is posted within the Mathematical Association of America
Mathematical Sciences Digital Library Loci Resources.

Curriki Algebra 1 includes free resources grouped in relation to each of the Common Core algebra standards. Projects, videos, interactive Web 2.0 tools, and real world examples are provided.

Dave's Short Course on Complex Numbers

Education Portal has four free courses in math, which feature video
lessons from experienced educators, transcripts of those lessons,
supplementary content, quizzes, and exams: **College Algebra **
(81 video lessons)**, College Mathematics **(90 video lessons),**
Precalculus **(63 video lessons)** and Calculus **(104
video lessons). Much of the content within those is also addressed in
high school courses. Learners can potentially earn college credit.

Elementary Algebra with Applications (3rd edition) is a free online text from GetMath.com and Bernard Klein Publishing for students who have not had a formal course in algebra. The entire document can be downloaded. Quizzes, tests, and answers are included. Written by experienced math teachers.

EngageNY/Common Core Curriculum for Algebra I and Algebra II: a full-year curriculum map with lessons organized in modules.

Get the Math is a multimedia project about algebra in the
real world, which is for middle and high school learners. Begin by viewing
videos that show how professionals working in fashion, videogame design, music production,
basketball, the restaurant business, and the special effects industry use
algebraic thinking. They present challenges connected
to their jobs to two teams of teens. "At that point viewers are encouraged
to try the challenges themselves using interactive tools provided on the *GET
THE MATH* website, before returning to the video to see the teams’ solutions.
Students can further explore the same algebra concepts through additional
interactive challenges on the website" (About section). The project
is made possible by the Moody's Foundation THIRTEEN in association with WNET New
York Public Media.

Glencoe McGraw-Hill Pre-Algebra Personal Tutor contains a series of "personal tutor" videos aligned with this text, which assist learners to review concepts and the "how-to's" of problem solving. These videos could also be used with any pre-algebra text/course.

IXL Algebra problem sets address the skills that students learn in Algebra 1. As students practice online and demonstrate mastery, the level of difficulty increases. Skills are organized in categories.

KarlFisch Algebra Videos are categorized by topic and deal with writing, solving, and graphing linear and quadratic equations and inequalities; systems of equations and inequalities, exponents and radicals, polynomials and factoring, rational expressions, and probability and statistics. These can also be downloaded.

Linear Algebra applets from the Department of Mathematics, Indian Institute of Technology, in Bombay, India. The list includes multiple applets addressing topics such as vectors, matrices, determinants, linear transformations, and more.

Lure of the Labyrinth is a free online adventure game for pre-algebra learners, also suitable for middle school. Math is embedded in a story line--it' in an eerie world where monsters want world domination at any cost. The only way students can prevent that from happening is by solving the puzzles. Students progress through three sections, or "wings." Each is related to a different math strand: proportions (including fractions and ratios), variables and equations, number and operations (including geometry, order of operations and modular arithmetic). Each wing includes three puzzles, and each of the puzzles has three levels, from easy to hard. There are resources for teachers to help implement the game and also a section for families. This game is a product of the Learning Games to Go project, funded by a Star Schools grant from the U.S. Department of Education, spearheaded by Maryland Public Television.

Mathematics as a Second Language
includes courses in Arithmetic, Algebra, and Calculus in Everyday Life, which
students and teachers can use to review or learn concepts in a self-paced mode.
Materials are available in different formats: text, videos, and slide shows.
The site's mission is to "replace mathematical concepts that are difficult to
understand with one or more simpler concepts that are easy to understand and
hard to forget and to make all materials available free-of-charge to anyone who
is interested in learning math." There are two algebra courses.
**Algebra By Example **includes text and a study guide.
**The Game of Algebra** "consists of
animated Powerpoint and video presentations. Because different students will
likely use different textbooks, the algebra videos are not linked to any one
textbook but do match the slide show presentations" (Algebra Course Overview section).

Math 101: College Algebra is a free online course from the Education Portal Academy. It consists of video lessons and self-grading quizzes that prepare students to earn college credit in the U.S. These modular lessons, each about 10 minutes long, are taught by subject-matter experts and are animated and edited by a professional video-editing team. Lessons have been reviewed by community college and university level instructors. No log-in is required, so anyone studying introductory algebra can benefit from the videos.

The Math Dude, available from Montgomery Public Schools (MD), is an award winning video series for helping middle and high school learners of Algebra I. Shows can be viewed on the Web, via podcasts through iTunes or RSS feeds, or downloaded to your computer. These are a must see!

Pre-Algebra from Prentice Hall contains valuable video tutorials for algebra to accompany each of the lessons in the text. You'll also find assessments and real world applications that accompany each chapter.

Purplemath is a "not to miss" Algebra resource with lessons on preliminary, beginning, intermediate, advanced topics, and word problems. An extensive list of Internet resources related to Algebra is also provided, including a link to online calculators. This site, developed by E. Stapel of Western International University, is award winning.

Schmoop Pre-Algebra is online and free. Content is arranged in six learning guides: basic operations, fractions and decimals, ratios and percentages, basic geometry, basic statistics and probability, and basic algebra. There are succinct explanations, diagrams, interactive examples, practice problems, and real-world applications. This site is designed to make learning fun. Content has been developed by educators and experts. Many are from Ph.D. and Masters programs at Stanford, Harvard, UC Berkeley (and other top universities). The vast majority of Schmoop's writers have taught at the high school or college levels.

The Lost Function from Advanced Training & Learning Technology provides pre-algebra game-based learning using content aligned to the Common Core standards, grades 6-8. The curriculum is comprehensive covering topics in

- Whole Numbers
- Integers
- Fractions & Decimals
- Exponents & Roots
- Rates, Ratios, & Proportions
- Equations
- Inequalities
- Graphing
- Probability & Statistics
- Geometry

An "in-game virtual math tutor monitors progress and teaches concepts when you need it most." The storyline involves a town that has lost its ability to do math and a character Pi who is the only hope to help the people re-learn their skills. The pre-algebra bundle comes with a price of about $25, with school volume discounts available.

Virtual Math Lab from West Texas A&M University concentrates on Beginning Algebra, Intermediate Algebra, and College Algebra. Each course has complete online tutorials (many with video explanations), practice problems with answers, and resources for extra help. Anyone can use these.

Virtual Nerd from Pearson Education, Inc. is an online, self-guided video tutorial site for math. There are over 1,500 video lessons covering grades 6-8 math, pre-Algebra, Algebra I, Algebra 2, and Geometry. Pearson has made these videos available for free. You'll also find alignments to Common Core, SAT Math topics, and ACT Math topics.

Wolfram Education Portal is a free website with math teaching tools and materials, organized by course. You'll find a dynamic interactive textbook, lesson plans aligned to the Common Core Standards, widgets, interactive demonstrations, videos, and more. Courses include algebra and calculus.

WOWmath.org features hundreds of videos developed by math teacher, Brad Robb, for algebra 1 and algebra 2. They correlate with 2008 Holt - Algebra 1 California Edition; and 2008 Prentice Hall - Algebra 2 California Edition.

Alge-Bingo for iPhone or iPad is a game "for Pre-Algebra and Algebra I students who are just learning how to solve equations as well as for Algebra II and even Precalculus students needing to review their equation solving skills." It has "13 different levels and it will allow you to choose from a variety of Bingo games from straight-line Bingo through multiple patterns of Bingo all the way up to Black-out Bingo if you choose. The game will keep track of your time and points to be able to compete with others as well as yourself." (Description section). There's a small fee for this app.

Algebra Genie by ZeGenie is a free app for iPhone or iPad use. It includes over "250 interactive and dynamic lessons, covering all the important Algebra topics." There are 14 units in the interactive textbook: Algebraic Expressions, Exponents, Linear Relations, The Pythagorean Theorem, Function Basics, Functions, Quadratic Functions, Absolute Function, Square Root Function, Step Functions, Exponentials & Logarithms, Factoring, Systems of Equations, and Conics (Description section).

Algebra Pro by Larry Feldman for iPhone, iPod Touch, and iPad has solvers, tutorials, and examples for topics found in algebra 1 and algebra 2. It is available for $0.99 in iTunes. Per its description, topics include: classifying numbers, points and lines, word problems, inequalities, direct and inverse variation, polynomials, zeros and end behavior, synthetic division, remainder theorem, factoring, domains and ranges, compositions of functions, inverse functions, piecewise functions, quadratic functions, quadratic formula, logarithms, logarithmic and exponential functions, rational functions, compound interest, graph symmetry, conic sections, arithmetic and geometric sequences, matrices, binomial expansion, complex numbers, rectangular and polar notation/conversions. Note: Developer Larry Feldman also has apps for other math subjects, including geometry, pre-calculus, calculus, trigonometry, probability, statistics, and so on.

Algebra Touch by Regular Berry Software LLC for iPhone, iPod Touch, and iPad ($2.99) offers users touch-based techniques to help them refresh algebra skills. It covers the following: "Simplification, Like Terms, Commutativity, Order of Operations, Factorization, Prime Numbers, Elimination, Isolation, Variables, Basic Equations, Distribution, Factoring Out, Exponent Product Rule, and Exponent Quotient Rule."

Algebra 1 Booster and Algebra 2 Booster from Homeworkhelp.com are compatible with iPad, iPhone, and iPod Touch and is available for a few dollars each in iTunes. Homeworkhelp produces several boosters. Also see Pre-Algebra Boosters: Part 1 and Part 2 and Probability and Statistics.

HMH Fuse contains Apple iPad technology. Houghton Mifflin Harcourt has developed Common Core editions of algebra 1 and algebra 2 exclusively for a touchscreen mobile device. According to HMH, the "mobile platform combines direct instruction, ongoing support, assessment, and intervention—enabling teachers to personalize instruction" (HMH Fuse Overview section). HMH Fuse: Algebra I and HMH Fuse: Algebra 2 apps are available in iTunes for free.

Khan Academy video apps by Ximarc Studios are free apps compatible with iPhone, iPad, and iPod touch.

- Khan Academy: PreAlgebra
- Khan Academy: Algebra 1 includes videos 1-36.
- Khan Academy: Algebra 2 includes videos 37-72.

Mathomatic from Gotow.net is compatible with iPad, iPhone, and iPod Touch. For a small fee ($1.99), you can solve a range of problems from middle school math through higher level calculus. For algebra, solve systems of algebraic equations showing steps, factor and simplify expressions, and work with complex numbers.

MathTerms is a glossary featuring almost 1,000 entries relevant to middle school math, Algebra I, Algebra II, Geometry, Precalculus, Calculus, and Statistics. The entries feature colorful, annotated illustrations along with complete, sentence-based definitions in both English and Spanish. This glossary was adapted from the Dana Center/Agile Mind online course resources, and is a free download for iPad available in iTunes (Description section).

Math Pentagon has a series of apps for iPad, iPhone, and iPod Touch for middle and high school learners, available in iTunes for a fee. Students can track scores and progress: Pre-Algebra!!! (also includes honors pre-algebra) and Algebra Pro!!! (includes Algebra 1 and 2).

Ventura Educational Systems iPad apps for algebra:

- Algebra Concepts
- Simultaneous Equations

Virtual Nerd Mobile Math by Pearson Education, Inc.

Platform: iPhone and iPad

Price: Free in iTunes

Target: Grades 6-12

Info: This app provides interactive tutorial videos aligned to Common Core and
other standards. Select videos by topic
or standard to review fundamental math concepts in middle school and high school
(pre-Algebra, Algebra 1, Algebra 2, Geometry). You can also search
by keyword and set video favorites. Users of one of Pearson's most
recent math programs (e.g., Pearson High School Mathematics Common Core
2015, Integrated High School Mathematics Common Core 2014, digits 2015) can
also search by the program's table of contents.

Wolfram Algebra Course Assistant can be used with your iPhone or iPad. It is available in iTunes for a couple of dollars. Per the description in iTunes: This app covers topics in Algebra I, Algebra II, and College Algebra. It can:

- Evaluate any numeric expression or substitute a value for a variable.
- Simplify fractions, square roots, or any other expression.
- Solve a simple equation or a system of equations for specific variables.
- Plot basic, parametric, or polar plots of the function(s) of your choice.
- Expand any polynomial. Factor numeric expressions, polynomials, and symbolic expressions.
- Divide any two expressions.
- Find the partial fraction decomposition of rational expressions."

Wolfram Pre-Algebra Course Assistant App assists with many topics covered in pre-algebra courses, including solving equations, simplifying expressions, and plotting equations on the coordinate plane. Powered by Wolfram|Alpha, the Pre-Algebra Course Assistant is optimized for iPhone, iPod touch, and iPad, and is available as a universal app in iTunes.

**Join a social network for teaching Algebra**

The Algebra 4 All (A4A) Social Network, sponsored by Michigan Learnport, is a community of educators committed to sharing resources and supporting one another in the practice of teaching Algebra.

**Do you need End of Course Exams for Algebra
1 and Algebra II?**

Some states require four years of high school math, including Algebra II or its equivalent. Ohio is among those beginning with the graduating class of 2014 (ODE Gradution Requirements section). Achieve's American Diploma Project Assessment Consortium has posted practice tests and released test items from the recent end of course exams for Algebra I and Algebra II, respectively. These would be test preparation assets to teachers who give end of course exams in these subjects.

Get your students moving with a little function aerobics. View this video for how the idea works "to kinestheticallly reinforce function shapes and simple translations" (Video description).

**Do you need Common Core
Performance Tasks for Math? Or Apps?**

The Mathematics Assessment Project has both summative tests or tasks and formative assessment lessons. Tasks available are for high school and middle school, also linked to specific Common Core standards. Tasks are grouped into novice, apprentice, and expert levels and "variously ask students to use their mathematics in routine or non-routine situations to design, plan, estimate, evaluate and recommend, review and critique, investigate, re-present information, explain, define concepts, and show their skills in routine technical exercises."

Common Core Quest by OpenEd

Platform: iPad and iPhone

Price: Free

Target: grades 6-12

Info: This app contains SBAC and PARCC
style questions within quizzes covering Common Core math and language arts standards for
grades 6 and up with some coverage of the elementary standards.

** Are your students ready for a formal course in geometry?** While they may
be in your geometry class, consider that their levels of geometric
thinking might not be sufficient to engage in formal proofs.

View the Mind Map of the van Hiele Model of Geometric Thought at GoGeometry.com. The Van Hiele model asserts that learners develop geometric thought sequentially through five levels of understanding. Numbered as levels 0-4, they are visualization, analysis, informal deduction, deduction, and rigor. Hopefully, students have progressed to the level of informal deduction by the end of grade 8. Read more on The van Hiele Levels of Geometric Understanding by Marquerite Mason for further implications for teaching.

**The
following resources should help your learners address geometry standards:**

Curriki Geometry uses a project-based approach to teach geometry. There are six Common Core State Standards (CCSS)- aligned projects from which to choose. They can be taught in any order. Each focuses on two or more of the eight CCSS mathematical practices and includes technology and Web 2.0 resources. Quizzes, exam suggestions, and project assessment rubrics are provided. Titles are:

- Selling Geometry
- Designing a Winner
- What’s Your Angle, Pythagoras?
- TED Talk: House of the Future
- The Art of Triangles
- How Random is My Life?

EngageNY/Common Core Curriculum for Geometry: a full-year curriculum map with lessons organized in modules.

Floor Planner can be used with learners in grades 6-12 to interactively design floor plans in 2-D and 3-D using point-and-click, drag-and-drop tools, and then share them with others. Consider scale drawings for studying geometric shapes, spatial relationships, and ratio/proportions, for example. Students can also show their creativity, and move elements of their design (e.g. furnishings) around as often as they like. Its free for personal use.

Geometry from the Saylor Foundation is a free math course aligned to the CCSS. Other free K-12 math courses include algebra, advanced calculus (AB level), and advanced statistics, which are also aligned to the CCSS. Saylor Foundation holds the copyright to its course outlines, but has "made them available under a Creative Commons Attribution license (http://www.creativecommons.org/licenses/by/3.0). As such, you are free to download, modify, and reproduce [the] course outlines for any purpose, as long as you provide credit (attribution) to the Saylor Foundation for the source material" (FAQ section).

GeoGebra is free, open-source, dynamic mathematics software that joins geometry, algebra, tables, graphing, statistics, and calculus. You can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards. You can enter equations and coordinates directly. GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum. It received several international awards, including the European and German educational software awards. In Dynamic Mathematics with GeoGebra, authors Markus Hohenwarter, principle developer of the software, and Judith Preiner (2007) said, "You can use GeoGebra both as a teaching tool and to create interactive web pages for students from middle school up to college level. Specifically designed for educational purposes, GeoGebra can help you to foster experimental, problem-oriented and discovery learning of mathematics" (Abstract section). They illustrate the basic ideas of the software and some of its versatile possibilities by discussing several interactive examples. Additional resources for using GeoGebra include:

- GeoGebraWiki--Educators can post their creations and view the work of others. You can also share your materials at Geogebra Tube.
- Exploring Geometric Transformations in a Dynamic Environment, posted within the Mathematical Association of America Mathematical Sciences Digital Library Loci Resources. You'll find eight activities and virtual manipulatives related to teaching reflections (also including reflections with parallel lines, intersecting lines, and concurrent lines), rotations, translations, and glide-reflections to middle and high school learners.

Geometry 3-D Shapes
is one of the **Interactives** from Annenberg Media's Learner.org. Students
can interactively learn about three-dimensional shapes, calculate surface area
and volume, and discover some of the mathematical properties of shapes. It comes
with tutorials and an interactive test.

Geometry Gallery from Saltire Software contains some excellent Java applets, which are easy to use and suitable for illustrating several theorems presented in a high school or post-secondary geometry course. These relate to angles in parallel lines, circular configurations, Pythagorean triples and incircles. Students can drag points to explore concepts. Four triangle calculators are presented. By entering SSS, SAS, or ASA, students can see the unique triangle that is formed, but if SSA is entered the resulting triangle is not unique (two are possible) or no triangle is possibly formed with given information. Some advanced geometrical configurations are explored. There is a geometry atlas of algebraic formulas with associated visuals. Saltire's work has been assisted with a number of National Science Foundation grants.

Geometry: The Supplement is by Dan Meyer, a high school math teacher who also blogs about math. This site is an entire year's worth of supplementary lessons in Geometry complete with handouts, PowerPoint demonstrations, pdf files, and keynotes.

Geometry Playground is a free Java applet from the Mathematical Association of America Mathematical Sciences Digital Library Loci Resources, which enables "ruler and compass" constructions in "Euclidean, Spherical, Projective, Hyperbolic, Manhattan and Conical geometries" (Description section). Potential classroom activities are included.

GoGeometry from the Land of the Incas, by Antonio Gutierrez, is an award-winning site that provides an eclectic mix of sound, science, and Incan history intended to interest students in Euclidean geometry. The site includes geometry problems, proofs, quizzes, puzzles, quotations, scientific speculation, and more. Best used with high school students and above.

Math Open Reference is a free geometry textbook with an introduction to trigonometry for high school learners by John Page. There are topics in plane geometry, coordinate geometry, and solid geometry (e.g., cylinders, cubes). Its focus is on using interactive materials and animations to develop concepts. Sections on constructions are excellent.

Poly is a shareware program for exploring and constructing polyhedra. You can manipulate solids on the computer and also print out the net of the polyhedra so that you can create the physical 3-D model. Poly is downloaded from Pedagoguery Software.

Virtual Nerd from Pearson Education, Inc. is an online, self-guided video tutorial site for math. There are over 1,500 video lessons covering grades 6-8 math, pre-Algebra, Algebra I, Algebra 2, and Geometry. Pearson has made these videos available for free. You'll also find alignments to Common Core, SAT Math topics, and ACT Math topics.

39-in-1 Geometry and Trigonometry Calculator by Charles Vu includes 33 solvers (actual calculators, not formulas) on topics in geometry and 6 solvers for trigonometry (radians to degrees, degrees to radians, law of cosines, law of sines, trig functions and inverses. It's available in iTunes for $0.99 and is designed for iPad, iPhone, and iPod Touch.

Apollonius is "Interactive Geometry Software for the iPhone and iPod Touch. It allows you to make geometric constructions (such as those made using a compass and straightedge/ruler) and move their parts smoothly using the device's touchscreen" (Description section). There is a small fee for the app; however, there is no additional fee for upgrades when they occur. It is also compatible with iPad.

Easy Geometry for the iPad is available in iTunes for a couple of dollars. With this app, developed by JMS Solutions, you can "explore the basic family of geometry shapes from the closed plane curves, quadrilaterals, triangles, polygons and geometric volumes. Learn the basic equations that describe each geometric shape as well as each shapes properties and interesting facts." The "Interactive Geometry Calculator allows you to choose the parameters used to solve the basic geometric shape" and explore the boundaries of each (Description section).

Euclid-Geometric Constructions by Gaussian Distribution is compatible with iPhone, iPod touch, and iPad. For just under a dollar, the app "introduces you to the world of geometric constructions in the form of a puzzle game. Starting from primitive tools, you'll learn to construct increasingly hard (and beautiful) patterns: bisectors, duplicators, regular polygons, lunes, etc. Solve puzzles to unlock more tools to aid in the construction of later puzzles. Free draw mode allows you to experiment with your own geometric patterns" (Description section).

Ezy Triangle by KGiants is for iPad and iPhone. This free app for high school learners and above is available in iTunes. It is designed to solve triangles given any 3 out of 6 quantities (sides and angles). It can handle an ambiguous case of multiple solutions and allows interacting with the triangle. It displays side lengths of the triangle, has auto-saving of the triangle and comes with a list of triangle-related formulas.

Geometry: Circumference and Arc Length requires Windows. It is a free interactive "Vbook" (e-book) app from Adaptive Curriculum that blends "a variety of instructional activities, from explorations and visual explanations to hands-on, web-based exercises." Learners are engaged with simulations. The app is aligned with Common Core standards.

Geometry Stash for iPad and iPhone is a quick reference guide for "commonly used theorems, postulates, and corollaries" with "descriptive, visual diagrams explaining each item" (Description section). It's available for just under two dollars in iTunes.

Geometry Pad by ZonMobile is a free dynamic geometry iPad app. "With the Geometry Pad you can create fundamental geometric shapes, explore and change their properties and calculate metrics. The shapes are displayed on a scrollable and zoomable workbook with a rectangular coordinate system" (Description section). Multiple tools are built into the application: move & scale, point, line, midpoint; parallel, perpendicular, tangent lines; angle, triangle, triangle lines, quadrilaterals, circles, radii, chords, polygons, arcs, sectors, conic sections, measurements and transformations tools, manual inputs, calculator functions, and more.

HMH Fuse contains Apple iPad technology. Houghton Mifflin Harcourt has developed a Common Core edition of geometry exclusively for a touchscreen mobile device. According to HMH, the "mobile platform combines direct instruction, ongoing support, assessment, and intervention—enabling teachers to personalize instruction" (HMH Fuse Overview section). HMH Fuse: Geometry is available for free in iTunes.

Khan Academy: Geometry videos by Ximarc Studios is a free app compatible with iPhone, iPad, and iPod touch.

Math Pentagon has a series of apps for iPad, iPhone, and iPod Touch for middle and high school learners, available in iTunes for a fee. Students can track scores and progress. See the full course for Geometry Pro!!!.

MathTerms is a glossary featuring almost 1,000 entries relevant to middle school math, Algebra I, Algebra II, Geometry, Precalculus, Calculus, and Statistics. The entries feature colorful, annotated illustrations along with complete, sentence-based definitions in both English and Spanish. This glossary was adapted from the Dana Center/Agile Mind online course resources, and is a free download for iPad available in iTunes (Description section).

Protractor by Silverview Consulting is compatible with iPhone, iPad, and iPod Touch. For a small fee ($.99), you can then find the number of degrees in angles using images on the Web or photos you've captured. Think of taking a picture of a roof, for example, and finding angles on it, or measuring the lean of an object.

Sketchpad Explorer is a free app available in iTunes for use on the iPad. "Drag, manipulate and animate visual mathematics to develop and generalize your understanding of fundamental concepts across elementary math, geometry, algebra, trigonometry, calculus, and beyond. Based on the award-winning software The Geometer’s Sketchpad®, the Sketchpad Explorer app allows you to interact with, and investigate, any document created in Sketchpad" (Description section).

Ventura Educational Systems iPad apps for geometry:

- Geometry Concepts
- Coordinate Geometry

Virtual Nerd Mobile Math by Pearson Education, Inc.

Platform: iPhone and iPad

Price: Free in iTunes

Target: Grades 6-12

Info: This app provides interactive tutorial videos aligned to Common Core and
other standards. Select videos by topic
or standard to review fundamental math concepts in middle school and high school
(pre-Algebra, Algebra 1, Algebra 2, Geometry). You can also search
by keyword and set video favorites. Users of one of Pearson's most
recent math programs (e.g., Pearson High School Mathematics Common Core
2015, Integrated High School Mathematics Common Core 2014, digits 2015) can
also search by the program's table of contents.

Algebra and Trigonometry (2nd edition), a Prentice Hall book by Robert Blitzer, contains 12 chapters of useful quizzes and tests in multiple choice format, which can be taken online. There are also graphing calculator tutorials and manuals for the TI-83/84 and TI-86 calculators. Learners can create profiles to email quiz/test results to the instructor.

Dave's Short Trig Course, by Dave Joyce of Clark University in Massachusetts, contains explanations, practice problems and answers. Best of all is that images are virtual manipulatives developed in Java so that learners can explore the properties presented. A brief background of the Pythagorean Theorem and similar triangles is included. See also Dave's Short Course on Complex Numbers

Demos with Positive Impact connects instructors of pre-calculus, calculus, and post-calculus courses with effective teaching tools to help them get important ideas across to students during their lectures. This project focuses on demonstrations that use some form of instructional technology (e.g., physical equipment or examples, graphical displays, simulations, computations, interactive modules, calculators, computers and computer software, the Web). Really first-rate!

Education Portal has four free courses in math, which feature video
lessons from experienced educators, transcripts of those lessons,
supplementary content, quizzes, and exams: **College Algebra **
(81 video lessons)**, College Mathematics **(90 video lessons),**
Precalculus **(63 video lessons)** and Calculus **(104
video lessons). Much of the content within those is also addressed in
high school courses. Learners can potentially earn college credit.

Exploring Precalculus, by Dr. William Mueller, provides an introduction to precalculus using Mathcad or StudyWorks as a calculation and visualization tool. Many interactive examples let you explore the concepts and practice the skills necessary for understanding calculus. Central themes (functions, rates of change, and accumulation) are developed intuitively, from several perspectives (algebraic, numeric, and graphic). Materials suitable for use in grades 9-12, and post-secondary.

FlashyTrig.com uses Flash accompanied by audio to provide an introduction to trigonometry based on the unit circle and the geometry of the right triangle. The flash animations for illustrating the development of the six trig graphs are particularly useful for classroom demonstration.

Interactive Talking Math Book is "designed to be a complete pre-calculus preparatory supplement. It is not recommended as a substitute for other scholastic math books, but as an additional learning tool that can assist all math students. Audio, animation and interactivity using web based multimedia brings the math to life. Mathematical concepts are solidified with verbal reinforcement using interactive animated presentations and audio tutorials" (Book Preface section). It was written by a technical mathematician, copyright is 2006, and published by Webovations Interactive Digital Media. There are six chapters: number, sets, basic math, functions, geometry, and trigonometry. A talking calculator is included, too.

Law of Sines and Law of Cosines are two applets that allow you to input sides and angles per each law and then each applet calculates the missing sides and angles in a triangle. Some practical applications of each law are also included. These are hosted by the Department of Physics and Astronomy at Georgia State University.

Pre-Calculus II is a full online trigonometry course posted at Saylor.org, which is free to use under a Creative Commons license. It was "developed through a partnership with the Washington State Board for Community and Technical Colleges." It includes the syllabus, readings, assignments, media, assessments, and final exam. Content addresses "families of trigonometric functions, as well as their inverses, properties, graphs, and applications. Additionally, [learners] study trigonometric equations and identities, the laws of sines and cosines, polar coordinates and graphs, parametric equations and elementary vector operations." The course features several Khan Academy videos.

The Math Page: Trigonometry addresses 20 trig topics and provides examples and problems. The section for Pre-calculus includes over 25 topics. The site is by Lawrence Spector.

Ezy Trigonometry by KGiants is for iPad and iPhone. This free app for high school learners, or others learning trigonometry, is available in iTunes and features an interactive unit circle to explore relationships between angles and trig ratios. It comes with an exact trig ratios table, a handbook of trig formulas, and settings for radians/degrees, related angles, and angle snapping.

Khan Academy video apps by Ximarc Studios are free apps compatible with iPhone, iPad, and iPod touch.

Math Pentagon has a series of apps for iPad, iPhone, and iPod Touch for middle and high school learners, available in iTunes for a fee. Students can track scores and progress. See the full course of Trigonometry Pro!!!.

MathTerms is a glossary featuring almost 1,000 entries relevant to middle school math, Algebra I, Algebra II, Geometry, Precalculus, Calculus, and Statistics. The entries feature colorful, annotated illustrations along with complete, sentence-based definitions in both English and Spanish. This glossary was adapted from the Dana Center/Agile Mind online course resources, and is a free download for iPad available in iTunes (Description section).

Trigonometry Pro HD by voi nguyen can be used on an iPad. It is available in iTunes for a couple of dollars. This "Interactive Trigonometry Calculator contains 13 Modules: Geometry, The Right Triangular Shapes, 30/60 Degrees Triangular Shapes, 45 Degrees Triangular Shapes, Oblique Triangular Shapes, Trigonometry/Geometry Equations, The Ultimate Units Converter, Animated Sin/Cos/Tan Graphs Functions, Interactive Central Angle, Interactive Unit Circle, Circle Functions, Lines, Point & Slopes Plot Functions and the bonus module is: The Arabic Numerals System."

Wolfram Precalculus Course Assistant is compatible with iPhone, iPod touch, and iPad and is available in iTunes for about two dollars. It covers topics in precalculus and trigonometry. It can:

- Evaluate any numeric expression or substitute a value for a variable
- Solve a single equation or a system of equations
- Plot functions on the x-y plane or draw a parametric or polar plot
- Determine the sine, cosine, and tangent of a specific angle in a right triangle
- Simplify, expand, or factor trigonometric functions
- Find the partial fraction decomposition of an expression
- Calculate the dot product, cross product, and magnitude of two vectors
- Identify the mean, median, mode, and standard deviation of a set of data
- Calculate permutations and combinations (Description section).

*Tired of using 3, 4, 5 right triangles?*

Generate your own sides of the right triangle. Using positive integers s and t with s< t, calculate legs x and y and hypotenuse z as follows:

Get a quick list of Pythagorean Triples using Nicholas Exner's (MSTE division of University of Illinois-Urbana-Champaign) simulation or learn more about Pythagorean Triples using several online calculators from Dr. Ron Knott of the University of Surrey in the UK.

*Are
your students ready for calculus?*

The University of California at San Diego has posted an online Calculus Readiness Test with answers. Many of the problems are challenging, but don't give up.

**Are your students ready for their AP exams?**

See resources, including released questions from prior exams, at the College Board AP Central for both the AB and BC exam levels.

University of Houston System has two online test prep courses for high school learners: Preparing for the AP Calculus AB Exam and Preparing for the AP Statistics Exam. Courses are provided for free and are offered via Coursera. Each runs for six weeks, and includes a practice test at the end of the course.

**The
following resources should help your learners address calculus standards.**

Applied Calculus: Everything! by Stefan Waner is a complete text beginning with a precalculus review, topics in finite mathematics, then study of differentiation and integration with applications. There are online tutorials, tools for exploration, review exercises, quizzes, and more.

Bates College Resources for Calculus and Linear Algebra. The value of this site lies in posted quizzes/exams/drill problems with answers for many, common errors, and additional links to resources.

Calc101 provides automatic solutions, powered by webMathematica, to derivative and integral problems found in beginning calculus. The site's goal is to help you to raise your grade by 10%. Get all the steps including the final answer. Step-by-step matrix operations and determinants are coming soon; linear equations will come later. Derivatives and polynomial multiplication and division are free. There is a small fee for integral solutions and linear algebra, but you can always see the first and last step.

Calculus (24 applets) from the Department of Mathematics, Indian Institute of Technology, in Bombay, India. These applets address major topics in calculus. The list includes, but is not limited to, the following: One-One/Onto Function, Composition of Functions, Derivative of a Function, Second Derivative, Rolles' Theorem and Mean Value Theorem, Differentiability, Vector Fields, Numerical Integration, Asymptotes, Linear Approximation, the Fundamental Theorem, and more.

Calculus (64 applets) by International Education Software helps students to learn about limits, derivatives, and integrals visually through precise definitions and a series of interactive applets.

Calculus is among math projects at the secondary education level posted at Curriki and the Global Education and Learning Community. Java is used for complete courses in AP Calculus at both the AB (Introductory Calculus 1 and 2) and BC (General Calculus 1 and 2) levels. Audio explanations accompany text equivalent and multimedia demonstrations of concepts are presented. Interactive practice problems are presented to test understanding. Feedback shows how answers are derived. This is truly worth using. Note Java modules might take a while to download.

Calculus I is a full online course posted at Saylor.org, which is free to use under a Creative Commons license. It was "developed through a partnership with the Washington State Board for Community and Technical Colleges." It includes the syllabus, readings, assignments, media, assessments, and final exam. There are five units. Unit 1 "provides a review of algebra specifically designed to help and prepare for the study of calculus." Unit 2 "discusses functions, graphs, limits, and continuity." Unit 3 delves into the concept of derivatives. "The fourth unit makes “visual sense” of derivatives by discussing derivatives and graphs. Finally, the fifth unit provides a large collection of reference facts, geometry, and trigonometry that will assist in solving calculus problems long after the course is over."

Calculus: Graphics for the Calculus Classroom by Douglas Arnold of the University of Minnesota. This site contains a series of graphical demonstrations suitable for first year calculus: a differentials and differences animation, computing the volume of water in a tipped glass, Archimedes' calculation of pi, animation of a bouncing ball, secants and tangents, zooming in on a tangent line, a trig limit, graphs for a discussion on limits, a nowhere differentiable function, introduction to the number e, and computing the volume of the region formed by two intersecting cylinders.

Calculus on the Web is partially supported by the National Science Foundation and is a project of G. Mendoza and D. Reich at Temple University. This site has interactive exercises with feedback and help sections that explain concepts. There are seven books online: Precalculus; Calculus 1, 2, and 3; Linear Algebra, Number Theory, and Abstract Algebra.

Calculus-help.com by M. Kelley has been recognized by The National Council of Teachers of Mathematics and The College Board. Don't miss Tutorials for the Calculus Phobe, an ongoing project to present topics from this subject in an easily understood audio-visual format. Flash animation is incorporated. Topics include limits, continuity, differentiation, with more to come.

Calculus: The Integrator from Wolfram Research is powered by webMathematica. Enter any expression and the integration is performed for you. Also read about the history and applications of integration.

Calculus Review by Linda Valdes of San José State University contains review sheets of key concepts followed by online quizzes for topics in limits, derivatives, and integration. Well worth using these.

Calculus Revisited, posted at the MIT OpenCourseWare initiative, features a series of videos and related resources on topics covered in an introductory calculus course.

Capsules for One-Variable Calculus is brought to you by the Mathematical Association of America within its digital library, MathDL. It is a compendium of learning capsules in different areas appropriate for anyone teaching one-variable calculus. Among topics are general tools, continuity, limits, differentiation, the Fundamental Theorem, antidifferentiation, integration, differential equations for calculus, function approximation, theoretical issues, and more. Content has been vetted by leading experts in the field of mathematics and math education.

Demos with Positive Impact connects instructors of pre-calculus, calculus, and post-calculus courses with effective teaching tools to help them get important ideas across to students during their lectures. This project focuses on demonstrations that use some form of instructional technology (e.g., physical equipment or examples, graphical displays, simulations, computations, interactive modules, calculators, computers and computer software, the Web). Really first-rate!

Education Portal has four free courses in math, which feature video
lessons from experienced educators, transcripts of those lessons,
supplementary content, quizzes, and exams: **College Algebra **
(81 video lessons)**, College Mathematics **(90 video lessons),**
Precalculus **(63 video lessons)** and Calculus **(104
video lessons). Much of the content within those is also addressed in
high school courses. Learners can potentially earn college credit.
Education Portal also has a specific collection of 125 video lessons for
AP
Calculus AB & BC.

*Elementary Calculus: An Infinitesimal Approach,
*On-line Edition, copyright 2000 by H. Jerome Keisler,
is a freely-available calculus text from the author. You can download the
whole book or selected chapters. This is one of the books selected as part
of the California Free Digital Textbook Initiative.

Mathematics as a Second Language
includes courses in Arithmetic, Algebra, and **Calculus in Everyday Life**,
which students and teachers can use to review or learn concepts in a
self-paced mode. Materials are available in different formats:
text, videos, and slide shows. The site's mission is to "replace
mathematical concepts that are difficult to understand with one or more
simpler concepts that are easy to understand and hard to forget and to
make all materials available free-of-charge to anyone who is interested
in learning math." **Calculus in Everyday Life** relates
"the basic concepts of Calculus to ideas we encounter in our lives every day.
[The authors] especially invite teachers to read these "lessons" as they will,
in turn, help you internalize how important a role basic arithmetic, algebra and geometry play
in helping students be adequately prepared to take topics in higher mathematics of
which calculus is one example. In addition, by the time you are finished
you will have a deep understanding of limits, derivatives and integrals."
There is also a second calculus course for the more traditionalist. (Calculus
Courses section)

Math 104: Calculus is a free online course from the Education Portal Academy. It consists of over 100 video lessons and self-grading quizzes that prepare students to earn college credit in the U.S. These modular lessons, each about 10 minutes long, are taught by subject-matter experts and are animated and edited by a professional video-editing team. Lessons have been reviewed by community college and university level instructors. No log-in is required, so anyone studying calculus can benefit from the videos.

MOOCulus is a MOOC (Massive Open Online Course) for Calculus One brought to you by Ohio State University. "This is a first and friendly introduction to calculus, suitable for someone who has never seen the subject before, or for someone who has seen some calculus but wants to review the concepts and practice applying those concepts to solve problems. One learns calculus by doing calculus, and so this course is based around practice problems" (Homepage intro section). The course is set up as an adaptive learning platform, created by Jim Fowler of OSU and his team, and provides 15 weeks of content. Video lectures, a downloadable textbook, online exercises, explorations, and a forum for discussion are provided. Read reactions to this MOOC in Learning to Love Math: How One Teacher is Using MOOCs to Transform Instruction, Inspire Passion in Students (Education Week, September 21, 2013).

Pre-Calculus and Calculus Online Tutorials from Harvey Mudd College in California are extensive and great for student review of concepts.

Video Calculus is a collection of over 50 lectures provided in two video formats that were developed by Selwyn Hollis of the Mathematics Department at the University of Houston. Videos cover a full range of topics for calculus I and calculus II.

Visual Calculus from the University of Tennessee Math Department at Knoxville is an excellent resource that includes tutorials, drills and programs for pre-calculus and calculus. Animations, interactive pages, step-by-step solutions and illustrations are included for topics ranging from functions, graphing with technology, solving equations, and conic sections to limits and continuity, derivatives and applications, integration and applications, sequences and series. As a veteran calculus teacher, Patricia Deubel highly recommends this site.

Wolfram Education Portal is a free website with math teaching tools and materials, organized by course. You'll find a dynamic interactive textbook, lesson plans aligned to the Common Core Standards , widgets, interactive demonstrations, videos, and more. Courses include algebra and calculus.

WOWmath.org features hundreds of videos developed by math teacher, Brad Robb, for AP calculus (AB level). They correlate with the 8th Edition Houghton Mifflin - Calculus of a Single Variable.

Calculus Pro by Larry Feldman is compatible with iPhone, iPod touch, and iPad; and Android. It has examples, tutorials, and solvers for the following topics: Limits, continuity, differentiability; RDQ, LDQ, SDQ and NDQ, equations of tangent lines, partial derivatives, implicit differentiation; the chain, product, and quotient rules; related rates, optimization (min/max), derivatives of inverse functions, Taylor and Maclaurin series, derivative tables, derivatives of integrals, arc length, Riemann sums, fluid problems, double integrals, integration by parts, definite and indefinite integrals, volumes of solids of revolution, cross sectional volumes, integral tables; position, velocity and acceleration; and theorems. Feldman also has apps for other math content areas.

Khan Academy video apps by Ximarc Studios are free apps compatible with iPhone, iPad, and iPod touch.

- Khan Academy: Calculus 1 includes videos 1-20
- Khan Academy: Calculus 2 includes videos 21-40
- Khan Academy: Calculus 3 includes videos 41-60
- Khan Academy: Calculus 4 includes videos 61-80

Mathomatic from Gotow.net is compatible with iPad, iPhone, and iPod Touch. For a small fee ($1.99), you can solve a range of problems from middle school math through higher level calculus. For algebra, solve systems of algebraic equations showing steps, factor and simplify expressions, and work with complex numbers. Also solve calculus problems such as indefinite integrals of polynomials, derivatives, Laplace transforms of polynomials, and Taylor series expansions.

Wolfram Calculus Course Assistant is compatible with iPhone, iPod touch, and iPad and is available in iTunes for about three dollars. It covers topics applicable to Calculus, AP Calculus AB, AP Calculus BC, Calculus I, and Calculus II.

**How Big is Infinity?**

Some learners first encounter a formal study of infinity when they take their first course in calculus. However, you don't have to wait to take calculus to get a sense of the size of infinity and to appreciate that there are some unanswered questions in mathematics. View the video at TED-Ed in which educator Dennis Wildfogel used fundamentals of set theory and basic knowledge of real numbers to explore How Big is Infinity? To stimulate conversation, there's also a quiz, questions to think about and to dig deeper. This lesson can also be used in a flipped classroom.

In How Many is a Googol? posted at Wonderopolis.org, students can get a better appreciation of the difference between a really large number and infinity. Even students just learning about powers of 10 will benefit from this.

Also at Wonderopolis.org is What is Infinity? Infinity is more than just a really large number. There can be different sizes of infinity! Really.

**Play the Derivative Matching Game!**

Barbara Margolius created this Derivative Matching Game posted at the Mathematical Association of America. Macromedia Flash is used. When you play you are presented a series of cards on which functions and their derivatives are presented. The goal is to select the function, then its derivative. You can customize to also include the second derivative and make the game more challenging my increasing the number of game cards presented. This is first rate for calculus students and their teachers.

Dennis Wildfogel also has a series of videos on
Making Sense
of Calculus. Several deal with the visualizing the
derivative of terms such as x^{2}, x^{3}, x^{-1}, x^{-2},
x^{1/2}, x^{1/3}, x^{-1/2}, x^{2/3}, x^{3/2}, then the Product Rule, and
more to come.

ChartGizmo is free online software that can be used to create several kinds of graphs: Pie, Bar, Line, Ring, 3D Pie, 3D Bar, 3D Line, Candle, Scatterplot, TimeSeries. A manual is available; chart settings can be customized. Several data sources are supported: static data, historical data, dynamic data from JavaScript code, dynamic data from html table on the page. Results can also be displayed on your own web pages.

Collaborative Statistics is an online textbook written by Barbara Illowsky and Susan Dean, faculty members at De Anza College in Cupertino, California. Per the Summary, "The textbook was developed over several years and has been used in regular and honors-level classroom settings and in distance learning classes. This textbook is intended for introductory statistics courses being taken by students at two– and four–year colleges who are majoring in fields other than math or engineering. Intermediate algebra is the only prerequisite. The book focuses on applications of statistical knowledge rather than the theory behind it."

Data Games from KCP Technologies are
free online games suitable for learners in grades 6-12. Each is linked to
Common Core Standards for Math. Lessons are designed to help learners to "analyze data, interpret and create linear and quadratic models, use
proportional reasoning, and understand probability." Student and teacher
videos, worksheets, and teacher facilitation guidelines for classroom use are
provided (For Teachers section). Data can be saved. KCP
Technologies is also the developer of such data analysis tools as *The Geometer's
Sketchpad, Fathom Dynamic Data* and *Tinkerplots Dynamic Data Exploration*.

DIG Stats is an excellent resource for integrating statistics and data visualization into mathematics and science courses in grades 9-14. Modules with student activities cover concepts from descriptive statistics, inferential statistics, two dimensional graphical analysis and three dimensional data visualization. Using Microsoft Excel and graphing calculators, students analyze real world data sets from a variety of scientific disciplines. Data were derived from WWW databases, university and government researchers, and student projects. Guides, quizzes, and answers are provided. This site is associated with the Central Virginia Governor's School for Science and Technology.

Online Statistics Calculators: There are five calculators available for measures of central tendency and dispersion, box and whisker plots, linear regression, correlation coefficients, and scatter-plots.

Online Statistics Education: A Multimedia Course of Study is a free online textbook by David Lane, project leader at Rice University, and his co-authors. Chapters also contain videos, simulations, and case studies. For example, there are simulations with instructor notes for sampling, measurement, box plots, summarizing distributions, describing bivariate data, probability, normal and sampling distributions, confidence intervals, testing means, power, linear fits, ANOVA, and chi-square. Educators who use the book can request additional materials, including an instructor guide and PowerPoint slides.

Probability Resources from the Math Association of America (MAA) is a vast collection categorized by topics suitable for high school and college learners. The section on basic probability can also be introduced to middle school learners. You'll find explanations, videos, interactive applets, problem sets, and so on depending on resource.

Research Randomizer, part of the Social Psychology Network, is a free tool that can be used to introduce students to random sampling and random assignment. Students in K-12 often conduct surveys, which is just one use of this tool. It includes tutorials.

STatistics
Education Web (STEW) provides online peer-reviewed lesson
plans for K-12 math and science teachers who teach statistics concepts
in their classrooms. These are organized by grade band and include
statistical concepts being addressed. "The statistical concepts
follow the recommendations of the
*Guidelines for
Assessment and Instruction in Statistics Education (GAISE) Report: A
Pre-K-12 Curriculum Framework*,* Common Core State Standards for Mathematics*,
and *NCTM Principles and Standards for School Mathematics*" (STEW
description section).

Statistics in Schools from the U.S. Census Bureau helps to educate K-12 students about the value and use of statistics in everyday life. The program provides resources for teaching and learning with real life data. Common Core standards-aligned, classroom-ready activities in math and history, data visualizations and more are included.

Statistics: Power from Data, a product of Statistics Canada, is a complete online text with exercises, answers, and lesson plans. Although intended for use with secondary students, the text could be used with middle school or post-secondary audiences. Chapters address data collection, data processing, problems with using information, organizing data, graph types, analytical graphing, measures of central tendency and spread, sampling methods, and more. There is a glossary.

Stat Trek has tutorials to teach yourself statistics and probability. Videos and practice quizzes are included. It includes the full AP Statistics curriculum.

What are the Odds? The Ins and Outs of Probability (by graduate students at University of Virginia) if primarily for secondary students. It contains lesson plans, an interactive quiz, an extensive data bank of questions relating to probability concepts sorted by topic, history, and additional resources on probability.

Khan Academy: Statistics videos by Ximarc Studios is a free app compatible with iPhone, iPad, and iPod touch.

Probability and Statistics from Homeworkhelp.com is compatible with iPad, iPhone, and iPod Touch and is available for a few dollars in iTunes. "Topics in this app include: Probability, Organizing Data, Distributions, Gathering Data, frequency Tables, Mean, Median, Mode, Random Variables, Inference, Correlations, Z-Score, Measures of Dispersion, and many more" (Description section). Homeworkhelp produces several boosters. Also see Pre-Algebra Boosters: Part 1 and Part 2; Algebra 1 Booster and Algebra 2 Booster.

Wolfram Statistics Course Assistant is for iPhone or iPad. It's available in iTunes for a couple of dollars. This app enables you to create a bar chart, histogram, or scatter plot of any set of data; find the mean, median, mode, standard deviation, quartiles, and interquartile range of a dataset; calculate normal probabilities and find information about the normal distribution; calculate binomial probabilities and find information about the binomial distribution; compute probabilities based on dice rolls and coin flips; find the best-fit line of a set of data points; select random integers or random real numbers. (Description section)

**Teach using data simulations for better
understanding of statistics and statistical reasoning.**

Data simulations can help learners answer particular research questions or to solve statistical problems. Get more information on how to teach using data simulations at Teaching with Data Simulations, which is within Pedagogy in Action, the SERC portal for Educators. Examples are included. Use simulation software such as Sampling SIM or Key Curriculum Press Fathom. Fathom is designed for learners grade 8 and up.

Web applets are also beneficial. Consider:

- The Mathematical Association of America Mathematical Sciences Digital Library has applets for statistics and probability exploration. You'll find a Mean and Median Applet, which can be used two ways: enter data and then investigate the mean and median; or, identify if a marker on a given distribution shows a mean, median, or both; a Law of Averages Applet; and the Probability Aquarium for sampling with or without replacement.
- Rossman/Chance Applet Collection
- WISE: Web Interface for Statistics Education from Claremont University
- Rice Virtual Lab in Statistics

See CT4ME's Math Manipulatives for other possibilities for exploring statistics and other subjects.

Analyze Math by A. Dendane (United Arab Emirates University) contains math tutorials and problems (with applets) for a various topics in algebra, trig, geometry, calculus, and more. Excellent explanations. Applets are found in the section called "Math Software."

Brightstorm.com has over 2000 online video lessons appropriate for middle and high school students. The math lessons are free and developed by a group of math teachers for algebra 1, geometry, algebra 2, trigonometry, precalculus and calculus. Additionally, the site offers a fee-based service to prepare for the SAT, ACT, and advanced placement courses.

Electronic Bookshelf is an outgrowth of the original project developed at Dartmouth College called Math Across the Curriculum (MATC). Now part of the Center for Mathematics and Quantitative Education at Dartmouth, it features materials with an interdisciplinary approach to mathematics topics. Among topics in the bookshelf topics are algebra, calculus and differential equations, physical science, art, computer sciences, pre-algebra, history and geography, literature and writing, biology/medicine/psychology, data analysis, statistics, business/economics, engineering, math and music, financial literacy, health QL, and discrete math. There's a little bookshelf of K-12 resources.

GeoGebra is free, open-source, dynamic mathematics software that joins geometry, algebra, tables, graphing, statistics, and calculus. You can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards. You can enter equations and coordinates directly. GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum. It received several international awards, including the European and German educational software awards. In Dynamic Mathematics with GeoGebra, authors Markus Hohenwarter, principle developer of the software, and Judith Preiner (2007) said, "You can use GeoGebra both as a teaching tool and to create interactive web pages for students from middle school up to college level. Specifically designed for educational purposes, GeoGebra can help you to foster experimental, problem-oriented and discovery learning of mathematics" (Abstract section). They illustrate the basic ideas of the software and some of its versatile possibilities by discussing several interactive examples. Educators can join the GeoGebraWiki to post their creations and view the work of others. You can also share your materials at Geogebra Tube.

Google: Exploring Computational Thinking (CT) contains classroom-ready lessons, worksheets, and examples illustrating how educators can incorporate CT into the K-12 curriculum. You'll find everyday applications, too, such as pattern recognition, algorithm design, and other computational thinking skills. Content was developed by teachers in collaboration with Google software engineers; math addresses pre-algebra, algebra, geometry, and calculus. Additional features include a Getting Started Guide, web resources, and a discussion forum to connect with other CT educators. Readers might be particularly interested in the conceptual frameworks and how-to papers listed in web resources for incorporating CT into every classroom.

Gooru is a free search engine for learning. Within its content are collections of multimedia resources, digital textbooks, videos, games, and quizzes created by educators within its community. Collections are aligned to standards and cover math and science for grades 5-12. With the section called "Library" you'll find extensive math in the areas of math fundamentals, pre-algebra, algebra, integrated math, geometry, algebra II, pre-calculus, and calculus.

Higher Bitesize Maths from the BBC-Scotland contains Revisions (means study help with examples and illustrations) and Interactive Practice Tests on a variety of topics for algebra (functions, logarithms, polynomials, quadratic theory, and recurrence relations), geometry (circles, straight lines, vectors), trigonometry (compound and multiple angles, radians and equations, wave function), and calculus (differentiation and integration). [Note: Bitsize Maths, also from the BBC-Scotland, contains materials at a standard level for maths 1 and maths 2, also with study help (revisions) and interactive tests. Maths 1 deals with number at the general level, Pythagorean theorem, trigonometry and angles; area and volume, algebra at the general level, and graphs. Maths 2 deals with trigonometry, algebra, and statistics.]

HippoCampus, a project of the Monterey Institute for Technology and Education, is a "free, public website for high school and college students that offers NROC multimedia correlated to most major textbooks." NROC is the National Repository of Online Courses for students and faculty in higher education, high school, and advanced placement. Among the full math textbooks online are Algebra (IA, IB, and Elementary) and Calculus (Introductory I and II, General I and II, AB and BC levels for advanced placement). Multimedia includes audio explanations, visuals, interactivity. Individual instructors can customize the Hippo website to better meet the needs for groups of students. Highly recommended.

Jim Reed's Math and Science Resources are hosted by the Argyll Centre of Edmonton Public Schools in Canada. You will find elementary and secondary topics in numbers/operations, patterns and relations, shape and space, probability and statistics, algebra, geometry, and trigonometry. Students can engage with interactive learning activities, see movies, and manipulate excellent Java math applets to study concepts. The site is also an excellent resource for the home-schooled student. Highly recommended.

Just Math Tutorials by math teacher PatrickJMT contains an extensive collection of his videos primarily for algebra, trigonometry, and calculus. They are free to use and are suitable for high school and above. There is also a section on miscellaneous topics such as probability and statistics, SAT test prep and the advanced placement AB and BC calculus test prep levels.

MangaHigh is devoted to mathematics game-based learning for middle and high school learners. Games are of a commercial quality, but are free. You'll find interactivity, more than one level of difficulty, and topics such as transformations, estimation with basic number calculations, order of operations, equation solving (linear, quadratic, cubic, plotting coordinates), basic shapes to advanced geometry (some trig, too), ordering of decimals, fractions, percents. Adaptive learning, which automatically adjusts difficulty level based on how students engage with the game, is featured in Prodigi where students work to master skills in number, algebra, data, and shape. Games are endorsed by well-known UK mathematician Marcus du Sautoy.

MathBits.com "is devoted to offering fun, yet challenging, lessons and activities in high school (and college level) mathematics and computer programming for students and teachers." Subjects include algebra 1, geometry (including materials for using Geometer Sketchpad), and algebra 2, principally.

MathsIsFun.com provides supplementary content for K-12 in topics ranging from working with numbers and basic operations to money, algebra (with an introduction to trigonometry), geometry (including some interactives for constructions and using a protractor), data, measurement, worksheets, games, puzzles, and an illustrated math dictionary. Tutorial information is provided within content.

Math Expression has excellent tutorials, including animated videos with audio explanations and text equivalents. Topics include graphing (e.g., using the coordinate plane, slope, writing equations of lines), basic algebra (e.g., variables, expressions, solving equations), exponents, fractions, symmetry, and volume. Math study tips and practice questions help reinforce concept development. There is also a Q&A Library. The site is by Wei Chong, and is well-worth investigating.

Maths for all provides interactive practice for concepts typically studied in grades 5-8, pre-algebra, algebra 1, algebra 2, geometry, pre-calculus, trigonometry, and statistics. You'll find an overview of the concept, worked examples, and interactive practice problems with solutions.

MathFLIX from Loyola University Chicago School of Education is a real find. It has 1000 free "instructional math movies covering a wide range of math concepts including Number & Operations, Algebra, Measurement, Geometry, Data Analysis & Probability, Connections and Technology. In addition to MathFLIX’s valuable video resources, the site also features 400 downloadable worksheets that reinforce concepts and provide valuable practice." Each movie is from 4-7 minutes long.

National Curve Bank, maintained at the Department of Mathematics at California State University, Los Angeles, contains highly relevant visual content and explanations for upper level mathematics--calculus, pre-calculus, trigonometry, geometry, fractals, and so on. JAVA applets are included. Some content includes practice problems on various topics. The developers "strive to provide features - for example, animation and interaction - that a printed page cannot offer." Geometrical, algebraic, and historical aspects of curves, "the kinds of attributes that make the mathematics special and enrich classroom learning" are included. Participants are encouraged to submit your best web animation as a "deposit" in the National Curve Bank. Highly recommended!

SAS Curriculum Pathways provides standards-based content in core subjects (English, Math, Science, History, and Spanish) for teachers and students in middle grades, high schools, and community colleges. Math content addresses introductory algebra through advanced algebra, geometry, trigonometry, and data analysis and probability. Activities include simulations, multimedia resources, innovative tools, and more; also included are web inquiries, additional web resources, and complete lessons. This award-winning content is free for U.S. educators.

S.O.S. Math from MathMedics has over 2,500 pages of short explanations in multiple subjects: algebra, trigonometry, calculus, differential equations, complex variables, matrix algebra, or mathematical tables. There's also a section for test prep: intermediate algebra, mathematics for the social sciences (I and II), pre-calculus (I and II), calculus, and differential equations.

The Math Page is by Lawrence Spector of the Borough of Manhattan Community College, The City University of New York. This site has several major sections. Practice problems illustrating concepts are presented throughout with answers.

- Skills in Arithmetic is a complete course, which includes addition, subtraction, multiplication, division; mental arithmetic, powers of 10, regrouping, decimals, parts of natural numbers, fractions, and percents.
- Plane Geometry
- Topics in Trigonometry addresses ratio and proportion, definitions of the trig functions, solving right triangles, the 30-60-90 degree right triangle, the law of sines, the law of cosines, the unit circle, radian measure, and arc length.
- Topics in Pre-calculus include formal rules of algebra, rational and irrational numbers, functions and graphs, logarithms, the binomial theorem, and mathematical induction.
- An Approach to Calculus
- The Evolution of Real Numbers.

Things to Remember from Oswego City School District's Regentsprep.org in New York consists of four-pages of facts and formulas summarizing key concepts addressed in Algebra 2 (pp. 1-2), Trigonometry (p. 3), and Probability and Statistics (p. 4). These are very useful for quick reference for test prep.

Zona Land's More Mathematics than Science is devoted to helping students understand algebra, geometry (including conic sections), trigonometry, and curve fitting. Don't miss the interactive manipulatives and virtual reality demonstrations of concepts.

Mudd Math Fun Facts. Fun Facts are designed for, and work best in courses where the typical student needs that boost to generate interest and arouse curiosity in math. The Fun Facts will cause students who are studying algebra, geometry, probability, calculus, number theory, and topology, for example, to look at mathematics differently!

Taken from the quick activity on multiplication by 11, can you figure out what's going on here?

253 x 11 = 2783

117 x 11 = 1287

532 x 11 = 5852

267 x 11 = 2937

**How about presenting the number of the day?**

Mathematical Association of America’s NumberADay blog. Every working day, the MAA posts a number and offers a selection of that number’s properties. For example, on September 17, 2008, the daily number was 54 --the number of squares on a Rubik's cube, and also the smallest number that can be written as the sum of three squares in three ways. 54 = 12 + 22 + 72 = 22 + 52 + 52 = 32 + 32 + 62

**What about a problem or riddle of the day?**

Visit MAA MinuteMath blog, which contains a problem of the day taken from the AMC-8, AMC-10, or AMC-12, provided by MAA's American Mathematics Competitions. Each problem is presented with an interactive version and solution. On October 6, 2008, the problem was to find the minimum product using three different numbers from the set {-8, -6, -4, 0, 3, 5, 7}. Find out by visiting the site.

Famous Problems in the History of Mathematics is from the Math Forum. "The purpose of this site is to present a small portion of the history of mathematics through an investigation of some of the great problems that have inspired mathematicians throughout the ages. Included are problems that are suitable for middle school and high school math students, with links to solutions, as well as links to mathematicians' biographies and other math history sites."

Good Riddles Now
**GRiN:)** has an entire section devoted to math riddles with
their answers. "All of these tricky riddles are based on real math
concepts and can be solved with purely math and logic."

**Or spice up your lesson with video.**

WatchKnowLearn.org contains a database of free educational videos gathered from across the internet and organized for kids, so that they can get extra "instant tutoring" on almost any topic taught when they need it. These are also good for classroom use. There are multiple content areas, including about 1000 videos in mathematics and over 100 on standardized test skills and math study skills. Explore numbers and shapes, applied math for young learners, arithmetic, measurement, algebra, geometry, precalculus, and calculus, the history of math, and more.

**How about presenting a math joke, comic of the day, a math song or a math rap?
Let your students also create their own.**

Aha! Jokes has thousands of clean jokes, funny pictures, cartoons, funny audio, funny videos, and more in numerous categories. The Math jokes feature humor about algebra, geometry, statistics, calculus, proofs, addition, and more.

MakeBeliefsComix.com allows anyone to create comic strips online, then print or email them. Students might create a comic to explain a math concept or a new vocabulary word, or how they solved a problem. They might share their feelings or a concern about their learning. It's a fun way to incorporate writing in math class.

Pixton is a social cartoon making website that anyone can join. The site includes safety tips, tips for parents, and a special section for school use. A site search reveals many math cartoons. You can leave comments, rate your favorites, exchange messages with friends, build a fanclub, print and download your own comics, email them to others, and post them on other web sites and blogs.

The Rappin' Mathematician explains How to Write Your Own Math Rap. Need an example? Visit his Mathraps.com, which contains a few free raps.

Songs for Teaching includes Math Songs: Teaching Math Facts & Concepts. Songs are categorized by addition, subtraction, multiplication, division, advanced math, algebra, geometry, trigonometry, and there is a section for miscellaneous math songs.

**How about relating your math education to culture?**

Culturally Situated Design Tools for teaching math and computing through culture comes with lessons and software for students to create their own cultural designs associated with African, African American, Latino, Native American cultures, and Youth Subcultures—even graffiti! Explore other ways to incorporate this area into your classroom using resources from the International Study Group on Ethnomathematics.

**You could present a mathematician of the
day or some facts relating to the history of mathematics.**

Egyptian Mathematics Numbers Hieroglyphs also includes some problems for kids and an Egyptian math worksheet generator. You'll also find a history of ancient Egypt.

The Greatest Mathematicians of All Time is an infographic created by studygeek.org. Get some quick facts on Newton, Archimedes, Euclid, Gauss, Pythagoras, Lagrange, Fermat, Descartes, and Leibniz.

Guide to the History of Mathematics from StartLocal (Australia) contains facts on math around the world, famous mathematicians, a timeline, and resources to learn more.

**HOT:**
Minds of Modern Mathematics is a free app for iPad from IBM
Corporation, which "takes you on an interactive journey that spans
nearly 1,000 years. It tells the story of mathematics and how it has
impacted almost every aspect of human progress, from science to
music, art, architecture, and culture. Minds of Modern Mathematics
is a digital recreation of a 50-foot-wide wall installation that was
part of the groundbreaking 1961 Mathematica exhibition sponsored by
IBM and designed by the legendary design team, Charles and Ray
Eames" (Description section). View the
IBM press
release of April 2012.

The University of St. Andrews (Scotland) provides a list of mathematicians who were born or died on the current calendar day, along with biographies. So, you can check this site daily.

Wolfram Research has an extensive list of mathematicians with their biographies. As students often know little of who discovered some of the concepts they study, such biographies might be of interest.

Biographies of Women Mathematicians is a project from Agnes Scott College in Atlanta, Georgia. The achievements of women in mathematics are sure to motivate. You also will find the woman mathematician of the current day.

**Students and Teachers:
Print Graph Paper for Free!**

Graph paper and grids from Mathbits.com: number lines, dot, rectangular, trig, polar, and blank calculator screens---over 30 different types.

GraphTablet is a freeware program that allows you to print Cartesian, polar, and semilog graph paper and notebook ruled paper. You can also choose the line spacing, width, and color and print one, two, or four grids on a page.

Print free graph paper! -- Cartesian, polar, logarithmic, and more.

Print graph paper of all kinds from Incompetech.com: square, triangle, rhombus, hexagon, octagon, trapezoid, circular, asymmetric, accounting, log, polar, number line, storyboard, perspective, note-taking, and so on. A terrific collection.

Hohenwarter, M., & Preiner, J. (2007, March). Dynamic mathematics with GeoGebra
[online]. *Journal of Online Mathematics and Its Applications, 7*,
article 1448. Retrieved from
http://www.maa.org/external_archive/joma/Volume7/Hohenwarter/index.html

National Mathematics Advisory Panel (2008). *Foundations for success: The
final report of the National Mathematics Advisory Panel*. Washington,
DC: U.S. Department of Education. Retrieved from
http://www.ed.gov/about/bdscomm/list/mathpanel/index.html

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See related topics: Math Manipulatives and Standardized Test Preparation.