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Math Resources:
Middle, Secondary, Post-Secondary Subject Specific Resources
Plus Apps for Mobile Devices

Showing Grade 1 Boy with A+ ScoreMath resources concepts Gif

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

Arrow: You are hereMath Resources (Page 2): Middle, Secondary, Post-Secondary: Subject Specific Resources also include apps for your mobile devices:

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, the Flipped Learning Model, and for Whiteboards, including tips on using whiteboards

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Middle, Secondary, Post-Secondary Math Resources

 

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)

 

Algebra and Pre-Algebra

What is Algebra?

Algebra --math of generalized arithmetical operations

Are you ready for algebra?

Question markStudents: Find out by taking an Algebra Readiness Test.  This test at algebra-class.com includes answers, and content to review theory.

Math Fortress offers six algebra 1 readiness quizzes to test if you remember the basics of arithmetic, including operations with integers, decimals, square roots, fractions, prime factorization, percents, geometry and evaluating expressions, and word problems.

The following Dr. Math books (2003) from The Math Forum will help you, too.

Dr. Math Gets You Ready for Algebra  Dr. Math Explains Algebra 

Educators: 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:

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

Educators might be interested in the Ohio Algebra I Blueprint, which is helpful for planning curriculum and guiding instruction.  It includes the standards for number, quantities, equations and expressions, functions, and statistics typical for an algebra 1 course.  These standards are eligible to appear in the state year-end standardized test for algebra 1.

 

Algebra Line JPGThe following algebra resources should help to achieve those recommendations.

HOT: AlgebraByExample "gives students practice solving problems and practice modeling, analyzing, critiquing, and articulating mathematical arguments."   There are 42 individual assignments within nine algebra 1 topics, including one for pre-algebra.  The assignments support the Common Core Practice Standards.  Problems are designed to address student misconceptions.  Each assignment contains a worked example accompanied by an analysis question.  Then learners solve a similar problem on their own.  The sets are free downloads from the SERP (Strategic Education Research Partnership) Institute.

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.

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 Aligned to the CCSS-M Standards is another algebra 1 offering designed for grade 9 learners with resources organized by Common Core math standards.  Per its description, "Both teacher facing and student facing assets are included and are of various forms, e.g. text, video, and animations."

Curriki Prealgebra Aligned to the CCSS-M Standards is designed for grade 8 learners with resources organized by Common Core math standards.  Per its description, "Both teacher facing and student facing assets are included and are of various forms, e.g. text, video, and animations."

Dave's Short Course on Complex Numbers

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.

HOT: Graspable Math is a free web-based tool to help students interactively explore and solve algebra equations by moving terms around on a screen--also a good tool for homework.  Several video tutorials are available to learn how to use mouse and touch gestures for algebra, such as the nature of the online "canvas" itself, the on-screen keyboard, simplifying expressions, working with the distributive and commutative properties, entering and solving equations, rewriting expressions, and more.  There are options to undo/redo, save, load, and share.  Teachers will value the additional materials and use of Graspable Math as a classroom presentation tool.  The development of this resource was supported by the Institute of Education Sciences.

Introduction to Algebra hosted by edx.org is a free online course that covers the standard high school Algebra 1 curriculum. Common Core alignment is indicated where applicable.  Learners can progress at their own pace, choose their own path within each lesson and jump between lessons to review earlier material.

IXL Algebra 1 and IXL Algebra 2 problem sets address the skills that students learn in Algebra 1 and Algebra 2.  As students practice online and demonstrate mastery, the level of difficulty increases.  Skills are organized in categories.

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.

MathBitsNotebook: Algebra 1 and Algebra 2-- Each of these sections includes a series of lesson and practice pages for high school Algebra. Materials cover all CCSS math standards and coordinate with SBAC, PARCC, and state assessments.  You'll also find two end-of-course interactive multiple-choice practice tests in each section related to CCSS Algebra 1 and CCSS Algebra 2.

Math 101: College Algebra is a free online course from the Study.com 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.

Math Planet includes four courses: Pre-Algebra, Algebra 1, Algebra 2, and Geometry.  Each comes with multiple chapters divided into several lesson topics.  Under each lesson, you'll find theory, examples, and video lessons.  Math Planet also includes ACT and SAT practice questions.  Content is free.

Open Course Library from the Washington State Board for Community and Technical College features multiple courses and course materials, free to use under a Creative Commons license.  Descriptions are provided for each course:

OpenStax open-source free online math textbooks, which are also peer-reviewed:

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.

Purple Math Logo GifPurplemath 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.

Science Friday: Explosion Math is a Common Core-aligned lesson in which students use linear equations (y=mx+b) to "to answer the age-old question of who would win between an Olympic sprinter, tortoise, car, you, and a volcano."  To find out, students calculate how much of a head start each would need to run away and escape from two different volcanoes that had erupted in 2018.  Brian Soash of Science Friday designed this lesson for students in grades 6-12.  It can be completed in 1-2 hours and includes all supporting materials and illustrative images for classroom use.

Study.com has online math courses in pre-algebra, algebra, geometry, statistics, precalculus, calculus, college math and more.  These feature video lessons from experienced educators, transcripts of those lessons, supplementary content, quizzes, and exams.  Learners can potentially earn college credit.

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

  1. Whole Numbers
  2. Integers
  3. Fractions & Decimals
  4. Exponents & Roots
  5. Rates, Ratios, & Proportions
  6. Equations
  7. Inequalities
  8. Graphing
  9. Probability & Statistics
  10. 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.

Why U contains animated videos in pre-algebra and algebra.  "Rather than focusing on problem solving, the objective is to give insight into the concepts on which the rules of mathematics are based."  Beginning with a humorous look at the Dawn of Numbers, the pre-algebra tutorial videos (over 30 of them) lay the foundation for high school and college algebra.  The algebra tutorial videos (over 70 of them) explain not only the how, but why the rules of algebra work.  Videos were written by Steve Goldman and animated by Mark Rodriquez.  The animations are outstanding.

Wolfram for Education has some free resources for teaching math.  For example, see Using Wolfram|Alpha in the Classroom.  This latter is useful for teaching topics in Algebra and getting step-by-step solutions to problems, including graphing.  Lesson plans are also available.

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.

Algebra on your mobile devices

Algebra Pro iconAlgebra Pro for iPhone, iPod Touch, and iPad has solvers, tutorials, and examples for topics found in algebra 1 and algebra 2.  It is available for free.  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, precalculus, calculus, trigonometry, probability, statistics, and so on.

Algebra Tiles app from BrainingcampAlgebra Tiles from Brainingcamp
Platform: iPad
Price: $1.99
Target: Grades 6-8 (Note: learners studying algebra 1 will also benefit.)
Info: These algebra tiles can be used to explore 1) adding, subtracting, multiplying, and dividing integers, 2) modeling and simplifying expressions, 3) substitution, 4) adding and subtracting polynomials, 5) solving equations and systems of equations, 6) distribution, 7) adding, subtracting, multiplying and dividing polynomials, 8) factoring, and 9) completing the square (Product description).  Brainingcamp also has other apps for iPads.

Algebra Touch by Regular Berry Software imageAlgebra Touch by Regular Berry Software LLC for iPhone, iPod Touch, and iPad (free) 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."

HMS Fuse iconHMH Fuse: Algebra I and HMH Fuse: Algebra 2 apps are available for free, designed for the iPad, with in-app purchases available.  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.  You'll find video tutorials, step-by-step examples, homework help, quizzes, tips, hints, and built-in manipulatives, text and audio note-taking, scratchpad, and a student response system for instant learner feedback.

Khan Academy imageKhan Academy app is free and compatible with iPhone, iPad, and iPod touch.  You can learn just about anything.  Multiple subjects are included.  "Learn using videos, interactive exercises, and in-depth articles in math (arithmetic, pre-algebra, algebra, geometry, trigonometry, statistics, calculus, linear algebra)."

Socratic app iconSocratic--Homework helper by Google is a free app compatible with iPhone, iPad, iPod Touch, and Android.  It's designed primarily for high school and university learners.  You can take a photo of your homework question, typed or handwritten, and get explanations, videos, and step-by-step help.  The app also includes voice recognition.  When you pose a question, "the app will find the best online resources for you to learn the concepts."  Supports Math, Science, Social Studies, Literature, and more.  For example, math supports algebra, geometry, trigonometry.

Ventura Educational Systems icon Ventura Educational Systems has a variety of iPad and iPhone apps for education.  For example, iPad apps for algebra:

Virtual Nerd Mobile Math iconVirtual Nerd Mobile Math by Pearson Education, Inc.
Platform: iPhone and iPad
Price: Free
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 iconWolfram Algebra Course Assistant is available for a couple of dollars. Per the description: This app covers topics in Algebra I, Algebra II, and College Algebra.  It can:

Wolfram Pre-Algebra Course Assistant iconWolfram 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.

 

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

Friendly reminder GifSome states require four years of high school math, including Algebra II or its equivalent.  Several state departments of education post released test items from prior end-of-course exams.  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.

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 app imageCommon 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.

 

 

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Geometry

What is Geometry?

Geometry--pure math of points, lines, curves, and surfaces

 

Are your students ready for a formal course in geometry?

Question markAlthough they may be in your geometry class, consider that student 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.

Students might also benefit from the following Dr. Math books (2004) from The Math Forum.

Dr. Math Introduces Geometry  Dr. Math Explains More Geometry

Educators might be interested in the Ohio Geometry Blueprint, which is helpful for planning curriculum and guiding instruction.  It includes the standards for congurence and proof, similarity and trigonometry, circles, and probability typical for a geometry course.  These standards are eligible to appear in the state year-end standardized test for geometry.

Byrne's Euclid

Euclid's Elements has been the foundation for hundreds of editions of geometry textbooks since he first created his 13 books dating back to around 300 BC.  In 1847 Oliver Byrne's edition of the first six books became noteworthy owing to his use of multiple colors in diagrams to help explain proofs and make geometry easier to understand.  In recent times, web designer and artist Nicholas Rougeux reproduced Byrne's Euclid as an interactive website staying true to its color scheme, graphics, and typography.  Rougeux's clickable shapes and links to cross-references in the text add value to exploring his project, which was launched in December 2018.  You can read how he did it in his blog post, Making of Byrne's Euclid.

interactive website of Byrne's Euclid

Geometry Area of Triangle JPGThe following resources should help your learners address geometry standards:

Curriki Geometry Aligned to the CCSS-M Standards "is a collection of resources for a high school course in Geometry. The resources are organized by Common Core State Standards for Mathematics and reflect a ninth or tenth grade course. Both teacher facing and student facing assets are included and are of various forms, e.g. text, video, and animations" (Content section description).

Curriki: PBL in the Geometry Classroom is a free curriculum that takes a project-based approach for learning.  There are several Common Core State Standards aligned projects, each of which focuses on at least two of the eight mathematical practice standards and includes a rubric for assessing mathematical practices.  Teachers can select to use all or just some of the projects.  Projects can be taught in any order and include technology and Web 2.0 resources "such as videos, documents, web pages, and dynamic geometry constructions, quizzes and exam suggestions for assessment, and other tools related to the project."  You'll find:

  1. Selling Geometry
  2. Designing a Winner
  3. What’s Your Angle, Pythagoras?
  4. TED Talk: House of the Future
  5. The Art of Triangles
  6. 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.

GeoGebra LogoGeoGebra 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:

Geometry 3-D Shapes is an interactive from Annenberg Learner intended for grades 6-8, although other grades can benefit.  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 a glossary, tutorials, and an interactive test.

Geometry Applet Gallery from Saltire Software contains some excellent applets, which are easy to use and suitable for illustrating several theorems presented in a high school or post-secondary geometry course.  Basic Geometry relates 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.  Saltire's work has been assisted with a number of National Science Foundation grants.

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.

Introduction to Geometry hosted by edx.org is a free online course that covers the standard high school geometry curriculum. Common Core alignment is indicated where applicable.  Learners can progress at their own pace, choose their own path within each lesson and jump between lessons to review earlier material.

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

MathBitsNotebook: Geometry "is a series of lesson and practice pages for students studying high school Geometry. These materials cover ALL standards stated in the Common Core State Standards for Mathematics (Appendix A, traditional pathway for Geometry), and coordinate with both SBAC and PARCC assessments" (Website description).

Math Open Reference includes 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.

Math Planet includes four courses: Pre-Algebra, Algebra 1, Algebra 2, and Geometry.  Each comes with multiple chapters divided into several lesson topics.  Under each lesson, you'll find theory, examples, and video lessons.

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.

Geometry on your mobile devices

Apollonius iconApollonius for iPad is free interactive geometry software.  "It allows you to create geometric constructions, such as those that can be made using a ruler and a compass, and subsequently move parts of your constructions to explore the relationship between the geometric objects on the screen. Its interface is especially designed for touch screen" use (Description section).

Arloon geometry iconArloon Geometry ($2.99) for Android and iOS devices is for students ages 11 and up.  It "features 3D models with Augmented Reality for most geometric shapes" and enables users to "View geometric shapes from all angles and see their sides unfold into flat figures. Geometry combines conceptual content (definitions and characteristics) with procedural content (applying formulae and calculus)."  Practice exercises are included: "guess the geometric shape, confirm characteristics and calculate area and volume."  It's available in English and Spanish.  The app is posted on edshelf.com, which includes other augmented reality math apps.

Ezy Triangle iconEzy Triangle by KGiants is for iPad and iPhone.  This free app for high school learners and above 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.

Adaptive Curriculum Geometry: Circumference & Arc Length imageGeometry: 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 Pad IconGeometry Pad by Bytes Arithmetic LLC 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.

Geometry Stash iconGeometry Stash by Apparent Etch is  for iPad and iPhone.  It 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.

HMS Fuse iconHMH 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: Geometry Common Core edition is available for free, but has in-app purchases.

iCrosss imageiCrosss by Oleh Yudin helps users learn solid geometry.  Per the app description, it comes with about 50 polyhedra, including " regular polyhedra (Platonic solids), Pyramids, Prisms, Anti-prisms, Archimedean solids and duals."  You can rotate the 3D solid in any direction, add a plane to visualize a cross-section of it, get information about the solid ("short description, number of faces, vertices, edges, web-link for more information and useful formulas"), separate it into parts in 3D cut mode, print and email it.  It's available for iPad, iPhone, and Android devices for $.99.

Isosceles-Geometry-Sketchpad imageIsosceles+: Geometry Sketchpad by Base 12 Innovations is designed for iPad and iPhone.  This drawing app can be used to create lines, circles, arcs, polygons, and conics and combinations of those for more complex drawings.  You can add text and math expressions, write and draw freeform, and switch between Cartesian, Classical, Isometric, and Polar forms.  At $4.99, it also comes with tutorials, geometric activities, practice questions.  A free version is also available, but it has fewer features.

Khan Academy imageKhan Academy app is free and compatible with iPhone, iPad, and iPod touch.  You can learn just about anything.  Multiple subjects are included.  "Learn using videos, interactive exercises, and in-depth articles in math (arithmetic, pre-algebra, algebra, geometry, trigonometry, statistics, calculus, linear algebra)."

Ventura Educational Systems iconVentura Educational Systems has a variety of iPad and iPhone apps for education.  For example, iPad apps for geometry:

Virtual Nerd Mobile Math iconVirtual Nerd Mobile Math by Pearson Education, Inc.
Platform: iPhone and iPad
Price: Free
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.

 

 

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Trigonometry and Precalculus

 

Trig Graphs JPGDave'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 precalculus, 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!

EngageNY/Common Core Curriculum for Precalculus and Advanced Topics: a full-year curriculum map with lessons organized in modules.

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 Adobe Flash accompanied by audio to provide an introduction to trigonometry based on the unit circle and the geometry of the right triangle.  The animations for illustrating the development of the six trig graphs are particularly useful for classroom demonstration.  The Flash Player is needed to view content.

Interactive Talking Math Book is "designed to be a complete precalculus 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.

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

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.

Open Course Library from the Washington State Board for Community and Technical College features multiple courses and course materials, free to use under a Creative Commons license.  Descriptions are per each course.

OpenStax open-source free online math textbooks, which are also peer-reviewed:

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

Study.com has online math courses in pre-algebra, algebra, geometry, statistics, precalculus, calculus, college math and more.  These feature video lessons from experienced educators, transcripts of those lessons, supplementary content, quizzes, and exams.  Learners can potentially earn college credit.

Trig and Precalculus on your mobile devices

Ezy Trigonometry iconEzy Trigonometry by KGiants is for iPad and iPhone.  This free app for high school learners, or others learning trigonometry, 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 imageKhan Academy app is free and compatible with iPhone, iPad, and iPod touch.  You can learn just about anything.  Multiple subjects are included.  "Learn using videos, interactive exercises, and in-depth articles in math (arithmetic, pre-algebra, algebra, geometry, trigonometry, statistics, calculus, linear algebra)."

Wolfram Precalculus Course Assistant iconWolfram Precalculus Course Assistant is available for about two dollars.  It covers topics in precalculus and trigonometry.  It can:

 

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:

Finding Pythagorean Triples

Get a quick list of Pythagorean Triples using a simulation from the University of Illinois Office of Mathematics, Science, and Technology Education.

Pythagorean Triplets Generator at MathBlog is quick.  Choose the maximum value of a hypotenuse and get possible Pythagorean triples!

 

 

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Calculus

 

Are you ready for calculus?

Students: Question markThe 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 you ready for your AP exams?

See resources, including released questions from prior exams, at the College Board AP Central for both the AB and BC exam levels.  Also see the AP Calculator Policy.  Graphing calculators are required for the AP Calculus exams and a list is provided on those that are acceptable.

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.

 

Calculus Integration JPGThe following resources should help your learners address calculus standards.

AP Calculus is a free online course (a MOOC) from DavidsonX, which addresses the "most difficult topics in your AP Calculus AB & Calculus BC courses."  Per the course description, "tricky topics are broken up into bite-sized pieces—with short instructional videos, interactive graphs, and practice problems written by many of the same people who write and grade your AP® Calculus exams."  It can be used in a blended classroom or as supplementary help and exam review for students.  Topics include limits, definition of derivative, chain rule, implicit differentiation, L'Hospital's Rule, Riemann sums, functions defined by definite integrals, modeling and solving differential equations, rectilinear motion, parametric equations, introduction to series, series convergence, and series manipulation.

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.

Calculus, posted at Curriki.org, "provides learning materials for all topics in the College Board’s course description for an AP Calculus AB course, as well as a small number of additional topics."  Broad categories in the collection include limits and continuity, derivatives, applications of derivatives, integrals, and applications of integrals.

Calculus Online Textbook by Gilbert Strang is posted at MIT OpenCourseware.  The book "is a useful resource for educators and self-learners alike. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. There is also an online Instructor's Manual and a student Study Guide."  Dr. Strang also has a series of videos "to show ways in which calculus is important in our lives."  They include "real-life examples to illustrate the concepts."

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 the section of 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 Online Integral Calculator from Wolfram Alpha.  Enter any expression and the integration is performed for you.  Also read about the history and applications of integration.

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 precalculus, 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!

Derivative Calculator is free and finds derivatives of functions.  "The Derivative Calculator supports computing first, second, …, fifth derivatives as well as differentiating functions with many variables (partial derivatives), implicit differentiation and calculating roots/zeros. You can also check your answers! Interactive graphs/plots help visualize and better understand the functions" (About section).

Elementary Calculus: An Infinitesimal Approach, On-line Edition, copyright 2000 and revised 2019 by H. Jerome Keisler, is a freely-available calculus text from the author.  You can download the whole book or selected chapters.

Integral Calculator is free and finds integrals and antiderivatives of functions.  It shows you the full step by step integration.  "All common integration techniques and even special functions are supported. The Integral Calculator supports definite and indefinite integrals (antiderivatives) as well as integrating functions with many variables. You can also check your answers! Interactive graphs/plots help visualize and better understand the functions" (About section).

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

Math 104: Calculus is a free online course from the Study.com 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.

Math Vault: Calculus contains a series of essays with explanations on a variety of calculus topics.  Examples include, but are not limited to the following:

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).

Open Course Library from the Washington State Board for Community and Technical College features multiple courses and course materials, free to use under a Creative Commons license.  Descriptions are provided for each course: Calculus I, Calculus II, Calculus III.

OpenStax open-source free online math textbooks, which are also peer-reviewed:

Phet Interactive Simulations: Calculus Grapher.  This simulation requires Flash.  Learners can draw the graph of any function and see graphs of its derivative and integral.

Study.com has online math courses in pre-algebra, algebra, geometry, statistics, precalculus, calculus, college math and more.  These feature video lessons from experienced educators, transcripts of those lessons, supplementary content, quizzes, and exams.  Learners can potentially earn college credit.  Study.com also has a specific collection of video lessons for AP Calculus AB & BC.

Teaching Calculus by Lin McMullin is a blog for high school teachers and students, especially Advanced Placement Calculus teachers and students.  You'll find comments, suggestions, hints, and observations on the topics for the course, and a video collection.  Of particular value is the section called "Thru the Year," which includes topics in an order that they typically taught month-by-month.

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 precalculus 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.

Why U contains animated videos in pre-algebra and algebra, topology, and infinite series.  The infinite series animation "makes complex concepts such as summation notation, infinitesimals, and convergence easily comprehensible."  Videos were written by Steve Goldman and animated by Mark Rodriquez.  The animation is outstanding.

Wolfram for Education has some free resources for teaching math.  For example, see Using Wolfram|Alpha: Examples for Calculus and Analysis.  This latter is useful for teaching topics in Calculus and getting step-by-step solutions to problems: derivatives, integrals, graphing, and so on.  Free unlimited practice problems are also available.

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 on your mobile devices

Aida Calculus by Pearson iconAida Calculus from Pearson Education, Inc. uses artificial intelligence to provide step-by-step feedback to calculus problems.  Users take a snapshot of their work using an iPhone or input their solution to a problem using the app's keyboard.  The app comes with interactive explainer videos, worked examples, suggested activities and relates the math to the real world.  This app is compatible with iPhone, iPod touch, and iPad.  It's free through 2019.

Calculus Pro by Feldman iconCalculus 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 iconKhan Academy app is free and compatible with iPhone, iPad, and iPod touch.  You can learn just about anything.  Multiple subjects are included.  "Learn using videos, interactive exercises, and in-depth articles in math (arithmetic, pre-algebra, algebra, geometry, trigonometry, statistics, calculus, linear algebra)."

Wolfram Calculus Course Assistant iconWolfram Calculus Course Assistant is available 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?

Play video 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.  Adobe 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.

More Supplementary Videos

Play video Dennis Wildfogel also has a series of videos on Making Sense of Calculus.  Several deal with the visualizing the derivative of terms such as x2, x3, x-1, x-2, x1/2, x1/3, x-1/2, x2/3, x3/2, then the Product Rule, and more to come.

At 3Blue1Brown, a YouTube channel, author Grant Sanderson provides "some combination of math and entertainment, depending on your disposition. The goal is for explanations to be driven by animations and for difficult problems to be made simple with changes in perspective" (About section).  Calculus students will benefit from his playlist on topics such as the Essence of Calculus, the Paradox of the Derivative, Derivative Formulas through Geometry, Visualizing the Chain Rule and Product Rule, Derivatives of Exponentials, Implicit Differentiation, Limits, Integration and the Fundamental Theorem of Calculus.  Other playlists are available.

 

 

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Statistics and Probability

What do we mean by Data and Probability?

Data--collection of facts and Probability--how likely something will occur

 

Statistics Deviation JPG

AP Statistics Resources from JMP software by SAS.  Per the website: "This collection of lab activities provides step-by-step instructions and guidance on the exploration and application of statistical methods covered in AP Statistics courses. These can be used as in-class activities or homework assignments."  Learn about regression, hypothesis testing, confidence intervals, probability and sampling distributions, exploring and describing data, and more.  There's also a learning library with tutorials and videos.

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.

HOT!: Crash Course Statistics is series of videos posted on YouTube that takes a closer look at how statistics play a significant role in everyday life "from batting averages and insurance rates to weather forecasting and smart assistants" and more.  The first of those was posted in January 2018.  You'll find episodes involving topics such as mathematical thinking, measures of central tendency and spread, data visualization, correlation, controlled experiments, sampling methods and bias in surveys.  This is a great motivational series and a math course, as well.

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.

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.

Meta-Calculator.com: With the Statistics Calculator you can find standard deviations, correlation coefficients, do regression analysis, and perform t-tests.

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: An Interactive 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.

Open Course Library from the Washington State Board for Community and Technical College features multiple courses and course materials, free to use under a Creative Commons license.  Per its description, Introduction to Statistics introduces "some of the more common tools for analyzing data of several types. ...The main goal of the course is to highlight the general assumptions and methods that underlie all statistical analysis. The purpose is to get a good understanding of the scope, and the limitations of these methods."  Learners also explore "assumptions behind the most common methods, in order to evaluate if they apply with reasonable accuracy to a given situation."

OpenStax open-source free online math textbooks, which are also peer-reviewed:

Probability & Statistics is a free online course from Carnegie Mellon's Open Learning Initiative designed to be equivalent to one-semester of a college statistics course.  The only prerequisite is basic algebra.  There are two versions: Probability and Statistics, which includes a classical approach to probability, and Statistical Reasoning.  Per the Overview: "Each course includes all expository text, simulations, case studies, comprehension tests, interactive learning exercises, and the StatTutor labs."  Further, "The course was designed to be used as a stand alone (with no instruction in the background) however studies have shown that it is best and most effectively used in the hybrid mode together with face to face instruction" (In-Depth Description section).

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.

Seeing Theory is a visual introduction to probability and statistics.  Multiple interactive visualizations include explanations and are provided in categories of basic probability, compound probability, distributions, statistical inference, and linear regression.  Students studying AP Statistics or introductory college statistics will value this site.

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.  Activities are available by grade band with an approximate time for completion of each.  Teacher and student versions are provided.  This is an excellent resource.

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.

Statistics and Probability on your mobile devices

Khan Academy imageKhan Academy app is free and compatible with iPhone, iPad, and iPod touch.  You can learn just about anything.  Multiple subjects are included.  "Learn using videos, interactive exercises, and in-depth articles in math (arithmetic, pre-algebra, algebra, geometry, trigonometry, statistics, calculus, linear algebra)."

Wolfram Statistics Course Assistant iconWolfram Statistics Course Assistant is available 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.

Friendly reminder GifData 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 Fathom Dynamic Data software, which allows you to create simulations.

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 is extensive for applets in data analysis, sampling distribution simulations, probability, statistical inference, and more. 
  • WISE: Web Interface for Statistics Education from Claremont University includes interactive tutorials and applets.
  • Rice Virtual Lab in Statistics

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

Help students to interpret data visualizations.

See What's Going On in This Graph? from The New York Times Learning Network.  In partnering with the American Statistical Association, the Times  posts graphs, charts, and maps each month and invites students to discuss them live.  These are valuable to help students gain statistical literacy and learn to read, interpret, and question data--a key 21st century skill.

 

 

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Multiple Subject Sites

Statistics Bar Graph JPG

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.

CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide.  In addition to its Flexbooks, there are online problem sets in multiple math subject areas.  There is a section for elementary math for grades 1-5 aligned to the Common Core standards.  It contains interactive math practice problems, videos, video hints, and full step-by-step solutions, all free.  Other sections include arithmetic for middle school, algebra, geometry, probability, statistics, trigonometry, analysis, and calculus.  Also see the CK-12 Foundation Dedicated Channel for videos on YouTube.  CK-12 has a section for college flexbooks, which include Elementary and Intermediate College Algebra and College Precalculus.

MATC LogoElectronic 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.

Flippedmath.com promotes "Flipped Mastery Learning: Mathematics Without Boundaries."  Free online courses in Algebra 1 (Common Core and Traditional), Geometry, Algebra 2 (Common Core and Traditional), Pre-Calculus, and AP Calculus include multiple videos.  They were developed to "allow students to demonstrate mastery of all concepts and progress at their individual pace."

GeoGebra LogoGeoGebra 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.  Visit Geogebra's Blog.  Geogebra also has a "Materials" section where you can also share and find materials for arithmetic, geometry, trigonometry, calculus, probability, algebra, functions, and statistics.

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.

HippoCampus Logo from Monterey Institute for Technology and EducationHippoCampus, 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.

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 secondary (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.

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.

MathsIsFun 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.

Mathispower4u by James Sousa contains a collection of over 6000 mini-lessons and videos organized by course and topic: number sense, arithmetic/prealgebra, algebra 1, algebra 2, geometry, math for liberal arts, trigonometry, calculus 1, calculus 2, calculus 3, linear algebra, differential equations.  The site also includes problem solving activities with video solutions.  Multiple Common Core topics are addressed in these.

MathVids.com: Numerous video lessons for middle school math, high school math, and college math subjects.

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.

Math Fortress offers free videos for algebra 1, geometry, algebra 2, pre-calculus, calculus (1, 2, and 3), linear algebra, and differential equations.  Depending on subject, you'll also find quizzes, worksheets, and equation sheets.

HOT!: Math Portal Online Calculators and Solvers by mathematician Miloš Petrović contains multiple math calculators and solvers grouped into 17 categories.  Use this site for problems including polynomial equations, rational expressions, systems of equations, vectors, matrices, complex numbers, plane and analytic geometry, statistics, trigonometry, calculus (limits, derivatives, integrals), sequences, series, financial, numbers, and more.  Per the description, "These calculators can help you with basic and advanced problems in high school and college math.  All of them are capable of performing exact computations.  Also, almost every of them will generate a step by step explanation."  The step by step explanations are exceptionally valuable for learners.  There are three other math sections on the site: formulas, tests and lessons.  Math lessons relate to algebra, calculus, analytic geometry, and linear algebra, and also include solved problems and examples for students to solve.

Math Warehouse has numerous animations, lessons, and interactive practice problems for algebra, geometry, trigonometry, and calculus.  There is also a collection of worksheets.

HOT!: New Jersey Center for Teaching and Learning: Course Materials include free STEM resources for K-12, including advanced level courses.  Mathematics courses for high school include algebra I and II, geometry, pre-calculus, AP Calculus at AB and BC levels, and Integrated Math levels I, II, and III.  They are also available in Spanish.  You'll find presentations, classwork-homework assignments (includes answers), lab activities, and assessments in unit resources, as appropriate for each course.

NRICH LogoNRICH from the University of Cambridge is devoted to enriching mathematics for learners in grades 1-12 with games, problems, investigations, and articles.  Resources are free.  The collection of activities for topics in secondary mathematics are designed to develop subject matter knowledge and problem solving skills.  These are organized by number, algebra, geometry and measures, handling data, thinking mathematically, and developing mathematical habits of mind.

OER Commons lists over 5700 open education resources for math, searchable by subject, standard, grade level, learning domain.

Patrick JMT: Just Math Tutorials by math teacher PatrickJMT contains an extensive collection of his videos for arithmetic, algebra, linear algebra, trigonometry, differential equations, discrete math, probability and statistics, and calculus.  They are free to use and are suitable for high school and above.  There is also a section on miscellaneous topics.

PBS LearningMedia includes free media resources searchable by grade level, standards, media type, and subject. Resources include video and audio segments, interactives, images, documents, lesson plans, productivity tools for teachers, self-paced lessons,and student-oriented activities. You'll find media from NOVA, Frontline, American Experience, and other public broadcasting and content partners.  The mathematics section features K-8 mathematics strands, and high school number and quantity, algebra, geometry, functions, statistics and probability.  Highly recommended.

Softschools.com provides free math worksheets, games, and online practice activities for preK-5, middle school, and high school (e.g., algebra 1 and 2, geometry, some trig and calculus) in multiple concepts.  These are organized by grade, subject, and topic.  Explanations of many concepts are provided.

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), precalculus (I and II), calculus, and differential equations.

Teachoo.com is "a free educational resource for mathematics which will help students in schools from Grades 9 to 12 with topics ranging from Real numbers and Polynomials to Integrals and Vector Algebra" per its developer, Davneet Singh.  You'll find conceptual explanations on numerous topics found in algebra, geometry, trigonometry, statistics/probability, pre-calculus, and calculus, including sample problems, and complete solutions.

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.

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.

 

 

Spice up your math class!

Friendly reminder Gif

Mudd Math Fun Facts GifMudd 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

The Cartoon Guide to Algebra The Cartoon Guide to Calculus The Cartoon Introduction to Statistics
Spice up your class with Larry Gonick's The Cartoon Guide to Algebra (2015), which covers all the algebra essentials. Demystify first-year calculus as your students delve into functions, limits, derivatives, and integrals.  The Cartoon Guide to Calculus (2011), also by Larry Gonick, is just what you need. Coauthored by Grady Klein and Alan Dabney, The Cartoon Introduction to Statistics (2013) adds humor to your introductory statistics course.

 

How about presenting a number of the day, or a math question or riddle of the day?

MathsStarters: Number of the Day

Mathopolis features a Mathematics Question of the Day for each grade 2-8 and high school geometry, algebra 1, algebra 2, statistics, and calculus.  Each has a worked solution.  These are great classroom starters.

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: Mathematics contains a database of free educational videos for math (and other subjects) organized by topic area and subcategories within each of those.  There are about 5000 videos in mathematics, and over 200 on standardized test skills and math study skills.  Videos are gathered from across the internet and are previewed by a team of professionals.  Teachers will find the database helpful, too, to get new ideas on how to approach material and videos can be used in their classrooms.  Highly recommended site!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.  Do some doodles.

Per Joan Boaler (2016) and colleagues at Stanford University,"Good mathematics teachers typically use visuals, manipulatives and motion to enhance students’ understanding of mathematical concepts" (p. 1).  "Drawing mathematical ideas helps mathematics users of any level, including mathematicians, to formulate ideas and develop understandings."  Students can represent their ideas in "a multitude of ways, such as through pictures, models, graphs, even doodles ... or cartoons" (p. 5).

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.

Doodles? See an extensive list of possibilities for Doodling in Math Class, posted on YouTube.

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 website: http://alexkajitani.com/the-rappin-mathematician/ 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.

See EmergingEdTech.com for 20 Free Tools for Making Comics and Cartoons for Teaching and Learning by Kelly Walsh (2014, December).  The tools are Web apps, iPad and Android apps.  As with any tool, try it out first, make sure it works in your classroom and ensure that the site contains appropriate content for learners' viewing.

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!

You could present a mathematician of the day or introduce the history of mathematics into the curriculum.

A Brief History of Mathematics hosted by the BBC in the UK contains a series of 10 podcasts in which Professor of Mathematics Marcus du Sautoy "reveals the personalities behind the calculations and argues that mathematics is the driving force behind modern science."  You'll learn about contributions of mathematicians and challenges they faced.  Those featured include Bourbaki, Hardy and Ramanujan, Poincare, Cantor, mathematicians who helped Einstein; Gauss, Galois, Fourier, Euler, Newton and Leibniz.  Each podcast is about 15 minutes.

Egyptian Math Hieroglyphs imageEgyptian Mathematics Numbers Hieroglyphs also includes some problems for kids.  You'll also find a history of ancient Egypt.

The Hundred Greatest Mathematicians of the Past created by James Dow Allen.  Get some facts on mathematicians such as Newton, Archimedes, Euclid, Gauss, Pythagoras, Lagrange, Fermat, Descartes, Leibniz, Hilbert, and more.

Mathigon's Timeline of Mathematics features the contributions of famous mathematicians from Thales in 600 BC to the present day. 

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.

The Mathematical Association of America: On This Day is a section of the site where you can click on any day to learn about a selection of historical events related to mathematics that occurred on that day.

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.

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 and More for Free!

Dynamic Paper imageDynamic Paper is a free applet from NCTM Illuminations.  With it you can design and print the following, as stated in the app instructions:

  • Nets – two-dimensional outlines of three-dimensional shapes, including regular polyhedra, prisms, pyramids, cylinders and cones
  • Graph Paper – coordinate graphs, polar coordinates, logarithmic graph paper
  • Number Lines – including positive and negative coordinates
  • Number Grids – hundreds boards and the like
  • Tessellations – tiling patterns involving triangles, quadrilaterals, and hexagons
  • Shapes – pattern blocks, attribute blocks, and color tiles
  • Spinners – up to 16 sectors, with adjustable sizes

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

Cartesian Graph PaperPrint 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.

Printable Paper includes more than Cartesian, polar, and logarithmic graph paper.  There's pentagon, hexagon, octagon graph paper, 3D graph paper, and paper templates in other areas beyond math.

 

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References

Boaler, J., Chen, L., Williams, C., & Cordero, M. (2016). Seeing as understanding: The importance of visual mathematics for our brain and learning. Journal of Applied & Computational Mathematics, 5(5). doi: 10.4172/2168-9679.1000325.  Retrieved from https://www.youcubed.org/evidence/research-articles/

Hohenwarter, M., & Preiner, J. (2007, March). Dynamic mathematics with GeoGebra [online]. Journal of Online Mathematics and Its Applications, 7, article 1448. Retrieved from https://www.maa.org/node/115964 

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 https://www2.ed.gov/about/bdscomm/list/mathpanel/index.html

 

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Binoculars GifSee related topics:  Math Manipulatives and Standardized Test Preparation.