Ian Byrd has created a wonderful collection of short videos for gifted students and their teachers in many different content areas, including math. He even has videos related to social and emotional needs of gifted students. Most of the videos introduce deep, thought-provoking questions and activities for use in the classroom. A couple of the videos are based on activities in the Advanced Common Core Math Explorations series: "Fraction Puzzlers" from the Fractions book and "Factor Scramble" from the Factors and Multiples book.
Mathematician Alexander Bogomolny describes his website as "Interactive Mathematics Miscellany and Puzzles." One of his stated goals is to enhance our appreciation of the beauty of mathematics. This site offers a cornucopia of mathematical puzzles, problems, games, explorations, and explanations. Most of the site is dedicated to material appropriate for students of at least middle school age. However, if you look at the "Simple Math" menu, you will find things for younger students as well. He began the site in 1997, and its update chronology suggests that he adds something significant nearly every day. You could spend years exploring the mathematical wonders in Cut-the-Knot!
GeoGebra describes itself as "open source dynamic mathematics software for learning and teaching at all levels." I suggest beginning with the geometry tool and exploring for a while. Play with points, lines, circles, polygons, and geometric transformations. Click and drag your creations and watch what happens! As you explore and learn more about GeoGebra's capabilities, you will be developing skill with a tool that will be a resource for open-ended explorations for your entire math career. To help you get started, there is a manual with interactive tutorials here.
Illuminations from NCTM
There is something for everyone in this extensive selection of engaging lessons, games, interactives, and brain-teasers from the National Council of Teachers of Mathematics. All tasks are built on best practices in math instruction and are searchable by grade band and content strand. You choose your area of interest and level of challenge.
Ken-ken puzzles are like sudoku with a challenging mathematical twist. The grid is divided into "cages," each of which has a target number that you must reach by using the operation defined for that cage. These puzzles develop not only computational facility but problem-solving skills, number sense, and a deeper understanding of number properties. Ken-ken can be enjoyed by anyone from early elementary students to research mathematicians. This website will generate ken-ken puzzles for you based on your choice of grade level, mathematical operations, the size of the grid, and general level of challenge. You may print them or use them online.
This site was built by a mathematician who loves creating great problems that are accessible to everyone but can be taken to virtually any level of challenge desired. The problems may be sorted by grade level or content area. They are presented through the use of engaging videos, and they develop students' mathematical creativity as well as problem solving skills. Read about his set of 13 "million dollar problems" for kids! An incredible website!
Problems of the Week, by Math Forum @ Drexel
The Math Forum offers a set of weekly math challenges that you may search by grade level, content strand, level of challenge, and Common Core content standards. Various levels of membership are available and may include current problems, solutions, access to archives of problems, mentoring, opportunities to submit solutions (and, of course, to view other's submissions), and professional development for teachers. The Other Problems and Puzzles tab on the small navigation bar near the top of the site will also take you to a long list of high quality web-based resources that are organized by grade level (a number of which are also included on this list).
Three-Act Math Problems, by Dan Meyer
This link takes you to the Author's Choice page of Dan Meyer's blog. Near the top of the page, you will see the Three-Act Math Problems. He begins by telling you what they are and how to teach them. Then he gives you a large bank of tasks to use. The type of teaching he is advocating is challenging but deeply rewarding for students and teachers alike.
The Year Game
This is another part of the Math Forum website (mentioned above in connection with the "Problems of the Week"). Participants use the digits in the current year along with a specified collection of mathematical operations to create the whole numbers 1 - 100. Students are encouraged to submit responses and some are posted. The details of the rules vary a little from year to year. Materials and support for implementing the challenge in your classroom are provided.
The activities on Yummy Math are accessible to all students and offer an opportunity to apply fundamental skills and concepts to interesting real-world situations. The focus is more on relevance and interest than on a high level of challenge. The activities are searchable by "genre" (real-world connection) and grade level and are aligned to Common Core content standards. The activities themselves are available for free. Solutions and other support materials are available by purchasing a membership at reasonable price.
Please contact me if you know of other excellent web resources for talented and curious math students that should be included on this page!