The Exploding Dots story comes in 12 short-and-swift explorations
What to do:
Start with Exploration 1 and move on through! Go as far as you and your students desire.
At each stage…
1.) Share and discuss the videos with your students
2.) Share the written materials and handouts with your students (Click here for non-English materials.)
3.) And perhaps guide your students to our interactive EXPLODING DOTS web app too
Or
You can just conduct your own classroom experience using nothing but a chalkboard following what you see in the videos.
EXPLORATION 1: A MYSTERIOUS MACHINE
Here we will introduce and interact with a mysterious machine that sets the scene for the entire EXPLODING DOTS story and the transformational mindset to mathematics
Topic: Discovering place-value
Grades: All
Written guide for this Experience:
Student Handouts for this Experience:
Interactive web app for this Experience:
EXPLORATION 2: A MATHEMATICAL INSIGHT
Here we figure out what these machines are actually doing and discover place-value. We now have a powerful visual understanding of this subtle concept that will profoundly help grade-school, middle-school, and high-school students alike.
Topic: Deeply understanding place-value. Understanding why we humans are drawn to base-ten
Grades: All
Written guide for this Experience:
Student Handouts for this Experience:
Interactive web app for this Experience:
EXPLORATION 3: ADDITION AND MULTIPLICATION
Here we make profound sense of the long addition and multiplication algorithms.
Topic: Long Addition. Multiplication.
Grades: All, but of particular interest to K-6.
Written guide for this Experience:
Student Handouts for this Experience:
Interactive web app for this Experience:
EXPLORATION 4: SUBTRACTION
See subtraction as the addition of the opposite. This insight, coupled with the Exploding Dots machinery, provides natural and deep insight into long subtraction.
Topic: Negative numbers. Long Subtraction.
Grades: All, but of particular interest to K-6.
Written guide for this Experience:
Student Handouts for this Experience:
Interactive web app for this Experience:
EXPLORATION 5: DIVISION
Long division made exceptionally clear.
Topic: Long Division. Long Division with remainders.
Grades: All, but of particular interest to K-6.
Written guide for this Experience:
Student Handouts for this Experience:
Interactive web app for this Experience:
EXPLORATION 6: ALL BASES, ALL AT ONCE: POLYNOMIALS
Here we discover that all the work of K-6 arithmetic is essentially high-school polynomial algebra.
Topic: Polynomials. Division of Polynomials. Synthetic division in disguise.
Grades: All, but of particular interest to 7-12
Written guide for this Experience:
Student Handouts for this Experience:
Interactive web app for this Experience:
EXPLORATION 7: INFINITE SUMS
Here we play with “infinitely long polynomials,” discover the geometric series formula, and more.
Topic: The geometric series formula.
Grades: All, but of particular interest to 7-12
EXPLORATION 8: DECIMALS
Another infinite process has introduced more boxes into the machines. We discover decimal numbers.
Topic: Decimals. Converting fractions into decimals. Constructing irrational numbers.
Grades: All, but of particular interest to 7-12
EXPLORATION 9: WEIRD AND WILD MACHINES
Now we are just playing. We discover fractional bases, negative bases, and more. Each usually comes with unsolved research questions.
Topic: Base-one-and-a-half and other fractional bases. Negative and other bases.
Grades: All who want to have wild intellectual fun
EXPLORATION 10: UNUSUAL NUMBERS FOR UNUSUAL MATHEMATICS
Is 0.9999… equal to one or does it not equal one? We explore this question and an even wilder one than that!
Topic: Convergence of infinite processes (limits). p-adic numbers.
Grades: All who want to have wild intellectual fun
EXPLORATION 11: GRAPE CODES AND NAPIER'S CHECKERBOARDS
Making Exploding Dots two-dimensional allows you to do algebra with polynomials of two variables: x and y. Scottish mathematician of the 1600s essentially did this with his multiplication checkerboard.
Topic: Napier’s historical approach to multiplication and division. Polynomial algebra with two-variables.
Grades: All who want to have wild intellectual fun
Written guide for this Experience:
Interactive web app for this Experience:
Go to this special site for this experience.
EXPLORATION 12: PUZZLES THAT CAN BE SOLVED WITH EXPLODING DOTS
Here is a growing collection of puzzles – some classic, some not-so-classic, some new – that can be explained with Exploding Dots. Puzzles of all levels for all ages can be found here.
Topic: Puzzles galore!
Grades: Absolutely everyone.
Written guide for this Experience:
Interactive web app for this Experience:
Go to this special site for this experience.