# More Widgets

Once I got started designing the Math Widgets series I realized it was going to be a lot of fun.  The study of mathematics is an immensely broad topic and for kids who are really into math it is wonderful if a teacher can provide enrichment experiences that introduce new ideas and concepts that go beyond “what’s on the test”.  The Math Widgets series is a collection of tools with these kids and teachers in mind. Math Widgets III explores some special concepts in number theory.  For example, the Multibase Abacus piques one’s interest in alternative number systems.  What if we only had 8 fingers instead of 10?  Would we use a base 8 system?  With the Multibase Abacus students can investigate base 8 and any other base from 2 to 10. On the Multibase Abacus, numbers are represented by tapping beads to add or subtract values from the columns of various place value systems.  In the example below we are showing a base 5 system.  Starting on the right we have the ones place labeled with 50.  Moving to the left we have 5’s, 25’s, 125’s, 625’s etc.  Tapping one of the column labels results in a bubble showing the value of the place in base 10.  In the quiz mode students are challenged to represent randomly selected numbers in randomly selected bases.  Quite a few skills come into play when trying to answer the questions posed by the Multibase Abacus, including: estimation, addition, subtraction, multiplication and using exponents. Clock Arithmetic is another topic in math that is perfect for an enriched curriculum.  Whenever we divide two integers there is always a remainder (sometimes it’s zero).  Clock Arithmetic is a widget that encourages student’s to explore this concept which is known as Modular Arithmetic.  In programming languages there is a special operator used to signify the modulus function.  If a=7 and b=3 then in computer code a%b returns 1 since one is the remainder when 7 is divided by 3.  Sometimes the word ‘mod’ is used instead of the “%” symbol to describe this calculation, for example 15 mod 12 is equal to 3.  In the example below, tapping the 2 is the correct response. Once in a while, but rarely, you might find a newspaper article about a mathematician solving a problem or finding a proof that has been elusive for hundreds of years.  For example, the conjecture  that no three positive integers ab, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known to have infinitely many solutions since antiquity.  First conjectured by Pierre de Fermat in 1637, this theorem had been the focus of study by mathematician for centuries and was finally proven in 1995 by Andrew Wiles.  Goldbach’s Conjecture has never been proven. It was first posed in a letter from Goldbach to Euler in 1742.  The conjecture posits that every even integer greater than 2 can be expressed as the sum of two primes.  Use the Goldbach’s Conjecture widget to explore this idea by finding the primes whose sum is the given even numbers. The Hexagon Arithmetic Widget is a six number system that uses emojis as the symbols.  Counting moves in a clockwise direction around the hexagon.  The object is to complete the addition table using logic to figure out the pattern. Math Widgets I, II and III are offered exclusively by the iTunes App Store and sell for \$1.99 each.  Educational discounts are available for schools.  Please visit our website for more information about these and other apps for education.