# Multiplying points in the plane

It is easy to add points in the plane “coordinatewise”, that is, by writing (x,y)+(z,w)=(x+z,y+w). This even has a very nice geometric interpretation: completing the parallelogram with first three corners (0,0), (x,y), and (z,w). But it isn’t easy to multiply points because the simple rule (x,y)*(z,w)=(xz,yw) doesn’t allow division. In this activity we looked for something better. Continue reading Multiplying points in the plane

# Counting simplified fractions

We all know that many fractions can be simplified, for example, 6/20 may be better written as 3/10. So how many fractions (less than 1) really use a denominator of 20? The answer turns out to be exactly eight… 1/20, 3/20, 7/20, 9/20, 11/20, 13/20, 17/20, and 19/20. But is there a rhyme or reason to this answer? Continue reading Counting simplified fractions

# Multiplication with a slide rule

Anyone can see how to add using a pair of rulers. Say we want to add 3+5. If you slide the “0” of one ruler underneath the “3” of the second ruler, then the “5” of the first ruler appears underneath the “8” of the second ruler. But can we multiply this way? Continue reading Multiplication with a slide rule

# Patterns in the Fibonacci numbers

This special discussion was led by Zach Teitler and concerned patterns in the famous Fibonacci sequence of numbers. This sequence begins with 1,1 and then each successive number is obtained by taking the sum of the previous two. Here are the next few in the list:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169

Which Fibonacci numbers are even? Which ones are odd? Which Fibonacci numbers are divisible by 3? In this discussion we answered these questions and many more. Continue reading Patterns in the Fibonacci numbers

# Ways to write numbers

How do we write our numbers? When ten units get together, they make one new thing. We keep this new thing in a separate box to keep track of its status. This is called the 1←10 rule. Some people think we use the 1←10 rule because we have ten fingers. But what if we had a different number of fingers? Or a different type of rule altogether? Continue reading Ways to write numbers