Quadratics

1.3 Can We Trust Patterns?

Lesson materials located below the video overview.

Here is one of my favorite puzzles:

 

EXAMPLE 11: CAN WE TRUST PATTERNS?
What are the numbers counting? What is the next number in the sequence? (Be sure to actually finish drawing the sixth diagram and to count whatever it is you should be counting!)

dots on a circle

This puzzle is counting the number of pieces formed by connecting lines between each and every pair of dots placed on the rim of a circle. One dot gives 1 piece. Two dots gives 2 pieces. Three dots yields 4 pieces. Four dots gives 8 and five dots 16 pieces.

 

We’ve been trusting patterns all along, so clearly six dots on the rim of a circle will yield 32 pieces. (We’re doubling every time it seems.)

But try as you might, you do not get 32 pieces!

If you a person who evenly spaces ones dots, then you see only 30 pieces. If your dots are not symmetrically placed you will see an extra piece at the center of the picture and count a total of 31 pieces.

 

MORAL … We can’t actually trust patterns! In the end we have to rely on logical thinking to decide whether or not a pattern we encounter might or might not be true. Mathematicians might be excited and motivated by patterns, but they will never believe them until they can be justified with reasoning.

 

QUESTIONAssume dots are always spaced so as to produce the maximum number of pieces possible. This yields the sequence 1, 2, 4, 8, 16, 31, ….

Show that the next number in this sequence, for seven dots on the rim, is 57.

What are the next few numbers in the sequence? Dare I ask … any patterns???

 

This “DOTS ON A CIRCLE” activity is written about in great depth in the COMPANION GUIDE to this QUADRATICS course – written as a handout to share with students. This handout goes deep into the mathematics of this puzzle. (And yes, it leads you and your students to a formula for numbers of the sequence!)

 

 

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