Fibonacci Sequence & a Cool Pattern

Image from ShutterStock.

Image from ShutterStock.


The Fibonacci sequence adds consecutive terms:

  • 1
  • 1
  • 1 + 1 = 2
  • 2 + 1 = 3
  • 3 + 2 = 5
  • 5 + 3 = 8
  • 8 + 5 = 13
  • 13 + 8 = 21
  • 21 + 13 = 34
  • 34 + 21 = 55
  • 55 + 34 = 89

Since the last two terms were 55 and 89, we would add these together to get 89 + 55 = 144.

Then you would add 144 and 89 to make 233, and so on.

I saw a cool pattern involving the Fibonacci sequence recently at the Mathemagical Site:

Fibonacci Triples via Mathemagical Site

This involves Fibonacci triples.

A Fibonacci triple consists of three consecutive numbers from the Fibonacci sequence, such as:

  • 1, 1, 2
  • 1, 2, 3
  • 2, 3, 5
  • 3, 5, 8
  • 5, 8, 13
  • 8, 13, 21
  • 13, 21, 34

As shown on the Mathemagical Site, the square of the middle number is always one less or one more than the product of the first and third numbers:


Here are a few examples:

  • (2, 3, 5): 3 x 3 = 2 x 5 – 1
  • (3, 5, 8): 5 x 5 = 3 x 8 + 1
  • (5, 8, 13): 8 x 8 = 5 x 13 – 1

Curious about this, I’ve been working through the algebra, and finally came up with an algebraic proof, which follows.

My proof is algebraic and not necessarily obvious, but since I worked it out, I thought I would share it here. 🙂

We begin with two facts about the Fibonacci sequence:


These are two different ways of saying that if you add two consecutive numbers from the Fibonacci sequence, you get the next number in the sequence.

Now solve for xn in each sequence:


Multiply these together:


Foil this out:


Recall that


Plug this into the first term on the right-hand side of the previous equation:




Two of these terms cancel (the remaining terms are rearranged):


Believe it or not, this basically concludes the proof. The remainder is basically interpreting this result.

This is a recursion relation that relates the square of the n-th term to the square of the previous term (xn-1 times itself).

Following is the Fibonacci sequence, labeling values of n:

  • n = 1 is 1.
  • n = 2 is 1.
  • n = 3 is 2.
  • n = 4 is 3.
  • n = 5 is 5.
  • n = 6 is 8.
  • n = 7 is 13.
  • n = 8 is 21.

Let’s plug in n = 3 and see what happens:


If instead you plug in n = 4, you get:


Now just plug in these two expressions (x3x3 – x4x2 and x4x4 – x5x3) into the previous recursion relation and you can prove that all of the Fibonacci triples satisfy one of the following relations:


That is, if x3x3 – x4x2 = 1 and x4x4 – x5x3 = -1, the previous recursion relation gives similar expressions for x5x5 – x6x4, x6x6 – x7x5, and so on.


Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

  • Algebra Essentials Practice Workbook with Answers
  • Trigonometry Essentials Practice Workbook with Answers
  • Learn or Review Trigonometry: Essential Skills

A Challenging Dissection Challenge

What a cool visual puzzle from Mathemagical Site. Better think out of the box for this one.

Cursive Handwriting for Math Lovers

Cover designed by Melissa Stevens at



Julie Harper is the author of a popular series of handwriting workbooks.

I recently had the opportunity to coauthor a handwriting workbook with her, geared toward math lovers.

This unique book combines math terms and concepts with handwriting practice.

Find this book here:

Find Julie Harper’s handwriting books:

Find Chris McMullen’s math workbooks:


Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Algebra with Roman Numerals

Algebra Roman Numerals


Aren’t you glad you don’t have to do algebra with Roman numerals?

The problem is: Xx + X = CX.

This translates to 10x + 10 = 110.

Subtract X from both sides: Xx = CX – X = C.

Translation from Roman numerals: 10x = 110 – 10 = 100.

Now divide both sides by the coefficient: x = C/X = X.

Not expressed in Roman numerals, it looks like this: x = 100/10 = 10.


Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Practice Arithmetic with Geometric Dice



My daughter recently received a huge pack of cool geometric dice in several different colors:

  • An icosahedron (20-sided polyhedron) with the numbers 1 thru 20.
  • A dodecahedron (12-sided polyhedron) with the numbers 1 thru 12.
  • A decahedron (10-sided polyhedron) with the numbers 0 thru 9, and another with the numbers 0 thru 90.
  • An octahedron (8-sided polyhedron) with the numbers 1 thru 8.
  • A cube (6-sided polyhedron) with the numbers 1 thru 6.

These dice turned out to be really handy for learning addition and multiplication facts.


You can easily practice addition facts and multiplication facts with these dice.

Here are some examples:

  • ADDITION/MULTIPLICATION. Roll two decahedra, marked 0 thru 9. Add or multiply the two numbers to practice addition or multiplication facts 0 thru 9.
  • SMALLER NUMBERS. Roll two cubes, marked 1 thru 6. Add or multiply the two numbers to practice addition or multiplication facts 1 thru 6. The cubes let students focus on the smaller numbers first, before working with 7, 8, and 9. (If you want more basic practice, find tetrahedra—4-sided polyhedra—marked 1 thru 4.)
  • FOCUSED FACTS. Roll one decahedron, marked 0 thru 9. For example, suppose you want to practice your multiplication table of 4’s. Simply multiply the die by 4. This lets you concentrate on a single number’s addition or multiplication facts at a time.
  • 11 THRU 20. Advance to dodecahedra or icosahedra to practice the facts 1 thru 12 or 1 thru 20.
  • SUBTRACTION. Practice subtraction facts using an icosahedron and a decahedron. Be careful to subtract the smaller number from the larger number; sometimes, the number on the 10-sided die will be larger. (Advanced students who are learning about negative numbers can use these to sometimes subtract the larger number from the smaller number.)
  • TENS. Multiply powers of 10 using one decahedron with 0 thru 90 and another with 0 thru 9. Or roll one decahedron and multiply that by 10 for more basic tens practice.
  • POWERS. Roll a tetrahedron and a decahedron together to learn about powers. Let the tetrahedron serve as the exponent.
  • FRACTIONS. Roll four decahedra to learn about fractions. These will give you the numerators and denominators of two fractions. Then you can add them, multiply them, divide them, compare them (figure out which is bigger), or subtract them (but first find out which is larger).
  • DICE WAR. If you have several dice to divide equally, you can play dice war with a friend. Each player rolls two dice. Either add or multiply the numbers (choose one before the game begins). The higher sum or product collects both dice.


Another cool thing about using a variety of geometric dice to play math games is that kids get to hold various geometric solids in their hands, see how they look, get a feel for them, and after much use remember how many sides each shape has.

Better than just being told or shown what a dodecahedron is… hold one in your hands, roll it, play with it for months. Then you’ll ‘know’ that solid when you hear its name. (It helps when someone learns and uses the correct names while using the dice.)

Many of these dice packages are sold with role-playing games in mind, but there is no reason that you can’t use them for math practice instead.


  • Polyhedron: a three-dimensional solid.
  • Polygon: a two-dimensional object, not a solid; it’s flat.
  • Polyhedra is plural, polyhedron is singular.
  • Dice is plural, die is singular.


Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Tessellations Hexagon Square Triangle Rhombus Trapezoid Star Patterns

Lattice Hexagons


A tessellation is a repeated two-dimensional geometric pattern, with tiles arranged together without any space or overlap.

Simple tessellations can be made by creating a two-dimensional lattice out of regular geometric shapes, like equilateral triangles, squares, and hexagons. Not any regular polygon will work, however. For example, it won’t work with pentagons.

Tessellations can also be made from irregular polygons. (A regular polygon is one with equal sides and angles.) All quadrilaterals can form tessellations. (Quadrilaterals are polygons with four sides.) Although regular pentagons don’t tessellate, some irregular polygons can (such as the pentagon made by placing an isosceles triangles on a square, as children often do to draw a simple picture of a house).

There are many other shapes that tessellate, such as stars combined with other shapes. Even arrangements of curved objects can tessellate. Some of the more extreme examples of this can be seen in M.C. Escher’s artwork.

The lattice structure below can be shaded in several different ways to create simple geometric patterns that tessellate:


For example, here is a tessellation composed of hexagons:

Lattice Hexagons

Here is another made from triangles:

Lattice Triangles

This one is made with squares:

Lattice Squares

The same pattern can make a tessellation with stars and hexagons:

Lattice Star

Here are a variety of basic geometric shapes that can tessellate from this same pattern, including a hexagon, triangle, square, trapezoid, parallelogram, pentagon (irregular), rhombus (diamond), and rectangle:

Lattice Shapes


Copyright © 2014 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Singing Arithmetic



Sound can be a useful learning tool.

I saw this firsthand a few weeks ago when I met a couple and their daughter at a local restaurant.

Their daughters were learning arithmetic, history, language, science, and other facts by singing.

They were using a special curriculum that included songs for many basic things that students learn in various subjects.

One of the girls sang a few different songs and it was quite impressive how much she had learned from singing.

My daughter enjoys singing, too. Recently, I heard her singing some of her multiplication facts.

You can simply sing facts in order, like the table of fours: 4 times 1 equals 4, 4 times 2 equals 8, 4 times 3 equals 12, etc. Or you can add an occasional phrase here and there, especially if it rhymes.

It’s good for patterns, too, like 5, 10, 15, 20, 25, etc.

Most children learn the alphabet song. There are many other songs available to help with learning.

I recall listening to the Schoolhouse Rock songs when I was a kid.

Most people learn better one way than another. So using a variety of teaching strategies helps each student learn through his or her strength.

But I think it’s also important not to let every student rely only on his or her strengths. It’s important to develop the other learning styles, too.

We become better students not just by focusing on improving our strengths, but also by improving our weaknesses.

Singing arithmetic isn’t for everybody, though. I’m sure nobody wants to hear me sing my multiplication facts—or anything else, for that matter. 🙂

Copyright © 2014 Chris McMullen, author of the Improve Your Math Fluency Series

The Particle Adventure



In chemistry, students learn about protons, neutrons, and electrons, but it turns out that there are many more particles than just these—and the protons and neutrons are themselves made up of even smaller particles called quarks.

There is an amazing world of particle physics. Unfortunately, particle physics isn’t one of the standard science classes. Biology, chemistry, physics, and astronomy are more common. An actual particle physics class is usually only an upper level course taught at universities, packed with mathematics.

Occasionally, a little exposure to particle physics is given through a physical science course, for example.

Yet much is known about elementary particles and can be taught without introducing mathematics into the picture.

The Particle Adventure provides a good interactive introduction to elementary particles:

Websites for high-energy colliders or particle accelerators sometimes have an educational page. For example, here is CERN’s page (home of the Large Hadron Collider) for students and educators:

Sometimes, it can be refreshing to learn something new, which isn’t in the standard curriculum.

Go on an adventure with Higgs bosons, gluons, quarks, muons, tau leptons, neutrinos, and more. Learn about the four fundamental forces of nature, fractionally charged particles, how particles get mass, and discover patterns in a different kind of periodic table—one of elementary particles.

Copyright © 2014 Chris McMullen, author of the Improve Your Math Fluency Series