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