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

1 + 1 = 10

1 Plus 1


Let me ask a simple question:

How can 1 + 1 equal 10?

Is your instinct to prove that it’s wrong?

If so, then you’ll fail to answer the question.

Because I was serious: 1 + 1 can equal 10.

No, you don’t have to break any rules of math.

At the end of this article I will explain how 1 + 1 can equal 10 without breaking any rules of math.

In the meantime, I wish to use this to illustrate a point.

We all know that creative problem-solving requires thinking outside the box, right?

Despite this, we often find ourselves not only thinking inside the box, but fighting to stay in the box.

When we close our mind to the possibility that the solution lies outside the box, there is no chance of finding such a solution.


How can 1 + 1 = 10 without breaking any rules of math?

If you think 1 + 1 can only equal 2, that’s because you’re in the habit of working with base 10.

Very often, when a difficult problem can only be solved through a creative solution, the difficulty lies in overcoming incorrect assumptions that we take for granted.

Let’s consider the binary number system (base two).

In the binary number system, the only digits are 0 and 1.

The first number is 1, followed by 10, then 11, then 100, 101, 110, 111, 1000, 1001, etc.

In this system, the number 2 doesn’t exist. The second number is 10.

So 1 + 1 = 10 in the binary number system.

The number 10 is the “second” number of the binary number system. It’s not the “tenth” number in this system.

The solution was simple, just as simple as 1 + 1 = 2.

If it seemed impossible or difficult, it was just a matter of looking beyond assumptions that we often take for granted.


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

Breaking It Down Doesn’t Mean Dumbing It Down



I’ve taught from both sides of the spectrum:

  • high-ability, highly motivated math/science students in advanced courses
  • students struggling to survive the most basic physical science classes

Yet, in both cases, I’ve seen two common factors:

  • even the brightest students benefit when the material is broken down
  • it really doesn’t help anyone to dumb it down


Breaking it down is different from dumbing it down:

  • Breaking it down means showing how a more advanced concept is built from more basic concepts.
  • Dumbing it down means accepting a simpler, but not as correct or complete, concept as a substitute for a more advanced concept.
  • (Dumbing it down sometimes also refers to the way that this is done, making the person feel inferior, i.e. unable to understand the more advanced concept.)

Breaking it down can be a very helpful skill for communicating effectively:

  • It’s crucial to master the fundamentals in order to understand more advanced concepts.
  • First, teach the fundamentals very clearly.
  • Then build up more advanced concepts from these fundamentals.

Even the most advanced students can better understand advanced concepts by understanding how they are built up from more fundamental concepts.

Very often, when an advanced student makes a mistake applying an advanced concept, it’s because the student doesn’t fully understand the underlying fundamental concepts or doesn’t fully understand the relationship between those fundamental concepts and the more advanced concept from which it is built.

Advanced concepts are more accessible to struggling students, too, when they are broken down into more fundamental concepts.

(I would go so far as to say that when you can’t explain an advanced idea in simpler terms, then you really don’t understand that advanced concept as well as you should. However, there are occasions when it would take an enormous amount of time to fill in the gaps between the simpler terms and the more advanced terms. For example, try breaking down string theory so that anyone can understand it without sacrificing the material in any way. It’s a formidable task, and a few authors have done an excellent job with this, but there sure isn’t a suitable five-minute explanation to be found.)


The keys to breaking it down without also dumbing it down are:

  • Don’t sacrifice the content. Let the fundamentals build up to the final product.
  • Don’t oversimplify the advanced concept, losing important aspects of the complete, correct formulation.
  • Don’t settle for a lower level of understanding in the final stage.
  • Don’t make the student feel inferior in the process.

My goal as a teacher is to make it possible for every student to understand the material well in its most complete, correct formulation.

Not every student may reach this level of understanding along the same path, however. The first step is to master the fundamentals, and then build more advanced concepts up from this foundation.

(Of course, learning also involves motivation and diligence on the part of the student. My goal is to make learning possible, not to offer students a path of no effort. Unfortunately, not every student will choose to follow the path to success as you suggest it.)


I will illustrate the distinction between breaking it down and dumbing it down with an example.

I will use Newton’s first law of motion to illustrate this.

Objective: Students understand Newton’s first law when it is phrased, “Every object has a natural tendency to maintain constant momentum.”

Unfortunately, it is common for physical science textbooks and courses to simply dumb this down. Here are a couple of examples of how it can be dumbed down:

  • Every object has a natural tendency to maintain constant speed.
  • An object in motion tends to stay in motion, while an object at rest tends to stay at rest.

I call this dumbing it down because these statements aren’t equivalent to the original. The original statement is more complete and more precise. In practice, the distinction is important.

The original statement can be understood well, without sacrificing the content in any way, simply by breaking it down.

The way to do this is to first teach the fundamentals and then show how to build the main idea up from the fundamentals:

  • Velocity is a combination of speed and direction.
  • Momentum is mass times velocity.
  • All objects have a natural tendency to maintain constant momentum.
  • For an object with constant mass (that’s most objects), constant momentum means constant velocity.
  • Constant velocity means traveling in a straight line with constant speed.
  • So, for most objects, maintaining constant momentum means traveling in a straight line with constant speed.
  • For a rocket, mass isn’t constant because it’s ejecting steam. Rockets have a natural tendency to maintain constant momentum, but not a natural tendency to maintain constant velocity. That’s why it’s more precise to say momentum than velocity. For most objects, however, this distinction isn’t important.

There isn’t any point in this breakdown that a basic student can’t grasp, which means there is no reason that every student shouldn’t be able to grasp the full meaning of Newton’s first law of motion.

If you’re wondering what’s wrong with, “An object in motion tends to stay in motion, while an object at rest tends to stay at rest,” it’s because “motion” is quite vague, and, in fact, includes situations that are inconsistent with Newton’s first law. Acceleration, for example, is a kind of motion, but objects in motion do not tend to have acceleration. It’s more precise to say constant momentum or constant velocity than to say “stay in motion.”

(Of course, illustrating the concept by showing how to apply it to a variety of examples also aids greatly in helping students understand it. Also, some students will need additional help fully understanding specific points on the list above. However, with effort and assistance, everyone can master Newton’s first law of motion.)


Imagine a teacher has two classes. One is a physical science class; the other is a physics class.

The instructor teaches the physics class the proper definition of Newton’s first law, but dumbs this down for the physical science class.

Now, imagine that one physical science student and one physics student get together and chat.

They happen to see another student pass by on a skateboard. The skateboard strikes a curb and the student falls forward.

The two students excitedly begin describing how this relates to Newton’s first law.

The physical science student describes how the skateboarder stayed in motion.

The physics students corrects the physical science student, describing how the skateboarder maintained constant momentum.

“What’s momentum?” wonders the physical science student, suddenly feeling dumb. But why? There is no reason that the physical science student can’t grasp this same definition in the same terms.

It’s true that they are enrolled in different classes. Physics will inherently involve much more math. That doesn’t make the physical science student stupid, of course. (In fact, I’ve found some very capable math students in my physical science classes, and sometimes there are physics students with very rusty algebra skills.) It’s just the difference in the nature of the courses, including their goals. Physical science is focused more on the concepts (and also includes some chemistry, and perhaps geology or astronomy, in exchange for less physics coverage), and may contain more range and less depth. Physics is strongly oriented toward how to apply mathematics to solve problems (which requires understanding the fundamental concepts very well). But pick any concept that’s common to both courses, and there isn’t anything that a physical science student can’t learn just as well (in principle).

Let me take this a step further and flip the table.

Instructor X teaches physical science.

Instructor Y teaches physics.

Instructor X teaches the proper definition of Newton’s first law. Instructor Y, feeling the need to put more time on the math, dumbs down Newton’s first law in the interest of time.

Now imagine the previous example with the skateboarder. The physics student won’t be happy feeling that the physical science student has come to understand this law on higher terms!


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

A Simple Way to Remember the Order of the Planets

Image from NASA

Image from NASA


There is a very simple way to help remember the order of the planets.

I’m not talking about a song or poem. There are a few of these, but then you’re just memorizing a song instead of memorizing the order of the planets. You’re still memorizing.

What I have in mind is a simple trick. It’s a mnemonic device to help remember the order of the planets in our solar system.


Our solar system begins and ends with S-U-N.

The beginning is easy: The sun is in the middle.

But it also ends with S-U-N. The last three planets of the solar system are Saturn, Uranus, and Neptune.

Take the first letter of Saturn (S), Uranus (U), and Neptune (N), and you see how it ends with S-U-N.

As an astronomy teacher, I see firsthand that many (university!) students easily forget whether Uranus comes before or after Neptune, and whether Saturn comes before or after Jupiter.

I explain how the solar system both begins and ends with S-U-N. Students who remember this invariably get the order of the planets correct.

You naturally study the order of the planets from beginning to end. This way, students tend to learn the beginning better than the ending.

Also, earth is near the beginning (third planet from the sun). Since earth is the most familiar planet, it’s easier to remember Venus and Mars because they are closest to the most familiar planet.

The biggest problem, in my experience, is how the solar system ends. Remember that it ends with S-U-N, and you’ll know that the last three planets are Saturn, Uranus, and Neptune.

Here is the complete order:

  • M-Mercury
  • V-Venus
  • E-Earth
  • M-Mars
  • J-Jupiter
  • S-Saturn
  • U-Uranus
  • N-Neptune

What happened to Pluto? Pluto turns out to be one of very many Kuiper Belt objects. There are many icy bodies in the Kuiper Belt, beyond Neptune, with comet-like compositions (in fact, many comets that we see from earth spend much of their time in the Kuiper Belt, or beyond that in the Oort Cloud). There is another belt, called the Asteroid Belt, filled with rocky bodies between Mars and Jupiter. Pluto is a large Kuiper Belt object, smaller than any planet. Pluto is one of the dwarf planets in our solar system. Pluto is not even the largest Kuiper Belt object. That honor belongs to Eris.

Image from NASA

Image from NASA

Here is how I teach the order of the solar system:

  • S-Sun
  • M-Mercury
  • V-Venus
  • E-Earth
  • M-Mars
  • A-Asteroid Belt
  • J-Jupiter
  • S-Saturn
  • U-Uranus
  • N-Neptune
  • K-Kuiper Belt
  • O-Oort Cloud

The Oort Cloud is a very large, spherical shape with occasional icy bodies that lies beyond the Kuiper Belt. The icy bodies that comprise the Kuiper Belt lie more in a plane that matches the solar system, whereas the Oort Cloud is spherical. The Oort Cloud is much wider than the Kuiper Belt.

Oort Cloud

Image from NASA

Copyright © 2014 Chris McMullen, author of math and science books (including An Introduction to Basic Astronomy Concepts)

Percentage Confusion



With percentages, the wording is very important.

Let me illustrate this with a couple of examples:

  • Fred has 10 candy bars. Sally has 8 candy bars. Fred has 25% more candy bars than Sally, but Sally has 20% fewer candy bars than Fred. In one case it’s 25%; in the other case it’s 20%. The distinction is important.
  • Jenny has 65 dollars saved. Mike has 50 dollars saved. Jenny has saved 30% more money than Mike, yet we could also say that Jenny’s savings is 130% of Mike’s savings. In one case it’s 30%; in the other case it’s 130%. The wording makes all the difference.


A percentage is an alternate way to express a decimal or a fraction.

100% corresponds to one unit.

So, for example, if Linda ate 50% of the donuts, this means that Linda ate half of the donuts, since 50% is half of 100%.

Here are a few more examples:

  • 200% means double, since 200% is twice 100%.
  • 25% is one-fourth, as 25% is a quarter of 100%.
  • 150% is one and one-half, since it’s 1.5 times 100%

The purpose of this article isn’t to teach percents, but to explain the importance of how it is worded. This quick review was intended just as a brief refresher to illustrate the basic concept.


Let’s look at the two original examples more closely.

(1) Fred has 10 candy bars. Sally has 8 candy bars.

When Fred compares his candy bars to Sally’s, Fred divides 10 by 8 to get 1.25, which equates to 125%.

(Recall that any decimal value can be converted into a percentage by multiplying by 100%, since 100% means one.)

Since 125% is 25% more than 100%, this means that Fred has 25% more candy bars than Sally.

However, when Sally compares her candy bars to Fred’s, Sally divides 8 by 10 to get 0.8, which equates to 80%.

Since 80% is 20% less than 100%, this means that Sally has 20% fewer candy bars than Fred.

The distinction here is that in the first case Fred used Sally’s candy bars for the comparison, so Fred divided by Sally’s number (8) to see how his compared to hers.

In the second case, Sally used Fred’s candy bars for the comparison, so Sally divided by Fred’s number (10) to see how hers compared to his.

In either case, divide by the number that you’re comparing with.

(2) Jenny has 65 dollars saved. Mike has 50 dollars saved.

This time, we’ll only compare Jenny’s savings to Mike’s savings, so we’ll definitely divide by Mike’s 50 dollars.

Therefore, we divide 65 by 50 to get 1.3, which equates to 130%.

This means that Jenny’s savings is 130% of Mike’s savings. That’s one way to put it.

There is another way to say the same thing. Since 130% is 30% more than 100%, we could instead say that Jenny has 30% more savings than Mike has.

In the second case, we used the word ‘more.’ If you follow the percentage by the word ‘more’ (or by the word ‘less’—or their synonyms, like ‘fewer’) you’re comparing the overall percentage (which we obtained by dividing the two values) to 100%.

Saying that Jenny has 130% of Mike’s savings means to multiply Mike’s savings by 1.3 ($50 times 1.3 equals $65).

Saying that Jenny has 30% more than Mike means to find 30% of Mike’s savings and then add that to Mike’s savings ($15 plus $50 equals $65).


Here are a few more examples:

  • 3 bananas is 75% of 4 bananas.
  • 4 bananas is 133% of 3 bananas. (Technically, it’s 133 and 1/3 percent, but I rounded.)
  • 3 bananas is 25% less than 4 bananas.
  • 4 bananas is 33% more than 3 bananas. (Really, 33 and 1/3 percent.)
  • $800 is 400% of $200.
  • $800 is 300% more than $200.
  • $200 is 25% of $800.
  • $200 is 75% less than $800. (Subtract 25% from 100%.)
  • 90 cents is 150% of 60 cents.
  • 90 cents shows a 50% improvement over 60 cents.

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


Creative Arithmetic

7 times 7


It’s wonderful to see young kids as they first learn new math concepts.

Very often, they see things in a different way than what we’re accustomed to, and different from how math is often taught.

I believe that such creativity should be encouraged. Isn’t math more exciting when you think of your own way to do something and it works out, compared to being told what to do and how to do it?

Sometimes, though, one method does have an advantage over another method. In that case, it may help to challenge the student with a situation where his or her method doesn’t work out. When the child sees this firsthand, he or she will be more likely to embrace a different method.

Other times, one method is much more efficient than another. Again, it helps when the child can see a specific example where his or her method is inefficient, and to understand why it matters.

Where possible, encouraging a little creativity may be a good thing, even in mathematics. That creativity can help to engage the student in math. Looking way ahead, if the child ever gets far in math, creative problem-solving can become a valuable skill. So it would be a shame not to foster some creativity along the way.


I’ve seen many young kids come up with neat math ideas. My daughter (six) recently thought of a neat way of looking at multiplication. I will use this as an example.

To start out, she was presented only with the definition of multiplication—rather than starting out with a table of multiplication facts. I wanted her to first understand what multiplication means on her own terms and then proceed from there.

She was given a few examples of what multiplication means. For example, 3 times 2 means to add three two’s together (or to add two three’s together). With a few examples like this, she quickly understood what a multiplication problem meant.

The next step was to give her multiplication problems and ask her to figure them out. So, for example, she figured out that 3 times 2 was six by adding three two’s together (and then she could see that adding two three’s together produced the same result).

Now here is where things became really interesting.

My daughter was asked, “What is 6 times 3?”

Surprisingly, after a while, she answered, “9 plus 9 is 18.”

I was obviously curious about this, but decided to be content that she answered the problem correctly (even though her solution seemed questionable) and see how this would play out before investigating this.

A couple of days later, she was asked another question. “What is 5 times 4?”

This time, she said, “10 plus 10 is 20.”

Evidently, it wasn’t a fluke. There must be some method to this madness. So now I asked her a few questions about this to figure out what she was doing.

It turns out that she was visualizing pyramids in her head. She was solving the multiplication problems by grouping numbers together in pyramids.

Below is a picture showing how she worked out 6 times 3.

6 times 3

She started with the bottom row: She knew that 6 times 3 meant to add six three’s together, so she started with six three’s. Then she grouped the three’s into two sets of nine’s, and added the nine’s to make 18.

The following figure shows how she figured out 5 times 4.

5 times 4

Again, she began with four five’s, regrouped them into two pairs to create two ten’s, and added the ten’s to make 20.

Eventually, she will learn and practice her multiplication facts to become fluent in multiplication (that’s the goal, anyway). She will also learn that her solution isn’t efficient if, for example, she wants to multiply 9 times 8—or, worse, 35 times 24. But for now, she has made multiplication her own, and her method works fine for simple multiplication facts.

My daughter doesn’t realize it, of course, but she has the basis here for factoring. For example, she’s writing 6 x 3 as 9 x 2, which both boil down to 3 x 3 x 2.

Copyright © 2014 Chris McMullen, author of the Improve Your Math Fluency Series (which, by the way, is focused on practice and drills—which build fluency—and not on the creative learning aspect that I described in this article)