# Tessellations Hexagon Square Triangle Rhombus Trapezoid Star Patterns ## TESSELLATIONS

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: Here is another made from triangles: This one is made with squares: The same pattern can make a tessellation with stars and hexagons: 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: ## CHRIS MCMULLEN

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

# 1 + 1 = 10 ## ONE PLUS ONE

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.

## 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.

## CHRIS MCMULLEN

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

# Percentage Confusion ## PERCENTAGE COMPARISONS

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.

## REVIEW OF PERCENTS

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.

## INTERPRETING PERCENTS

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).

## COMPARISON EXAMPLES

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

## CREATIVITY IN MATH

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.

## MULTIPLICATION EXAMPLE

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. 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. 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)

# Mathematical English

## MATHEMATICAL ENGLISH

You don’t have to love both math and English to appreciate mathematical English.

Though it may help if you at least like one or the other. 🙂

Mathematical English can be fun all on its own, but can also be helpful in a couple of ways:

• Use it to help get math lovers interested in English.
• Use it to help get language lovers interested in math.
• Integrating math and English may help use both sides of the brain together—a valuable word problems skill.

Here are a few entertaining activities where mathematics and language intertwine:

(1) Victor Borge’s inflationary language: The concept here is to add one to any number used in ordinary speech or writing. For example, “forever” would become “fivever” and “tension” would become “elevension.” You can find video footage of this on YouTube, for example, or more information about Victor Borge at http://en.wikipedia.org/wiki/Victor_Borge.

Here are a few examples:

• Twice upon a time, a big bad wolf nine four little pigs.
• Five the life of me, I can’t understand three-halves of that lecture.
• How ofeleven can two fiveget where two’s keys are?

(2) Mathematical word games combine math and English in a fun way.

Here are some examples:

• Anagrams involve permuting letters to form new words, as in “listen” and “silent.”
• In cryptograms, each letter is replaced by a different letter to make a secret code.
• Scrabble assigns a numerical value to every letter and players try to form words from the letters to make the highest possible score.

Such games combine mathematical thinking or logic with language skills.

(3) Poetic verse that involves numbers:

• William Shakespeare used iambic pentameter when writing plays or sonnets, which follow a mathematical pattern (rhythm).
• Japanese Haiku is a short form of poetry following the mathematical pattern 5-7-5.

(4) Chemical words combine chemistry’s periodic table with word scrambles.

For example, the chemical word ThInK is composed of the symbols for thorium (Th), indium (In), and potassium (K).

Most words, like ‘quiz’ or ‘test,’ can’t be made just from symbols of the periodic table, but it turns out that there are thousands of chemical words.

My favorite chemical word is ScAtTeRbRaIn, which combines 6 elements that all have 2-letter symbols to make a 12-letter chemical word.

There is such a thing as a chemical anagram, too, as in VErBOSe and OBVErSe (but note that the word ‘obvserve,’ which is an anagram for ‘verbose’ and ‘obverse,’ isn’t a chemical word at all, as there is no way to make it end with -VE using symbols from the periodic table).

Some chemical words can also be made more than one way. For example, compare GeNiUS with GeNIUS—one has Nickel (Ni), the other has nitrogen (N) and iodine (I).

Here are a few chemical words that relate to math and science:

• AlGeBRa
• PHYSiCs
• ThErMoDyNAmICs

A fun way to make chemical anagrams is with VErBAl ReAcTiONS (the name of this game is made of chemical words). That is, you put the ‘ingredients’ on one side of what looks like a chemical reaction and the chemical word on the other.

Here are a couple of VErBAl ReAcTiONS:

• 2 C + N + 2 I + P → P I C N I C
• 2 C + U + 2 S + Es → S U C C Es S

(If you like these puzzles, you might check out author Carolyn Kivett, my mom, who coauthored some puzzle books with me—she did most of the hard work.)

Chris McMullen, author of the Improve Your Math Fluency series of workbooks