My Brother’s Face

As my daughters learned about exponents, they waded through a sea of definitions and formulae. I taught exponents in algebra class, and it really surprised me that exponents were such a difficult topic. There are only a few rules to remember; it seemed straightforward to me. Rules about when to add exponents, when to multiply them, and when to change their sign.

But this is after years of being a mathematician. It’s not that the exponent rules are so ingrained in me that I never make a mistake simplifying expressions. It’s that exponents have a big set of associations for me. I recognize them in different guises. Much like my brother’s face.

For a long time, my brother’s face looked like this:


Now he looks rather different.

But the nice thing about knowing someone for a long time, and through many changes, for me anyway, is that all the essentials are still there. To me his gray hair is not his basic hair color. It’s a recent dusting of frost. He’s still the same. It’s the same with all the faces in my family.

What I’m saying is: when you first learn about exponents, your understanding is like my brother’s face above. Your understanding will grow, and change, and become unforgettable and recognizable in many guises. You might still mistake that face once in a while. But with time you can know it, very well.

Choice of theme: Up to 100

We’ve started a blog for elementary school math practice and the theme for this week is Up To 100.

This theme is about pairs of numbers that add up to 100. Why this theme?

– Many money systems are based on units of 100. The dollar, for example, is 100 cents. If you buy something that costs less than a dollar, and pay for it with a dollar, the change returned to you is the number that pairs with the cost to add up to 100.

It’s important to get correct change!

– This theme is full of useful addition and subtraction facts.

– This theme builds up the idea of the “100s – complement” of a number. The 100s-complement of 70, for example, is 30. The 100s-complement of 85 is 15.

– Percentages are typically in the range 0 – 100 and it helps to understand the meaning of a percentage and its 100s-complement. A “confidence” of 95% means an uncertainty of 5%. A 15% discount means you are paying 85% of the full price.

– This theme has some surprises. It’s easy to make little errors such as “100 = 75 + 35” because 100 = 70 + 30. But the correct sum is “100 = 75 + 25”. With practice, you can create a sense for yourself of correct and incorrect. You realize that the 100s complement of 71 has to be less than 30, because 71 is more than 70.

– Later, in triangle geometry, the ideas of the “90s – complement” and “180s – supplement” are used often. So the “100s – complement” is a good foundation. You can quickly figure out 90s-complements from 100s-complements.

Math study skills – tests

Something we get asked about often is how to prepare for a math test. While there is some variation, here are some general guidelines:

1. Know exactly when your test takes place. Three days before the test, do the following things.

2. Know exactly what topics will be on the test. Know the names of topics, techniques, rules, theorems. 3 days in advance!

3. Know the definitions of all of the vocabulary words in the test topics. Usually it’s best to use the definition that is in your text book, not a google definition. Why is this important? Because you might know the word “polynomial” in one context, but then see it on the test in a slightly different context. If you do not know the definition well, you might misunderstand the usage on the test.

Make sure you can explain clearly to yourself what each word means, or give an example of what it is. 3 days in advance!

4. The problems in the textbook are arranged from easy to hard. If you want to do well on a test, you have to do at least some problems from the end of the section in the textbook, not the beginning; in other words, it’s not enough to do the first ten problems. Attempt questions that throw you a curve – are not obvious. 3 days in advance!

5. Spend some time puzzling over problems you cannot answer – but not too long. How long to puzzle depends on your level of math. At the highest levels it’s appropriate to think for days and days! In elementary school, at least a few minutes; in middle school, a longer time. In high school, longer still. But after a certain amount of time, ask for help. First look again at your book, ask a classmate, ask your teacher, ask a tutor. 3 days in advance!

To do this successfully 3 days in advance, you have to attempt some hard problems.

6. Mix up your studying: do problems in your head; write problems and solutions down; tell a friend some math; watch a video. In this way, you are using your mind, eyes, ears, speech, and hands to do the math. This helps you to understand the material better and more fully, and access more of your problem solving ability.

Some people learn best while walking or kicking a ball; others while perfectly still; others with music. Find out what works for you – and mix things up. 3 days in advance!

7. Three days before the test is an approximate number. For some subjects, you need more time. You need enough time to attempt hard problems, find out what you don’t know, and get some help if you need it.

8. Two or three days before the test, write a summary of the material you need to know. Give examples of problems you need to know how to solve. Refer to topics, rules, types of problems by name.

Knowing math is a mixture of memorization and how-to. Some people emphasize one or the other. I prefer both. Memorizing the vocabulary can help.

Sequence Photos: Call for entries

What does a sequence mean to you?

Professor Bear is pleased to announce a new fortnightly math magazine, coming out soon via The topic for the first issue is: Sequences.

We’d love to publish your sequence photo in our inaugural issue.

A sequence is an ordered list of terms. For some people, what matters in a sequence is the pattern followed by its terms, or whether there is a pattern:

1, 2, 3, 4, …
0, 1, 0, 2, 0, 3, …
1/2, 1/3, 1/4, …
sunny, cloudy, rainy, sunny, partially cloudy, …
1/2, 1/4, 1/8, …

For some people, what matters is the “destination” or whether there is a destination:

3, 3.1, 3.14, 3.141, 3.1415, 3.14159, …
-1, 1, -1, 1, -1, 1, …
duck, duck, goose
… -2, 0, 2, 4, ….

If you have an interesting photo of a sequence, and would like it included in our magazine, please send it to:

If your photo is chosen, you’ll have a photographer byline. Creativity encouraged! How about a sequence of sequins?

Discrete and Continuous

Another aspect of the two paintings posted in Why Art is that they illustrate the difference between discrete and continuous, at least in terms of color.

In Keith Haring’s painting, every point is either yellow, red, green, or purple: four discrete possibilities. In Wayne Thiebaud’s painting, there are many continuous transitions from one color to another. Notably, for me, a field that changes from gray to brown to indigo. If you look at the body of work of these artists, you would see that they each stay pretty much in one color camp. Haring in the discrete camp, Thiebaud in the continuous.

In mathematics the discrete and continuous camps exist too. People typically pick one or the other. What camp they choose is an individual matter probably much as it was for Haring and Thiebaud. Is one camp “easier” or “simpler” than the other? Not necessarily. It’s tempting to think that the discrete camp is easier – all whole numbers, no messy extended decimals. No Zeno’s Paradox.

However, the continuum can be comforting because it is filled with solutions. We know, for example, that two non-parallel lines in the plane definitely intersect, which means that the system of equations defining two such lines always has a solution. Ask a similar question in the discrete world – does a given line intersect any lattice points in the plane? And that question is harder.

(By lattice points in the plane, I mean points with whole-number coordinates in the X-Y plane, such as (2,1) and (3, 56). These points form a regular grid of dots.)

Then there is the famous Diophantine equation (love that name, Diophantine, makes me think of an equation in a flowing white dress with a long trunk):


In the continuum are many solutions. But if you ask about discrete solutions, you get Fermat’s Last Theorem, which stumped mathematicians for centuries.

On the other hand, the continuum is a very mysterious place. That’ll have to be another post.

Why art?

I feel a need to say more about A Study in Scarlet (Rectangles). What place does a painting have in a math club?

I think it’s worth pointing out that the kinds of thinking people use when they think about art are similar to mathematical thinking. In fact, from my perspective, it’s all just thinking.


Before jumping in to some obvious observations about colors and numbers let’s look at two paintings. I think color is important in both of them:

25_89 by Keith Haring

25_89 by Keith Haring

One of Wayne Thiebaud's landscapes

One of Wayne Thiebaud’s landscapes

When I look at paintings, I ask myself, what is this painting about?
If I think it’s about color, I ask myself, in what way is it about color?
This is similar to what happens when I look at math problems. A math problem can seem impossible, or like I’m stumped, until I start to ask myself, what is this about? Is it about numbers or geometry? etc.

Back to the paintings, Keith Haring’s painting uses four distinct colors. Two colors are allowed to drip over the other colors. I think the bold colors express strong or bold feeling, and the dripping shows us different relationships between colors; red next to yellow is different from red next to green.

Wayne Thiebaud’s painting shows he put a lot of thought and care in mixing colors. Shadows and outlines are highlighted in slightly nonrealistic colors, lots of purples and greens, like reflections in abalone shell. When I see this use of color it is both familiar and surprising to me; yes, there are purples and turquoises in the shadows, but no, I don’t always see them.

Mixing colors is well known to be challenging. Just rendering something like a green sweater is not a simple matter of green paint and maybe gray for the shadows. Maybe the lighter parts are more yellow, maybe the dark parts more brown. Like numbers, colors can be “added together”. But unlike numbers, which have one dimension, a size, colors are said to have three dimensions: hue, saturation and value.

It’s easy to visualize the set of all numbers as a line. What would the set of all colors look like? Would it be fully three dimensional like a cube? Would it extend in three directions, or have boundaries? Would parts of the color space fold in on itself? (There are many ways to make gray, for example.)

If you look at a mostly monochromatic painting like the one in A Study in Scarlet (Rectangles) you can see how the interplay of value and saturation can play a big role if the hue is mostly left alone.

Here is more about Keith Haring.