For a while, a long while, I let this math blog just sit. I didn’t want to delete it. I spent too much time on it and I knew it was full of great stuff that was still being used. But I wasn’t in love with writing on it.
In my lethargy, the hackers and jerks managed to infiltrate and completely take it down. That was on October 23, 2025.
Today is December 30, 2025. I rescued all the posts and will gradually get all the good ones reposted. But I have to start over, creating the whole website from scratch.
Luckily the wayback machine (internet archive) has a nice snapshot of the images and layout. It’ll take me a while, but I’ll get it all back up. Maybe even with some improvements!
If you are looking for a specific post – something you remember from long ago or something you want to know about – leave it in the comments. I’ll hunt it down and get it posted.
I’m looking forward to having all those great math stories, thoughts and tips live again – I hope you are too.
The union is on the left and the intersection is on the right.
I discussed empty sets in a previous article. I hadn’t yet listened to the recent podcast of The Math Dude, Jason Marshall, in which he introduced sets and subsets. Such a curious coincidence; I thought I would continue down our united path.
This week, the sweet Math Dude talked about union and intersection of sets. So I’m heading there too – and incorporating the empty set.
Union and intersection have symbols.
These symbols represent “operations” (like addition or multiplication) that you do to a pair of sets. They are called the cup and cap and look like, well, a cup and a cap. Or a U and an upside down U. The Union is the U and the intersection is the other one – that’s how I remember it and teach it.
For the sake of illustration, I’ll use Daughter’s handy SnackTraps like in the picture above.
The union is everything.
The sets I’m using are
L ={green cube, blue half circle thingie}
R = {green cube, red triangular prism, orange rectangular prism}
I’m using the letters L and R for left and right in the picture.
The union of the two sets results in all the pieces from each set, all crammed together. (And if there is a repeated item, you only take it once.)
The union of the two sets is everything combined (without duplicates).
Let’s take the set of all states that border Texas and the set of all states that border Oklahoma, like this:
T = {Louisiana, Arkansas, Oklahoma, New Mexico}
O = {Arkansas, Colorado, Kansas, Missouri, New Mexico, Texas}
Then the union of the two would be “The set of all states that border both Texas or Oklahoma or both” and would be T U O = {Louisiana, Arkansas, Oklahoma, New Mexico, Colorado, Kansas, Missouri, Texas}
Notice we don’t keep duplicates.
The intersection is only the common stuff.
Using the same L and R sets from above, I can take the intersection.
The intersection is only the stuff they have in common (we get rid of the duplicates here, too).
Let’s go back to our set of all states that border Texas and Oklahoma:
T = {Louisiana, Arkansas, Oklahoma, New Mexico}
O = {Arkansas, Colorado, Kansas, Missouri, New Mexico, Texas}
Then the intersection of the two would be “The set of all states that border both Texas and Oklahoma, at the same time” and would be T ∩ O = {Arkansas, New Mexico}.
What happens with the empty set?
When you start throwing the empty set into the mix, you follow the same rules.
For the union, you throw everything in both sets into one bag. Well, since there’s nothing in the empty set, “everything” is the original other set!
The union of the empty set with anything… is that same set back!
For the intersection, you take only the stuff that’s in common between the two. Since the empty set has nothing – there’s nothing in common.
The intersection of the empty set with any other set it just the empty set – there’s nothing in common at all!
Think about it…
Does this remind you of anything? What similarities do you see between this and addition or multiplication (or subtraction or division)? Ask your children these questions too.
Get out some snack traps and blocks (or a map) and go for it! Let me know how it goes in the comments.
Most parents aren’t professional mathematicians. But there are a few. This is the first in a series of interviews with mathematician parents with the goal of helping parents integrate math teaching into parenting. See the list of interviews here.
I had the privilege of interviewing David Chandler, a physics and math teacher. He runs Math Without Borders, an ongoing project to supplement high school math textbooks with Home Study Companions, to make them useful for homeschoolers and adults for self-teaching.
Last week I published an article about mazes that was inspired by some of his work with homeschoolers. He does some really cool things!
MathFour: Thanks for taking the time to answer some questions, David. First, what’s your degree and career? And how long have you been in math?
David: I have a BS in physics (1970), an MA in education (1975), and an MS in mathematics (applied math) (1997). I have taught physics and math (high school, Jr. college, some K-12) since 1972 and done other math-related projects on the side (inventions, design projects, publishing, curriculum innovations).
MathFour: Tell me about your family – how many kids do you have and how old are they? Are any of them more or less interested in math than the others in the family?
David: Two daughters born in 1974 and 1979, making them currently 37 and 32. B1, my older daughter, went through Alg II. She had some excellent teachers and some rather poor teachers. She would freely come to me for help and discussion. My younger daughter, B2 didn’t want the discussion. She tended not to come to me because she just wanted to know how to do the immediate problem at hand and got irritated when I said any more. As an adult she went back for more math as a prerequisite to other course work (involving logs, trig, etc.) and got tutoring from me over the phone. She values the explanation thing more now. My wives have not been mathematicians.
MathFour: Did you have any worries about your daughters academically? In particular, did you think they would do better in math than in other subjects because of your influence?
David: I think they both got the point that math is about understanding and problem solving and creativity. Both are academically well rounded.
MathFour: How did you play with your girls? How do you play with your grandchildren? Did/do you view playtime as different in any way than other “non-mathematician” parents?
David: I engage in a lot of banter that is probably math-influenced. When they say to me “Happy Birthday” I say back to them, “Happy Birthday to you too,” in the sense that she can be happy on my birthday. This year B1 sent a Facebook message, “Happy Birthday to us all.” That’s not explicitly mathematical, but it is the kind of thing Sheldon might say. I came up with a song that eliminates the asymmetry of l-m-n-o-p so you could sing the alphabet both forward and backward and taught it to my grandkids. Every rainbow we see is a physics excursion into dozens of phenomena. Theme park rides are, of couses, a learning experience. We have lots of “shared learning experiences.” Whenever something triggers a new insight in me, they are definitely exposed to the excitement it generates. They know what it is like to experience the world through eyes informed by physics/math awareness.
Screenshot from David’s Algebra II with Trigonometry Home Study Companion
MathFour: Do you think you speak with your daughters or behave differently than other parents because you have a math background?
David: I don’t talk down to them, but I don’t overwhelm them in jargon either. I wonder about a lot of things out loud. I critique the world with a quantitatively informed (and order of magnitude informed) crap detector in their presence. I model inquisitiveness and observation and show appreciation when it is a shared endeavor. I think that attitude has rubbed off on both of them, possibly genetically as well as environmentally.
MathFour: Have you ever had either of the girls express negative thoughts about math?
David: Not really, except for B2’s impatience for my explaining in too much detail, and a tendency to go on too long when her attention has already shifted.
MathFour: Have you ever disagreed with one of your daughter’s math teachers?
David: Yes. B1 took Alg II in India where I was teaching at an international school. Her teacher was a young, very traditional Indian woman who spent a week or so teaching them how to read log tables, including interpolation, funky work-arounds for logs of numbers less than 1, etc. It was a total turnoff to the class, all of whom had calculators by then, and seeing this drove me up the wall.
MathFour: What happened; how did you handle it?
David: I discussed with B1 how all of this was now outdated and how using logs to solve exponential equations, using log paper, modeling perceptual phenomena with log scales (dB’s, astronomical magnitude scale, octaves, etc.) was really where it was at. I approached the teacher about this issue, as tactfully as I knew how. She listened politely, and didn’t react in an overtly negative way, but she may have been overwhelmed by me. She didn’t change what went on in class. I taught B1 a lot of things not in the book, and better ways to do things, but this didn’t generally lead to direct interactions with the teachers.
MathFour: Now to change direction a little to a more worldview of math. What do you see as the biggest challenge in math education today?
David: Getting kids to quit memorizing things and start digging for understanding. Also seeing math not so much as a body of knowledge as a mindset.
MathFour: What do you see great happening in the world of math education?
David: Internet communication for discussions like these. New insight-generating computational tools (Geogebra, Geometer’s Sketchpad, Tracker, spreadsheets, etc.) and their use in the classroom. What goes on in my (and presumably a lot of other) classrooms despite regressive influences like high stakes testing, minimum standards which become maximum standards, etc.
MathFour: What advice can you give to non-mathematician parents that might help them raise their kids to like and appreciate math.
David: Learn some math, engage in problem solving, problem posing, observation of the world, cultivating a sense of wonder, taking curiosity a step further and digging a little, model these for your children, etc.
MathFour: Thank you for taking the time to answer our questions, David!
How about you – do you have any questions for a mathematician parent? Share them in the comments – we’ll see if we can get David in here to answer them!
Last week’s article on the order of operations included a quick mention that division is the same as multiplication – but different. Now’s the time to explain that a bit.
The only thing mathematicians like to do more than create, is destroy. That’s how we get imaginary numbers, dividing by zero and raising things to infinity.
Once we create with multiplication, we want to know what happens when we destroy with the opposite of multiplication.
Enter Division, stage right.
Multiplication is created from the need to quickly add a bunch of numbers that are all the same. They each must be equal to make this work:
6 + 6 + 6 + 6 + 6 is shortcutted to 5 X 6.
Division is the breaking up into pieces that are all equal.
Technically we can break 30 up into these 5 pieces: 4, 8, 7, 6 and 5. But “division” requires (or implies) that we are dividing equally. So 30 would have to be broken up into 5 equal pieces of 6 each.
Of course this is in a purely mathematical world. When you get into a toddler world, things will be different.
How division is the same as multiplication.
We represents multiplication with little x‘s or dots or stars. Like this:
3 X 5 or 3 • 5 or 3 * 5
We represent division with this cute little symbol:
But we can write it in many more ways. Notice the cute symbol looks like a fraction with dots on the top and bottom. That’s not a coincidence. A fraction means division.
is the same as .
Remember the poem:
When dividing fractions Don’t bat an eye Just flip the last And multiply!
Well, you can change this to:
When dividing numbers Don’t bat an eye Just flip the last And multiply!
And “flipping” the last just means taking the “assumed 1” that’s underneath it and putting it on the top.
Now go forth and divide…
So when you’re teaching the MD in PEMDAS, the order of operations, remember that D is the same as M. And if things get a little confusing, demand some parenthesis before doing the problem.
I just finished Dan Pink’s book A Whole New Mind in which he discusses the difference between a maze and a labyrinth. It made me think of this wonderful video of dissecting a maze into two walls to solve it:
From the discovery that David and his class of Math Explorers made, I learned that you can take two colors and create a maze. So off I went…
I’ve been dying to use this elementary school graph paper I got the other day so out it came. I grabbed a purple marker and started:
I immediately began creating the “rules” of what would be good maze design. And almost as quickly I began seeing that I was all wrong.
After a while I got out the orange marker:
More rules… more “…no, that’s not really a requirement” thoughts from me.
After a while I just gave up on making rules and decided to make sure that my purple and orange would come out at some point together.
Then with a little photoshop magic, I made it all black. You can download it here and play it!
Okay, your turn!
This might not be something to teach, but rather something to do at home together. It’ll be a learning experience for the whole family.
Grab anything with lines or a grid on it and two different color pens or markers and give it a shot. Let me know how it goes. You can even post a link to the pictures of them in the comments!
Are you teaching sets? Some of the stuff that you talk about in set theory is pretty clear – you have a basket { } and you put stuff in it {♥, ✂, ☎, ✿}. No problem.
But what’s the deal with the basket with no stuff in it?
That’s the empty set. It’s written like Ø or simply as the “basket” like this: { }.
But kids might be tempted to write {Ø} – and then they get it wrong. BUT WHY?
The empty set is an object!
Suppose you have four empty sets. Do you have nothing? Nope – you have four containers. Just check your cabinets – I’m guessing you have a handful of empty sets in there with the Rubbermaid® or Tupperware® logo on them.
And {Ø} is an object inside a set.
Supposed you put one inside the other, like this:
This is the same as {Ø}. Would you be willing to put leftovers inside the bigger one while the smaller one is there? It would be something like this:
Instead, wouldn’t you be more likely to remove one empty set from the other and have just Ø and Ø (no curly brackets) like this:
What do you think? Does this help you understand the empty set? Will it help you teach it?
The Order of Operations can be boiled down into three “real” operations.
Parenthesis are merely a way to group things – they aren’t a real operation. So they doesn’t count as a real operation.
Since division is really just multiplication turned upside down, we don’t need to include it separately, either. Likewise, subtraction is addition on its ear. So we throw him out, too.
Now we have only three: exponents, multiplication and addition.
You see that 3 x 3 x 3 x 3 is 34. The exponent is the shortcut for multiplication.
Multiplication is the shortcut for addition.
Likewise, when we get overwhelmed with adding up the same numbers over and over, like in the video, 4 + 4 + 4 + 4 + 4, we can use multiplication to shortcut it: 5 x 4.
Follow the shortcut evolution.
The shortcut evolution is like this
Addition came first.
Then we created multiplication to make addition easier.
Then we created exponents to make multiplication easier.
So when you do arithmetic, we should do the “recent” shortcuts first (exponents) and then the “older” shortcuts (multiplication) and then the “regular” arithmetic (addition). Remember that subtraction is addition and division is multiplication.
And keep in mind that we need to watch the grouping or anything isolated with parenthesis, absolute value bars or in a fraction. The grouping/isolation tools always trump the other operation rules.
What if your answer isn’t the same as the back of the book?
Some textbooks actually have multiplication done before division. This means that the problem 9 ÷ 3 x 2 will show a different answer (3/2) in that textbook’s solution pages than what you would get following the OoO I’m describing here (6). These texts are rare, but I’ve seen them. So keep your eyes peeled.
The Order of Operations is a set of rules that we’ve agreed on. Which means that as long as a textbook clearly defines their order that they will follow, they can do things like this.
If you get confused as to which to do first, demand parenthesis. (Yes, you can do that.) Or put parenthesis into your child’s textbook to help them out.
The problem sets should be there to enforce, not confuse.
What do you think? Does this help or hinder the way you’ve always viewed the Order of Operations? Share your thoughts in the comments.
In the Story Portfolio, there’s a suggestion to create a mini-saga. It’s a story in exactly 50 words, not counting the title.
“What a great idea,” I thought, “Can it be done with math?”
Well, of course it can. But what that will look like, I’m not sure. So I’m launching 50 Word Friday – a Friday series for these mini-sagas.
The first is about me:
When Life Happens, Maybe It’s Happening Right
A girl spent years in college. Learning, then teaching, math. Mostly because she didn’t want to get a job. Then she joined corporate life as a choice. She soon realized that the previous unchosen path was more appropriate for her than a life in corporate American. And here she is.
Do you have a mini-saga? Tell your story in exactly 50 words and share it in the comments!
See these other 50 Word Articles from MathFour.com
Would you like to teach math everywhere you go? Well, here’s one from the table!
Daughter enjoys playing with our salt-and-pepper shaker holder at dinner. She takes out the salt, then takes out the pepper, then replaces the salt, then replaces the pepper.
The order in which she does these four operations vary. Including switching the salt and pepper.
She’s slowly putting together the pieces that will one day become the commutative property.
She’s also practicing substitution…
She’s learning that the salt and pepper can be switched (commutative). And she’s learning that one can be interchanged for the other (substitution).
…and the associative property!
She attempted to put her small milk cup into the holder. It fit, but only with pushing. She then removed the milk cup and attempted to put it in the other side. (At her age the things grown-ups understand are not obvious to her.)
Although non-equality isn’t part of the associative property (which is if a=b, b=c then a=c), the comparison of three things is.
Here are the things she’s learning from this dinner session:
This fact she discovers from interchanging them in the holder.By putting them in the holder in a different order, she learns that the equality is commutative.Since the milk cup won’t fit into the spot the salt was just in, she learns this.And trying to shove the milk cup in the other side yields this fact.
So pull out the stops – give the children everything. And let them explore. If they have the gift of language, you can hint at some of these properties, but be careful not to go into a full “lesson” at dinner. Teaching math at the dinner table should be fun.
Where have you seen math properties in your world? Share your stories in the comments – or link back to your story on your blog!
Thanks to Molly for this question she left in the comments, “Why is a negative times a positive, a negative?”
“Sign – One Way” by Andrew Schmidt
It’s common to hear a child say, “They created this rule just to make me mad.”
And on the heels of that usually comes a frustrated teacher’s comment of, “No, this has a purpose.”
When then faced with, “Okay… so what’s the purpose?” a teacher or parent might become more frustrated. It’s not always clear the purpose of things like the rules for multiplying differently signed numbers.
It may also be that the grown-up secretly believes that this was all created just to make us mad. After all, when the last time you had to multiply a negative number times a positive number in the real world?
I, personally, can’t ever remember a time when that case has come up outside of a math class or tutoring session.
Then why create the rules at all?
The truth is that many math rules are created because we need a rule to continue working – something to agree on among peers working on the same project. We need everyone on the same page.
This agreement somehow gets filtered down to the kiddos as “this is the absolute way it is – do it because we said.”
So it very much looks like we created rules to make them mad.
Good deduction, kids.
How do they pick the rules?
“Street Chess” by Petr Kratochvil
Some math rules are created arbitrarily just so we have something (anything) to agree to. Like the rules of a game. The order of operations is one such situation.
There is no natural reason why we’ve chosen to do multiplication and division before we do addition and subtraction. No more so than the people in the US have decided to drive on one side of the road whereas the people in England, Australia and the Bahamas drive on the other.
We’ve agreed to it.
It keeps us from crashing.
Sometimes, we agree to a rule because it is “less bad” than any of the other possible rules we can think of. The result of any number raised to the zero power is one of these. 30 feels like it should result in either zero or 3. But if you mess with all the other rules (in particular the one about subtracting exponents here ) you’ll see that 30 really needs to be 1. If it were anything else, it would mess up the rule of .
Why is a negative times a positive equal to a negative? Is it because it messes other things up or is it an arbitrary rule? Instead of explaining the reasoning to a child, allow them discover it.
Have a “King for a Day” day where the students get to rewrite a particular math rule. Allow them to sit as a class or family and decide on the rules for multiplying signs numbers.
Give them plenty of time to experiment and see if the new rule messes anything up. (If your administrators or math curriculum specialists fret about you allowing students this time, note to them that the students are behaving like real mathematicians. They are inventing a concept or rule and testing it out to see what happens.)
If the students discover that the new rule has dire consequences when applied to the rest of our number system, they then see the reason for the rule. And they’ve learned through experience and solidified the rule in their minds!
If the students discover that there are no consequences of the new rule, they can then determine that the rule was created arbitrarily and agreed upon for convenience. It is a rule for rules’ sake.
You can discuss with them how things would be different if they kept the new rule and everyone else in the world followed the old rule. Again experience and discussion will solidify the “real” rule.
So the next time you’re faced with “why?” regarding a math rule, give it a shot. Let the children be the rulers of their own math world!
Will it work? Did it work? Share your thoughts in the comments!
