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.
It inspired me to draw the graphs of Celsius in terms of Fahrenheit and Fahrenheit in terms of Celsius.
The conversion from Celsius to Fahrenheit and back again can be strange. One way to understand it is on a graph. And you can use this to teach some linear algebra too!
Notice these two intersect at (-40, -40). Which means that -40°F is -40°C!
Use it to convert temperatures.
The x values on the red line are Celsius – so find the °C you have and then look at the y-value to convert to °F.
It’s just the opposite on the purple line.
Okay, fine. This isn’t the greatest way to convert – but it’s exciting to see it graphically. And it might be easier to convert this way for someone who’s more visual.
Use it to teach math!
These two lines are inverses of each other. So the coordinates of one are switched to make the other.
Also, they mirror image across that 45° line. I marked the line with dashes and wrote $latex y=x$ on it.
And if you’re into this, their functional composition (both ways) is… x! (not factorial)
Compare and Contrast…
Take a look at the way J.D.Roth did it and then look at the graphs I have. Let your students find the way they like the best. And encourage them to create new ways!
Oh, yeah – and share what happens in the comments!
Years and years ago there was a math education model. It went like this:
Children would read a section or chapter of a textbook the night before class. They would come to school the next day ready to ask questions and do hands on practice with a teacher close by to help.
This was a “normal” classroom situation.
Over the course of many years, textbook publishers have squeezed more and more topics into textbooks. Thus, they have squeezed more and more detail out. Which has gotten rid of much of their value.
As math texts had their details extracted to fit more topics, children began having difficulty comprehending them. Even if they attempted to read the section the night before, the teacher would have to fully explain it during class the next day.
Thus the first flip happened!
Kids gave up altogether on trying to read math textbooks. They soon realized that the teacher’s instructions to “read the section” as homework was just an empty request. The teacher would explain all of it the next day anyway.
This model, the first flipped classroom, has sustained for quite some time.
Enter web-based video.
Fun to watch and easy to rewind, online videos allow kids to flip back the model. Because videos aren’t regulated, nor monetarily driven, anyone can make them – including people who are really good at it.
Yes, a pair of teachers at Woodland Park officially did the first flip. But check out what kids have done without teachers. They found Cousin Sal’s videos before it was ever <cue music>Khan Academy.
They inherently knew that there could be more and that there was a better way to learn. They started watching videos at night after they didn’t get everything they needed during the day. And they started going to school the next day asking questions of their live teachers: “I saw this video on YouTube and he did it this way… can I do it like that?”
We didn’t flip anything.
Yes, we think we’re really clever and have this great “flipped classroom” model. Guess what? We adults haven’t done squat – we’re just writing about it and making cute infographics.
Let’s give credit where credit’s due, shall we? Textbook companies drove the first flip. And kids are driving this one.
Last night was taco night and my job was to grate the cheese.
I didn’t get too far in before I noticed some math.
For some reason I always start grating on a corner. Then I rotate the block so I’m grating on another corner. After doing this a few times I noticed the angles I was creating:
And what exactly are the shapes, anyway? What is the shape of the grating holes of the grater? And what is the resulting shape of the cheese sliver?
How much cheese is in the pile after you “fluff” it by grating it? What’s the volume of fluffed cheese compared with stuck-together-in-a-block cheese?
More importantly, is there enough for two grownups and a toddler? (When one of the grownups loves cheese!)
I’ve been itching to get into some basic abstract algebra goodies. With the help of the Cuisenaire Rods, Simply Fun Sumology number tiles and the Discovery Toys Busy Bugs, I’m able to do that.
Start with wrap around addition.
This type of math is officially called “modular arithmetic.” We are only going to use the numbers 0, 1 and 2.
It begins as regular addition. And since we are only using those three numbers, all our answers have to be either 0, 1 or 2. So when we add 1+2, we wrap around.
If we were to count in our system, we’d say: “0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, …”
The addition table looks like this:
(Notice you could do this with numbers from 1- 12 and it would be clock addition!)
Now things get buggy.
Switch out all the number tiles with some pretty color Cuisenaire Rods. They don’t have to be the “right” rods. We’re only looking at the colors. Here’s the progression I did:
The end result is a very abstract chart!
You can “bug” two things together.
Like this:
(I know – a spider isn’t a bug. But run with me on this, okay?)
Notice that each of these are directly from the “spider table” above.
You can read this as, “Purple spider green equals green,” just like you would say, “Zero plus one equals one.”
Over the past ten years or so I’ve been hearing this word “creationism.” It seems that it’s the opposite of “evolutionism.”
No problem – until I read about people trying to “prove” creationism. And articles trying to refute it.
I’m not sure what the big hubbub is about. 20 years ago I heard a guy make a simple statement about it all. He proved creationism in 30 seconds.
And it was a mathematical argument.
Before I give you that 30 second super-statement, let’s chat a little about what a real mathematical proof looks like.
Math starts with definitions.
We say, “Okay, here’s the deal. Let’s define a nebino as a number that’s greater than all prime numbers,” or something of that nature. (And yes, you get to make up your own words if you want.)
No math – none – ever starts out with confusing terms. If it does, someone jumps in and says, “You’ve got stuff that isn’t defined clearly.”
Everything shuts down until that gets resolved.
Math assumes… well… assumptions.
Once you have your definitions clear, you get to set up what you assume. This actually might come before the defining part. And often it isn’t said out loud at all.
Which is one reason that scientists sometimes think that they can do math. They’re always assuming the world (i.e. reality). Mathematicians don’t cotton to such vast and willy-nilly assumptions.
Then you get your hands dirty.
You’ve got definitions and you know what you’re assuming. You’re foundation is down. Now you build.
In other words, you create some math.
But things don’t always work out like you planned. So…
If it doesn’t work, you change the definitions or assumptions.
Yep – sometimes we really want something to work, so we just go back and tweak some of the starter points. Which means we change a definition or add (or delete) an assumption.
(Which means if you’re using someone’s math, you have to make sure you’re working with the same definitions and set of assumptions.)
And that’s the best argument for Creationism.
The statement I heard from this Creationist was, in essence, this:
We don’t have to use any evidence of science to prove God created the world in 7 days, 6,000 years ago. God planted the fossils and created all sorts of nifty things like DNA that would contradict the Bible. It was all meant to test our faith.
Voila! Creationism proven.
Brilliant! Change the assumptions, and you’re there.
Beliefs are just that: beliefs.
Which means there’s just no proving them. Kinda like my thoughts on the real line. I don’t believe in it – to the chagrin of my Twitter friend Colin.
So if you want to prove something, change the rules. Or ignore them. Mathematicians do it all the time.
*If you’re really really interested in my beliefs, I’ve shared them here.
Now that school’s officially in session, I’m really thinking about all the homeschooling moms who are taking on teaching math. I can’t help but see the similarities to something they’ve already done – given birth.
For each, I noticed that…
It’s darn scary.
Giving birth is pretty freaky. Especially if you’ve never seen or done it before. If you have the privilege of seeing another arrive in this world, it makes it easier. (I watched both my sisters and my best friend give birth!)
Teaching math might be a whole new world – and pretty strange if you’ve never done it. Dig back to some good math experiences you had as a child and draw on those.
It’s gonna happen no matter what.
Kids get born. They must. And they do. So you might as well make the best of it and enjoy the process the best you can – with a happy and positive pregnancy!
Kids learn math. They must. And they do. So you might as well make the best of it and encourage them in the best way – with a happy and positive math environment!
You get to choose how you do it!
You can give birth at home, in a hospital, with a doctor, with nobody, with drugs, without drugs… pretty much any way you want.
We’ve got more freedom now that we ever had. In Texas, where Betsy and I live, a homeschool is an independent private school. We get to teach in an unschooling way with lovely things like Math on the Level.
It may not go the way you want,
You’ve got a great plan. You know what to do and how to do it. You’ve taken the classes and my goodness! You’re in hard labor for 32 hours and you’re still at labor station -4.
You might think you have the best curriculum in the world. You’re all prepared. You start teaching an holy cow! Your sweet student couldn’t be less interested in everything you’re presenting.
…so be flexible.
Yes you’re frustrated. Remember the goal – a healthy child. Put away the frustrations and go for the C. Make it up by being a breastfeeding champion.
Likewise, you’re wanting a happy, healthy child. Choose a different curriculum. Let your child choose a different one. Experiment. Make it up by taking the crew to a great museum.
There are professionals for this.
Get a good ob/gyn, doula or midwife. You don’t have to go-it alone.
Get a tutor, enroll in a program like Kumon, join a homeschool co-op. You don’t have to go-it alone.
There is lots of information online.
With books like Betsy Dewey’s Birthright and sites like this one, there’s no reason to go into either blind.
It’s hard,
Giving birth is no piece of cake. However you choose to do it, you’re gonna have pain – in some way. Not to mention the nine months of puberty-like hormone swings!
Teaching math is a challenge. It’s not about conveying information, it’s about following a child’s natural curiosity and pointing out the math that happens along the way. Which means having to recognize math and the learning/curiosity style of another human. Not an easy task.
…and it’s worth it.
AH, the prize! When you are presented with a slimy little mini-human, you can’t help but think how cool it is!
And when a munchkin comes to you and says, “Did you know…” your heart fills with the glee of knowing that she extrapolated to get that – not just memorized it.
Are there more?
Any other similarities that I missed? Share them in the comments!
Betsy Dewey, my cousin, was a huge help and resource when I was pregnant with Daughter. She was pregnant with her second at the same time – that made it even cooler!
Betsy is an advocate of natural home birth and homeschooling. This article is written in honor of her and everyone else that listens to the beat of their family’s drum – and marches to it!
I’m fascinated by how the simple is really the most complex. Just proving that the two assumptions
3 + 0 = 3
and
3 + x = 3
will force x to also be 0 is a serious and important proof. And harder to do than you think.
When we get even more basic and talk about equality and comparison, things get impossibly complex.
And that’s the fun in the challenge inspired by 2nd grade math teacher, Suzanne Weider. She wrote:
Another idea I have been looking into has to do with the equal sign: teaching it as a symbol that means “the same as” as opposed to the sign that comes before the answer.
Equality and comparison are everywhere.
Some people are obsessed with fairness. Almost all people have some sense of fairness. This means that things need to be roughly equal, if not the same, for all situations in life.
We go into every situation assessing what’s the same and what’s different. Which means a judgment call on equality.
The beauty of this is that you can tap into how our natural comparison connects with the comparison tools of mathematics. These are tools like the greater than, less than and equals signs. This can set your children up for some hard-core success in algebra, geometry, calculus, and Radon-Schure-Greenlee Mathematics. (I made that last one up, but I mean crazy everything math.)
Equality and comparison are so complex that a single article won’t hold all there is to write about it. So this is the first in a series.
![[50 Word Friday] The Original Flipped Classroom](https://mathfour.com/wp-content/uploads/2026/01/image-199.png)
