Here’s a FREE Activity Packet to read and share with your children to get them thinking about how numbers got started.
Somewhere in the past we recognized that we have these “digits” on the ends of our hands. Using these, we created numbers, adding, subtraction, multiplication, division and even fractions!
I was at my dad’s house the other day and decided to pull out my new Math’d Potatoes game to see how my super-gaming family liked it.
The kids in the house were too young to play, so my sister and I asked Aunt Linda and our stepmom to play with us.
They quickly claimed they were “math Neanderthals” but agreed to play anyway. My dad, an engineer, was asleep.
The game has simple rules.
You play Math’d Potatoes by drawing a card, rolling five dice and making an expression that “satisfies” the card.
The card requests various types of “answers”:
Even or odd
Equal to a certain number
Between two numbers
Less than/greater than a certain number
Everybody got into it.
Aunt Linda and Louise (my pet name for my stepmom) both agreed that it was a fun math game. This is in spite of the fact that neither one of them like math, and Aunt Linda doesn’t even like to play games at all!
My dad saw the game the next morning.
I had intentionally not waken up my father to play with us the night before. My decision was validated the next morning.
My dad is an engineer, and as such tends to use the phrase “all you have to do is,” and the word “just.” He’s a very smart man, and I’ve learned lots from him through the years. And one of those lessons is: “Keep an engineer away from sensitive math learners.”
Sure enough, when he saw the game, he eagerly said, “What’s this? Are we going to play it?”
When I explained we played the night before he responded with, “Why didn’t you wake me? I totally would’ve won.”
Math learning is slowly build, and quickly destroyed.
When we were playing, Aunt Linda and Louise were both starting to warm to the idea of math. They were enjoying the game. My sister and I were holding back just a little to give them an opportunity to discovery their own skills. (We both experienced the engineer–math–dad super push growing up.)
So by the end of the game that night, they were excited, confident, and enjoying themselves.
Had I woken up my father to play the game, he certainly would have won. He might’ve turned it into a competition, or he might have tried to help a little too much.
Either way they would’ve lost interest. Their confidence would have been destroyed. And two beautiful, smart and happy women would have their, “I’m a math Neanderthal” thoughts validated.
You can use this with your children.
If you or your spouse are in a math related field, or was “always good at math,” be aware of your potential intimidation factor. Hold back. Don’t help. Allow discovery and confidence to come at its own slow and natural pace. Your children will learn math, in their own time.
Don’t force it, or you might destroy it.
Note: They sent me this game for free. This is not a review, per se, but still – you should know how I got it.
We had a fun little discussion via email at the day-job. Someone changed the code on the combination lock and the discussion around combinations, permutations, cryptography and all things sneaky-math cranked up.
It reminded me of this neat fact I learned about credit card security.
Not all digits on a credit card are created equally!
A credit card is made up of 16 numbers. But only the first 15 are “random,” according to NRich Math. The last number – the 16th number – is the “check digit.” It’s calculated using a fancy formula using the first 15 digits.
This check digit helps a computer system quickly determine if a credit card number is fake. Now, it can’t confirm that it’s real – it just can help determine if it is fake.
The digit is created using a method called Codabar.
First, let’s define some things.
An odd digit is a digit in the credit card number that is in an odd position. For the photo of the card here, the number is: 5184 8204 5526 6423:
The odd digits are 5, 8, 8, 0, 5, 2, 6, 2
An even digit is a digit in the credit card number that is in an even position. For the photo of the card here, even digits are 1, 4, 2, 4, 5, 6, 4, 3
The check digit is the last digit – it gets calculated and isn’t considered in either odd or even position. It’s a super special digit.
The method is weird – and kinda fun!
Here’s what you do:
Add up all the odd digits, or the digits in the odd positions: 5 + 8 + 8 + 0 + 5 + 2 + 6 + 2 = 36
Double that number: 36 × 2 = 72. Save this number, 72, for later…
Add the even digits together, or the digits in the even positions: 1 + 4 + 2 + 4 + 5 + 6 + 4 + 3 = 29. Save this number, 29, for later…
(This is a freaky step.) Count the number of odd digits that are greater than 4. The odd digits are 5, 8, 8, 0, 5, 2, 6, 2. Of these, 5, 8, 8, 5 and 6 are bigger than 4. There are 5 of those total. Save this number, 5, for this next step…
Add up the results you saved from steps #2, #3 and #4. 72 + 29 + 5 = 106
Take the answer from step #5 and figure out how many you need to add to get it to the next multiple of 10 (or the next number that ends in zero). The number after 106 that ends in zero is 110. So we need to add 4. 106 + 4 = 110. 4 is the super important number.
That final super important number – well, that’s the check digit! That’s the last number that should be on the credit card – if it were a real card!
For the picture, the check digit should be a 4. And it’s a 3. So I can tell instantly that this is a fake credit card number.
I’ve pondered this a bunch since then and decided I like the idea, but the playing cards are too cumbersome. I ran across a game called Sumology (from Simply Fun) at the Texas Home School Coalition Convention. The heavens parted and angels sang.
Or at least my heart started beating and my head started spinning.
So here’s the same teaching method, but with a little more pizzazz and a couple of free downloads:
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 order of operations, the “Exponents” rule represents a bunch more than just superscripts or tiny numbers flying up and to the right of things.
Roots are exponents, too!
Tree Roots by Linda Allardice
Not the ones from trees, but things like square roots and cube roots. Consider . You do the square root first because it qualified as an “exponent.”
But if you had , the 9 + 2 is under the radical sign (the square root sign) so it’s bound together in the “Parenthesis” rule.
This one isn’t that hard with arithmetic, but when you come to algebra and start “undoing” these things – it’s important to remember that roots fall into this category.
Fractional exponents are exponents.
This one seems pretty “duh” so it’s easy to see how they fall into the “E” of the order of operations. But what are fractional exponents really?
So fractional exponents are the same as roots.
Note that some fractional exponents are roots and “plain” exponents all mixed up. Like this one:
This is a big fat full concept that needs a little more explaining. I’ll write more on these in another article.
Logs fall under the E.
Axe In Stump by caroline steinhauer
As my algebra and computer math teacher in high school, Mrs. Kelley, used to tell us – logarithms are exponents. It took me a long time to figure out what the heck she meant. But when I did, I thought it was brilliant.
Notice who the exponent is in this: 32 = 9: 2 is the exponent. And 2 is the same as because the equals sign in means “is the same as.” So the logarithm is the exponent 2.
Still with me? Either way, it’s okay. It’s a weird concept that I can go into detail in a video soon.
The thing to remember here is that logarithms fall into the “Exponents” rule of the order of operations.
So if you have , you have to do the first and then add the 7 after.
Want more on exponents?
In the meantime, you can check out more than everything you always wanted to know about exponents on the Wikipedia Exponents page. Rebecca Zook created a great video on logarithms. And check out this explanation and problems to work on fractional exponents.
And let me know what you think. Did I miss something?
I mentioned in the introductory article for this series that “the order of operations isn’t best practice for expressions involving variables.”
When you involve a variable, you can’t just “do” the arithmetic. Like in the case of 2(3x + 4)=11. You can’t add 3x and 4 to get a result before moving on. You have to use the distributive property.
And even the distributive property won’t work sometimes – as in the case of absolute values shown below.
Parenthesis mean isolation.
When we say “parenthesis” in the Order of Operations (OoO for short), we mean anything that’s grouped together and isolated. This could mean with actual parenthesis. This could mean [square brackets] or {curly brackets}.
Although grownups seem to know that square and curly brackets are the equivalent of parenthesis, children don’t. This has to be said out loud.
“Parenthesis” in the OoO can also mean |absolute value bars|. This one’s not so clear.
When you try to solve the equation 4|2x+3|=20, you have to start with dividing by 4 to isolate the absolute value chunk. There’s not a “rule” for distributing the 4.
(Although, it would be worth it, and fun, to see if your children can come up with some rules for distributing within absolute value bars. This would be some real mathematics at work for them – experimenting and discovering.)
Tops and bottoms of fractions are implied parenthesis.
It’s also the case that the numerator (top) and denominator (bottom) of a fraction are isolated places. These fall under the OoO as parenthesis.
Check out this older video I did. It shows how this works with fractions:
Parenthesis are for deviation from the other rules.
David Chandler of Math without Borders commented this in the previous article of this series:
The rule is to do higher level operations first. Use parentheses whenever your intention is to deviate from this rule.
If you can focus on this instead of a mnemonic device, you can get students to internalize what’s going on with the OoO. It’s important, however, to make sure they remember about other bracket shapes as well as isolation.
Let us know your tips and thoughts on the P in PEMDAS!
The Order of Operations (OoO for short) is used everywhere in mathematics because it encompasses many of the foundational rules that we’ve agreed to follow.
Alas, students have been given the cheap and dirty version of it for years. “Here, memorize this thing about your Dear Aunt Sally!” What the heck?!
There are subtleties in the Order of Operations that every person over the age of seven should know.
The series begins today.
The order of operations is a set of rules – like the drivers’ handbook for math. If everyone follows the rules, we’ll all be safe. But if someone makes a bad turn, we could be looking at a crash.
But the Order of Operations is only a set of rules for arithmetic! It isn’t even the best practice when it comes to expressions involving a variable like x. I’ll cover what I mean in this weekly series.
Here are the proposed articles:
Intro and mnemonics
Parenthesis
Exponents
Exponents, Multiplication and Addition
Multiplication and Division
Addition, Subtraction and Conclusions
Exponents of Negative Numbers
Another Reason to Ban PEMDAS (aka parenthesis aren’t an operation)
Mnemonics for PEMDAS
Well, there’s one: PEMDAS (pronounced just like it looks). That’s what the cool kids in high school always said. It was the same kids who said “soh-cah-toa” – which I thought sounded really goofy.
And then there’s “Please Excuse My Dear Aunt Sally.” And of course “Piranhas Eat Mostly Decayed Antelope Skin.”
