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 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!
The March 2012 edition of Highlights magazine for kids fell in my lap this morning. Thumbing through it, I found a fun little algebra puzzle.
After looking at the equations and discovering my solution, I began to wonder some things. Then I realized that these ponderances of mine were good for kids to consider as well.
Let the curiosity flow!
Here are the extra questions you can offer your children in case they’re curious:
Did you use all four equations to decide your answer?
Could you have used three, two or only one of the equations to decide your answer?
Try to use only one equation to decide an answer. Did you discover anything?
Try to use only two equations. What did you discover now?
How about three – what happens when you choose three equations?
Can you make a puzzle like this of your own?
How about you?
As a grown up, what do you discover about math in this Highlights puzzle? What do you see your children doing with these extra questions?
The order of operations includes two types of rules: those that are based on the way the operations work, and those that are arbitrary. My friend @harrisonalg from the Twitter chat #mathchat and I have been discussing this.
You can explain the truly arbitrary elements of PEMDAS (the left to right of AS and MD) through an experiment. Allow students, independently, to do these two problems any way they want, ignoring any stupid arbitrary rule they might have previously memorized:
3 – 2 + 8 – 3 + 4
2 x 7 ÷ 2 x 6 ÷ 3
The idea is that they will come up with many different answers:
3 – 2 + 8 – 3 + 4 could be any of -14, 0, 10 or others
2 x 7 ÷ 2 x 6 ÷ 3 might be 7/18, 7/2, 14, etc.
If they were on a team building a bridge with these calculations, things wouldn’t work so well.
Enter the Stupid Arbitrary Rule (SAR).
Because we need to all come up with the same answer, we need a rule to follow. Really, it can be any stupid arbitrary rule (SAR). But we agreed, at some point in history, to all follow the “left to right” thing once we were down to addition & subtraction or multiplication & division.
It’s important to note that kids didn’t get to be part of that agreement we made. Just like they don’t get to vote in elections.
Is it fair? Probably not. They would probably do a better job of choosing leaders as well as determining the order of operations. But that’s the way things likes SARs work.
PEMDAS doesn’t cover it all!
To throw a few more kinks into it, check out how wikipedia explains some special cases of the order of operations. . Factorials aren’t covered in PEMDAS, neither are the order of exponents.
Here’s another fun one: even calculators do things differently (and arbitrarily). Check out the way the the TI-92 and the TI-30XII handles exponents. Clearly you get to make up your own order of exponents!
Let them play king: they create the rules!
Let your students determine the order of operations for addition & subtraction and multiplication & division – just for the class. Make sure to write the rule down and tack it to the wall so everyone remembers what they decided. In your class, and your class only, they are to follow that rule.
Remember the back of the book, as well as your answer keys or teacher edition textbook, will now be totally wrong. But it’s worth it.
Doing it this way they might start to understand which pieces of PEMDAS are dependent upon the way the operations are constructed () and which pieces of PEMDAS are arbitrary (left to right).
Once the class determines their special arbitrary rule, practice creating expressions whose result is some important number. Some options could be:
The age of one of the kids
Someone’s favorite number
Your age (be careful)
The age of some other grown-up they know
The last four digits of their phone number
The number part of their street address
Create the expression using the Class Rule as well as the stupid arbitrary rule. Then translate and have a good laugh. I just did my age with my “Bon Rule.”
Bon Rule: addition and subtraction go from left to right, and multiplication comes before division.
My age is 120 ÷ 2 x 3.
If I did this according to the SAR, I’d be dead. Or really famous.
Your turn!
Can you teach the order of operations like this with your kids? If you classroom school, how much trouble are you going to get in when you become the math teaching rebel?
I mailed out the bulk of the thank you notes for Daughter’s 2nd birthday party gifts. And wouldn’t you know it, it involved algebra!
I have so many stamps from the last 20 years, in every denomination you can imagine. I’m trying to use them up because I only save full sheets of stamps. (I’m a bit of a stamp collecting snob.)
Rule #1: No more than the required postage on any letter.
It has to be exact.
As of this writing, postage in the US is $.44 per letter.
I have $.33 stamps, $.37 stamps, $.41 stamps, etc. And I’ve purchased $.01, $.02, $.03, and on up to supplement and make my postage exact.
Rule #2: No more than 2 stamps of the same kind on a letter.
I like variety.
With these requirements, here are some of the combinations that I’ve come up with:
So how is this algebra?
Here is an algebra equation from the stamps in the picture:
10x + 17y = 44
The equation means x number of 10 cent stamps plus y number of 17 cent stamps will give me the exact total of 44 cents! So x=1 and y=2.
And here’s another:
10x + 24y = 44
The equation means x number of 10 cent stamps plus y number of 24 cent stamps will give me the exact total of 44 cents! So x=2 and y=1.
It would make it a better set of algebra problems if I allowed more than two types of stamps. But I’m sticking with my variety rule. 😀
How about it – will it make writing thank you notes with your kids more fun? More challenging? More annoying? Share your thoughts in the comments!
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.”
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.
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.
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!
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!
