Our discussion on fractions and pizza on #mathchat yesterday reminded me of a story.
Image via Wikipedia
My best friend lost on this deal back in 1978. But she gained a valuable lesson.
My friend, let’s call her Linda…
She was a terribly terribly slow eater. Like annoyingly so.
We would sit down to a Totino’s pizza on a summer afternoon. I would finish my half while she was still working on her first piece!
One day I suggested a different method.
“Let’s NOT divide the pizza in half – equally,” said I in my sweetest voice. “We’re best friends, and dividing food up is so primal. Let’s just eat like normal people.” (Okay, I’m sure I didn’t say, “primal,” but you get the point.)
She agreed. Even though she could manipulate the heck out of me, I certainly had my share of the talent.
She got two pieces.
A whopping 1/4 of the pizza.
She still tells this story.
And guess what? She totally knows the difference between her 1/4, our 1/2 and my 3/4.
What about you? Do you have any lessons you learned from friends in math? Share them in the comments.
Please note that my bottom still reflects this bad pizza eating habit. Perhaps writing it down for the world to see will help me knock of the over-pizza-eating habit. That and the LoseIt! iPhone app.
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.
A mom on the Living Math forum was asking for advice on how to teach adding numerators and not adding denominators in fractions. So this topic is again in the front of my mind.
First note, that you can add both numerators and denominators – this is called piecewise addition – but then you are getting into a whole new world of stuff. Actually, you’ll be experimenting with some goodies from Abstract Algebra (my field of study). We’ll talk about how and why this is good in a few paragraphs.
What is adding fractions, anyway?
For us in the “real world,” adding fractions is adding values that can’t be counted with fingers. It’s adding bits of things together – maybe whole things with bits, even.
We have agreed to use things that look like
to represent fractional pieces of a whole.
When we add we have said that our total is
Total: 3/4 of a “pizza” is blue.
Quantity over value is important to young children.
I remember taking money from Little Brother when we were kids. You might think I was stealing, but it was more of conning the innocent. He would have 3 dollar-bills. I would have 20 pennies.
He would quite willingly trade his 3 monies for my 20 monies. For him, this was a matter of quantity. I knew the value, though. (Bad girl – sorry, LB I owe you. xo)
This measuring of quantity, rather than value was important in LB’s development. Instead of someone teaching him how to measure value, he learned it through experience.
This is why we should allow kids to add the denominators.
Let’s use the example above and add the numerators and the denominators to see what that means.
Adding means that our total is
Notice here you can’t “reduce” the fraction – because this isn’t the value as we know it.
If you teach adding fractions using manipulatives or toys and allow adding denominators, soon your children will see that your (on the top in the picture below) is very different than having their, on the bottom.
We have six total pieces.
This will inspire them to work toward equality – and invent fractional addition (correctly) on their own!
Do you struggle explaining to your students what “simplify” means? Do you wonder why they won’t reduce fractions to their lowest terms?
Here’s how to explain and teach some of the top confusing terms in mathematics – without throwing you or your students into a tizzy!
Where
Really! “Where.”
As in “Where do the lines intersect?” or “Where is the highest point of the graph?”
“Where” in math almost always means “at what x-value?” Every now and then “where” can mean “at what point?” In the first case, “Where do the lines intersect?” can be just as easily satisfied by a point (x, y) as the x-value only.
When you get into calculus, the word “where” is used everywhere (pardon the pun). “Where is the slope of the tangent line equal to zero?” and “Where are there discontinuities?” among others.
Keep your eyes peeled in the textbooks for this tricky term.
Reduce
Typically this means to reduce to lowest terms. Which isn’t really helpful, since you still need the definition of “terms.”
Numerators and denominators (tops and bottoms of fractions) that have one or more common factors are not officially “reduced.” Once they have no common factors at all (except 1), they are said to be reduced or reduced to lowest terms.
A reduced fraction helps in many ways. You can “feel” the value more easily: 2/5 is a lot easier to picture than 14/35. Also, when you add fractions, starting out with all the numbers in their reduced form will often help to find common denominators.
Zeros, Real Zeros and Roots
Zeros, real zeros and roots all mean the same thing. It is where a graph passes through the x-axis. We can write “where are the zeros” meaning at what x-value (“where” from above) does the graph pass through the x-axis? We also will write “what are the zeros” and mean the same thing.
To find the zeros or roots, we set a function (or any equation solved for y that involves x) equal to zero. Then we solve for x. This will give the result that we want – the x-values where the graph will go through the x-axis.
We call them zeros because they are where y = 0. We call them real zeros because there can be complex or imaginary zeros that don’t exist on the graph.
Finding the zeros or roots of a function help when graphing it. If you know all the places it goes through the x-axis, you can then determine where it’s above or below the x-axis. Then you can roughly sketch the graph.
Simplify
Of all the terms in mathematics, this one is the most slippery. This meaning of this term is always in context to the book or situation. What it means is “manipulate the expression.” The end result may or may not be “simpler” than the starting expression.
Better textbooks use phrases that are clearer, like “multiply the polynomials” and “combine like terms.”
If “simplify” is the instruction to a problem set in a textbook, consult the examples to see what type of manipulation to do.
In later courses, students have to recognize when a situation requires a different form of the expression. Practicing various forms of manipulation (or “simplifying”) gives them tools to manipulate the expression to satisfy their needs.
Do you run into other terms in mathematics that are hard to explain to students? Hard to understand yourself? Share them in the comments!
I mean: “What is three times a number (that number is four), plus two?”
The domain is all the possible questions:
What is three times a number (that number is five,) plus two?
What is three times a number (that number is six,) plus two?
What is three times a number (that number is seven,) plus two?
What is three times a number (that number is eight,) plus two?
<how long will I have to do this – Egad!>
Not only do the questions go on forever, but they also have all the fractions and decimals in between. And all the negative versions of those numbers (and zero).
So, what’s the point, you might ask. Looks like the questions go on forever and you can just pick any number.
The domain might not include all the numbers.
The two sticky points for the definition of “function” are bolded:
A function is a question with only one answer to a valid question.
The “valid question” part is where the domain comes in.
The numbers that make “valid” – meaning we actually can get some answer – are the numbers that aren’t negative.
Many functions have “all real numbers” as a domain. There are no limits on the things you can put in, other than numbers that aren’t imaginary or alligators.
For the most part, there are only two places where you have to be careful of limited domains. Those are
Numbers that cause a zero to turn up in the denominator
Numbers that cause negatives to turn up in square roots.
Here are two videos tackling each:
What do you think? Is this easier to teach when you consider “analyzing” the function rather than “solving” it? Share your thoughts and tips in the comments!
Do you keep struggling to teach common denominators? Do your kids just not “get it”?
Well, it’s time to quit.
Yep! You can teach (and do) fractions without ever finding a common denominator.
The key is in the definition of addition for rational numbers. If you have two numbers and the sum of them is .
Granted there is the technical issue of reducing, but using a little prime factorization will get you there without an issue. (And you don’t really HAVE to reduce, which I will discuss in a later post.)
Here is an example:
Here’s one with an obviously easy common denominator that works just fine with this method. In it I explain a little about why you would want to do it this way:
You might argue that finding a common denominator is an important learning experience. And you’re right. In the same way that learning about death through the loss of a pet is an important learning experience. But if you can avoid all that pain, why wouldn’t you?
If you teach this method first, kids will get annoyed with having to do so much reducing and discover the common denominator method for themselves. And that’s really what learning math is all about.
What do you think? Can you teach fractions this way? Share your thoughts and experiences in the comments.
Thanks to @padgets for our conversation about teaching fraction on #mathchat a few weeks ago. You inspired this post!
I previously posted about prime factors and about using them in multiplication. Now we’ve got fractions to handle.
Before we get in too deep, first let me note the main reason why we do this with arithmetic. In algebra, students will be asked to take a rational expression, factor it and reduce it like this:
If they are familiar with how factoring works with numbers, this will be normal to them.
I also find that reducing factors in this neat and clean way helps a lot. In this video I showed the first few steps of reducing the fraction a sloppier way after I did it the “neat” way:
Here’s one that’s a little more complex:
Will be helpful to show your kids? Is there something I’m missing? Share with us in the comments.
Square Roots (aka radicals) are part of the multiplication world. They have the same properties, but because they look weird – living in little “houses” – it is hard to grasp what they can do.
This is how the “houses” (the radicals) can “move”:
After that, you can work with multiple radicals:
It is important to note that this little trick works because it’s all multiplication, division and roots. If you have addition or subtraction in there, you’ve got yourself a challenge. Watch this to see how to deal with addition in the mix:
Thanks to jrobbo at MyMathForum.com for the inspiration for this post.
The fancy answer is, “For all numbers a and b, ab = 0 implies a = 0 or b = 0.”
The less fancy, and still strange answer is: If two things multiply to get zero, one or both of them better be zero. Like this:
Too much pixie dust, for my taste. What doesn’t work, for sure, is this:
So why does it work with zero and not three?!? To answer this, we need a little background on fractions and zero. And pizza – pizza’s always a good thing to add to math:
Using the fact in the video above, the “foorp” in this video shows why it’s true:
Special thanks to GFC at MyMathForum.com for inspiring this post.
