Big Ideas in Mathematics

You open a math textbook and see the following problem:

1/3 + x = 1/8x + 1/5

There’s a lot going on.  Maybe you remember a teacher telling you about “two-step equations”, “getting x alone”, and “getting rid of the denominator”.  Maybe you remember what to do first.  Or maybe you don’t.

It’s hard to memorize a bunch of random facts and procedures.  That’s why when I think about math, I think about big ideas first.  Then, I ask myself how the little details fit inside these big ideas.  In the next few paragraphs, I’ll talk about some big ideas that students are likely to encounter in math class.  See if you can find at least one of these big ideas in each topic you study.

Big Idea 1: Equal things are interchangeable (they can be switched).

Robert went to the park to catch a Dragonite.”  Robert has a nickname, Bob.  So I could just as easily write “Bob went to the park to catch a Dragonite”, and the meaning of the sentence doesn’t change.

In math, we see problems like this:

Find the sum of 1/5 and 1/10

Like Robert, “1/5” has a nickname.  Actually, 1/5 has an infinite amount of nicknames!  One of these nicknames is “2/10”.  With this, we can change how the problem looks like, without actually changing the problem:

Find the sum of 2/10 and 1/10

Now that the problem looks easier, it can be more easily be solved: when you add two “tenths” and one “tenth” together, you have three “tenths”.  And so the answer is 3/10.

If two things are the same, you can replace one of them with the other.  Many math problems boil down to cleverly replacing parts of a problem with something equal to make the problem easier to figure out.

Big Idea 2: If two things are equal and you do the SAME thing to both of them, they stay equal. If you do different things, they don’t stay equal.

Doublemint Gum and Dubble Bubble Gum cost the same amount of money.  One day, the cost of both brands of gum increased by 50 cents (atrocious!).  Do both brands of gum still cost the same amount of money.  Absolutely!

 In math, the problems look like this:

Solve for x:

3x + 1 = 2

The equal sign says that “3x + 1” and “2” are equal.  As long as I treat them the same, I can do anything I want to them and they will stay equal.

Let’s just play around by adding the number 5 to each expression.  We get this:

3x + 1 + 5 = 2 + 5

3x + 6 = 7

Was there any point to adding 5 to both expressions?  No, not really!  It didn’t help us solve for x.  All it did was show that the same value of x which makes “3x + 1” equal to 5 also makes “3x + 6” equal to 7.

But was there any harm in what we did?  No!  Unlike a chemistry experiment where you might burn your eyebrows off if you mess around too much, you can goof around in math, take back anything you’ve done, and try again.

Let’s try the original problem again.  To solve for x, we’re going to be clever and make the equation look easier by subtracting both expressions by the number 1.  Here’s what happens:

3x + 1 – 1 = 2 – 1

3x = 1

Ah, this equation looks much simpler!  (Ask yourself: has the problem actually changed at all?)  And now that the problem looks much simpler, we can use common sense to figure out that x has to be 1/3!

If two things are the same, treating them the same means that they stay the same.  If you treat them differently, they don’t stay the same and you’ve changed the problem completely.  

Many math problems boil down to manipulating two equal things so that the problem stays the same but looks easier to solve, which makes it easier to solve.

Big Idea 3: The same thing can be represented in multiple ways.  In other words, two things can look different but be the same thing. 

In Big Idea 1 and Big Idea 2, we played around with symbols that look different but are actually the same.

For example, 1/5 and 2/10 look different but they represent the same amount of pizza, or brownies, or whatever you’re breaking up into a fraction.

Likewise, getting a 85% on a test is the same as the following:

  • 85/100 correct
  • 0.85 correct
  • 8.5/10 correct
  • 1 – 0.15 correct
  • 15% wrong
  • 100% – 85% wrong
  • Eh, you get the point

Here’s another example: 3x + 1 = 2.  Even though “3x + 1” and “2” look very different, they represent the same quantity when x is equal to 1/3.

Here’s a final example: choose any number, and double it.  For example if we chose 5, we would get 10.  Or if we chose 13, we would get 26.  Represent this idea using a table, graph, and equation.  Notice that each representation looks very different but is actually the exact same idea!

The same thing can take on many different forms.  Switching from one form to another can make problems easier to think about and solve.