On Subtraction

Today's lesson is broadly applicable. I present subtraction the same way to my kindergarten daughter as to my 7th grade math class -- and yes, 7th graders still TOTALLY need to revisit this topic, as do many 8th graders, and as may you.  

Most of us are familiar with the Take-Away method of subtraction.  It's popular.  It's comfortable.  If you stopped taking math as soon as you possibly could and vowed never to think too hard about it again, this is likely the method you use.  I use take-away when my daughter poses a question like, "How many cookies will be left in the box after dessert tonight?"

You could use the cookies themselves, or beads:

Make a pile.

Count it.

Take some away.

Count what remains.

Wonderful!  You're subtracting.  

Taking-away subtraction runs into a wall at zero.  So do cookies!  Right this second, my family can only eat as many thin mints as exist in our pantry, and no more.  If I want another cookie, TOUGH LUCK.  I'm going to have to wait.  Possibly a long time, seeing as girl scout cookies are a seasonal treasure!

But this is math, and numbers don't run out like cookies. 

So I introduce negatives early and use them often. 

Negatives aren't so hard, really.  We have this concept of opposites built into all of our classic stories and lore and many world philosophies.  People are enchanted with ideas of doppelgangers, light vs. dark, day vs. night, up and down, good vs. evil.  I strongly encourage you to build symbolic or poetic meaning into your numbers.  I like a yin and yang approach to my positive and negative numbers, as I love all my numbers so the more Western construct of good vs. evil just doesn't seem fair to the evil half.  But you do you!  Because every story we make builds a visceral understanding.  Art and story ABSOLUTELY supports math learning. 

Two ways I like to teach subtraction to people of all ages: 

1) THE NUMBER LINE

Zero in the middle, regularly spaced numbers stretching in both directions.  Negatives to the left, positives to the right.  

I painted this number line way back when Elise was just a little girl (because I am a bit extra when it comes to my math). Maybe your kid is into arts and crafts and wants to beautify a number line.  Go for it!  You should also feel free just to pull out a ruler and mark up a couple pieces of paper. You can make a game out of it by adding a set of dice and putting a + and - on a coin, flipping and tossing to see where you go.  Have your child start at zero, and march her little counter rightward for addition and leftward for subtraction, then note where she lands!  

Better yet, make it HUGE.  Mark out a number line down a long hallway of your house or along the sidewalk with chalk (if you're still doing sidewalks), and make those kiddos work for their learning!  Masterful teachers build kinesthetic learning into each day, and many children love it.  Simon Says works great for this game.  If they have too much energy, make them hop their steps like a frog or walk them like a bear on 4 straight limbs.  Build that OT into math.

PLUS means take this number of steps towards the positive end of the number line. 

MINUS means take this number of steps towards the negative end of the number line.  

Call out your orders, let the kids count their steps, then look down and read off where they are.  The lesson is that we can go one way by adding and the other way by subtracting.  We can pass zero as many times as we like!  And if we need a longer number line, we can make it longer. 

Believe it or not, I make my middle schoolers get up and march across my classroom.  Eyes roll, but they do it, and it helps them learn.  And if it's absolute mutiny, I let them stay seated and I do my own little number line dance for their entertainment.  It gets a laugh every time. 

Just be careful how you compare these numbers.  Leftward is MORE NEGATIVE, LESS POSITIVE, or LESS THAN ( < ).  Rightward is MORE POSITIVE, LESS NEGATIVE, or GREATER THAN ( > ).  Stay away from "bigger" and "smaller" as these words have ambiguity in their mathematical meaning.

2) OPPOSITES

Subtracting is the exact same thing as adding.  Subtracting is adding a negative.

You're going to need two colors of beads or pebbles or coins for this exercise.  One color for positives (blue).  The other color for negatives (orange).  The rules of the game are this:

One positive cancels one negative.  If we want to subtract, we add that many negatives, pair them off, and clear the perfect pairs away.  For instance: 

12 - 8 

becomes: 

12 + (-8)

So I gather 12 blue beads and 8 orange beads and put them altogether.  

Organize them like yesterday, separating positives from negatives and grouping by 10 where applicable:

Make your pairs! 

Each perfect pair: one blue, one orange, equals zero.  This is such a small detail, but some children will do best if you have a zero place to put them all: 

And other children will do better if you sweep the zero pairs all the way out of sight so that you can concentrate on what is left behind.  For many children, all the business of those zero pairs hanging out in the visual field is confusing and overwhelming.  Follow your child's lead.

Either way, you have subtracted by adding negatives!  The magic of this method is you can get ANY answer!  Positive, negative, or zero!  If your answer is blue beads, it's a positive answer.  If your answer is in orange beads, it's a negative answer, and if the field is clear, it's zero. 

Take aways: 

Subtraction IS addition.

Introduce the concept of opposite numbers (positives and negatives) early and reinforce it often.  Build comfort and familiarity with these concepts by playing games.  Kids learn the rules to games fairly quickly and easily, and practice like this builds INTUITION around positive and negative numbers.  (We do not have to be born with math intuition. Like anything, we can develop it through practice.) 

Number lines are one way to think about numbers, and open up kinesthetic learning opportunities. 

Opposite pairs are another way to think about positives and negatives. 

One of these will probably resonate more with you and with your kids than the other, and that's fine.  It still pays to play around with both methods to develop fluency. 

Wishing you lots of fun with subtraction!