“SORRY, OUT OF ORDER”

A Facebook friend of mine posted this problem for folks to solve:

90 – 100 ( 6 + 3 ) = ?

Answers included 0, -90, 810 and -810. The correct answer is -810. Some of you are sniggering at the errors — quit that! 

Now, if you didn’t get -810, hang on …

Why do people have problems solving math equations? It’s not that they’re stupid, but that:

  • they get confused;
  • are anxious;
  • the teaching was boring;
  • they’ve moved around and have missed bits here and there;
  • they’ve learning difficulties;
  • the teachers are trapped following the text and the text is a mile-wide and an inch deep and not in sensible order;
  • the teaching made no sense or was based upon “just memorise how to do this process” instead of understanding why or when to use what methods;
  • … and sometimes people have problems for several reasons.

Hey folks, don’t feel badly if you got it wrong. I had trouble with the maths in school, too! I didn’t even learn all my multiplication tables until 8th grade. You know what? It’s not fatal; I slowly went through some pre-College Algebra classes at my local community (junior) college, and filled in the confused bits, gained confidence, and eventually went on to introductory Calculus.

And I still have to pause and think on some of my multiplication facts, and still have days when I’m prone to reversing numbers. But those difficulties don’t detract from the fact that I am able to learn math, and they don’t mean I’m stupid. (“Take THAT, ‘Mr. Dull’!” she says, shaking her fist at a middle-school algebra teacher.)

But now I work with students in 7th – 12th grade math, and you know what? Good news! It makes a lot more sense when you go back and review it as an adult! You can fill in the parts you missed or didn’t understand, and get a better idea of how it all fits together. Honestly.

Math no longer terrifies me, even though my brain still has that glitch that prevents me from memorising the quadratic formula. But I never use the quadratic formula in real life.

I DO use ratios in real life, for example, adjusting a recipe, figuring how much stuff to put on my garden, planning travel time… And I’ll show you how to do those really easily, without getting all tangled up in multiple steps, and you don’t need some mysterious “intuitive feel for how to set the problem up”.

.~#~.

MEANWHILE, In our problem above we use Order of Operations. I tell my students, “You use Order of Operations every day! You put your tee on before you put on your shirt, and you put on your jacket last.”

The problem above is solved like this:

90 – 100 ( 6 + 3 ) = ?

6 + 3 = 9

100 ( 9 ) = 900

90 – 900 = -810

The abbreviation for remembering the order of operations is known to many as PEMDAS, one mnemonic being, “Please Excuse My Dear Aunt Sally”.

Or as Randall Munroe suggest in his XKCD comic, “Please Email My Dad A Shark” or “People Expect More Drugs And Sex” (er, won’t share the latter in school). See the bottom of the post for the cartoon.

PEMDAS stands for:
1. Parentheses
2. Exponents
3. Multiplication & Division
4. Addition & Subtraction
You do operations in that order, and left to right.

So, you add the stuff in Parentheses first ( 6 + 3 = 9 ),
when you have something outside the parentheses and there’s no other sign, that means Multiply 100 ( 9 ) = 900,
then Subtract 90 – 100 = -810.

“Mn, beer…”

Here’s another problem, hopefully something that’s easier to relate to. Now, if maths makes you anxious, you might usually just grab a spare piece of scrap paper and start counting on your fingers and writing down columns of numbers to add up. But frankly, it’s easy to lose track of who you have counted.

So. “B” is for Beers, and “C” is for Cases — there are 12 bottles in a case.

How many cases of beer will we need if your friends (the 4 gamers, and the 6 historical re-enactors with their assorted significant others), and my family (my 3 relatives, and my kids with their peeps, that’s, :: counts fingers :: 8 of them) all come over? Your friends drink more beers (3B) than my family (2B). Oh, plus some beers for us!

3B [ 4 + 6( 2 ) ] + 2B ( 3 + 8 ) + 4B   = C

                             12

WOAH!

What the heck did she do, throwing in those square brackets?! Well, we have parentheses ( ) and square brackets [ ]. You do what’s in the parentheses, then what’s in the square brackets. Brackets are just outside-the-parentheses parentheses.

Breeethhhe … It’s like cleaning up a big mess: tackle one bit at  a time.

Importantly, write down the whole problem each time! Do your little step for the small answer, and then write down what’s left of the problem around it. That keeps you from forgetting stuff.

1. Parentheses

6 ( 2 )  means 6 * 2 = 12 historical re-enactors with their assorted significant others

and ( 3 + 8 ) = 11 of my extended family

Easy-peasy. Now, write those back into the equation — you can leave out the parentheses around the number 12 because there is still more stuff going on inside the square brackets, BUT you cannot leave off the parentheses around the 11 because you’re going to be multiplying it by 2B!

3B [ 4 + 12 ] + 2B ( 11 ) + 4B   = C

                       12

More parentheses.

2B ( 11 )  means 2 beers times 11 people. See, those silly variables (letters) aren’t that strange to work with; that’s 22 beers for my family, of course! 2B ( 11 ) = 22B

3B [ 4 + 12 ] + 22B + 4B   = C

                      12

Okay, we’ve used up our round parentheses. The brackets are just outside-parentheses, so we do what’s inside them next. (When I’ve used up all the curved parentheses and have nothing but square brackets left, I like to write the brackets as curved parentheses because they are easier to write, and it helps me remember, “Oh yeah, multiply what’s by the parentheses”. You have to figure out how to work around your own brain.)

[ 4 + 12 ] = 16 gamers and historical re-enactors with their assorted significant others

3B [ 16 ] + 22B + 4B   = C

                12

3B * 16 =

(Um,

3 * 6 is 18,

3 * 10 is 30,

and 30 + 18 is 48.)

3B * 16 =  48B  That’s 48 beers for your guests.

48B + 22B + 4B   = C

           12

2. Exponents

Gee, we don’t have any! That was easy. Move along, move along.

3. Multiply and Divide

We do this from left to right.

Now, this particular problem has a big line of stuff on the top and a number on the bottom. It’s a division problem, which is the same thing as a fraction.

(Seriously, 1/2 means 1 ÷ 2. 10/8 means 8 pieces of an 8-slice pizza and 2 slices left over from another; you have 1 2/8 pizzas, or 1 1/4 pizzas. I like pizza analogies better than pie — do you know what a blueberry pie looks like if you try to cut it into eight pieces and eat two? You don’t have 6/8 or 3/4 left, you have a mess left!)

The problem can also be written like this, using a forward-slash instead of a division underline, and it will mean to do the same thing:

( 48B + 22B + 4B ) / 12   = C

OR, just write a division sign already:

( 48B + 22B + 4B ) ÷ 12   = C

WHAT THE HECK?! Where did she get those parentheses from?

Because when we re-write the whole problem on a single line (instead of as a fraction-like division problem), we have to explain that we are doing everything that was on top first, and then dividing by 12.

Why? This is because the 12 helps us figure out how many cases, and we can’t do that until we know the total number of beers!

That may be obvious here, but it’s easy to forget in other situations, especially when you’re just working abstract problems that are all numbers and variable letters, with no connection to reality.

Parentheses:  ( 48B + 22B + 4B ) =

(Um, 48 + 22 is

8 + 2 is 10, 40 + 20 is 60 + 10 is 70, 70B, don’t forget that B, Andrea!

and 70 + 4 is 74, so 74B. Wow, 74 beers.)

Divide: 

74B ÷ 12 = C

Um, 5 * 12 = 60, so 6 * 12 is 60 + 12 that’s … 72, and 74 – 2 = 2.

That means we have 6 with 2 left over.

NOW LISTEN UP, FOLKS!

This isn’t some silly problem in your maths book. You are NOT going into the liquor store and asking for “Six cases of beer with a remainder of 2″. Do you want to buy 6 cases and be short 2 beers? Of course not! You’re gonna buy 7 cases of beer so you have enough, and because leftover beer is not a problem.

Of course, trying to figure out what kinds of beers to get is your problem. (I’m not even going to touch the “what kinds of pizza toppings” issue.)

Amazingly, there are some math assessment tests that have a similar question, and students can get the WRONG ANSWER. Why?  Because they’re too used to doing just “book” math, not real-world math.

For example: A school bus can carry 52 students. How many busses do you need to take 130 students* on a field trip?

An answer of “2 with a remainder of 26″ is silly, what are you going to do, leave people standing on the sidewalk? You need 3 busses!

I tell my students, “Always put your answer into a sentence — it should make sense!” No, I don’t make them write out the sentence, just explain it to me if we’re practicing, or tell it to themselves.

(* Funny, they don’t include all the adults chaperoning those students on the field trip in that number…)

FEELING SHARP? TRY THIS PUZZLER:

Grab a piece of paper and pencil.

EXPONENTS: If you haven’t done Exponents in æons, then remember that 5^2 means “five squared” or “five times five”:  5 * 5 = 25.

(It does not mean five times two! It’s easy for students to get confused when they have 2^2, because that’s 4, just like 2*2 = 4.)

The first number is the “base” or the amount you’re using. The second number is the “power” or how many times you multiply the base times itself.

4^3 means 4 * 4 * 4, which is 4 * 4 = 16, and 16 * 4 is … 64.

(Um, 16 + 16 = 32, and 32 + 32 = 64. Or, you could multiply, 16 * 4 : 4 * 6 = 24, and 4 * 10 = 40, 24 + 40 = 64. Whatever. There’s often more than one way to do things, so use whatever method is easiest for you!)

Remember, PEMDAS.

I’m filling up some raised garden beds with soil. How many cubic feet of soil, S, do I need?

One bed is a large L shape (we’ll turn it into a 4×8 rectangle plus a 4×3 rectangle), and there are four smaller beds, including two 4×6 beds, and a square 3×3 bed. One of those small beds is a right triangle with two sides the same length, so the area = length squared, divided by 2.

The sides are 8″ high, but I’m only filling them 6″ deep. Normally the volume would be length * width * depth, but the length and width are in feet, and the depth is only half a foot. So that’s the total area times 1/2, or area / 2.

Worked-out solution is at the very end.

[ ( 4 * 8 ) + ( 4 * 3 ) ] + 2 ( 4 * 6) + 3^2 + (4^2)/2  = S

2

______

The whole reason we have Order of Operations is because if we don’t all do the operations (add, subtract, multiply, divide) in the same order, then we will all get different answers! 

Remember all those incorrect answers to the problem my friend posted? Confusion on the Order of Operations is why folks ended up with different answers.

90 – 100 ( 6 + 3 ) = ?

If you ignore  the Parentheses first, you get:
90 – 100 = -10  and -10 * 6 = -60  and  -60 + 3 = -63.
Which, nobody messed up. (-:
If you ignore Multiply after Parentheses, you get:
90 – 100 = -10  and  -10 ( 9 ) = -90.
If you ignore the left to right, you get:
100 ( 9 ) = 900  and 900 – 90 = 810
I think the answer of 0 was a mis-read of 100 as 10:
10 ( 9 ) = 90  and  90 – 90 = 0.

_____

More funny mnemonics (don’t forget to hover your mouse cursor over the comic and pause, to see the mouse-over text):

_____

Here’s our garden bed puzzler:

[ ( 4 * 8 ) + ( 4 * 3 ) ] + 2 ( 4 * 6) + 3^2 + (4^2)/2  = S

2

1. Parentheses:

4 * 8 = 32    4 * 3 = 12    4 * 6 = 24    and  4 ^ 2 = 16

But 4^2 is an Exponent! Yes, but it was in parentheses.

[ 32 + 12 ] + 2 ( 24) + 3^2 + ( 16 )/2 = S

                        2

More parentheses, those square brackets:

44 + 2 ( 24) + 3^2 + ( 16 )/2 = S

                           2

2. Exponents:

3^2 = 9

44 + 2 ( 24) + 9 + ( 16 )/2 = S

                       2

3. Multiply and Divide, left to right:

2 ( 24 ) = 48  and  ( 16 ) / 2 = 8

44 + 48 + 9 + 8 = S

              2

Well now, we certainly can’t divide that stuff by 2 yet, can we? Obviously, we need to add the stuff on top first, and if we want, can re-write this:

(44 + 48 + 9 + 8) / 2 = S

44 + 48 = 92  and 9 + 8 = 17  and 92 + 17 = 109

109 / 2 = S

54 1/2 = S, or we need 54 1/2 cubic feet of soil (or you can write that as 54.5 ft^3).

If you’re buying compost or potting soil in bags, it’s 2 cu. ft/bag, so you would need 26 bags, and be just a wee bit short. Or, you could buy 27 and have extra.

What if we’re ordering it delivered by bulk? Well, smart gardener suggests putting tarps on the driveway (to keep it clean) and having another one to keep the soil dry because wet soil weights a mucking ton!

A yard is 3 feet, so a cubic yard is 3^3 or 3 * 3 * 3 = 27 ft^3.

Amazingly, 27 + 27 = 54, which wasn’t planned, but is nice. Two cubic yards. (-:

 

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4 Comments

  1. Mados said,

    27 August 2012 at 7:23

    Hello Andrea.

    I have enjoyed reading your blog for a while and have nominated it for the Versatile Blogger Award.

    Kind regards

    Mados

  2. B said,

    3 August 2012 at 11:42

    I like this TED talk about teaching mathematics: http://www.ted.com/talks/lang/en/dan_meyer_math_curriculum_makeover.html
    It has optional subtitles in several languages.

  3. Jeni Bate said,

    2 August 2012 at 22:47

    I agree that putting problems in real-world perspective really helps! I knew a substitute teacher who worked in the ghetto – he subbed for a math teacher one time and the kids couldn’t do decimal places to save their souls. Once he told them ‘think about this like it was money’, all their sums were suddenly correct – and super fast!

    • 4 August 2012 at 17:34

      Precisely!
      I do the same thing with decimals. Well, adding and subtracting.

      Multiplying and dividing works slightly differently when there is more than one number with digits after the decimal, and you count the number of digits after the decimal point* and use that quantity to place the decimal. (* But not including any zeroes after other numbers, only zeroes that come before other numbers.)

      andrea


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