Friday, January 31, 2014

14: Negatives, Squares and the Mistaken Calculator

From Kenneth Tilton via email (
If -3 is a non-zero number and a non-zero number squared is positive,
why does $-3^2 = -9$?

I went to and typed it in:

I know I've got my TI around here:

That can't be right.  Wolfram is always the definitive source, right?^2

What the heck is going on?

Thursday, January 30, 2014

13: Strange Triangles

Triangle ABC is congruent to triangle DEF.
In triangle ABC, side AB measures 9, side BC measures 3x+18, and side CA measures 7.
In triangle DEF, side DE measures 9, side EF measures 2x+26, and side FD measures 7.

What equation would help you to solve for the side length of BC and EF?

(Actual question in an on-line geometry course.)

What would you do to fix this question?

Wednesday, January 29, 2014

12: Trapezoids ...

from Kevin Shonk (via email)

What makes for a better definition?
  • A trapezoid has at least 1 pair of parallel sides 
    • OR 
  • a trapezoid has exactly one pair of parallel sides? 
 Categorize it:
  • Is a parallelogram a type of trapezoid or is a trapezoid a type of parallelogram?

Tuesday, January 28, 2014

11: More Exponents

Can you show if this is true?

$2^6 * 2^6 = 2^{11} + 2^{11}$

Can we generate more like this one ... simple and elegant?

How about one with 3 as a base?


10: Negative Numbersssss.

Can anyone explain in a clear and concise way, in English that is understandable by middle schoolers, why a negative number times a negative number is a positive number but a negative number plus a negative number is always negative?

e.g., ( -41 )( -10 ) = +410  but  ( -41 ) + ( -10 ) = - 51 


Monday, January 27, 2014

9: Order of Operations is Wrong

I've always liked the version to the right as it displays the order of operations in a hierarchical fashion, which helps to eliminate a lot of the issues that students have.

They memorized PEMDAS and assume that M comes before D, such that $30 \div 6 * 5 = 1$, when the intent was for $30 \div 6$, then $5*5$.

Here's another: " $10^2 - 5^2 \div 5$.

The order of operations is most appropriately a way to ensure that everyone can communicate in a consistent fashion, that there isn't really a "natural reason" as much as there was a human one. "It's just an arbitrary rule" was one of the most-viewed posts on MathCurmudgeon, relating the story of an expression  that only 30% of over 300,000 Facebook users got right. (And because Facebook is what it is, they were all certain that they were correct.)
$6-1*0+2 \div 2$
 ... but here is an interesting viewpoint. What is your take on it?

If you went to elementary school in the United States (or much of the rest of the world) you almost certainly learned about something boringly called the "order of operations" - a set of rules for whether or not you should do multiplication before addition or addition before subtraction to get the right answer on your math test.
Except, you don't always get the right answer, or even, one answer - I mean, is 8-2+1 equal to 5 or 7? and is 6/3/3 equal to two thirds or six? - the problem is, focusing on the order of operations can lead to ambiguity and obscures the real, underlying, and often beautiful mathematics.

A mathematician will tell you that 8-2+1 is really 8+(-2)+1, which is unambiguously equal to 7, even though the so-called "order of operations" standard in the US tells you the answer is 5. If you want five for your answer, then you really need some parentheses! But why is this ambiguity even possible? Because fundamentally, all of these operations are simply different procedures that start with two numbers and combine them in some way to give you a single number. Each operation takes two numbers as input - two, and no more. If you want to be entirely unambiguous (and pedantic), every single pair of numbers in an algebraic expression should be inside parentheses, and then there's no need to know ANY order of operations - just evaluate the innermost parentheses first, and work outwards, collapsing them down pairwise like a championships bracket.

But it turns out that's not the only way - it's possible to CHANGE the order in which you do the operations, to rearrange the parentheses, as long as you know what the underlying mathematics IS. For example, if you want to add (3+4) and then multiply the result by 5, you can either do the addition first and get 7 times 5 = 35, or you can do the multiplication first as long as you know that multiplication "distributes" across all the terms in the parentheses... that is, (5*3) plus (5*4) = 15 plus 20 = 35 - the same answer! And how do we know multiplication distributes? One way is to draw rectangles... but I've done that before.

The same rearranging can be done for exponentiation and multiplication: (3*2) all squared (or 6^2=36) is the same as 3 squared times 2 squared - 36; It even works for addition and subtraction: 5 minus (1+2) is (5 minus 1) minus 2.
So, the true order of operations is this: use parentheses, and learn what exponentiation, multiplication, addition and the rest are REALLY doing - then you can proceed however you want.

That's not to say that we don't have a conventional order of operations in mathematics... I mean, deciding that we evaluate multiplication before addition allows us to get rid of lots and lots of redundant parentheses, and noticing that (1+2)+3 = 1+ (2+3) and 2*(3*4)=(2*3)*4 removes a ton more... but the point is that those parentheses are still there, still implied - just like how 3 minus 4 is secretly implying 3+ negative 4 and 3 divided by 4 is really 3 times one fourth. But any time the result might be ambiguous, you really need to use parentheses. Then you can proceed in whatever order you want.

The order of operations learned in school, however, is different - it's a mechanical set of instructions that dictates just one of many ways you can do algebra - it locks you into a single path through the beautiful mathematical landscape, which, while necessary for a computer whose goal is merely to give you the right answer, doesn't really give any insight onto WHAT IT IS that you're doing when you do algebra.

So, while, the order of operations isn't technically wrong, because most of the time it'll give you the right answer, it's morally wrong, because it turns humans into robots.

Sunday, January 26, 2014

8: More Exponents

From Dan Meyer (via email):

Which is bigger: $60^{63}$ or $63^{60}$?

The beauty of this problem is that both sides are beyond the capability of a TI-84 so a reduction of some kind is called for. has no difficulty but the answer is startling.

Find Dan on Twitter @ddmeyer, too.

7: Exponents.

What's bigger?  $2^{100} or {100}^2$ ?

Saturday, January 25, 2014

6: Why 360?

Everyone knows that there are 360 degrees in a circle.

We can also take it as a given that people who did math back in the days of Babylon didn't mess around ... they must have had really good reasons for choosing that number.  They were logical and just as smart as we are now, even if we sometimes don't admit that.

How many good reasons can you think of for 360° in a circle?
How many ridiculous reasons can you think of for 360° in a circle?

Should we change to another number for doohickeys in a circle?

Friday, January 24, 2014

5: Is 1 prime?

What makes a number prime?

Is 1 prime?

2 is prime. 3 is prime. 5 is prime. What should the definition of "prime" be? 

Thursday, January 23, 2014

4: Percent Increase

On the Praxis Exam I took some years ago, there was a question that ran something like this:

The parking lot at the college was a square 100 yards on a side. The lot was increased by paving enough land to make a new square with a side that was 400% bigger. The size of the new lot was what percent of the old lot?
a)  400%
b)  800%
c)  1200%
d)  1600%

So ... what's wrong with this question and those answers?

Wednesday, January 22, 2014

3: 0/0 = 1, or 0, or undefined?

Consider the following:

0 ÷ 4 is clearly 0.
4 ÷ 0 is undefined.
How should we deal with 0 ÷ 0?

Is 0 ÷ 0 = 1 because anything divided by itself is 1?
Is 0 ÷ 0 = 0 because 0 divided by anything is 0?
Is 0 ÷ 0 indeterminate because you can't determine it?
Is 0 ÷ 0 undefined because you're not Neo?

Tuesday, January 21, 2014

2: Improperly Canceling Fractions

Simple fraction reduction ...  $\dfrac{26}{65} = \dfrac{2}{5}$ ... Wait a minute. Reduce by canceling the 6s?

Is this a fluke or is there something more sinister at work here?
Does this trick work in any other fractions?

If the Latex isn't working, this problem is 26/65 = 2/5.

Monday, January 20, 2014

1: Is zero Odd or Even?

I'd say even ...
  • to fit the pattern. 6, 4, 2, 0
  • to contradict the formula that an odd number can be written as 2n + 1, n ∈ integer.

Any others?
Any reasons to call it odd?

Wednesday, January 1, 2014

NULL: LaTex test.

Consider the sequence:  $1, a, a^2, a^3, ... , a^n$

What is the median of that sequence? is it $a^{\frac{n}{2}}$?

How about if a is a fraction?
How about if a is negative?