“Observe and make use of structure”. Observe would be a start. A tale from the chalkface.

Here’s my little story:

It was a class of day release students on a Higher National Certificate course in engineering. I reached a point in one class with a relationship between p and q, p = kq, with k a constant. “What’s its graph look like”, I asked. Deathly silence. “Ok, let’s try x and y”. Result y = kx. Same question, same response. “Well, what about y = 3x ?”. Same question, same response. So I wrote y = 3x + 2. Their eyes lit up, and they unanimously shouted “It’s a straight line!”.

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CCSS SMP 7 Look for and make use of structure. Sums of powers of the natural numbers

The following is unreadable. Use the browser zoom or click the picture, or download the .doc file from https://howardat58.files.wordpress.com/2014/10/sums-of-powers.doc

The last performance – non mathematical in the formal sense.

In three hours from now (5.00pm PR time) takes place the last performance of Nana Badrena’s Ballet “DRACULA”, at the Teatro Yaguez in Mayaguez Puerto Rico. Created in 1999 in Cuba this ballet has been performed many times, in Cuba, Mexico, Pennsylvania, South America, and here. The ballet is based on the film from 1973 by Francis Ford Coppola.

This is our Ballet Company, Western Ballet Theatre, and we have a website http://www.wbtpr.com

(so it’s not just math !)(and theatre is the English spelling)
Dracula seduces Lucy Westenra
and then moves on to Mina, the reincarnation of his dead bride of 400 years earlier
The vampires in Dracula’s castle.
Jonathan, after Renfield grabs his valise
The dramatic bed scene, as the drugged Jonathan wakes up.

Halloween fun

Found on the net. With avocados raining down on my porch roof this seems a good use for them!

“To hell with pumpkins, try the Guaco-lantern

Fractions, at home and away.

So one day recently I was bored, and then the following rushed onto the page:

Half of a big pizza is equal to two small pizzas – rewrite this in as precise way as possible.

Is this 3 hours or 1/4 of a pizza?

How many hours equals 1/4 of a pizza?

Apologies to those for whom a clock face is a historical artifact.

How do I know it’s a pizza and not a cake?

Does half a day include half the night?

There are four feet in our yard, mine and my sister’s.

Would you prefer 1/2 of a round pizza or 1/2 of a square pizza?

Are ratios numbers or fractions (or neither) ?

Fractions are parts of the same whole. OK, I’ll have 5 quarters of that pizza (5/4 is a fraction, isn’t it?)

You cut, I choose !

This year fractions are parts of a whole. Next year fractions will be numbers. I guess the other party won the election.

2/3 and 4/6 are equivalent fractions. Equivalent to what ?

The word “fraction” has the same root as “fracture”. So something got broken. I think it was my faith in common sense.

The Common Core test question asked – How long is 3.25 hours. This could be 3 hours and 25 minutes or 3 hours and 15 minutes. I guess it depends on the grade level.

Back to the heavy stuff next time !

Subtraction in algebra – let’s use algebra !

I have seen some heavy handed ways of explaining the identity

a – (b + c) = a – b – c

Let us use algebra. Give the left hand side a name, say d .Then

a – (b + c) = d

This is an equation, so add (b+c) to each side and get

a = d + (b + c), then a = d + b + c as the parentheses are now superfluous.

Now subtract  b  from each side

a – b = d + c

Now subtract  c  from each side

a – b – c = d

so  a – (b + c) =  a – b – c

or is this too simple ? Look, no messing with p – q = p + -(q) stuff,

and no appeal to the famous distributive law.

You can do this, and other stuff, with numbers as well.

Calculus without limits 5: log and exp

The derivative of the log function can be investigated informally, as log(x) is seen as the inverse of the exponential function, written here as exp(x). The exponential function appears naturally from numbers raised to varying powers, but formal definitions of the exponential function are difficult to achieve. For example, what exactly is the meaning of exp(pi) or exp(root(2)).
So we look at the log function:-

Weather puts math aside !

It’s a nice breezy day but just check the weather map!

Fractions as parts of a whole, again !

This was found on “talking math with your kids” as an example of the “strange” stuff that kids bring home and cause mystification in their parents.

“The whole is 8. One part is 8. What is the other part ?”.

Just what exactly is this supposed to mean?
That the whole always consists of two parts?
Since when did numbers have parts?
What is the definition of “part”?
Even if we are talking about 8 things, then the natural AND logical answer is “There isn’t another part”.
If I want to see ways of creating 8, using adding, then what is wrong with
8=1+7 8=2+6 8=3+5 … 8=7+1 and 8=8+0 for completeness’ sake.

To call zero a part of 8 is going to lay the groundwork for a feeling that math hasn’t got a lot to do with real life, which is a crying shame. This feeling can arise at any stage, we should give reality a chance at this level. To conclude “What a stupid question!”.

Double negatives, or the meaning of -(-2)

In the extended number system of signed numbers, that is, the positive and negative numbers I see a lot of heart searching over the meaning of -(-2). This can be put to rest in one or both of two quite satisfactory ways:

1: Signed numbers are directed numbers, used for position, temperature, voltage etcetera. The basic question is “How far apart are the two numbers  A  and  B ?”, or more useful in a practical situation “How far is it from  A  to  B ?”.

This is a subtraction problem with direction and the answer is  B – A
For A=3 and B=7 we get
Distance from A to B = B – A = 7 – 3 = 4
For A=-3 and B=7 we get
Distance from A to B = B – A = 7 – (-3) = ???????????????
But a quick look at a number line shows that the distance is 10
So 7 – (-3) = 10
But 7 + 3 = 10 as well
Conclusion: -(-3) = +3

2: A simple and more abstract approach:
Starting with 7 – (-3) = ??????????????? we give a name to the unknown answer. Call it D.
Then using the basic fact that 12 – 4 = 8 is equivalent to 12 = 8 + 4 we have
7 – (-3) = D is equivalent to 7 = D + (-3)
7 = D + (-3) is equivalent to 7 = D – 3
7 = D – 3 is equivalent to 7 + 3 = D
which says that D = 10
So subtracting -3 is the same as adding 3

A meaningful example is as follows:
My friend from Anchorage calls me and says “It’s cold here this morning, -5 degrees”.
Down here in Puerto Rico it’s 68 degrees this morning.
How much warmer is it here than in Alaska?