Directed Numbers

Analogies and Directed Numbers

Directed Numbers – positive and negative numbers – is abstract, particularly the negative thing.  Bank statements aside, it is rare for us to see negative numbers. We don’t even have sub-zero temperatures here in Sydney; not in our waking moments anyway.

While I’ve taught my year 7 class about ‘two negatives make a positive”, this remains an elusive concept to understand.  At least they remembered what I said.

Being an ESL-speaker myself, I spend a fair amount of time with literacy within the context of maths. Revising Directed Numbers today, we went through words that were associated with Positive and Negative. One of the most useful ones was ‘not’ for negative.

not down – up

not left – right

not negative – positive; i.e. minus negative is a plus (positive)

On the last period on a warm Friday afternoon, cheerful “I get it” from students is like a cooling G&T or glass of iced water (non-alcoholic option).

I guess this post is more about my teaching than learning.

I’d like to say the ‘not’ thing was completely random. The truth is, I used ‘not’ a lot in my computer programming days. If you think of computers as a collection of switches (which is a very simple but apt analogy), all you’ve got is on and not on (off). I still use ‘not’ in my SQLs. Sometimes it’s a lot easier to say NOT (A or B), particularly when you want the rest of the 24 letters, or something like that. Example is interpreted as Not A AND Not B; the not applies to the OR as well. But this is going into Boolean Algebra…oops, algebra again.

Polya, GGSC and Algebra

Polya and Algebra

I introduced Polya to my year 7 maths class when we started Algebra. Well actually, all I said was Polya stated that “if the problem is too hard, try a simpler one“. A bit of googling and I learned that Polya said it more elegantly –

If you can ‘t solve a problem, then there is an easier problem you can solve: find it.

For some reason, this has really struck a chord and nary a lesson would pass that Polya is not mentioned, by me or a student. I reckon it’s because the class knows he was real. It’s rare we acknowledge the great minds behind maths as a body of knowledge. I guess we all need a hero and in Algebra, more so. I can’t wait to introduce Descartes!!!!

GGSC and Algebra

GGSC is my concoction, an adaptation of Polya’s steps to solving mathematical problems and the KWHL mentioned in a previous post on learning strategies. One day I will tell my class who the inspiration really was. Anyway, GGSC is supposed to help students deal with word problems, regardless of the strategy used (yes, we did a quick revision of the strategies they’ve learned: guess-check-refine, working backwards, drawing a diagram, solving a simpler problem, etc). The acronym stands for Given, Goal, Solve, Correct, i.e.:

1. What is Given?

2. What is the Goal?

3. How will you Solve it? This is where the chosen strategy is used.

4. How do you know you are Correct?

My process was to unpack the following word problems step-by-step, first using words, and then using Algebra.

1. Five years ago, John’s age was half of the age he will be in 8 years. How old is he now?

2. The sum of the least and greatest of 3 consecutive integers is 60. List the 3 integers.

First mistake, using a fraction and distributive property of multiplication in the first problem. Combined with the algebraic pronumerals/variables and the students forgot everything they already knew!

Second mistake, showing all the steps at once too soon (I should have kept my step-by-step presentation animation used in example 1 for example 2).  As soon as the students saw the algebraic solution, they became bamboozled – seriously…”what is an integer?” let alone differentiate it from a decimal. At least they all agreed  an integer is a number.

In the end, I had to quickly wrap up – abandon ship, in a way – and focus more on GGSC itself. I challenged them to think of a memorable mnemonic. They enjoyed this activity which gives me hope that I can mention GGSC again, even if not with Algebra. The last mentioned before the bell rang was ‘Good Girls Stay Cool’.

Speaking of mnemonics (a useful learning strategy), I once again draw inspiration from Polya

How I need a drink, alcoholic of course, after the heavy chapters involving quantum mechanics

This is his mnemonic for π (pi). The letter count represents the digits 3.14159….. Brilliant!