The end marks the beginning!

Exams are over. A new cohort is ready to start along the path to Higher Maths. It’s June.
What topic do we use as an introduction?

Come August some of the original students will have gone, and some more will have joined the course. The latter set will prove problematic if we have already covered a pivotal topic.

A common solution to this problem is to develop the notion of ‘gradient’ as covered in the National 5 course and to lead into the idea of the ‘gradient at a point’ and hence to Newton’s derived function.

The laws of indices will be revised; historical notes add to the interest; the student gets a mixture of the known and the novel.

An alternative to this, also involving the laws of indices, is to introduce the concept of logarithms as a calculating aid.

Currently the topic of logarithms is seen by the Higher student as difficult.
Why? Most likely because he is introduced to the idea as a fully formed field of maths with its function, graph, laws and inverse.

Fig1

In the pre-calculator days every student used logs as an essential calculating tool from the age of 12 – even the student not destined for the Higher course. In the old O-Grade Arithmetic and Maths exams, each student was issued with log tables. Questions were set knowing that these log tables were accessible. The inverse of the log of a number was the ‘antilog’.

When the Higher student was introduced to the algebra of logs there were no surprises. When it was explained that, for historical reasons, the antilog was referred to as an exponential function, 10x in the context of the story, it didn’t cause a conflict.

I’m advocating reintroducing ‘the use of log tables as a calculating aid’ .

An historical tale of their invention by Napier and their immediate improvement by Briggs acts as a good introduction. Scientists, astronomers and the likes toiled endlessly with lengthy calculations – and progress in their field was slow as a consequence. A good entrepreneur, Napier saw a need and addressed it.

His original concept was to match the real numbers in a one-to-one relation with another set of real numbers, the logarithms:  these would be designed so that multiplying numbers meant adding their logs, dividing meant subtracting their logs, etc.

Give students log tables and some numbers to crunch and you’ll be surprised how quickly the laws of logs become ‘obvious’.

Towards that end, you’ll find a set of log tables (base 10) that can be printed off here

They are only 3-figure logs, but good enough for the purpose.

For example: 2.14 × 3.61 mapped to logs gives 0.330 + 0.558 = 0.888 which corresponds to a number between 7.72 and 7.73.

The SQA advocate that the place of scientific notation in the context of logs be explored.
Investigation: Given log102=0.301 and log103=0.477, how many numbers can you find logs of?

Take care,
Eddie Mullan

Eddie Mullan taught maths in Scotland for over 35 years. He was part of the SQA examination team for Higher, and Chairman of the Scottish Mathematical Council. Since 1985 Eddie has led the Maths in Action team, developing maths resources for Scottish schools. He now leads several successful CPD courses.

 

Leave a Reply