Dividing by Zero

I found this on Pinterest and now use it to help my pupils remember the rules regarding division and zero.  It seems to work!

I also like to show them a silly meme like the one below to help the rule stick in their heads.  A search for 'divide by zero' on Google Images brings up loads of these. I do apologise to my pupils in advance for the bad language, which I think makes it more memorable.

I was very pleased when one of my pupils recently wrote this in a test. It's nice to see that some of them pay attention!

Most calculators give the message ‘Maths Error’ when a number is divided by zero.  It’s crucial that pupils understand why. For teaching ideas, I recommend this excellent blog post 'Why can't you divide by zero?'.

We use this understanding when solving equations like this:  x² - 5x = 0. A common mistake is to divide by x to get x - 5 = 0. This incorrectly gives only one solution (x = 5). Knowing that we shouldn't divide by x because x could be zero (unless we're told otherwise) allows us to understand why the correct method is to factorise ie x(x-5) = 0, giving the two solutions x = 0 and x = 5.

To stretch your students, try this interesting activity about the value of 0⁰, Evaluating a Special Exponential Expression from Illustrative Mathematics.

It's interesting that throughout history, many great mathematicians (including Euler) have argued that division by zero gives infinity. Most modern mathematicians take it as undefined.  If you want to do some background reading on the ‘history’ of dividing by zero, I recommend this article - it's an enjoyable read.

Logs

I like teaching logarithms but my students find them really hard. There's a wide variety of C2 exam questions on logs so pupils actually have to understand the topic, they can't just learn a set of rules to get by. This is a good thing of course.

James Tanton's take on logarithms is an excellent resource, highly recommended for anyone who is teaching logs for the first time. For example he suggests starting off by writing the following on the board, perhaps in silence...

power₂(8) = 3
power₅(25) = 2
Your turn!
power₃(27) = ___
power₁₀(100) = ___

Give students a lot more examples to complete and then congratulate them for their cleverness on having taught themselves logs. Then go through and cross out the word power in each example, replacing it with log. As James says, "Taking the time to do this in a showy way brings home the point that logarithms are just powers -  whenever we see the word log we are to think power.". Read the rest of James' paper for more ideas.

This blog post from Sarah Hagan has absolutely loads of activities for teaching logs and her subsequent post has some follow-up activities.

There's a wide range of teaching ideas in this Logarithm Functions booklet from Mathematics Vision Project, which starts with an ordering activity:

Mathematics Vision Project

STEM Centre provide four resources designed to practise the basics and extend understanding of logarithms, plus some open ended questions for discussion or a mini-whiteboard activity. I've found the true or false exercise particularly effective for identifying misconceptions in the past (longer version with answers here).

Susan Wall - via STEM Centre

Here are some more of my favourite resources for teaching logs to Year 12:

• Teachitmaths.co.uk has a selection of logs dominoes - I particularly like these. 
• SRWhitehouse on TES always provides excellent A level resources. She has a great selection of resources for logs.
• Slides - 'An introduction to logarithms' by Teachit Maths.
• C2 Target Board - Logs and Indices - hannahlees on TES.
• Student activities on introducing logarithms - Project Maths 
• Become Friends with Logarithms - notes and exercises by Scott Surgent.
• RISP 31 - building log equations.
• Standards Unit A13 - Simplifying Logarithmic Expressions.

Matrix Multiplication

When I teach matrix multiplication I find that pupils can get a bit lost in their calculations.  I found a fantastic new method on this blog.

Let's say we have matrix A and matrix B.  We want to calculate AB.  Write down the two matrices side-by-side.  Then move matrix A down. Now write your answer in the space at the bottom right.  I think the next step is self-explanatory from the picture below, but read the original blog post for a full explanation.

I love this method - it's so simple to remember and now almost impossible to get lost when multiplying matrices.

And - on the subject of matrices - this made me chuckle: