# A Level Real Numbers

Discussion in 'Computing and ICT' started by Penny10p, Aug 18, 2019.

1. ### Penny10pOccasional commenter

I have some rather specific questions about how to teach real number representation at A Level. I hope some of you more experienced teachers can help. It seems to me that the progression in terms of difficulty is: binary fractions to denary and vice-versa; floating point positive numbers; floating point negative numbers; normalisation positive; normalisation negative. I am intending to teach the topic this year over at least 4 lessons and come back to it several times during the year. My questions are: do you agree with the above progression? Is 4 lessons enough initially? What do I do about differentiation? In my (limited) experience some students are never going to be cope with all of it, so how do I differentiate for them? And just to preempt certain posters, any replies to the effect that people without a CS degree should not be teaching the subject are not helpful!

2. ### SundaeTrifleOccasional commenter

4 x 1hr lessons sounds about right. You may find you can go quicker.

Once taught I used to use a few floating pt conversions as a settling in activity on entry to the lesson. (I’m retired now.) So students had lots of practice over the two years.

Start with small number of binary digits to learn concept, say 5-bit mantissa and 3-bit exponent.

Positive mantissa and exponent -> +ve mantissa and -ve exponent -> -ve mantissa and +ve exponent, -ve mantissa and exponent.

You will find the students who do maths and physics will pick it up very quickly and will be willing to help other students. There will be a spread of ability with less mathematical students needing more support from you.

3. ### SundaeTrifleOccasional commenter

I used to just make up the exercises and have the worked solutions to hand so that students could mark their own.

4. ### SundaeTrifleOccasional commenter

A couple of common questions from the exam board:

If you have a 10-bit mantissa and 6-bit exponent (or whatever) what is the largest/smallest positive/negative number you could store?

Why do computers use floating point representation of numbers?

5. ### Penny10pOccasional commenter

Thank you Sundae Trifle. I have taught this before in a rather haphazard way and this year I want to teach the topic more effectively, especially, as you rightly say, students who are not so good at Maths find it difficult. I like the suggestion of giving floating point questions as starters in other lessons. I do have a bank of past paper questions that I could use for this.

I'd start off with a short quiz, to try and gauge the hill you may have to climb e.g.
1. Give an example of a binary number
2. Give an example of a denary number
3. Convert the base 2 number 10010100 to base 10
4. What is a floating point number ?
5. Describe ,with an example, what is meant by the mantissa of a number
Provided you don't get "a type of fish" for number 5 - you might actually get somewhere

7. ### DorsetdreamsOccasional commenter

I start by extensively reviewing 2’s complement (covered the year before), in particular making a big fuss about being able to add / remove leading 1’s to a negative number without affecting its value.

Then I introduce FP, going both ways and normalising, in order: large positive numbers, small positive, large negative and lastly small negatives (‘small’ being less than 1). This sounds just like what you're doing. I get the students to do lots of examples of each before moving to the next. I make sure students to do a ‘sanity check’ on every answer – is their answer large or small, positive or negative? This is a steady, confidence inspiring approach. By the end they are happily normalising small negatives. Four lessons wouldn’t be quite enough.

8. ### harpplayerOccasional commenter

There was a fabulous booklet we used to cover all the A Level stuff on floating point numbers on www.theteacher.info which my last UK school used. It was five years ago but might be worth checking.