# The power of practice

Dear M,

You are making great progress with algebra, I think. The crucial link between the written symbols and what they actually MEAN is not easy to make, but you are putting the hours of practice that you need, in order to become confident.

Here are some more questions. I know we discussed the idea that I could build an interactive game for you that would check each piece of your working, and your answers, but on reflection, I think you are getting sufficiently fluent with algebra now, that this level of feedback would actually be unhelpful, like bolting stabilizers back onto a bicycle after you have begun to balance it by yourself.

You are already in the (excellent) habit of keying the question into your calculator, keying in the answer, and comparing the two, What I would like you to focus on this week, is actually writing down the “easy version” and, if the question and answer DON’T match, then key in these workings out as well to narrow down the problem.

The huge advantage of you and the calculator (rather than the computer) doing the checking, is that you are able to use your calculator throughout the IGCSE exam, so you are practicing really useful skills, not using a tool that will be taken away from you.

After working on the last worksheet with you, we both realised that the sums were too densely packed on the page, and a bit dazzly. I hope this version is easier. I have left gaps for you to write the “easy version” out directly below the sum.

As you work, it may be worth being aware of the commonest errors you were making in the lesson (which are errors most people make, you are not alone 🙂  )

• If there was a minus sign in front of the first bracket, you were not always “seeing it” clearly when you keyed expressions into your calculator
• If there was a minus in front of the bracket as well as one inside it, you sometimes made the wrong decision (although you had pretty much stopped doing that by the end of the lesson, errors have a nasty habit of creeping back in a few days later!)
• You made more errors when you did everything mentally, and fewer when you jotted the “easy version” as a working step.

Jargon-wise, what we are doing is called “multiplying out brackets and simplifying”. That’s the terms the examiner would use (instead of “writing down the easy version and writing down the answer”.

So here are three more worksheets:

multiply out and simplify 1

multiply out and simplify 2

multiply out and simplify 3

# A very elegant piece of algebra

I set this problem today, during a lesson in which we had been looking at simultaneous equations with 2 unknowns:

A + B = 92

A + c = 57

B + C = 59

M came up with this answer:

POSTSCRIPT –

M then set me this problem:

A-B = 25

B+C = 151

A+C = 176

which turned out to have an unexpected twist in it….. Have fun!

# Nth term – finding algebraic rules for number sequences

Mathematical Language Covered

• Function
• Simplest Form
• Mathematically equivalent
• Sequence

You will need:

• One Scientific Calculator each  (I use Casio fx-83GT Plus)
• A4 Card – the best type has one coloured side, the other white – cut into tenths. You will need 3 or 4 pieces per pupil.
• A group which is already reasonably good at supporting one another
• Class Whiteboard

Introduction:

1. Walk the class slowly and carefully through the key presses needed to put the calculators in TABLE more. (Mode 3) – like all ICT inputs, some will inevitably get lost and panic – this is where it’s important for them to be used to getting support from each other.
2. Once they can all see F(X)=, ask them to input 10 =
3. For “Start”, stick to 1 all lesson.
4. For “End”,  input 10 =
5. For “Step” input 1 =
6. Ask them to describe what they can see. Including the column headings. Remind them to “mouse” down to see the bottom section of the table. Write the F(X)=10 and the sequence on the board.
7. Press AC to clear the table and start again. This time try F(X)=12. Ask how they could make it more interesting, let them experiment. My lot came up with larger numbers but still the whole sequence was all the same number.
8. How can you make the numbers change? Someone might already know – discuss and try F(X)=2X. This is achieved by keying in 2ALPHA (= because the ( button has X above it.
9. Describe and write on the board the sequence this creates.
10. Try 6X-1, describe, discuss.

Creating the Puzzles

Give out 2 pieces of card each. Create your choice of sequence, write it on the coloured side of the card. On the back write F(X)= and the function you used.

Discuss boundries – numbers under 10. (I should have set more – no decimals and no division yet. But it did make life interesting!

As they produce the cards, check the function – if it isn’t in simplest form, then simplify it with them and write that version underneath. Point out that their friends won’t be able to guess any more complicated version of the rule. eg if the rule they used was F(X)= 2X+3X+1 then the friends will guess 5X+1

If you think any of them are quite difficult, give them one or two stars. This will let the pupils self-differentiate.

Solving the puzzles

Put the cards down SEQUENCE SIDE UP and the pupils choose which one they want to do. Try to guess the F(X), key your guess onto the calculator, can you make that exact sequence? Once you have made it, turn over and check the rule on the card. Then initial the front of the card to note that you have done it.

Plenary:

What was easy? What was more challenging? Do we need another lesson on this (My group decided they found the minus ones harder so we pencilled in a lesson on handling negative numbers.