Introduction to Ratio Grids

Ratio Grids can be used whenever you have a problem that starts with 3 numbers, and you have to times and divide. They are a clear way of laying out your working, and knowing which order to do the calculation.

You will need a calculator to do the practice questions, but first, here are 2 videos. The first shows you how to solve a small ratio grid, the second deals with the larger ones:

Larger Ratio Grids

ratio grids questions v2 Download

Click here for asnwers

Your comments are very welcome – please use the box below.

Now you can use Ratio Grids to solve a Maths Problems, here are some types of problem you can try:

Game – Prime Number Recognition

You will need:

2 dice and a pen and paper.

Take in turns to:

Throw the dice
From the dice, construct 2 or perhaps 3 numbers. For example, if you throw a two and a three, you can make 5, 23 and 32 (3+2=5, two followed by three is 23 and three followed by two is 32)
Score one point for each prime number you have made (so this example scores one for the 5 and one for the 23, scoring two points in total).
If you need to use a calculator, then a Casio fx-83GT PLUS can tell you whether a number is prime. This is *not* cheating – students will soon start to recognise the primes they need, rather than having to check using the calculator!

The Winner Is:

The person who has the most points.

Things to discuss:

Why are two even numbers always such bad news? (even+even=even and the only even prime number is two)
Is it possible to score three points with one throw?
If there is a six in your throw, what happens?

This sample space diagram may help:

	Sample Space Diagram for 2 Dice
	1	2	3	4	5	6
1	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	(6,1)
2	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	(6,2)
3	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	(6,3)
4	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	(6,4)
5	(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	(6,5)
6	(1,6)	(2,6)	(3,6)	(4,6)	(5,6)	(6,6)

How to Factorise a Number (Or Check that is Prime) using a CASIO fx-83GT PLUS calculator

On the CASIO fx-83gt PLUS factorising is done like this:

Entering the number,
press equals,
SHIFT and ., ,,, (this has “FACT” written above it in yellow).
The Prime Factor Form is displayed as the answer.
If the number is Prime, then the number itself is displayed.

This is a natural way to introduce what indices mean, because the CASIO gives the answers in index form eg 3⁴ rather than 3x3x3x3

Pythagoras and Dyscalculia

This approach:

Uses a calculator.
Uses just one single approach for all problems (long side or short)
Reduces the number of steps by rooting and adding in the same step.
Keeps to an absolute minimum, the need to compare sizes, because Dyscalculia makes this challenging.
Uses a mnemonic to recall the steps.

Method:

Normally in a problem involving Pythagoras, you are given 2 of the sides of a right angled triangle and asked to find the third. You need to Root the Add of the Squares (R.A.S, end of PythagoRAS). Always key in the larger number first….

This will give the answer 8.0622..

but Perhaps You Takeaway (PYThagoras) so mouse back to the plus (+) and change it to a take away (-)… and this gives 5.7445..

Now it’s decision time – using the picture, which answer looks most sensible, 8.0622 or 5.7445? The length of the 3rd side in this example is clearly longer than 7 so the correct answer is 8.0622.

How to display the 18x table on a CASIO fx-83GT PLUS

This video covers:

How to put your calculator into TABLE mode
How to display the first 20 terms of the 18x table on the screen
How to mouse up and down the table

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

Substitution into a formula (calculator method)

Using formulae to solve problems can be fraught with difficulties if some of the numbers are negative. If the calculator is given the values of the letters, then the formulae is keyed in using those letters, accurate results can be achieved every time. The work is also easier to check.

Click here to see a film of the calculator being used to solve this problem.

To set the letter A to the value 12 , B to -3.6 and C to 15

2 Shift RCL(Sto) (-)A
7 Shift RCL(Sto) ..”’ B
-12 Shift RCL(Sto) hyp C

Once that is done you can type entire algebra phrases into the calculator and find their values. Here is an example of using the calculator to use a formula. You will see that the formula is written with capital letters to match the way the calculator sees the alphabet.

How to use your calculator to understand Standard Form.

These instructions work on a CASIO fx-83GT PLUS, which is widely used in schools and will, if you learn to drive it with confidence, do a tremendous number of different sorts of Maths.

If you want the calculator to give all its answers in standard form, follow this sequence:

shift mode 7
it will offer you a choice of Sci 0~9 and this means how accurately it will display the answers. 3 is OK, but you might want to experiment with 4 and 5 as well, to see for yourself what difference it makes.
Try it, type in 23×3000= and you will get 6.90×10^4 (sorry! the ^ means to the power!!! The calculator is designed to display Maths properly and computers aren’t)
To make your calculator display your answers normally again, shift mode 8 1 will do the trick. mode 8 is “Norm”, normal mode.

To type in a number that is in standard form, for example 4.5 x 10^12, use the button with x10x on it (it’s in the middle of the bottom row of buttons, and I will use square brackets to mean button) so the keystrokes you need, are 4.5 [x10x]12

Factorising Race

A game for 2 players. The winner is the player with most points at the end.

Starter (optional)

Play a game with the 13x table. This is almost completely unfamiliar and the kids will be intrigued. The KEY fact is that 7×13=91 because 91 is a terribly prime-looking number but it isn’t. Pupils who are confident with their other tables will, by learning this factorisation, have completed the full set of skills in factorising numbers under 100.

Introduction:

Make sure the players can factorise a number using their calculator. On the CASIO fx-83gt PLUS this is done by entering the number, pressing equals, SHIFT and ., ,,, (this has “FACT” written above it in yellow). The Prime Factor Form is displayed as the answer.

Each turn:

Roll 2 (or 3) dice, and choose which number to build. For example, a 5 and a 1 could be 15, 51 or 6. Factorise your chosen number. The score is the number of prime factors. eg 15 would score 2 for 3×5. 8 would score 3 because it is 2 cubed. 24 scores 4 because it is 2x2x2x3. 71 scores 1 because it is prime. ** You may need to explain the “Index Form” that the calculator displays. This is a very important notation anyway, which a lot of students misunderstand.

What Maths is learned:

To choose the best number, ideally the players have to mentally factorise both numbers. They will make repeated use of the standard divisibility tests (for 2, 3, 5 and 9) and probably invent a few more (this evening my pupil realised 357 and 217 must both be in the 7x table, just by looking at them.)
Once they know a number will divide, they have to actually DO it mentally. Practice makes perfect here!
A printed tables sheet may be a help.
In their enthusiasm to win they are stretching their own mental maths to the limit. If they don’t fully factorise both numbers, they may miss a high score!
A younger pupil may want to try all the possible numbers with the calculator – this is good practice anyway and reinforces the correct factorisations.
An enthusiastic player will start to memorise some of the factorisations – this is really helpful knowledge.

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:

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.
Once they can all see F(X)=, ask them to input 10 =
For “Start”, stick to 1 all lesson.
For “End”, input 10 =
For “Step” input 1 =
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.
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.
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.
Describe and write on the board the sequence this creates.
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.