Factorising Quadratics

Click here for the answers

Currency Conversion using Ratio Grids

Check out the Introduction to Ratio Grids if you are not already an expert!

When you are asked to do a currency conversion, you will usually be given an exchange rate first, as in this question:

“George changes some pounds into Euros. The exchange rate is £1=€1.18. If he changes £350 into Euros, how many Euros will he receive?”

Draw a simple Ratio Grid and give headings of £ and €. Put the numbers 1 and 1.18 in the correct columns. Then, the final number from the question, £350, goes in the £ column under the 1. Putting the numbers into their correct places is easy if you have headings! Put a ring around the empty space – that is where your answer will go.

In this grid, the 2 multiplying numbers are the “1.18” and the “350”, and you will divide by 1. So the sum will be 1.18×350÷1.

The answer is 413, and it is clearly in the Euros column, so give your answer as €413.

It’s easy if you get the headings right!

If you want lots of practice, Mr Corbett has a great set of currency conversion questions.

For lots of simple questions click here and the answers are here

Exam-style questions are here** and the answers are here. (Mr Corbett doesn’t use Ratio Grids so his methods won’t look the same, but the answers are all the same, obviously!)

** Some of these questions are really challening – they may require 2 ratio grids, so think carefully! If an exchange rate changes, you will need a new grid for the new rate.

Introduction to Percentages

Finding a percentage of a number using Ratio Grids

The left hand column of a percentage Ratio Grid will be headed “%” and the right hand one will be for the numbers that you are working with. For example if you are asked to “Find 65% of £40”, you are starting with £40, so the right hand column will be labelled “£” and the number 40 will be on the top line, beside the 100%:

“100 Percent” is a phrase everyone knows and uses, and that’s great because when you solve a percentage problem using a Ratio Grid, you will need to put the number 100 into one corner of the grid.

Once the numbers are all in the right places, (assuming you know how to complete a ratio grid!!) , you will know the sum you need to do is 40×65÷100 and the answer is 26. Because the answer is in the column headed “£”, then it must mean £26. Headings are really useful in Ratio Grids because they will always help you to put Units in your answers.

Questions about Percentage Increase and Decrease are easy if you use Ratio Grids. For example:

“A tree increases from 15m to 17m in height over one year. What is the percentage increase?”

The 2 columns of the ratio grid will be headed % and m. The first percentage will be 100, which is for the start of the story. In the story about the tree, it starts at a height of 15, so put 15 beside the 100, in the metres column. It grows to 17m, so that goes underneath the 15.

The missing number is found from this ratio grid by working out 100×17÷15, on a calculator, which is 113.33333333… It’s usually fine to round percentages to one decimal place, so the missing number is 113.3%.

Because the question is asking for a percentage increase, you need to find the difference between 100% and 113.3%, so the answer is 113.3%-100%=13%.

Here is a worksheet you can try, and here are the workings and answers.

Solving Ratio Problems

If you don’t yet know how to work out the 4th number in a ratio grid, watch this video first!

The videos on this page use the worksheet “Ratio Problems” which you can download:

Simplify a Ratio using a calculator
Split Money using a ratio
Wordy Ratio problem about socks
Wordy ratio problem given one of the values and not the total
Ratio problem sharing sweets

Answers

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:

Using Ratio Grids
Larger Ratio Grids

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:

Mathematical christmas cake

This cake contains algebraic sequences, inequalities, and geometric sequences, a sum, a square root and a 3D shape you have never heard of before. If you loved the Da Vinci Code, you will love this…. and if it is almost Christmas, then this is great. You can make it from start to finish on Christmas Eve, if you like. It will keep fine for up to 10 days.

And, it is easy, and vegan.

Shopping List:

  • Self Raising Flour
  • Golden Caster Sugar
  • Mixed Spice
  • Ground Ginger
  • Bicaronate of Soda
  • Plain Flour
  • Teabags
  • Dried Mixed Fruit
  • a pack of marzipan
  • some plain jam without pips (apricot is best)
  • Pack of ready made white icing
  • A Mixing Bowl and another bowl
  • Wooden Spoon
  • Sieve
  • Teaspoon measure
  • Weighing Scales
  • Loaf tin
  • Silicone tin liner or greaseproof paper
  • rolling pin (or clean glass bottle) and clean flat surface
  • Oven
  • the number “200”

Arithmetic Sequence

An arithmetic sequence is a list of numbers that goes up (or down) the same amount, all the way through.

The first sequence for this recipe starts with the number 200, like this:

  1. Weigh 200g of dried mixed fruit into a bowl
  2. Make a nice strong mug of tea with the teabag (or use leftover tea without milk)
  3. Put the bowl of mixed fruit on the scales and weigh in 180g of the tea.
  4. Mix together and leave to soak for at least 160 minutes. (you are going to need the number 160 again, so don’t forget it).
  5. When the fruit has soaked, turn on the oven at 160 degrees (fan), and weigh 160g of plain flour into a sieve that is sitting on top of a mixing bowl.

Geometric Sequence using a teaspoon

A geaometric sequence is made, by repeatedly multiplying by the same number. In this sequence, the multiplier is 0.5 which means that every number is half the size of the previous number.

The cake is very spicy, a little bit gingery, and not very light. So measure into the sieve:

  1. Two teaspoons of Mixed Spice
  2. One teaspoon of Ground Ginger
  3. Half a teaspoon of Bicarbonate of Soda

Tap the sieve carefully until all these dry ingredients have fallen into the mixing bowl.

A Sum

40+50 = …

… 90g of golden caster sugar, mixed in with the dry ingredients.

Square Root and a Prismoid

You have 90g of sugar… what is the square root of 9?…

THREE teaspoons of rapeseed oil (often labelled “vegetable oil”). Add this to the mixed fruit and tea, stir in, then stirthe wet mixture into the flour and put it all, carefully, into the loaf tin.

A loaf tin is an example of a prismoid. This is such an obscure word that it does not even have its own Wiki page! Poor little prismoid! It just gets a measly mention on the page for prsimatoids. A psimoid is a bit like a cuboid, except that it has sloping sides. A loaf tin has sloping sides, because if they were straight, it would be really difficult to get the loaf out!!!

Upper and Lower Bounds of Cooking Times

Put the tin in the oven and set the timer. Rememeber the sum 40+50=90? You have used the 90, for the sugar. The 40 and 50 are the upper and lower bounds of the cooking time t.

40 <= t <= 50

If you take a look after 40 minutes, it might be cooked. If it sinks as you look at it, it isn;t cooked. If you slide a knife into it, and the knife comes out smeary with mixture, it isn’t cooked. If it isn’t cooked, give it another 10 minutes.

Cool it in the tin for 5 minutes, then turn it out onto a rack to completely cool.

Sprinkle some icing sugar (about a heaped teaspoon) onto the clean surface and roll out enough marzipan to drape right over the cake.

Brush the top and sides of the cake with jam

Drape the marzipan over the jammy cake and press it gently to stick. Trim off any extra.

Roll out the ready made icing and drape over the marzipan.

Voila. Ready to eat. No fuss.

And next year, will you still remember the recipe?

… 200, 180, 160 …. (160 160)

… 2, 1, 1/2

… 40+50=90 … square root of 9 is 3

marzipan and icing and munch.

Happy Christmas

Confidence in KS1 Maths

Rosalind very kindly agreed to work with my son, who was much younger than her usual age group. He was really struggling with KS1 maths, and was developing a fear of numbers which for a boy who was ahead of his peer group in all other subjects was causing him some confidence issues. Rosalind thought outside of her usual comfort box of teaching tools and adapted her reliable tried and tested teaching methods, for her younger audience – the result was fun! A whole host of maths games that my son really enjoyed and actually asked to play at home. These games helped him to gain confidence with numbers and combatted his fears. The result was a happy son, who by the end of KS2 was a strong all round learner who passed the grammar examination and obtained a place at Grammar School. We thoroughly recommend Rosalind as a tutor, and as my son approaches the commencement of his KS4 GCSE courses and now that Rosalind has returned from her sabbatical, we will definitely be seeking her additional support for revision and maths resilience.

Innovative approach to teaching

My daughter was coached by Rosalind for five years from Y7 through to the end of Y11. When she started her confidence and self belief were low. Through her innovative approach to teaching, Rosalind built her self confidence and caused her to fall in love with Maths. She was more than a tutor, acting as a learning mentor which helped boost her confidence in all areas of the curriculum. My daughter went on to achieve a level 9 at GCSE Maths and A* in Maths and Further Maths at A level. She is now studying at Cambridge.

Dominic