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

My Child Is Struggling with maths

I am very sorry to hear it. That’s a really tough thing for you to watch.

You may be thinking “I wish I could do this Maths myself, then I could help”. There are several reasons that this can come about: Perhaps you are “rubbish at Maths” yourself, perhaps you were OK at Maths at school, but just rusty with most of it, or perhaps your child is doing a topic (or doing something in a particular way) that you have never seen before. But you may be surprised to hear, that parents who are very good at Maths themselves, may well find it very difficult to support their struggling child, and often approach me and ask me to help them.

What you need at this point, to help your child, is lots of understanding of “how to help”. Not, lots of understanding of Maths. I hope this encourages you. Remember all the times you have successfully helped your child to learn something? That’s the skill.

Good

If you had phoned me up, and asked me for help, saying “my child is struggling with Maths”, I would have replied “Good!”. I don’t mean “I am glad your child is miserable and frustrated”. I mean, “I am glad your child is still struggling, still making an effort, I am really glad your child has not just shut down”.

The very fact your child is still struggling with Maths, means they still want to succeed. That is a huge positive.

Does he want to be helped?

This is really important. A person can struggle and still really want to solve the problem for themselves. It is incredibly rewarding to solve a puzzle, to finish a jigsaw, to reach a new level in a game. Badly timed help can take all the satisfaction out of it. Even a child in tears, may just want a hug, and then to succeed on their own. Remember that time you walked into a clothes shop and looked around and an assistant zoomed over to you and said “Can I help you?”. Often, the answer is, “No, I am just looking”. Your child’s response may be “no, I just want…” So listen to that, and help the way THEY want to be helped.

The right time and place

So, you’ve checked, and your child says yes, they want some help. You need a time and a place where you will be in a slightly different mode from normal. You are struggling with lockdown. Working from home. Coping with more than you can actually cope with. You don’t find it easy to do this Maths Teacher thing. So try to give yourself the maximum chance of success. Decide a time and a place when you will help them. Ideally, you can use a table and chairs, so you can spread some paper and pens out. Probably, you will need a computer to show you the question they are stuck on. Or, It might be on paper or in a book.

My son asked me for some Maths help Year 11. I was astonished. He had maintained, doggedly, that he was not going to let me help him. I am good at Maths, his elder sisters are good at Maths as well, and he was determined to be himself, a Maths-hater, whose Maths grade was less than the other subjects, the ones he actually liked. But his resolve broke when he saw his Mock GCSE results, and the Maths grade stuck out like a sore thumb against all the others. He admitted defeat and asked me to help him.

I offered to get him a proper paid tutor (Yup, I do understand it’s hard to teach your own child!!!), and he refused, on the grounds that then he would have to do a set hour each week, and he wanted to be able to get small lessons, ad hoc, when he wanted them. On his terms. I was happy to accept his terms, as long as he accepted mine. We would use the table and chairs in my teaching room, and he would behave himself. No loud sighing. No laying his head on the desk. He said OK then, as long as he could stop a lesson when he wanted to, with no argument from me.

Rules established, we started our very first Maths Lesson. Within 30 seconds his head was on the desk. I reminded him of the agreement and we were on track. He “drove”. In other words, he came to the mini-lessons with a question in his hand (now, it would be on a screen), and I answered his questions about it. When he had had enough, we stopped.

I don’t remember, now, how many lessons we did like that. But his GCSE grade in Maths was nice and harmonious with all the others, and now he is a primary teacher, teaching Maths himself, alongside everything else. If you had suggested to me back then, that my 16 year old son, who loved his X-box and his guitar, would be a teacher, I would have been incredulous. Honestly. It is so hard to imagine the adult that our child is going to become. That adult is properly invisible, most of the time.

I am rubbish at Chinese

I think I can imagine what you mean, when you say “I am rubbish at Maths”. Have you failed Maths exams? Have you struggled with school Maths? Does Maths freak you out, even now? Does my suggestion of you sitting beside your child, looking at a Maths problem together, just make you want to cry?

First things first. However bad you are at Maths, you can still help. You have already made the decision to try to help, I know that because here you are, reading. You are still trying and struggling to help, you have not shut down.

You can help your child with Maths.

I know that, because I helped my son with Chinese.

My son had the very unusual chance, at school, to study Chinese in Year 8 and if he wanted to, to take it to GCSE. He liked languages, so he had a go at Chinese. Hand on heart, I am rubbish at Chinese, myself. I spent quite a lot of time and effort helping his with his Chinese homework and I have retained nothing at all. I think that’s a pretty convincing way of proving I am rubbish at it…

But I don’t think I was rubbish at helping him, or he would have stopped me from trying. He would come home from school with 10 or so characters that he had to learn for a test. And he had no idea, really, how to do that. So I would say….

Show Me

… and he would show me the first character. And it was Chinese, so it meant nothing at all to me. But we would discuss the character. This one has a little thing that looks a bit like a fishing net. That one, it has 2 legs. And if you compare those ones, see they have the same top bit? And our discussion would have the effect of making HIM really, properly, look at each character and start to fix it into his memory, ready for the test. This was a memorisation exercise (I know Maths is often more about understanding). For memorisation, I would make cards, and he would draw each character onto the front of the card, and write the meaning on the back. And then I could test him, and he could test himself, and we would know when to stop, because he could recognise all the characters correctly, on the cards.

Don’t recycle all those cardboard boxes to quickly

You can make cards, for memorisation, out of empty cardboard boxes. Cereal packets, pizza boxes, whatever is available. Put the thing to learn on the blank side and the answer on the printed side.

You learn, too

By the end of one of those sessions, I would have been able to score a few marks in his Chinese tests too. I was happy for him to test me and to discover I was “rubbish” at it, that encouraged him and made him feel good. If I was un-rubbish and able to remember the meaning of a difficult character, I would let him into my secret “see the little pair of whiskers there, I’m thinking cat, and this one means …”, so I was giving him a clue that might help him. Or it might not.

I didn’t need him to learn MY way of memorising pictures, I wanted him to do well in his lesson, that was all.

It’s nice for a child to win a competition with the parent. I am honest and stopped “trying to lose” games once they were old enough to notice, but often your child will sail effortlessly past you in a skill, they have a young brain and it is really, really good at learning new things. That moment, when they cruise past you. That is what success feels like. You may feel small, and not like it very much. Classroom teachers can find it challenging too, but it is a sign that you are teaching really, really well.

Can you do a bit of the question?

I am imagining you are looking at a Maths problem together. Your child can’t do it. Perhaps you can, or perhaps you can’t. That isn’t relevant to the next step, actually. However tempted you feel, don’t grab the pen and start doing the question yet. The danger is, you will reinforce their belief that they are rubbish at Maths. This question does not need to be done. It can wait.

If you are lucky, all they needed was a bit of encouragement to get stuck in, and in 5 minutes they will be happy and smiling and the question will be finished.

No? Then, click here and read a really excellent page of advice, on how to help your child with Maths