# 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)
• 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.

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.

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.

# More interesting Maths Papers (Spire Maths)

https://spiremaths.co.uk/printable-paper/

I have downloaded some of the Excel Spreadsheets onto my computer and it has not exploded (yet!!) – can’t wait to teach some lessons using this gorgeous set of papers! Some pdfs of examples to download below, if you don’t want to generate your own from EXCEL:

# Game of Rectangles

If you know a student has rather problematic times tables, and/or is not confident working with areas, then this game will help you to assess what the scope and nature of the problem is. Please read the notes about what to say before each round – if you “teach” all the strategies in advance, the game won’t be fun any more!

## You Will Need

• 3 or 4 sheets of printed hundred squares
• a pen each
• 2 dice (or even better, 2 each)
• 5 reward counters (coins? buttons? toy dogs? be imaginative)
• A page of pre-printed tables in the student’s favourite layout.

## Before you start

“Here is your new game board – by the way how many small squares is it made up of?”

Most will count the first row with their finger. If they do it without a finger, they may get to 9 or 11. If that does happen, I suggest saying “oops, I think you need to check that”. Wait until they have “10”.

Do they then count the number of rows next? Or count each square in the next row? Or count down the rows saying 10, 20, 30?

Wait until they have 100.

“So the maximum score for you will be 100.”

This step will tell you what strategy your pupil is comfortable using to find areas.

1. If they counted all the squares from one to 100, then they still need practice doing that, and they will be doing so for the rest of the game. Concentrate on accurate strategies for counting, and celebrating correct answers. Using a pen to “dot” each counted square is usually enough. Confirming that the answers at the end of each row are 10,20,30 will avoid some of the errors.
2. Most students can chant “10,20,30” and will be confident to do so for this task.
3. If they count to 10 then count 10 rows and say “100”, they are demonstating that they are confident with the link between areas, repeated counting, and tables.

## Round 1

“Shall I start so you know what to do when it’s your turn?”. (This avoids the need for too many words!).

“Throw the dice, and use the 2 numbers to draw a rectangle. I’ve thrown 4 and 5. 4 and a 5 give you a 4×5 rectangle. Your score would be 20 for that, because it has 20 squares inside. That’s the area of the rectangle.” (You’ve explained it without putting them on a back foot by asking them for any of the information. You are only telling them how to play, not how to win.

Take turns to throw 2 dice and draw a rectangle. If a dice goes on the floor, say “Oh, it doesn’t count if it goes on the floor. You’ll have to roll it again”. This keeps the game calm!

How do they find the areas? Always count? Count correctly? Make mistakes with counting? Sometimes say “5,10,15,20”, sometimes say “5 5s are 25”?

This observation is *key* to what they may learn today. Choose the lowest level of skill (if they can’t confidently count, don’t worry about tables!!).

You should model at and just above their secure level of skill.

• They count badly? You use dots.
• They count well in ones? You count in 2s and 3s
• They know some of the tables facts? You use others
• They look up some facts on their chart? You look up all of them to reinforce this is a good strategy.

Once one of you has “blocked” most of the board, you will both need to draw 3 lives and each time you have a dice throw you cannot draw, you will lose a life. Once you are dead, the other player continues until they are dead.

Once both are dead, total up the scores you each have.

As they add their score, notice how they do it? Are they correct? Do they want you to do it? Can they add the numbers silently in their head? Do they want to jot the sum? do they want a calculator?

If they struggle and are unhappy, be helpful. Addition can be worked on with a different game on another occasion.

## Round 2

Did they sensibly squeeze the rectangles onto the board? Or spread them out and waste space? Are they ready for a “nudge” on strategy, and start being more efficient? Or are they still overwhelmed by the skills needed for this game?

If you decide to nudge, give specific advice like “why not draw this one in the corner here, to leave room for big ones later?”. This is simpler than trying to explain in an abstract way.

If you decide not to nudge, then aim to lose by spreading your rectangles out. if they make a comment, you can say “I’m trying a different approach this time to see what happens”.

Aim for an understanding of some strategies work better than others rather than one being more “right” or “clever” than another.

## Winning…

The winner of each round gets a counter. Play best of 5 games.

## Extension ideas

If this whole game is too easy, then draw triangles instead. These may be all right angled or for very advanced version, allow scalene triangles. Discuss areas in either case. Use a ruler!

# Skill or no skill? A Game for 2 or 3 players

Objective:

The winner is the player who has the highest total score at the end of a round

You Need:

• A set of playing cards
• A dice (or one each)
• A spare piece of card or paper you can cut up

Before You start:

• Agree how many score cards you will have, whether you are playing skill or no skill. Make enough score cards (about 2cm square is fine)

Example of working out your score:

If your dice says “3” and your playing card says “3d+1” then your score is 10, because 3×3+1=10

Playing “No Skill”

• Suppose you have agreed to 5 score cards each. Deal the players 5 playing cards each, which they put in a line in front of them (no cheating!!)
• Players take turns to roll the dice, and work out the score from the card, writing that score on the next small score card they own.
• When everyone has finished using their cards, total up the score cards and see who won.

Playing “Skill”

• Suppose you have 5 score cards. Agree a higher number of playing cards and deal those out.
• Each time a player rolls their dice, they can choose which playing card to use this turn, to work out their score. Once a playing card is used it is turned over. You won’t be able to use it again.
• You will find that some cards work well with large dice scores, some work well with smaller ones. Some cards (we called them “golden cards”) work brilliantly with sixes and can score as much as 49!!!

Making the playing cards:

It’s up to you how hard you make the algebra, but here are some ideas to get you started. You need about 20 cards.

• 2d+1
• d²-5
• 4-d
• d-3
• (d-1)²
• 3d+1

# Part 2 – More Revision and Practice, changing the subject of an equation

These examples are a bit more difficult that in my previous post.

• M=2P  (Who is older, Mo or Polly?)

The easiest way of working this out is to give P an imaginary age, and work out Mo’s age. The equations says Mo is double Polly, so if Polly is 20, Mo would be 40. So Mo is older.

• M = 2P (What is Polly’s view of the relationship, in other words, write P=…..)

Try these….

1. X=2Y. What is Y’s point of view?
2. C=Y+12.What is Y’s point of view?
3. D=2Y-1.What is Y’s point of view?

To check your answers, give the person on the right hand side of the equation an age, work out the age of the “subject” person, and then work backwards in your answer to check you get back to where you started.

Here are some more:

1. X=2Y+4. What is Y’s point of view? If Y is 20, how old is X?
2. C=3Y+2.What is Y’s point of view? Again, pretend Y is 20.
3. D=2Y-5.What is Y’s point of view?
4. A=2Y-6.What is Y’s point of view?

# Points of view (changing the subject of an equation)

This post is practicing simple equations. In 3 days there will be another one to change the subject of more simple equations. If you write your answers in the box at the bottom I will check them and reply to you. I won’t publish them on the page though!

• “I am forty-one years older than you” is my point of view. In algebra language I would write R=C+41
• What’s your point of view?
• C=R-41

Looking at some of the people in my family then,

• R=D+26 (I am 26 years older than D, my oldest daughter)
• What’s her point of view?
• D=R-26

Now let’s look at another relationship in my family,

• R=G+28
• Who is older? R or G?
• R is older than G, by 28 years.
• so how would G see it?
• G=R-28

Now here’s something you can work out,  given I am 54 at the moment,

1. How old is C?
2. How old is D?
3. How old is G?
4. One more person in my family is T. R=T (that’s my point of view) How old is T?
5. What is T’s point of view, in other words write T=…