Direct and Inverse Proportion Read More »
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]]>To start with, you will need to be confident in two other skills, namely basic rearranging of formulae and solving basic linear equations.
In its most basic form, it is how one amount changes as another changes. So, for example, (and it is a classic example!) if you want to make twice as many cakes, you need twice as many ingredients. On that level, it seems to be obvious. However, there are many ways in which the difficulty can be ramped up in GCSE maths.
Here is an example of a question at the first level of difficulty:
A recipe requires 3 eggs to make 12 cupcakes. How many eggs do I need to make 36 cupcakes?
Here we are looking at how many times more cupcakes we are looking at making :
So, 3612 = 3
This means we need 3 times more eggs, 3 x 3 = 9
OR to put it another way :
12 Cupcakes 36 Cupcakes
Needs Needs
3 Eggs 9 Eggs
This is an example of DIRECT PROPORTION – as one increases, so does the other at the same rate, or in the same proportion – think how many times more?
HOW TO FACTORISE AND SOLVE QUADRATIC EQUATIONS
Next up, we get the less obvious increases, again, in direct proportion, but with slightly more difficult numbers involved, but only because the number of times each is increasing isn’t so obvious. It is easier to go through this using a worked example:
A recipe states that you need 200g of flour to make 8 pancakes. How much flour do you need to make 30 pancakes?
This is about adopting a slightly different strategy, based on working out how much flour one pancake needs in order to then multiply it back up by 30. Ask yourself, “How much do I need for one?”
From the first part of the question, we can see that 200 ÷ 8 = 25g of flour per pancake
So, it follows that 25 x 30 is how much we need to make 30 pancakes = 750g.
Now let’s have a look at an example of a decrease, remaining in direct proportion. In this case, we are looking at as one decreases, so does the other at the same rate, or in the same proportion – think how many times less?
Using the first question we looked at, let’s see how this might work:
A recipe requires 3 eggs to make 12 cupcakes. How many cupcakes can I make if I only have 1 egg?
We have 3 times less in terms of the number of eggs, so we can make three times fewer cupcakes
3 Eggs 1 Egg
Makes Makes
12 Cupcakes 4 Cupcakes
3 REASONS WHY PARENTS BOOK PRIVATE TUTORS
Once again, let’s do the same with some less obvious numbers, using the “How much for one?” approach.
A recipe for making 16 pancakes uses 800g of flour. How much flour do we need to make 10 pancakes?
So for this – let’s look at the “How much for one?” approach :
If 16 pancakes need 800g of flour, it stands that 1 pancake will need (800÷16)g = 50g
We now know we need 50g per pancake, so 10 pancakes will need 10 x 50 = 500g.
Once you have got to grips with the “how many times more / less?” and “how much for one?” ideas then we can finish with a look at the most common way these questions are framed in a GCSE assessment – the ‘Best Buy” question.
Go to any supermarket and you will see all kinds of offers and similar products from different brands. Best Buy questions are all about value for money and being able to compare two very similar products or offers.
There are two main types of question at level 4 / 5 and are as follows:
What we need to ask ourselves here is “how much for one?” – but that leads to “one what?”
We need to identify a common unit of volume in this question that can be applied to both buying options – given the amounts in question, I would suggest finding out how much one-litre costs in each case:
Multipack 6 x 330ml = 1980ml = 1.98 litres
Cost of this is £1.70
So cost per litre = £1.70 ÷ 1.98 L = £0.85858585……, so rounded to 86p per litre.
For the bottle, we have 1.5 litres
The cost of this is £1.50
So cost per litre = £1.50 ÷ 1.50 L = £1.00 per litre.
From the two calculations, we can now compare the costs and can see that the multipack offers better value for money as the cost per litre is less.
INTEGRATION IN MATHS: THE PADDINGTON BEAR APPROACH
Another classic example is the “special offer” scenario and involves the use of percentages and other skills:
Laundry liquid normally costs £6 per bottle.
Two different supermarkets have competing offers as follows:
Tesdi supermarket – 30% off all laundry liquid.
Snozbury’s supermarket – buy 3 get one free on laundry liquid.
Which offers the best value?
For Tesdi, we now have a new price of £6 – 30% = £4.20 per bottle.
For Snozbury’s we are now buying 4 bottles, but only paying for 3.
Total cost = 3 x £6 = £18
This is now spread over the 4 bottles, so cost per bottle is £18 ÷ 4 = £4.50 per bottle
So from this, we can see the 30% off offer from Tesdi is the better value.
For these types of question, the two questions that we need to consider are: “how many times more/less?” and/or “how much for one?” in order to do a direct comparison and are all linked directly to proportion.
David is a PGCE qualified maths teacher with many years of experience teaching maths. Having taught maths from years 7 to 11, he has also set up and taught Primary Maths workshops.
He was also responsible for writing, and then assessing, a BTEC course in money and finance.
You can enquire about tutoring with David here
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]]>Improving number confidence – A fraction at a time Read More »
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]]>
I often stated that multiplying fractions together is one of the easiest things to do when working with maths. By simply multiplying the numerators together and multiply the denominators together we do not have to find the lowest common multiple. It is understood at an early age that the word ‘of’ means times and that a ‘half of a half’ of a pizza is the same as a half times a half, which is a quarter:
Cross canceling is often taught as a way of saving time for more challenging fraction work:
Can be cross-canceled in the diagonal going from 18 to 36 by dividing
each number by 18 – striking through 18 and writing 1 and striking through 36 and writing 2.
Sim
ilarly, for the diagonal going from bottom left to top right, we can divide through by 11 to obtain 3 on the bottom and 1 at the top. We then have:
which is simply equal to 1/6
HOW TO REVISE: 5 STUDY TIPS THAT REALLY WORK
Dividing fractions is also very easy if we remember the strategy of ‘Flip, Kiss, Keep’ which involves inverting the second fraction, replacing the divide with a times (the kiss), and keeping the first fraction the same. Therefore:
2/3 divided by 4/5 becomes
2/3 times 5/4 which is 10/12
which simplifies to 5/6.
Fraction multiplication and division can also successfully achieved by first changing the mixed number into a top-heavy fraction. For example,
2 ½ x 3 ¾ can be calculated by first converting each mixed number;
I have found that using highlighter pens makes this process a lot easier for younger pupils if they are asked to shade the whole number and the denominator of each fraction first to remind them that the process involves multiplying the big number by the denominator and adding the top number to obtain:
5/2 x 15/4
which becomes 75/8
The most memorable breakthroughs with teaching fractions have happened when large class sizes have considered how best to calculate:
1/3 + ¼
I will always remember the looks on the faces of my first ever adult students in Warminster who gave up two hours on a Monday evening for an academic year to achieve a cornerstone qualification with maths. Students told me that during their school days, they got lost with fractions, and the feelings of failure carried on from there with other maths topics.
WHY SHOULD YOU BOOK A MATHS TUTOR?
I instinctively remembered that I had a twenty dollar bill in my wallet as well as a ten pound note. With both notes on the table, I asked my adult students to say how much money was on the table. Inevitably, the students quickly responded that we needed to first use the exchange rate between pounds and dollars and suddenly there was a fantastic lightbulb moment for the entire class. This was especially helped by the exchange rate being one pound equals two dollars at the time and the answer of twenty pounds in total was quickly understood.
We then decided to write the first few times table for 3 and the first few times tables for 4 to find the first number in both lists (12) – hence finding the lowest common denominator became easy. I also mentioned the gifted Classical Greeks had found a ‘Golden Rule’ for fractions even before the birth of Christ:
“A fraction remains unchanged if we multiply (or divide) the numerator and denominator by the same number.”
Hence 1/3 became 4/12 and ¼ became 3/12. Using curvy arrows between the numerators and the denominators really helped. As we now had a common currency of twelfths, the answer of 7/12 quickly became understood. After that, it was a fantastic feeling to see every student in the group achieve their target grade. In addition, this led to career opportunities for the nurses and army trainees in this group; this even matched my feeling of seeing lots of ‘A’ level success in future years as a subject leader.
I wonder how the adult evening course students are doing now….
Chris is a freelance tutor who has 28 years of full-time teaching experience. He was Head of Department in a large Somerset college from 1992 to 2014.
He holds a B.Sc. in Maths & Science and a B.Ed. in Mathematics.
With an enthusiasm and interest in mathematics, Chris can raise self-confidence and belief by making maths fun. He concentrates on setting the language in context and building clear scaffolding in topics based on his insight and understanding of the subject.
You can enquire about tutoring with Chris here
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]]>Probability – A Master Class Read More »
The post Probability – A Master Class appeared first on The Tutor Team.
]]>The fascinating topic of probability is more relent now than ever. We take probability choices every day of our lives and make critical decisions to enhance our safely and our well-being. For example, “The chances are if I wear a facial mask, I have a reduced risk of contracting covid”…..”the chances are if I am respectful to the people I meet in life, they will be respectful to me in return”.
Students learn at an early age that probability values range from zero (impossible) to 1 (certain) with a myriad of events in between.
If we consider probability work in year 7, 8, 9, 10 or 11, it is fascinating to hear the range of responses when you pose the question “If I roll two dice, what is the chance that both dice come down as a 6”. The most popular incorrect answers include 2 in 12, 1 in 12 and 2 in 6!
The dream solution to finding a way in to probability came unexpectedly and fortuitously when I accompanied a group of Frome College students to a mathematics Master Class at the University of Bath which was run as a series of Saturday mornings. This was the first time I had seen the impact of having six dice available and a score sheet for the game of ‘ZILCH’.
WHY SHOULD YOU BOOK A MATHS TUTOR?
Two students rolled one dice each and the highest score would go first.
The points scoring was quickly understood by the students:
The first player to reach 2000 or more points was the winner of ZILCH.
Firstly, each player rolls all six dice at once and places aside the scoring dice. For example, a score of 2,2,2, 3, 4 and 5 scores 200 points (three twos) plus 50 points (one five) to give 250 points.
Then the player makes a decision to stick with this score or to carry on and ‘gamble’. To gamble, the player would throw again the non-scoring dice, in this case the 3 and the 4.
However, if a player decides to gamble and then does not score in the rolling of the other dice, they score ‘ZILCH’ or as they say in America – Zilch means nothing.
After that, Zilch became a beautiful way in to my probability work for the next twenty years. ZILCH World Cups were a joy to observe with winners coming up to the whiteboard to put their names on the board before playing someone else.
INTEGRATION IN MATHS: THE PADDINGTON BEAR APPROACH
After a few games, we would discuss the ZILCH dilemma using a ‘Sample Space Diagram’ where a player has two dice remaining and has a score or three hundred or so points. Should I gamble or should I stick? Because of the points given for One 1 or One 5, students could see that there were 20 possibilities out of 36 for improving their score if a risk was taken.
Students could also clearly see that the chances of rolling a six and a six with two dice were one in thirty six! In additions, we would give a round of applause to the first 1,2,3,4,5,6 in a lesson that was witnessed.
I even, on just one occasion in twenty years, witnessed a win on the first go. 1, 1, 1, 1, 1, 1!
In conclusion, ZILCH was instantly one of my favourite ways into teaching a topic in mathematics. After all, it simplified a topic that many students did not at first find easy to understand. I will always be grateful to the University of Bath for running such sparkling sessions for pupils on an inspired Saturday morning workshop.
Chris is a freelance tutor who has 28 years of full-time teaching experience. He was Head of Department in a large Somerset college from 1992 to 2014.
He holds a B.Sc. in Maths & Science and a B.Ed. in Mathematics.
With an enthusiasm and interest in mathematics, Chris can raise self-confidence and belief by making maths fun. He concentrates on setting the language in context and building clear scaffolding in topics based on his insight and understanding of the subject.
You can enquire about tutoring with Chris here
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]]>The post Standard Form appeared first on The Tutor Team.
]]>Standard Form is a way to write really large or really small numbers. It is a way to write these numbers in a format that everyone agrees to.
Although the idea is to write the numbers into a set format, the name of the format is not always agreed on. You may hear terms such as Standard Index Form, Scientific Notation, Standard Form and SIF. However, they are all the same. The one to watch is Engineering notation which is similar but does not follow the same rules.
INTEGRATION IN MATHS: THE PADDINGTON BEAR APPROACH
A famous mathematician and Greek philosopher called Archimedes was thought to have used them initially. He used them to help him try to calculate the size of the universe.
As we get more and more advanced we are often dealing with very large numbers. This is not just bank balances! Science is also a key area where large numbers are used. For instance, when talking about the distance between planets or even galaxies we often need to use large numbers.
Equally, as science explores the smaller world such as atoms, particles and quantum physics, then we are often using ridiculously small measurements and we need a way of writing these numbers down.
If we just write these large or small numbers normally (often referred to as ordinary form) then it is often difficult to count or quickly see the number of 0’s. If we miscount or accidentally miss one-off, then it could make a dramatic difference to any calculations we are trying to do. Just imagine if you were expecting £100 but were given £10, you might not be too happy. When numbers are 30 or 40 digits off then it is easy to make a mistake.
A number is Standard Form has to follow a set of 3 rules which everyone uses.
You can see that from this that the first part is a number between 1 and 10. This number can be 1 and go up to but not include 10. We sometimes write this as 1≤n<10. The number can be an integer (whole number) or a decimal.
The second part is always the “x10” and this is always the same. This is because our place value system is based on 10 so by multiplying by 10 we are moving the digits one place left. For example, 8 becomes 80 and 1.23 becomes 12.3 .
The last part of the number is the power. This is sometimes called the index or exponent. This is placed on the “x10”. This has to be an integer but can be positive or negative. If it is positive, it means we are multiplying by 10 lots of times. So, for example, a power of 3 would multiply the number by 10 three times. If the power is 24, then we are multiplying by 10 24 times (which will make a very large number).
If the power is negative, then we are effectively dividing by 10 that number of times. For example, a power of -6 would suggest we are dividing by 10 six times.
So in the example above we are starting with the number 4.2 and multiplying it by 10 nine times. This means we are really doing
4.2 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x10
or
4.2 x 1,000,000,000 which is 4,200,000,000
HOW TO FACTORISE AND SOLVE QUADRATIC EQUATIONS
For GCSEs, you need to be able to convert numbers to and from Standard Form. If you can multiply and divide numbers by 10 then you should be able to do this without a calculator.
You also need to be able to compare numbers in Standard Form. This is done by looking at the power. The bigger the power, the bigger the number. Remember that small numbers will have a negative power.
Complete calculations with Standard Form. You need to be able to complete simple calculations with Standard Form. This could be both with and without a calculator. Your calculator has a special button on it for Standard Form.
Want your child to succeed? Why not try one of our expert tutors. You can read why this helps here: Why Should I book a Maths Tutor
Phil offers tutoring up to secondary school aged students, up to GCSE, in Maths. Having taught GCSE maths for many years, he can help both Foundation Level and Higher-Level students improve their grades.
He is currently a teacher in a secondary school in Colchester and has been teaching maths for over 22 years. He is a fully qualified teacher with a degree (BSc) in Maths and Computer Science.
You can enquire about tutoring with Phil here
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]]>Integration in Maths: The Paddington Bear Approach Read More »
The post Integration in Maths: The Paddington Bear Approach appeared first on The Tutor Team.
]]>A quick count of the marks in a recent (November 2020) A Level paper showed that 99 out of the 200 marks came from them. That’s nearly 50% of all the marks. Now whilst most pupils can cope with differentiation, nearly everyone finds integration in maths tricky. But why?
The big difference is that differentiation is really the starting point and there are fixed ways of doing it. If you have a function in a particular form there is really only one way to differentiate it. These are nicely learnable. A product uses the product rule, a compound function uses the chain rule etc… There are even nice easy ways to remember them.
Product rule: “Keep Differentiate + Differentiate Keep” what could be simpler?
Integration, on the other hand, has lots of different methods and, whilst each method on its own is very learnable and not at all difficult, not one of them ALWAYS works – there is no silver bullet.
After all, just learning a few different methods isn’t the end of the world. The issue is that spotting which method to use can be very hard. Two very similar looking functions can need very different ways of approaching them. Changing to , just by sticking in an innocent looking little “squared” changes it from a very simple, just write it down, integration problem to one that requires some serious university level maths.
To make it worse, there can often be more than one method that does work. This sounds great, but the different methods can lead to different forms of the answer achieved. Many an A Level pupil will have spent some considerable time trying to work out where they have gone wrong when their answer doesn’t match the one in the back of the book or the “show that” in the question, only to discover that their answer was correct, just written in a different way.
HOW TO FACTORISE AND SOLVE QUADRATIC EQUATIONS
The main thing, as in any A Level maths question, is to remember that you are doing an A Level maths question. This means that it will definitely work and will only need the methods that
you have been taught. So DON’T PANIC!
Next, start the question by employing the Paddington Method – give it a really hard stare! Yes, seriously….
If you look really hard at the function and think “if I differentiated something and ended up here, what must I have started with?” this can often get you most of the way there.
You are used to doing this. No one, probably(!), actually divides numbers – if asked “What is 42 divided by7?” we all think “what do I have to multiply 7 by to get 42”
So can you see where you must have started from?
Now I am not saying that this method is easy, but it is very useful and it can be practiced. You can get a set of ones that work like this from here and work your way through, seeing how many you can do in 5 minutes. Keep coming back to it and trying again so that you get better. The time spent on this now can save you lots of time and grief in an actual exam.
DEMYSTIFYING FORMULAS IN ALGEBRA
You’ve eaten all the marmalade sandwiches, tried again and it still doesn’t work. Well, in this case the reason why it doesn’t work will tell you what to do next.
A product that didn’t come from chain rule will need differentiation by parts or a fraction that isn’t from ln or chain rule needs to be flipped with a substitution or separated with partial fractions.
And don’t forget, rule number 1 – it does work, you are doing A Level questions! It’s not like real life where there are times when we can’t solve things. So, if it seems to be going wrong and taking too long you have either made a mistake or are doing it the wrong way…
Practice your Paddington stare and then just apply some common sense. And like all good Paddington stories, it will work out in the end…
Practice Paddington questions Practice Paddington answers
Want your child to succeed? Why not try one of our expert tutors. You can read why this helps here: Why Should I book a Maths Tutor
Andy is incredibly good at breaking down maths into easy to understand chunks, whether it be number bonds with primary pupils, the dreaded fractions with KS3 or the scarily hard Further Maths and undergraduate topics.
With a firm belief that you can teach anyone anything as long as it is approached in the right way, Andy has led a class of “sink” pupils in an inner city comprehensive through a lesson where they all “discovered” Pythagoras’ Theorem (and still remembered it 3 months later!) and has had year 8 pupils working through Year 13 algebraic proofs using multilink cubes.
You can enquire about tutoring with Andy here
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]]>Demystifying formulas in algebra – a journey into a bright forest. Read More »
The post Demystifying formulas in algebra – a journey into a bright forest. appeared first on The Tutor Team.
]]>Perhaps the reason algebra may seem so challenging for so many is that the letters used need to be placed in context so that students can see that each letter stands for something in everyday life. X could represent the blood sugar level in a diabetic patient and Y could represent the heart-beat of a patient receiving specialist care.
I have found that algebra can be demystified for students in the years leading up to GCSE exams by considering a taxi metre as a starting point. Therefore, I set the scene by imaging a taxi journey from Paddington Station to a location in the city of London:
C = 380 + 70n
C represents the cost of the journey in pence and n represents the number of miles. Before the journey even begins, there is a fixed charge of 380 pence.
The layout of clear, concise method work is immensely helpful with ‘flushed’ equals signs and sequential working:
For a journey of five miles:
C = 380 + (70×5)
So, C = 380 + 350
Add the sum to get:
C = 730 pence
C = £7.30
It is also very helpful to gain experience with ‘working backwards’ with a taxi meter formula, for instance, finding the number of miles given the total cost:
Cost = £8.70
870 = 380 + 70n
Subtracting 380 from each side:
490 = 70n
Dividing each side by 70:
7 = n
Hence the taxi journey costing £8.70 is seven miles.
HOW TO FACTORISE AND SOLVE QUADRATIC EQUATIONS
Another very helpful formula to bring a clear context to formulae in algebra is the relationship between degrees Fahrenheit and degrees Centigrade:
C = 5/9 (F – 32)
I have found that I can achieve great breakthroughs have been made when we post the question, what is the temperature in Centigrade when the temperature in Fahrenheit is 32 degrees. Students quickly realise that 5/9 times zero is zero and future calculations are easily made.
I also find that this formula is very helpful for teaching rearranging of formula to make F ‘the subject’
C = 5/9 (F – 32)
Multiplying each side by 9 gives:
9C = 5(F – 32)
Dividing each side by 5 gives:
1.8 C = F – 32
Adding 32 to each side gives:
F = 1.8 C + 32
I build confidence with algebra, immeasurably, by asking students to reflect for a moment on the huge number of words that they have learned in the thousand weeks or so that they have been alive on the planet; the universal language of algebra is just a small addition to our amazingly successful brain processing skills.
Want your child to succeed? Why not try one of our expert tutors. You can read why this helps here: Why Should I book a Maths Tutor
Chris is a freelance tutor who has 28 years of full-time teaching experience. He was Head of Department in a large Somerset college from 1992 to 2014.
He holds a B.Sc. in Maths & Science and a B.Ed. in Mathematics.
With an enthusiasm and interest in mathematics, Chris can raise self-confidence and belief by making maths fun. He concentrates on setting the language in context and building clear scaffolding in topics based on his insight and understanding of the subject.
You can enquire about tutoring with Chris here
The post Demystifying formulas in algebra – a journey into a bright forest. appeared first on The Tutor Team.
]]>Why should you book a maths tutor? Read More »
The post Why should you book a maths tutor? appeared first on The Tutor Team.
]]>Maths and English are in my view the most important subjects to do well in at school. There are many reasons for this but mainly because you will be barred from entry to University unless you have at least a grade 4 in both subjects, which is a very good incentive to do well in both subjects.
Some students can be successful in English but often fail in Maths or vice versa. I have taught many students who have failed several attempts to pass Maths GCSE before contacting me. Some students who are taking GCSE only make contact a few months before their exam in the last year of A level. Then their whole future rests on getting the grade 4 in Maths giving themselves only a short time to sort out problems. An important lesson here is to book a maths tutor in good time!
My advice to parents is if your son or daughter shows any problems in Maths and their school report indicates failures in Maths then seek help from a personal tutor as early as possible.
Students may have problems in learning mathematics. Rectifying this as soon as the problem arises pays the best dividends. A personal tutor can help students in their work by trying to improve confidence in the subject and then they can make progress in the more difficult areas.
I have taught many students privately in my time and I know what a difference it makes to have that one-to-one session with a tutor who can listen, recognise problematic areas, develop skills, and help students to become more confident in a subject they have never found easy to understand.
Having taught many pupils from as young as 7,8,9 years old, it is my experience that if you can sort out any misconceptions from an early age via a personal tutor, then the child develops confidence and progress can be transferred to the classroom. A personal tutor can set solid foundations that will support mathematical pathways for future life.
In age 11 to 16 progress in Maths is like being on a conveyer belt where it feels as though you cannot get off to consider and absorb all the information being taught. The pace of lessons is rapid and students often start to struggle in class and especially when it comes to being alone with homework and revision. A personal tutor can teach the work again at the students’ pace and start to make progress with both classwork and revision.
Sometimes, as students start A level mathematics, they find difficulties with the work as there is a huge step up in the understanding of the subject. If, as a parent you identify there is a problem then you should get in touch with a personal tutor as soon as possible. A tutor will help your child gain more confidence, improve their understanding in the subject thus leading to better progress, and give you peace of mind as parent that they are best prepared for the future.
Dave is an experienced, PGCE qualified teacher of Mathematics. He has taught maths in schools for over 30 years and has been Head of Maths in the largest department in Somerset. Dave is passionate about teaching at all levels and inspires his students to work for better results. In the last 12 years he has privately tutored over 800 students.
You can enquire about tutoring with Dave here
Dave specialises in pre-GCSE, GCSE, A level Maths including Pure Maths, Mechanics and Stats. He has taught all syllabi in Key stage 1, Key Stage 2, Key stage 3, GCSE and A level using Edexcel, AQA and OCR schemes of work.
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]]>HOW TO FACTORISE AND SOLVE QUADRATIC EQUATIONS Read More »
The post HOW TO FACTORISE AND SOLVE QUADRATIC EQUATIONS appeared first on The Tutor Team.
]]>It has been my experience that students overcomplicate the topic of quadratic equations.
There are a number of different ways to solve quadratic equations in addition to factorising, such as: the ‘difference of two squares’ (Level 6); the quadratic formula (Level 7), plotting the graphs of quadratic functions (Levels 5 – 7) and completing the square (Level 8). However, in this blog I am focusing purely on factorising.
My advice is to get familiar with what a quadratic expression looks like and the steps I have shown below. There are things you need to be familiar with first and they are covered in the next section. Most of the errors I come across are where the student has either missed or got muddled with the rules concerning negative values or made a mistake rearranging the equation.
A quadratic expression is one where the highest power of x in the expressions is x squared (x^{2}). Quadratic equations can be written in the form of:
ax^{2} + bx + c = 0
where a, b and c are real numbers and a is not equal to zero. This means that the values of a, b and c can be positive as well as negative. They can also be decimals or fractions.
The solutions to a quadratic equation are known as roots and there are generally two to find.
Note: The roots are also the points on a graph where the quadratic curve cuts the x-axis.
Step 1: Rearrange (if necessary)
Make sure the quadratic equation in the form ax^{2} + bx + c = 0 when factorising.
This may mean you need to rearrange the equation you are given before you start.
Example: X^{2} +2X = 15 rearrange to X^{2} + 2X – 15 = 0
Step 2: Prepare Brackets
Note the values of a, b and c. and write down the initial brackets (as factorising is putting the brackets in).
Example: In the expression X^{2 }+ 2X – 15 = 0 a = 1, b = 2, c = -15. As a is = 1, the brackets will be:
(X )(X ) = 0
Step 3: Find the values
When a is equal to 1 (as in this example), we can use the following method:
X^{2} + ( j+k )X + ( jxk )=0
where j and k are two real numbers.
The middle term (“b”) is the sum of j and k.
The third term (“c”) is the product of j multiplied by k
Example: X^{2} + 2X – 15 = 0 where a = 1, b = 2 and c = -15
j + k = 2
j x k = -15
so…….. 5 x -3 = -15 and 5 + -3 = 2
Therefore the factorised quadratic expression is ( X + 5 )( X – 3) = 0
Step 4: Check
It is important to check your answers are correct by expanding the brackets and simplifying the equation to make sure you get the original expression.
Example: (X + 5 )( X – 3 ) = X^{2 }+ 5X – 3X – 15
= X^{2} + 2X – 15
Step 5: Solve
By setting each bracket to zero
Example:
( X + 5 ) = 0 so X = -5
( X – 3 ) = 0 so X = 3
Example: X^{2} – 10X = 0 which will factorise to X(X-10) = 0
Solutions are X=0 and X=10
Example: Factorise 3X^{2} + 32X = – 20
Step 1: Rearrange to 3X^{2} + 32X + 20 = 0
Step 2: Prepare brackets. a = 3, b = 32, c = 20 so……….(3X )(X ) = 0
Step 3: Find the values:
j x k = 20 and j + 3k = 32 so………2 x 10 = 20 and 2 + 3×10 = 32
( 3X + 2 )( X + 10 ) = 0
Step 4: Check
( 3X + 2 )( X + 10 ) = 3X^{2} + 30X + 2X + 20
= 3X^{2} + 32X + 20
Step 4: Solve
( 3X + 2 ) = 0 , X = -2/3
( X + 10 ) = 0, X = -10
Karen is an experienced, PGCE qualified teacher of Mathematics at Key Stage 3 and above, including the International GCSE. She has been an Examiner and Moderator for Edexcel and therefore has good insight into what examiners are looking for and with exam techniques.
You can contact Karen about tutoring here
Karen is a skilled creative thinker and problem solver with a passion for identifying untapped talent in individuals so they are able to make a lasting transformational change into the best version of themselves. She provides a safe environment that is non-judgemental, encouraging, positive and challenging, with the freedom to experiment.
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]]>Lockdown Learning: 6 ways to help your teen cope Read More »
The post Lockdown Learning: 6 ways to help your teen cope appeared first on The Tutor Team.
]]>2021 has arrived and with it the return of lockdown learning and home-schooling. Whilst that has resulted in an enormous amount of juggling for parents (especially if you are working from home yourself) and teaching staff, it’s your teenager who’ll be feeling the effects – not seeing their friends or sharing classroom learning experiences with their peers – most keenly.
According to consultant psychologist and founder of the Chelsea Psychology Clinic, Dr Elena Touroni, young people have had their world turned upside-down this past year and explains that “limited social contact with peers, fears about COVID and worries around what the future holds post-pandemic could all be taking a toll on their mental wellbeing.” Not the best backdrop for learning, I’m sure you’ll agree.
So, how can you as a parent help your teenager cope in an upside-down world, especially if they are struggling or are feeling isolated or anxious? How can you make lockdown learning less stressful for all concerned?
This is one of the best ways to help a teenager especially when they are feeling anxious. Take time to ask them how they are feeling and really listen to their replies. Remember to be open with your child as this will help them realise their fears, anxieties and worries are nothing unusual. Do also remember not to pressurise them, they will talk if they want/need to.
If they are using their bedroom to study, try to encourage them out once and a while to engage with the rest of the family. Left alone, it’s easy for a young person to get distracted or start brooding over worries.
Encourage them to connect with friends as often as possible. Yes, it will be via a screen but it will be a lifeline for your teen at the moment. You can always monitor the amount of time they spend chatting to make sure it’s not impacting their schoolwork.
Whether they like it or not, your teenager needs a routine and works best when expectations are clear. Now, while online learning at home can’t possibly replicate the school day, you can still put a structure in place to help them feel like they’re at school; factoring in their timetable, regular breaks and chats with friends.
Continuing pre lockdown routines like getting up at the same time, sharing breakfast before school, making a packed lunch and chatting to them about their day, is a great way to build some structure. Do ensure there are plenty of breaks for refuelling – it’s especially important for your teenager to keep hydrated. It’s also important to factor in some quiet time, time for exercise and for having fun! If you can, try to encourage a regular bedtime and for them to turn off screens an hour before going to sleep.
There has to be an upside to home learning so remind your teenager that as they’re not in a classroom they don’t have to sit in one place or be in one room. Give them the option to move around the house or even outside if the weather’s fine and it’s okay to do so. Find whatever works best for your child and the rest of the household (bearing in mind that some rooms in the house might be off-limits due to working from home).
The most important thing is not to beat yourself up because your child’s online learning experience doesn’t look like anyone else’s. Remember, there are no rules for how home-schooling should look. We’re all making it up as we go along and doing the best we can. All that matters is that you and your child are happy, and schoolwork is going as well as possible under the circumstances.
One of the main drawbacks of online learning is the amount of time your teenager will spend staring at a screen. That’s why it’s important for them to take plenty of breaks during the day and, where possible, split their schoolwork into bite size chunks.
One of the advantages of online learning is that your teen won’t necessarily have to complete the work in one sitting, or in a particular order, and there is no reason why they have to stick to school hours or keep working on something that’s frustrating them. They have the luxury of working to a schedule that suits them – providing they’re attending their live lessons, meeting the deadlines for submitting work and it’s not an antisocial one for the rest of the household!
If your teen likes studying with others – and it’s appropriate for them to do so – why not encourage them to set up a virtual study group with classmate friends where they can replicate a classroom feel, discuss tasks, bounce ideas around and keep each other accountable. This is good for mental health and, providing they’re not just using the time to have a good old chat, can lead to an increased understanding of the subject or task and some seriously creative problem solving.
The current situation is difficult for everyone, but if your teenager is struggling to cope with learning online you can always reach out to their school, or individual teacher(s) for some one-to-one help or advice. Teachers have your teen’s best interest at heart and will be happy to talk to you (and them) about any learning worries or stumbling blocks they may have. Once teachers are aware, arrangements can be made for additional learning support, guidance or resources to help your child. If you still felt your child needed additional support, after that, you could always look at the services of a private tutor.
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]]>9 books your teenager should read in lockdown Read More »
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