MEI C1 Study Resources Core1 The Language of Mathematics 1-2

MEI Core 1 The Language of Mathematics Section 1: Problem solving Study Plan Background At G.C.S.E. you would encounter problem solving in many areas especially when doing investigative work. You will have followed a process of simplifying, looking for patterns, testing, predicting etc i.e. a systematic approach. You need similar skills when approaching real life problems and the flow chart on page 143 is a useful guide to this.

Detailed work plan 1. Revise the work that you did in chapter 1 so that you are comfortable with basic algebra and generalising, then read the text up to exercise 6A. There are some further notes in the Notes and Examples. 2. Exercise 6A Try questions 1, 2, 5*, 6*, 7, 8*, 10*

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MEI Core 1 The language of mathematics Section 1: Problem solving Notes and Examples These notes contain subsections on  Solving mathematical problems  The modelling cycle

Solving mathematical problems This section is about using mathematics to solve problems. This is, of course, an important part of everyday life. You are using mathematics to solve problems when you work out how long a car journey is likely to take you, or how much money you need to take on holiday. Throughout your learning of mathematics, you will have carried out investigations, including for GCSE coursework. Sometimes these deal with real-life problems, involving collecting data for example, but sometimes they are purely mathematical problems. The mystic rose problem in the textbook is an example of a problem which is presented as a purely mathematical problem. (However, as suggested by the Discussion Point on page 138, it could represent a real-life situation such as airline routes connecting cities.) The approach to the mystic rose problem suggested on pages 139 – 140 is quite a common one. This involves looking at simpler versions of the problem and finding the pattern, which can then be generalised and expressed algebraically. However, with some problems it is possible to deduce the solution simply by thinking about the problem. In the case of the mystic rose, this is shown on page 141 by the argument: “There are p points. Each is joined to (p – 1) other points, giving p(p – 1) lines. However each line is counted twice, so there are 12 p( p  1) lines.” Even if you can see a solution like this straight away, without investigating different sizes of mystic rose, it is still worth checking that it works for some simple examples. You might have made a mistake in your reasoning. For example, if you had failed to realise that each line is counted twice, you might have come up with the formula N  p( p  1) . If you then checked this against some simple examples, you would quickly realise that your formula gave twice the correct answer in each case, and you would probably be able to see why and correct your reasoning.

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MEI C1 Language of maths Sec. 1 Notes & Examples The modelling cycle Solving problems using mathematics is an important skill in almost all areas of life! When you need to solve a real-life problem involving mathematics, the first step is to express the problem in mathematical terms. This usually involves making simplifying assumptions. Look at the example on page 142 about Katie’s walk. Katie has made a number of assumptions to allow her to make her calculation. She has assumed:   

the distance she will walk is 874 km she will walk at an average speed of 4 mph she will walk for 24 hours a day every day.

It is clear from common sense that the third assumption is not realistic! Katie changes her third assumption to 

she will walk for 10 hours a day every day

The difficulty with applying a mathematical model to a real-life situation is that it may not be immediately obvious whether the assumptions are realistic or not. It is usually helpful to have some kind of experimental result to refer to. In Katie’s case she has found out how long other people have taken to do the walk. This suggests that her assumptions are unrealistic. She might need to do some further research or experiment to make her assumptions more realistic, such as to find out the route taken by other people, or to test how long it takes her to walk 10 miles. As part of your AS level Mathematics you are, or will be, studying some of the applications of mathematics (Mechanics, Statistics, or Decision Mathematics). You need to use modelling in all these applications, and you will usually have to make simplifying assumptions about your work. You should always think about how the assumptions that you have made might affect your results. For example, if you were investigating the best angle to throw a ball so that it goes as far as possible, it would probably be realistic to assume that the effect of air resistance can be neglected. However, if you were looking at the time take for a parachute to descend, air resistance would be a very important factor, and to assume that it can be neglected would probably give you a very inaccurate result. Most of the examples in Exercise 6A involve very simple mathematical models, and to do these questions you do not need to worry too much about making assumptions and using the modelling cycle. However, these are ideas which will be important throughout your study of mathematics.

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Core 1 The language of mathematics Section 1: Problem solving Crucial points 1. Check any formula you have found When you find a mathematical expression to describe a particular situation, make sure that it is correct for a simple example. 2. Think about assumptions in modelling When modelling a real life situation, always think about the assumptions that you are making, and whether they are realistic.

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MEI Core 1 The language of mathematics Section 1: Problem solving Multiple Choice Test 1) The length of the hypotenuse, c, of a right-angled triangle whose shorter sides have whole number lengths a and b is given by the formula

a, b  c  a 2  b2 Which of the following statements is true for all values of c? (a) c  (c) c  (e) I don’t know

(b) c  (d) c 

2) The quantities x, y and z are linked by the formula xy  z x, z  , x  0. The set of possible values of y is (a) + (positive members of )

(b) 

(c)  (e) I don’t know

(d) 

Questions 3, 4 and 5 are about a mobile phone with tariff as follows: 15p per minute for the first five minutes per day 5p per minute thereafter

3) The formula for the cost, in pence, of calls on a day on which n minutes of calls are made (n ≥ 5) is (a) 15n + 25 (c) 5n + 15 (e) I don’t know

(b) 5n + 75 (d) 5n + 50

4) The first call of the day lasts 20 minutes. The cost of this call is (a) £1.15 (c) £1.00 (e) I don’t know

(b) £1.75 (d) £1.50

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MEI C1 Language of maths Section 1 MC test 5) The number of minutes call time provided by a £5 top-up at the start of the day (assuming that the whole £5 is used in one day) is (a) 90 (c) 85 (e) I don’t know

(b) 100 (d) 97

6) Kevin is wallpapering the walls of a room. The room is l metres long, w metres wide and h metres high. The width of the wallpaper is 60 cm. The total length of wallpaper Kevin needs (ignoring the door and window) is given by the formula

0.6(l  w) h 2h(l  w) (c) 0.6 (e) I don’t know

lwh 0.6 2(l  w) (d) 0.6h

(a)

(b)

7) One of the formulae below gives the sum of the squares of the first n natural numbers. Which one is correct? (a) 16 n(n  1)(n  2)

(b) 16 n(n  1)(2n  1)

(c) 12 (5n2  7n  4) (e) I don’t know

(d) 12 (7n2  17n  8)

Questions 8 and 9 are about the series of diagrams shown below.

n=1

n=2

n=3

8) One of the formulae below gives the number of small squares in each diagram. Which one is correct? (a) n² + 2n – 2 (c) 2n² – 2n + 1 (e) I don’t know

(b) n² + n – 1 (d) n² + 3n – 5

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MEI C1 Language of maths Section 1 MC test 9) The number of small squares in the diagram for which n = 15 is (a) 239 (c) 252 (e) I don’t know

(b) 421 (d) 265

10) In an investigation, shapes are drawn on square dotted paper. The number of dots on the perimeter of the shape, p, the number of dots inside the shape, i, and the area of the shape, A, are noted for each shape.

For example, in the shape above, p = 8, i = 2 and A = 5. One of the formulae below gives the correct relationship connecting p, i and A for all such shapes. Which one is it? (a) A  12 p  i  1 (c) A  p  2i  1 (e) I don’t know

(b) A  12 ( p  i) (d) A  p  i  1

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MEI Core 1 The language of mathematics Section 2: Notation and proof Study plan Background Mathematicians like to be very clear and precise in what they say and write so as to avoid misunderstandings and misconceptions. Hence the language and the notation used are very important to clear communication. For you to do this you need to learn the meaning of the symbols used and the mathematical usage of the words necessary and sufficient. Everyday use may not be so precise.

Detailed work plan 1. Read pages 149-151 carefully. There are some further examples of mathematical statements in the Notes and Examples. 2. Exercise 6B Try all the even numbered questions. Talking about them to someone else might help you clarify your thinking. 3. Read pages 152-153. There are some additional notes in the Notes and Examples. 4. Exercise 6C Try all the even numbered questions. Again, talking about the questions with someone else may be helpful. 5. Mathematicians also like to be sure of their ground. They like to show that facts are true all of the time not just some of it if they can. They want to prove or disprove conjectures to be sure of their facts. Some of the methods available for doing this are on pages 154 to 156. Read them carefully. There are further notes and another worked example in the Notes and Examples. 6. Exercise 6D Try about half of the questions but include 3*, 5*, 8*, 11*

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MEI Core 1 The language of mathematics Section 2: Writing mathematics Notes and Examples These notes contain subsections on  Mathematical language  The converse of a theorem  Proof

Mathematical language In this section you are introduced to the symbols ,  and . You have probably seen these used before – they are used throughout the AS Pure Mathematics textbook and in most mathematics textbooks at this level. You will probably find using the symbols ,  and  fairly straightforward. The use of the words “necessary” and “sufficient” may be a little harder to understand. Here are a few examples which may help you. You may need to read them through more than once! 1.

A number ends in 5  the number is divisible by 5.

“A number ends in 5” is a sufficient condition for “the number is divisible by 5”, since there are no numbers which end in 5 which are not divisible by 5. However, it is not a necessary condition, there are numbers which do not end in 5 which are divisible by 5 (numbers which end in zero). You can express this the other way round: A number is divisible by 5  the number ends in 5. “A number is divisible by 5” is a necessary condition for “the number ends in 5”, since all numbers which end in 5 are divisible by 5. However, it is not a sufficient condition, as not all numbers divisible by 5 end in 5.

2.

A number is even  the number is divisible by 4.

“A number is even” is a necessary condition for “the number is divisible by 4”, since all numbers which are divisible by 4 are even. However, it is not a sufficient condition, as not all even numbers are divisible by 4. Again, you can write this the other way round:

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MEI C1 Language of maths Sec. 2 Notes & Examples A number is divisible by 4  the number is even. “A number is divisible by 4” is a sufficient condition for “the number is even”, since there are no numbers which are divisible by 4 which are not even. However, it is not a necessary condition, as there are even numbers which are not divisible by 4.

3.

A number is divisible by 10  the number ends in a zero.

“A number is divisible by 10” is a necessary and sufficient condition for “the number ends in zero”. All numbers which are divisible by 10 end in zero, and all numbers which end in zero are divisible by 10. Another way of expressing this is the statement “A number is divisible by 10 if and only if the number ends in zero.”

In the table below, all the statements shown in the same column are equivalent to each other. A is a necessary condition for B

A is a sufficient condition for B

A is a necessary and sufficient condition for B

If B is true, then A must also be true

If A is true, then B must also be true

A is true if and only if B is true.

AB

AB

AB

A is implied by B or A follows from B

A implies B

A implies and is implied by B

You can often use the symbols ,  and  when you are writing out a solution to a mathematical problem. For example, if you want to write the expression y = x² + 4x + 1 in the completed square form, you might write:  

y = x² + 4x + 1 y = (x² + 4x + 4) – 4 + 1 y = (x + 2)² – 3

In fact, you could use  instead of .  

y = x² + 4x + 1 y = (x² + 4x + 4) – 4 + 1 y = (x + 2)² – 3

People often use  when they could use , if they are only interested in the logical steps in one direction.

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MEI C1 Language of maths Sec. 2 Notes & Examples The converse of a theorem It should be clear from the examples above that just because a theorem, or a statement, is true, does not necessarily mean that the converse is also true. If a theorem can be written using , then both the theorem and its converse are true. For example: A triangle has equal sides  the triangle has equal angles. Both this statement and its converse are true. The statement An equation is linear  the equation has exactly one real root is true, since all linear equations have exactly one real root. However, the converse of this statement would be An equation has exactly one real root  the equation is linear which is not true, since there are many examples of quadratics, cubics and indeed polynomial equations of any order which have exactly one real root. This statement cannot be written using .

Proof Proof is a very important aspect of mathematics. If you go on to study A2 Mathematics, you will look at proof in more detail. At this stage you are expected to have an idea about what proof involves. The most important thing to realise is that checking lots of cases does not prove that the result is true. In the example of the mystic rose at the start of chapter 6 in the AS Pure Mathematics textbook, a formula was proposed which satisfied all the cases which had been drawn. This does not prove that the formula is true. However, on page 141 a mathematical argument is given which holds for mystic roses of any size, and leads to the formula. This is a proof. As a very simple example, think of three consecutive numbers and add them up. You should find that this sum is divisible by 3. Suppose you want to prove that the sum of three consecutive integers is always divisible by 3. You could test quite a lot of sets of numbers yourself, or you could program a computer to test a very large number of sets of numbers. The computer could keep checking numbers up to astronomically large numbers, but you would still not

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MEI C1 Language of maths Sec. 2 Notes & Examples have checked every single number, and you never can, since there are an infinite number of sets of three consecutive integers! At this stage you could feel sure that the conjecture is in fact true, but to prove it you need to show that it is true for all possible sets of numbers. Fortunately, this is very easy to do.

Example 1 Prove that the sum of any three consecutive integers is divisible by 3. Solution Let the first number be n. Then the second number is n + 1, and the third number is n + 2. The sum of the three numbers is n + n + 1 + n + 2 = 3n + 3 = 3(n + 1) 3(n + 1) is divisible by 3 for all values of n.

This is an example of direct proof, or proof by deduction. In this case the proof consists of a set of logical steps. Other methods of proof include:  proof by exhaustion, where there are a limited number of possibilities which can all be tested  proof by contradiction, in which the conjecture is assumed to be false, and then a set of logical steps lead to a contradiction or an impossible conclusion. The conjecture is therefore shown to be true. To disprove a conjecture, all you need is to find a single instance where the conjecture is not true. This is called a counter example. The AS Pure Mathematics textbook shows several examples of proofs.

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Core 1 The language of mathematics Section 2: Notation and proof Crucial points 1. Be careful with notation Use the symbols ⇒, ⇐ and ⇔ carefully. Make sure that you only use ⇔ when one condition is true if and only if the other is also true. 2. Think carefully about the meaning of mathematical statements Remember that if a statement is true, this does not necessarily mean that its converse is true. A ⇒ B does not mean that B ⇒ A. If the converse is true, then you can write A ⇔ B. 3. Make sure that a proof really is a proof Remember that to prove a result, you must show that it is true in all possible cases. If it is not possible to test all cases, then you need to generalise.

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MEI Core 1 The language of mathematics Section 2: Notation and proof Multiple Choice Test 1) P is any number with two or more digits. Which is the correct symbol to be inserted in the gap in the statement “P is divisible by 4…….The number formed by the last two digits of P is divisible by 4”? (a)  (c)  (e) I don’t know.

(b)  (d) none of these.

2) Which is the correct symbol to be inserted in the gap in the statement “ ( x  3)2  1........3  x  4 ”? (a)  (c)  (e) I don’t know.

(b)  (d) none of these.

3) If n is an integer, which is the correct symbol to be inserted in the gap in the statement “ x  n  n  1 n  2  .........x is divisible by 3 ”? (a) none of these. (c)  (e) I don’t know.

(b)  (d) 

4) For the following say whether statement one is necessary or sufficient for statement two (or both or neither): 1. There are thirteen people in the room 2. Two people in the room have a birthday in the same month (a) (c) (e)

sufficient both I don’t know.

(b) (d)

necessary neither

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MEI C1 Language of maths Section 2 MC test 5) If the statement “All the cars in the car park are red” is false, given that there is at least one car in the car park, we can deduce that (a) None of the cars in the car park are red (c) At least one car in the car park is red (e) I don’t know.

(b) At least one car in the car park is not red (d) None of the above

6) Which is the invalid step in the following proof that “ 1  0 ”?

x  0  x 2  x  0  x( x  1)  0  x  1  0  x  1 (i )

(a) (c) (e)

( ii )

iii ii I don’t know.

( iii )

(b) (d)

( iv )

i iv

7) Let x be a real number. For the following say whether statement one is necessary or sufficient for statement two (or both or neither): 1. x is not equal to zero 2. There exists a rational number y such that xy  1 (a) (c) (e)

sufficient both I don’t know.

(b) neither (d) necessary

8) Which of these statements are correct? (i) xy  x  y  1 (ii) x  2  x2  4

(a) (c) (e)

both (i) only I don’t know.

(b) (d)

neither (ii) only

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MEI C1 Language of maths Section 2 MC test 9) Consider the following statement: P:

n is an even number  n² + n is an even number

Which of the following is true? (a) P and its converse are both true (c) P is false but its converse is true (e) I don’t know.

(b) P and its converse are both false (d) P is true but its converse is false

10) The solutions of the equation x  3x  10  4 are (a) 2 (c) 2 and 3 (e) I don’t know.

(b) 3 (d) None of these

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