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Mathematics Teaching Self-evaluation Tools

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Here you can see a summary of the areas in which you are confident and those in which you are less confident; there are some ideas and suggestions which may help you in your professional learning.

Tidy up presentation by showing arrows between each pair of square numbers as follows

The conjecture would seem to be true

To prove the conjecture, let the two consecutive square numbers be x^{2} and (x+1)^{2}

You need to prove that the difference between x^{2} and (x+1)^{2} is an odd number

The difference can be written as
(x+1)^{2} − x^{2}
= (x^{2} + 2x + 1) − x^{2}
= 2x +1

Since the number 2x is always an even number then 2x + 1 must be an odd number so that the difference between two consecutive square numbers is always an odd number.

NOTE The conjecture can also be shown to be true diagrammatically as follows

Taking this mathematics further

There are many well known conjectures which mathematicians have tried to prove or disprove over the years.

Two such conjectures are:

Goldbach's Conjecture which says that "Every even integer greater than two can be expressed as the sum of two primes." For example 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 or 5 + 5 ...

The Twin Prime Conjecture which suggests that there are an infinite number of twin primes where a twin prime is a pair of prime numbers that have a difference of two. For example 3 and 5 , 5 and 7, 11 and 13, 17 and 19…. are all twin primes.

A conjecture is a statement which would appear to be true, but has not yet been formally proven. When a conjecture is proved beyond doubt then it is called a theorem.

Until recently, the most famous conjecture was the (misnamed) Fermat's Last Theorem, which stated that if an integer n is greater than 2, then the equation an + bn = cn has no solutions (providing a, b, and c are not zero). .

The conjecture was finally proven in 1994 by Andrew Wiles, an English research mathematician. See here for further information.

Complete the table for the 8x8, 9x9 and 10x10 arrangements. What generalisations can you make?

Answer 1

Size

L

T

+

3

4

8

4

4

4

12

9

5

4

16

16

6

4

20

25

7

4

24

36

8

4

28

49

9

4

32

64

10

4

36

81

From the table
L = 4
T = 4(n+1)
+ = (n+1)^{2
}where n is the number of sides

In the circumstances, the generalisations should be tested and, subsequently, proved (usually using geometric or algebraic properties). A generalisation for the total number of spacers (n+1)^{2} might also be offered.

Question 2
Esther is working out the angle sum of polygons with different numbers of sides.
She divides the polygons into triangles and uses the fact that the angles of each triangle add up to 180^{o}
She produces a table of results as follows:

Table 1

Shape

Diagram

Number of triangles

Angle sum

Triangle

1

1 x 180 = 180^{o}

Quadlilteral

2

2 x 180 = 360^{o}

Pentagon

^{ }

Hexagon

Heptagon

Octagon

Complete the table and use this to create a generalisation for the angle sum on any n sided polygon.
Can you find a generalisation for the internal angles of any n sided regular polygon?

Answer 2

Table 2

Shape

Diagram

Number of triangles

Angle sum

Triangle

1

1 x 180 = 180^{o}

Quadrilateral

2

2 x 180 = 360^{o}

Pentagon

3

3 x 180 = 540^{o}

Hexagon

4

4 x 180 = 720^{o}

Heptagon

5

5 x 180 = 900^{o}

Octagon

6

6 x 180 = 1080^{o}

From the table, a generalisation for the angle sum on any n sided polygon is (n - 2) × 180^{o}.

A generalisation for the internal angles of any n sided regular polygon is

n - 2 x 180^{o
}

4

Taking this mathematics further

Generalisations can be found in many areas of mathematics and the topic might be expanded by looking at proofs which offer some justification for these generalisations.

Explore mathematical proof and examples of different types of mathematical proof; for example here.

Other comments on problem solving and proving by generalisation can be found here.

Making connections

Generalising is the process of providing an overarching or general rule to cover a number of specific cases.

A hierarchy of generalising is identified in the National Curriculum levels as follows:

Learners look for patterns and relationships.

Learners draw simple conclusions of their own and explain their reasoning.

Learners begin to give mathematical justifications, making connections between the current situation and situations they have encountered before.

Learners justify their generalisations, arguments or solutions, looking for equivalence to different problems with similar structures.

Learners examine generalisations or solutions reached in an activity and make further progress in the activity as a result.

Learners use mathematical justifications, distinguishing between evidence and proof

Question 1
Christine says that all numbers have an even number of factors.
Give a counter example to show that Christine is wrong.

Answer 1
Trying different examples
Factors of 12 are 1, 2, 3, 4, 6 and 12 (6 factors)
Factors of 13 are 1 and 13 (2 factors)
Factors of 14 are 1, 2, 7 and 14 (4 factors)
Factors of 15 are 1, 3, 5 and 15 (4 factors)
Factors of 16 are 1, 2, 4, 8 and 16 (5 factors)

The latter example shows that Christine is wrong.
You may wish to investigate what other numbers have an odd number of factors. Can you generalise?

Question 2
Bronwen says that the difference between two consecutive cube numbers is always a prime number.
Give a counter example to show that Bronwen is wrong.

Answer 2 Trying different examples
2^{3} − 1^{3} = 8 − 1 = 7 which is prime
3^{3} − 2^{3} = 27 − 8 = 19 which is prime
4^{3} − 3^{3} = 64 − 27 = 37 which is prime
5^{3} − 4^{3} = 125 − 64 = 61 which is prime
6^{3} − 5^{3} = 216 − 125 = 91 which is NOT prime

The latter example shows that Bronwen is wrong.
You may wish to investigate other answers.

Taking this mathematics further

Leonhard Euler discovered the formula x^{2} + x + 41 which appears to generate prime numbers very effectively. Challenge learners to find a counterexample which proves it does not generate a prime number for every integer value of x.

Making connections

A counter example is an example which disproves a statement or conjecture. For example, the statement "All prime numbers are odd." is not true because 2 is a prime number and 2 is not odd, so this is a counter example.

Counter examples are particularly useful in dealing with common errors and misconceptions. For example, learners generalise that

all prime numbers are odd

division always makes numbers smaller

squaring numbers make them bigger

quadratic equations always have two roots

(x + y)^{2} = x^{2} + y^{2}

the larger the perimeter, the larger the area

Such misconceptions can easily be dispelled by the well chosen use of a counter example

The resulting formula is the same as Pythagoras’ theorem which states that, for a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

In this case the angle at A is 90^{o }so the triangle is right angled. Pythagoras’ theorem is a special case of the cosine rule where the angle is 90^{o.}

Taking this mathematics further

For a proof that 0! = 1

Consider the fact that
n! = (n − 1)! ( n
So (n − 1)! = n! ( n

Substituting n = 1 you get

0! = 1! ÷ 1 = 1 ÷ 1 = 1

An interesting debate on zero to the power zero can be found here and here.
In calculus, the power rule is not valid for n = 1 at x = 0 unless 0^{0} = 1.

Identities such as and are not valid for x = 0 unless
0^{0} = 1.

Making connections

There are many other examples of special cases in mathematics illustrating particularly interesting links or else answering anomalies. For example:

2 is a special case of a prime number since it is the only prime number that is even

The square is a special case of the rectangle since it obeys all the properties of a rectangle but all of the sides are equal.

The rhombus is a special case of the parallelogram since it obeys all the properties of a parallelogram but all the sides are equal.

An isosceles trapezium is considered to be a special case of the trapezium since it obeys all the properties of a trapezium but the non−parallel sides are equal.

0! Is a special case since it does not follow naturally from the definitions of factorial. For further information see here.

x^{0} is often considered to be a special case since it is always equal to one.

0^{0} can be considered to be a special case of x^{0} but it is arguable over whether it should equal one or zero

The following questions, all make use of mathematical reasoning:

Question 1 p is an odd number.

Is 2p + 1 an odd number, an even number or could it be either?

Is p^{2} + 1 an odd number, an even number or could it be either?

Answer 1

If p is an odd number then 2p is an even number and 2p+1 is an odd number.

If p is an odd number then p^{2 }is odd (O x O = O) and p^{2} + 1 an even number

Question 2
Show that the product of n(n+1)(n+2) must be a multiple of 6

Answer 2 Looking at the set of counting numbers, you should notice that

Every other number (2, 4, 6, 8, ...) is even, and is therefore divisible by 2.

Every third number (3, 6, 9, 12, ...) is a multiple of 3, and is therefore divisible by 3.

Since every combination of three consecutive numbers n(n+1)(n+2) will include an even number which is divisible by 2 and a multiple of three which is divisible by 3 then the product will be divisible by 2 and by 3 and therefore the product will be divisible by 6.

Question 3
Find the angle sum of a decagon

Answer 3
A decagon can be divided into 8 triangles as shown

Angle sum of decagon = 8 x 180^{o} = 1440^{o}

Question 4 The mean of a set of 100 numbers is zero. All but two of the numbers are 1. What is the mean of the other two numbers?

Answer 4
The problem can look quite daunting at first but we can reason that the total of the 100 numbers must be zero.

If all but two of the numbers are 1, then 98 of them must be one and the sum of these 98 numbers is 98.

To give a total of zero then the remaining two numbers must total −98.

If two numbers total −98 then their mean must be −98 ÷ 2 = −49

Taking this mathematics further

Learners are likely to be familiar with Sudoku puzzles. Encourage them to discuss the mathematics of these puzzles. Many will think that the numbers make it mathematics, but you could do a Sudoku puzzle using pictures of nine different fruits for example. It is the reasoning that makes it a mathematical puzzle − not the numbers.

The 'Bridges of Königsberg' is a well−known puzzle, the solution of which had eluded the residents of Königsberg for many years. The town of Königsberg, in Prussia, was split into four regions by seven bridges as shown in the diagram here:

The challenge sounded simple: to find a route which crossed each of the bridges once and only once. In 1736 the great mathematician Leonhard Euler showed that there was no solution by a brilliant piece of logical reasoning. Challenge learners to research the solution to this problem, and also to find other examples of reasoning in action.

Making connections

Reasoning is an important feature of mathematics. A progression map for reasoning includes the following hierarchy:

Explain why an answer is correct.

Understand a general statement by finding particular examples that match it.

Try out ideas to find a pattern or solution.

Make general statements, based on evidence produced, and explain reasoning.

Solve problems and investigate in a range of contexts, explaining and justifying methods and conclusions; begin to generalise and to understand the significance of a counter−example.

Draw simple conclusions and explain reasoning; suggest extensions to problems; conjecture and generalise.

Use logical argument to establish the truth of a statement; begin to give mathematical justifications and test by checking particular cases.

Present a concise reasoned argument, using symbols, diagrams, graphs and related explanatory texts.

Show some insight into mathematical structure by using pattern and symmetry to justify generalisations, arguments or solutions.

Appreciate the difference between mathematical explanation and experimental evidence.

investigate a problem systematically?(show/hide all)

What this might look like in the classroom

Question Three whole numbers, greater than zero, can be used to form a trio.

For example (1, 2, 2) is a trio whose sum is 1 + 2 + 2 = 5
and (2, 1, 2) is a different trio whose sum is also 5

How many different trios are there who sum is 5
How many different trios are there who sum is 6

Investigate further

Answer
To find the number of trios whose sum is 5 it is important to investigate systematically to ensure that none of the answers are missed or repeated

Trios for 5
113
122
131
212
221
311
There are six trios whose sum is 5

Trios for 6
114
123
132
141
213
222
231
312
321
411
There are ten trios whose sum is 6

Trios for 7
115
124
133
142
151
214
223
232
241
313
322
331
412
421
511
There are fifteen trios whose sum is 7

By now, the importance of being systematic will be obvious.

Continuing you will find that:

There are twenty one trios whose sum is 8
There are twenty eight trios whose sum is 9

You should now be able to generalise the number of trios for different numbers

Taking this mathematics further

Making an orderly list is one of George Polya’s problem solving strategies. Some of his other strategies appear in other sections of this audit. Find out more about Polya’s work in this field and its importance.

The ice−cream scoops problem is an example of a problem involving permutations and combinations. Rather than listing all the possible outcomes, use of the mathematics will give you the answer more quickly. The same can also be true of problems where a geometric or algebraic solution leads to the answer quickly.

Here is a problem you might like to try using proof by exhaustion and some algebra:
Prove that every integer that is a perfect cube is either a multiple of 9, or 1 more, or 1 less than a multiple of 9.

Making connections

Other examples of tasks where it is important to be systematic include statistics (e.g. collecting samples) and probability (identifying outcomes).

Sample spaces and tree diagrams are useful tools to ensure that information is collated systematically and all possibilities are considered.

distinguish between verification and proof?(show/hide all)

What this might look like in the classroom

Question Prove that the sum of the angles of a triangle add up to 180^{o}

Answer
The most common response to this question in the mathematics classroom is use a protractor to measure the angles or else confirm that the three angles when placed together form a straight line and that the angles on a straight line add up to 180^{o}.

However, these two methods are verification (or demonstration) and are insufficient to be called a proof. The following examples offer two proofs of the fact that the angles of a triangle add up to 180^{o.
}

Proof 1 (Euclid)

Let ABC be a triangle, and let one side of it BC be produced to D. Draw CE parallel to AB

ACD = BAC + ABC (exterior angle = sum of opposite interior angles)

ACD + ACB = 180^{o} (angles on a straight line)
BAC + ABC + ACB = 180^{o} (as ACD = BAC + ABC)
So the sum of the angles of a triangle add up to 180^{o} Proof 2 (The Pythagoreans)

The Pythagorean proof is even simpler.

A line parallel to the base BC is drawn through the vertex A creating two sets of alternate angles (between parallel lines)

CBA = DAB (alternate angles between parallel lines)
BCA = EAC (alternate angles between parallel lines)

DAB + BAC + EAC = 180^{o} (angles on a straight line)
CBA + BAC + BCA = 180^{o} So the sum of the angles of a triangle add up to 180^{o}

Taking this mathematics further

Research some of the different proofs of Pythagoras’ Theorem. Pythagoras’ Theorem is actually an interesting case as the fact itself was known long before the time of Pythagoras (by the Babylonians who lived 1500 years earlier). While Pythagoras was the first person to prove that this fact was true for all right−angled triangles (and hence had theorem was named after him) it was arguably the very first mathematical proof.

Given a right−angled triangle with integer side−lengths, inscribe a circle within it, prove that the diameter is also an integer. Is the radius always an integer?

Making connections

Proofs can take a variety of forms including

Direct proof
Proof by induction
Proof by contradiction
Proof by exhaustion
Visual proofs
Computer−assisted proofs etc

Visual proofs can be useful to demonstrate algebraic relationships. For example, the following visual proofs are offered for (a + b)c = ac + bc and (a + b)^{2} = a^{2} + 2ab + b^{2
}

state constraints and assumptions when deducing results?(show/hide all)

What this might look like in the classroom

Question
The quadratic equation

ax^{2 }+ bx + c = 0
has solutions
x =

-b±√b^{2}-4ac

2a

Write down one assumption about this equation

Answer The assumption is that a ≠ 0
If a = 0 then x = −c ÷ b but the formula will give an error as you are attempting to divide by 0

Furthermore, the number of real roots is constrained by the value of b^{2} − 4ac.

If b^{2} − 4ac > 0 then the equation will give two real roots
If b^{2} − 4ac = 0 then the equation will give one real root
If b^{2} − 4ac < 0 then the equation will give no real roots

Taking this mathematics further

The website offers some interesting statistics statements and asks whether they are always, sometimes or never true.

Making connections

An assumption is something which is taken for granted. Assumptions can be unjustified or justified, depending upon the circumstances.

A lot of mathematics makes use of assumptions. For example, in statistics there is an assumption that the sample is representative of the population.

In mechanics, assumptions are often made about fiction or resistance (which are often taken to be negligible)

Students select a word problem from a GCSE examination paper which has at least 3 marks allocated to its solution.
Ask them to write a plan of how they would go about answering the question. There is no need to actually work it out.

How many steps have been identified?

Taking this mathematics further

Look at the steps needed to achieve a solution to in the example.

Example 1
Bob buys a computer priced at £780 by paying a 20% deposit followed by 24 equal monthly instalments. How much does he have to pay each month?
Here, you need to calculate that the deposit is £156, leaving £624 to pay at £26 per month for 24 months.

Consider whether the order of the steps is fixed or whether there is the possibility of achieving a solution with the same operations applied in a different order.
How might this link with work on function machines?

Making connections

Multi step word problems (money) appear in the curriculum at lower KS2, so this will not be a new experience for your students. What will vary will be the number of steps and the mathematical skills required to reach a solution.

Related information and resources

from the NCETMfrom other sites

Problem-solving is considered to be one of the higher order cognitive skills and this is in part why mathematics is a core subject. In Maslow's hierarchy of needs, problem solving appears in the top tier (self-actualisation).
Research how and why the Polish mathematician George Polya defined a structure for approaching problem-solving in mathematics.

Have a look at finding the inverse of a 2 by 2 matrix.
Multiplying a matrix by its inverse gives you the identity matrix in a similar way to multiplying a number by its multiplicative inverse gives you 1.

Making connections

Working backwards can be both a problem solving strategy and a method of checking a solution. For example, to solve a problem relating to planning a journey you are likely to work backwards from the time you need to be somewhere to work out the time you need to leave. You can then check in reverse by starting the journey at the suggested time and checking what time you would arrive. Reverse percentage problems are also very suited to being solved then checked using the inverse.

Other checking strategies could include seeing whether the answer makes sense in the context of a question and seeing whether the answer is the correct order of magnitude using an approximation.

The notion of inverse operations, and later of inverse functions, is a powerful one in mathematics. Primary aged pupils are used to the concepts of addition and subtraction and of multiplication and division being the inverse of one another. Later, learners will meet operations such as reciprocal which are self−inversing. Some operations have limits on the applicability of the inverse operation − for example, we cannot divide by zero. Using inverse operations is an essential precursor to learning to solve linear equations both by using function machine to ‘undo’ operations but also to solving by transforming both sides of an equation.

ways in which mathematics is used and applied in different occupations?(show/hide all)

What this might look like in the classroom

Task
Finding non examples.

Ask students to think of an occupation where there is absolutely no mathematics involved.

Encourage them to think about skills such as estimating, visualising shapes etc as well as calculations.

Taking this mathematics further

Find out what Keynesian economists think would be the best approach to solving an economic crisis. Think about the mathematics they might use.

Read the 2004 report by Professor Adrian Smith, ‘Making Mathematics Count’, in particular Section 1, ‘The Importance of Mathematics’. In is all highly relevant reading, but pay particular attention to ‘Mathematics for the Workplace’ within section 1. An extract of it can be found here and the full report is here.

Making connections

Be careful not to force the issue. While it is certainly true that some of the mathematics encountered by learners will be directly useful in some jobs, it is the qualities of a mathematician that are valued so highly in the workplace. This includes logical thinking, problem−solving and reasoning for example − and these are developed through the entire mathematics curriculum. Share this with learners when the opportunity arises.

Problem:
Choose two newspapers, a tabloid and a broadsheet. Your task is to compare the two.

Comments:
You will first need to decide how you will compare them and perhaps formulate a hypothesis or write a question to answer. For example:
Are tabloids easier to read than broadsheets?
There are more photos and advertisements in a tabloid.
The breakdown of national, international and human interest stories is different in the two papers.
Next you will need to make a plan and then decide how to collect the data. Decisions will need to be made about how to sample data from the two papers − will you count sentence length, word length, picture area, or something else?

For the following stage you will need to choose the most appropriate diagrams and calculations to display your data.
Use these to write about what your data shows. Discuss your findings and link what you say back to the original problem.
The final stage is to evaluate your results and to think about how successful you have been in achieving your aim and how you could have improved your project.

Taking this mathematics further

The data handling cycle is similar to the PCAI model for a statistical investigation: Posing the question, Collecting relevant data, Analysing the data and Interpreting the results.

Statistical investigations provide a particularly rich context for cross−curricular work, perhaps with science, humanities, geography, social science and PE. These can provide learners with a high level of motivation and relevant contexts to work within.

Find some statistical studies on the Internet. How far have they followed the data handling cycle? How convinced are you by the findings? Think of a problem, find some data and carry out a statistical investigation of your own.

Making connections

It is important that statistical work is learnt in meaningful and purposeful contexts. When carrying out a statistical investigation the data handling cycle provides a model for students to learn how to carry out a statistical investigation. It provides students and their teachers with a useful focus for considering each stage of solving a problem in turn. It might also provide a template for writing up a statistical project.

Such investigations help students to see the relevance of statistical ideas outside the classroom, to develop their statistical skills, to see statistics as a set of tools offering a way of interpreting the world around and encouraging critical thinking.

Generally, students spend much of their time when learning statistics on collecting and presenting data. They spend much less time on the other, important, parts of the cycle so using the cycle encourages a focus on all the statistical processes.

some famous mathematical problems and puzzles?(show/hide all)

What this might look like in the classroom

Problem 1
A female bee does not need a male bee to reproduce. An unfertilised egg results in a male bee developing, while a fertilised egg results in a female bee developing. Draw up a family tree for a male bee. What do you notice about the number of bees in each generation?

Solution 1
The number of bees in each generation follows the sequence 1, 1, 2, 3, 5, 8, 13, … This is just one of very many examples of the Fibonacci sequence appearing in nature.

Problem 2

Consider the triangle here, which has a base of 10 units and a height of 12. What is its area?

Now rearrange the pieces into this triangle, which has the same dimensions, but two squares missing in the middle. Does 58 = 60?!

Solution 2
This famous construction was created by Paul Curry in the 1950's. The trick is that neither of the shapes are actually triangles so it is incorrect to use the area of a triangle formula. The true area can be found by finding the area of each of the six pieces individually.

Why do Buses come in Threes? (Rob Eastaway and Jeremy Wyndham)

Making connections

Famous problems and puzzles in mathematics are likely to be discussed in primary school: many learners will have heard of Fibonacci numbers and Pascal's Triangle before KS3 for example. Take the time to research how these fit into the bigger picture of mathematics though.
Fibonacci (real−name Leonardo de Pisa) travelled extensively in Asia, and in the same book that he first published the famous 'rabbit puzzle', he also introduced Arabic Numerals (i.e. 1, 2, 3, 4, …) to the Western world. Until that point in the early 1200's, Roman Numerals were still used.

Pascal's Triangle might be named after the 17th century French mathematician, but only because he used it so much in his work on probability. The pattern itself had been known for centuries, and the earliest known use of it is in China in 1261.

While algebra developed over centuries (and is still developing now in some sense), the word itself comes from the title of a book written in the early 800's: Kitab al−Jabr wa−l−Muqabala (by the Persian mathematician al−Khwarizmi).

The conjecture that the sum of the squares of the shorter sides equals the square of the hypotenuse (in a right−angled triangle) was well−known before the time of Pythagoras. Indeed, the 3, 4, 5 relationship was used to measure right angles accurately. But Pythagoras was the first person to actually prove it was true for all right−angled triangles. In doing so he simultaneously promoted the notion of mathematical proof as important. It is for this reason that the theorem is named after him.

Task
Find out why Egyptian surveyors were called rope stretchers.

Taking this mathematics further

Supposing we considered other (mathematically) similar shapes on each of the sides of a right angled triangle. Would the areas of the two smaller still add up to the area of that on the hypotenuse?

How many different proofs of Pthagoras’ theorem can you find? Which of them are accessible to your learners?

Research the mathematician Pythagoras. Other than his famous theorem, what other contributions did he make (consider music, fractions, proof, …)?

Making connections

Pythagoras’ theorem is an ideal opportunity to explore the ‘Applications and Implications: History and Culture’ part of the new secondary curriculum, due to its significance in terms of the development of proof and the intriguing stories that surround the mathematician Pythagoras.

Pythagoras’ theorem can be seen as a special case of the cosine rule. Can you find out why?

Activity 1
Investigate the ratio of the circumference to the diameter of a variety of circular objects. What do you notice?

Solution 1
If you are measuring accurately, to the nearest millimetre, you will end up with a series of values very close to ?. The values can be averaged to provide an even better approximation for?.

You can use string to measure the circumference of the circles, but a more accurate result can be found by rolling the circle one full turn along a piece of paper and measuring the line

Some tins have a circumference of 220 mm and a diameter of 70 mm (to the nearest mm), which provides a good example of the well−known approximation for π,

22

7

.

Once the result has been established learners can research some of the history of pi, from Babylonian times through to modern supercomputers, and including the contribution made by a Welshman called William Jones.

Challenge 1 Celebrate International Pi Day on March 14th (3.14).

Challenge 2
Hold a pi recital championship.

Taking this mathematics further

Complement your International Pi Day celebrations with an acknowledgement of International Pi Approximation Day on 22nd July (

22

7

). Investigate other approximations of π?.

Pi is probably the best known irrational number. Other examples of note are e, Ø (phi − the golden ratio), and √2. Pi is also a transcendental number − find out what this means.

The phrase ‘trying to square the circle’ has its origins in the fact that it is impossible to construct (using a straight edge and compasses and in a finite number of steps) a square that has exactly the same area as a given circle. Trying to do so was a popular pastime in ancient Greece.

Read 'The Joy of Pi' (David Blatner)

Making connections

Learners will need to be confident in their use of rounding, either to decimal places or significant figures. While more than one trillion decimal places of pi are now known, even physicists and astronomers never need to use more than the first twenty!

An interesting link to probability exists with Buffon’s needle experiment. Find out more about this.

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