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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.

Showing all next steps for the selected topic.Click on a question to show more information.

explain and illustrate the definitions of natural number, integer, directed number, rational number, irrational number, surd and real number?(show/hide all)

What this might look like in the classroom

Question 1

√10 1 3.5 −3 0

1

3

π

From the list write down

a natural number

a negative integer

the largest rational number

the smallest irrational number

Answer 1

1

−3

3.5

√10

Question 2
For each of the numbers tick all of the boxes that apply

Natural number

Integer

Directed number

Rational number

Irrational number

Surd

Real number

2.5

-8

0

√5

π

1

10

0.3^{0
}

10^{10
}

3.142

2

Answer 2

Natural number

Integer

Directed number

Rational number

Irrational number

Surd

Real number

2.5

-8

0

√5

π

1

10

0.3^{o
}

10^{10
}

3.142

2

Taking this mathematics further

Explore the origins of zero.. When was it invented?

Find out about other irrational numbers of note; e.g.

Ø – phi, the golden ratio

e – sometimes known as Euler's number

Explore definitions of other types of number; e.g.

complex

imaginary

transcendental

Making connections

In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction

p

q

, where p and q are integers. The best known irrational numbers are π and √2.

There is a natural progression in the build up of the different types of numbers, and it can be linked to different types of equations that need to be solved (bear in mind that scientific and technological advances have in the past required ever more complicated equations to be solved). Assume that we just have the counting numbers (or natural numbers) as a starting point. Each subsequent equation requires a new type of number to be 'created'.

a + 2 = 17. No problem here. a must be 15.

a + 11 = 5. This is a problem: the smallest number we know is 1. Never mind, let's invent negative numbers ... a = -6. Along with the natural numbers we now have a set of integers

a × 3 = 12 so a = 4. But if a × 6 = 3 we are stuck again. Now we need to invent fractions (or rational numbers, after 'ratio'). Conveniently enough, all the numbers so far can be written as fractions too.

x^{2} = 4 tells us that x = 2 (or x = -2), but x^{2} = 2 is once again pushing beyond our number system. The solution cannot be written as a fraction (rational number), so we will call it an irrational number.

But what about x^{2} = -3? How can that be possible? We'll just have to imagine some more numbers – so let's call them imaginary numbers! The ones that we had before must therefore be the real numbers as we can mark them on a number line.

If the idea of imaginary numbers seems hard to comprehend consider the fact that it wasn't until the time of the Industrial Revolution that mathematicians universally accepted negative numbers. Before then some people had even called them absurd numbers!

explain why sometimes only a fraction, surd or irrational number can express an exact answer?(show/hide all)

What this might look like in the classroom

Question A square has an area of 10 cm^{2 }

Write down the perimeter of the square exactly

Write down the perimeter of the square using an appropriate approximation

Answer

If the area is 10, then the side length of square = √10. Therefore the perimeter of the square = 4√10 cm

4√10 = 12.64911064067351732… An appropriate approximation would be 12 or 12.6 cm

Taking this mathematics further

Work on repeating decimals is a fruitful development of this area with an interesting investigation on recurring decimals to be found here.

Making connections

Accuracy is generally defined as the degree of closeness of a measured or calculated quantity to its actual (true) value.

In mathematics, accuracy will often depend upon the circumstances. For example, in calculating the area of a circle of radius 5 cm, the formula
A = πr^{2} can be used to give an area of π x 25 or 25πcm^{2}.

However, the accuracy of this statement will depend upon the measurement of the radius and how accurate this was ... any number from 4.5 up to 5.5 could be rounded to 5 cm.

Similarly the accuracy will depend upon the value of π which is used π = 3.14 will give a better approximation to the answer than π = 3. The use of

multiple, common multiple and least (lowest) common multiple(show/hide all)

What this might look like in the classroom

Question
Write down the least common multiple (LCM) of 3 and 4

Answer
The multiples of a number are the products of the multiplication tables.

Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, …
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, …

The common multiples of 3 and 4 are 12, 24, 36, …

The least (lowest) common multiple (LCM) is the least (lowest) multiple which is common to all of the given numbers.

The least common multiple (LCM) of 3 and 4 is 12.

Taking this mathematics further

Program a spreadsheet to find the lowest common multiple of any two numbers entered.

Making connections

When adding, subtracting or comparing fractions, it is helpful to find the least common multiple of the denominators (also called the lowest common denominator).

To compare fractions, you write each of them as an equivalent fraction with this denominator.

To find

2

5

+

3

7

you convert each to equivalent fractions where the denominator is 35 (since 35 is the LCM of 5 and 7)

So

2

5

+

3

7

=

14

35

+

15

45

=

29

45

The LCM of two numbers can be found by first finding the prime factor decomposition of each number and then filling in a Venn diagram where each of the two rings contains the all the prime factors of one of the numbers (the intersect containing each prime factor that is common). The LCM is then the product of all the numbers that appear in the diagram.

Related information and resources

from the NCETMfrom other sites

Types of Errors: considers concerns over the use of subject specific language

Factors & multiples: offers a useful game on factors and multiples Multiples: offers some useful resources on multiples LCM: can be used to find the LCM of two or more numbers Worksheets: will allow you to create worksheets Factors: offers some useful resources on factors

factor, common factor and highest common factor?(show/hide all)

What this might look like in the classroom

Question

Write down the highest common factor (HCF) of 8 and 12

Answer
The factors of a number are the natural (counting) numbers which divide exactly into that number (i.e. without a remainder).

Factors of 8 are 1, 2, 4 and 8.
Factors of 12 are 1, 2, 3, 4, 6 and 12.

The common factors of 8 and 12 are 1, 2 and 4.

The highest common factor (HCF) is the highest factor which is common to all of the given numbers.

The highest common factor (HCF) of 8 and 12 is 4.

Taking this mathematics further

Factors are used to define

Perfect numbers

Abundant numbers

Deficient Numbers

Amicable numbers

Research the definitions of these numbers

Research the link between HCF and Celtic knots.

The Greek mathematician, Euclid, invented an algorithm for finding the highest common factor.

Making connections

The ‘highest common factor’ is the same as the ‘greatest common factor’; which is used in other countries.

A knowledge of factors is useful for finding prime factors

Another method for finding the highest common factor is to list the prime factors, then multiply the common prime factors.

For the numbers 8 and 12

Prime factors of 8 are 2 x 2 x 2
Prime factors of 12 are 2 x 2 x 3

Notice that the prime factors of 8 and 12 both have two 2s in common.

To find the highest common factor you multiply the common prime factors i.e. 2 x 2 = 4

The highest common factor (HCF) of 8 and 12 is 4.

The HCF of two numbers can be found by first finding the prime factor decomposition of each number and then filling in a Venn diagram where each of the two rings contains the all the prime factors of one of the numbers (the intersect containing each prime factor that is common). The HCF is then the product of all the numbers that appear in the intersection of the two rings. Try this for the example above.

Factors & multiples: offers a useful game on factors and multiples Factors: offers some useful resources on factors Worksheets: will allow you to create worksheets

Question
Write the number 140 as a product of its prime factors

Answer
The prime factors of a number are found by successively rewriting the number as a product of prime numbers in increasing order (i.e. 2, 3, 5, 7, 11, 13, 17, … etc.).

where each number is written as a product of factors until they are all prime factors.

Taking this mathematics further

Prime factor decomposition is very important to people who try to make (or break) secret codes based on numbers. Find out more by searching "encryption" or "cryptography".

Making connections

A prime factor is a factor which is also a prime number. All natural numbers can be written as a product of prime factors.

33 can be written as 3 × 11, where 3 and 11 are prime factors.
60 can be written as 2 × 2 × 3 × 5, where 2, 3 and 5 are prime factors.

Prime factor decompositions are often written in index form where appropriate; for example, 60 = 2 × 2 × 3 × 5 = 2^{2} × 3 × 5

the number 136 162 cannot be a square number because the last digit (the units digit) is a 2 and no square number ends in 2

Focusing on the last digit (the units digit) only

number

1

2

3

4

5

6

7

8

9

0

square

1

4

9

6

5

6

9

4

1

0

So a square number cannot end in 2, 3, 7 or 8

Taking this mathematics further

Investigate the square numbers on a 100 grid

What do you notice?

The square number 36 can be written as the sum of two prime numbers; 36 = 13 + 23. The square number 49 can be written as the sum of two prime numbers; 49 = 2 + 47

What other square numbers can be written as the sum of two primes? What square numbers cannot be written as the sum of two primes?

The sum of the first n squares is given by
1^{2} + 2^{2} + 3^{2} + …..n^{2} =

1

6

n(n+1)(2n+1). Such numbers are known as tetrahedral numbers.

According to Waring’s problem, every positive integer can be written as the sum of four positive squares or fewer. This upper limit of four squares cannot be reduced because, for example, 7 cannot be written as the sum of fewer than four positive squares; 7 = 2^{2} + 1^{2} + 1^{2} + 1^{2}

Making connections

Square numbers are produced when numbers are multiplied by themselves; e.g. the square of 7 is 7 × 7 = 49, so 49 is a square number.

Using index numbers you can write the square of 7 as 7^{2}
7^{2} ← the power or index
So 7^{2} = 49

The square root of a number such as 64 is the number which when squared equals 64; i.e. 8 or –8 (because 8 × 8 = 64 and –8 × –8 = 64).

The sign ^{2}√ or (more simply) √ is generally used to denote the positive square root; √64 = +8. This can cause confusion as the equation x^{2} = 64 has two solutions, 8 or -8 (this is sometimes written ±8)

Using index numbers you can write the square root of 64 as . So = ±8

Squares are used in many areas of mathematics specifically in terms of area and also for calculating Pythagoras’ Theorem.

cubes and cube roots, and their index notation?(show/hide all)

What this might look like in the classroom

Question 1 Assad is digging a hole to hold a time capsule. He wants the hole to be a cube with volume 5m^{3}. Find the length of the sides of the hole correct to 2 decimal places.

Answer 1
3√5 = 1.71 (3sf), so the lengths of the side will be 1.71m (3sf)

Question 2
Bronwen says that the difference between two consecutive cube numbers is always a prime number.
Find a counterexample 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

The sum of the first n cubes is given by 1^{3} + 2^{3} + 3^{3} + …..n^{3} = where is the nth triangle number squared.

According to Waring’s problem, every positive integer can be written as the sum of nine positive cubes or fewer. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes; 2^{3} = 2^{3} + 2^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3} + 1^{3}.

Making connections

Cube numbers are produced when numbers are multiplied by themselves, then multiplied by themselves again; e.g. the cube of 6 is 6 × 6 × 6 = 216, so 216 is a cube number.

The first 5 cube numbers are 1, 8, 27, 64 and 125.

Using index numbers you can write the cube of 6 as 63
6^{3 ←} the power or index
So 6^{3} = 216

The cube root of a number such as 125 is the number which when cubed equals 125; i.e. 5 (because 5 × 5 × 5 = 125).

The sign ^{3}√ is used to denote the cube root; ^{3}√ 125 = 5

Using index numbers you can write the square root of 125 as . So = 5

Unlike squares, cubes do not have a small number of possibilities for the last digit.
Focusing on the last digit (the units digit) only

number

1

2

3

4

5

6

7

8

9

0

cube

1

8

7

4

5

6

3

2

9

0

The last digit of the cube number can be used to identify the last digit of the cube root (using the table above). Furthermore:

If the number is divisible by 3, its cube has digital root 9;

If it has a remainder of 1 when divided by 3, its cube has digital root 1;

If it has a remainder of 2 when divided by 3, its cube has digital root 8.

Cubes are used in many areas of mathematics specifically in terms of volume

Interesting debates about 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

You can write a^{0} a different way, by choosing two numbers, say 5 and 5 that differ by zero:

how to represent a number in standard form?(show/hide all)

What this might look like in the classroom

Question 1
The mass of the Earth is about 5,973,600,000,000,000,000,000,000 kg.
Write this number in standard form

Answer 1
5.9736 × 10^{24} kg

Question 2
The mass of an electron is about 0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg.
Write this number in standard form.

Answer 2
9.109 382 2×10^{-31} kg.

Question 3
If a = 3 × 10^{3} and b = 6 × 10^{5} calculate
(a) a + b
(b) a × b
(c) a ÷ b
Give you answers in standard form.

Answer 3 (a) a + b = 3 × 10^{3} + 6 × 10^{5}

Since the numbers do not have the same powers
you need to convert the numbers to ordinary form

= 3000 + 600000
= 603000
= 6.03 x 10^{5}

(b) a × b
= 3 × 10^{3} x 6 x 10^{5}
= 18 x 103 x 105
= 18 x 10^{3+5 } Using the powers of indices
= 18 x 10^{8}
= 1.8 x 101x 108
= 1.8 x 10^{1+8}
= 1.8 x 10^{9}

how to calculate using numbers in standard form using real-life examples?(show/hide all)

What this might look like in the classroom

Question 1
The table gives the diameter, in metres, of planets in the solar system. The diameters are given to an accuracy of 3 significant figures.

Planet

Diameter (metres)

Earth

1.28 × 10^{7}

Jupiter

1.43 × 10^{8}

Mars

6.79 × 10^{6}

Mercury

4.88 × 10^{6}

Neptune

4.95 × 10^{7}

Saturn

1.21 × 10^{8}

Uranus

5.11 × 10^{7}

Venus

1.21 × 10^{7}

The diameters are given to an accuracy of 3 significant figures.

Which planet has the smallest diameter?

Which planet has the largest diameter?

Write down the diameters of the following planets in order (smallest to largest) Earth, Saturn, Venus

Write down 5.11 × 10^{7 }in ordinary form

What is the diameter of Mercury in kilometres?

Give your answer in standard form.

Answer 1

Mercury

Jupiter

Venus, Earth, Saturn

51100 000

4880 km (as 1000 m = 1 km)

Question 2

The mass of one atom of hydrogen is 1.67 × 10^{–24} grams.

The mass of one atom of oxygen is 2.66 × 10^{–23} grams.

One molecule of water has two atoms of hydrogen and one atom of oxygen.

Calculate the total mass of one molecule of water. Give your answer in standard form.

Calculate the number of molecules in one gram of water. Give your answer in standard form.

Answer 2

(a) One molecule of water has two atoms of hydrogen and one atom of oxygen

2 × 1.67 × 10^{–24} + 2.66 × 10^{–23}

= 3.34 × 10^{–24} + 2.66 × 10^{–23}

= 3.34 × 10^{–24} + 2.66 × 10 × 10^{–24}

= 3.34 × 10^{–24} + 26.6 × 10^{–24}

= 29.94 × 10^{–24}

= 2.994 × 10 × 10^{–24}

= 2.994 × 10^{–23} g

(b) Number of molecules in one gram

= 1 ÷ (2.994 × 10^{–23})

= 1 ÷ 2.994 × 10^{23}

= 0.334 × 10^{23}

= 3.34 × 10^{22}

Taking this mathematics further

Research examples of large and small numbers that your learners might find interesting; for example:

The mass of a red blood cell is 9 × 10^{-14} kilograms

The human brain contains about 1 × 10^{10} cells (neurons).

The age of the Earth is 1.6 × 10^{17} seconds

The distance to the Andromeda galaxy is 2.1 × 10^{26} metres

Sir Arthur Eddington estimated the number of particles in the Universe at 1 × 10^{79}.

Certain multipliers in a standard form representation have a specific prefix. For example, a multiplier of 10^{-3} has a prefix of 'milli', and a multiplier of 10^{6} has a prefix of 'mega'. Find out about other prefixes; e.g. 'tera', 'giga', 'nano', 'zetta'

Making connections

Standard form is a short way of writing very large and very small numbers.

Standard form numbers are always written as A × 10^{n}, where A lies between 1 and 10 and n is a natural (counting) number.

Standard form is also known as scientific notation or exponential notation.

Scientific notation is useful in allowing comparison between sizes, so it is easier to compare 1.6726 × 10^{−27} and 9.109 382 2 × 10^{-31} than 0.000 000 000 000 000 000 000 000 001 672 6 and 0.000 000 000 000 000 000 000 000 000 000 910 938 22

which fractions produce terminating decimals?(show/hide all)

What this might look like in the classroom

Question
Without using a calculator, which of the following fractions are equivalent to recurring decimals?

1

150

2

150

3

150

4

150

6

150

15

150

Give a reason for your answer

Answer

1

150

,

2

150

, and

4

150

are equivalent to recurring decimals.

The denominator 150 can be written as 2 x 3 x 5 x 5, so numerators of 3, 6 and 15 will ensure that the 3 in the denominator is cancelled, leaving only 2s and 5s, and therefore resulting in terminating decimals.

Taking this mathematics further

The period of the repeating decimal

1

p

, where p is prime, is either p - 1 or a factor of p - 1

Examples of fractions in the first group include:

1

7

= 0.142857 has 6 repeating digits

1

17

= 0.0588235294117647 has 16 repeating digits

1

19

= 0.052631578947368421 has 18 repeating digits

1

23

= 0.0434782608695652173913 has 22 repeating digits

1

29

= 0.0344827586206896551724137931 has 28 repeating digits
The list continues to include

1

47

,

1

59

,

1

61

,

1

97

,

1

109

, etc.

Making connections

Denominators containing only 2 or 5 as factors result in a terminating decimal

1

2

= 0.5

1

4

=

1

2 x 2

= 0.25

1

5

= 0.2

1

8

=

1

2 x 2 x 2

= 0.125

1

10

=

1

2 x 5

= 0.1

1

25

=

1

5 x 5

= 0.04 etc

Denominators containing only 3, 7, 11 or higher prime numbers as factors, and no 2s or 5s, result in recurring decimals.

1

3

= 0.3

1

7

= 0.142857

1

9

= 0.1

1

11

= 0.0909

1

13

= 0.076923

1

17

= 0.0588235294117647

Denominators containing 3, 7, 11 or higher prime numbers as factors and 2s or 5s result in recurring decimals with static and recurring parts

how to convert a recurring decimal into a rational fraction?(show/hide all)

What this might look like in the classroom

Question 1 Convert 0.7 to a fraction
Answer 1
Let F be the fraction that we are looking for. Then:

10 x F = 7.77777777... (multiply both sides by 10)
and
1 x F = 0.77777777...

Now, (10 x F) − (1 x F) gives 9 x F = 7

So, 1 x F =

7

9

(divide both sides by 9)

Therefore, 0.7 =

7

9

Question 2
Convert 103.45 to a mixed fraction

Answer 2
Let A be the fraction that we are looking for. Then:

100 x A = 10345.45454545... (multiply both sides by 100)
and 1 x A = 103.45454545...

Now, (100 x A) − (1 x A) gives 99 x A = 10242

So, 1 x A =

10242

99

(divide both sides by 99)
A =

1138

11

(cancelling down)
A = 103

8

11

Therefore, 103.45 (45 recurring) = 103

8

11

Taking this mathematics further

Program a spreadsheet to convert a given recurring decimal to a fraction in each of the following cases:

0.a

0.ab

0.abc

Making connections

For the second question in the previous section, it might be easier to write 103.45 as 103 + 0.45 and convert 0.45 to a fraction before rewriting as a mixed number.

Recurring decimals are all rational numbers as they can all be expressed as fractions in the form

Question
Terry is using his calculator to find the square root of 13. He gives the answer as 3.605 rounded to 3 decimal places
Explain why Terry’s answer is incorrect.

Answer
Terry has truncated the answer to 3 decimal places

To round the answer to 3 decimal places, he needs to consider the fourth decimal place which is 5. So the number is closer to 3.606 than 3.605.

His answer should read 3.606 rounded to 3 decimal places

Taking this mathematics further

Read the following article, which discusses truncation errors arising from computer representations of numbers.

Making connections

Truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.

To truncate this number to 4 decimal places, you consider the first 4 digits to the right of the decimal point

0.2173

To truncate this number to 5 decimal places, you consider the first 5 digits to the right of the decimal point

0.21739

As these answers show, the truncated answer is not always the same as the rounded answer

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