Algebraic Summary

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GCSE Mathematics – Basics of number

SLIDE NUMBER 1

May 2019

Claire Roberts

Session Objectives

The purpose of the session is to:

Understand notation, vocabulary, positive and negative integers and symbols

Calculate using formal written methods of the four operations including brackets, powers, roots and reciprocals

Recognise and apply positive integer powers and associated real roots (square, cube and higher) powers of 2, 3, 4, 5

Describe estimation of calculations and apply the concept to round numbers and measures

Define factors and multiples and use Prime Factor Decomposition to identify HCF and LCM

Define the product rule for counting (combinations)

Perform a range of calculations using fractions, decimals and percentages

SLIDE NUMBER 2

May 2019

CONSIDER…

At this point you should consider the list of session objectives and ask yourself:

How many of the session objectives am I confident with

Could I explain these objectives in relation to teaching and learning

SLIDE NUMBER 3

May 2019

Session Objectives

The purpose of the session is to:

Understand notation, vocabulary, positive and negative integers and symbols

Calculate using formal written methods of the four operations including brackets, powers, roots and reciprocals

Recognise and apply positive integer powers and associated real roots (square, cube and higher) powers of 2, 3, 4, 5

Describe estimation of calculations and apply the concept to round numbers and measures

Define factors and multiples and use Prime Factor Decomposition to identify HCF and LCM

Define the product rule for counting (combinations)

Perform a range of calculations using fractions, decimals and percentages

SLIDE NUMBER 4

May 2019

Key vocabulary

In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting

are cardinal numbers and words used for ordering are ordinal numbers.

The symbol for the set of all natural numbers is shown

An integer is a number that can be written without a fractional component. The set of integers consists of the positive and negative natural numbers and zero.

21, 4, 0, and −2048 are integers, but 9.75, 5 ¹⁄₂, and √2 are not.

The symbol for the set of all integers is shown

(originating from the German word zahlen)

Basics of number

SLIDE NUMBER 5

May 2019

Key vocabulary

The type of number we normally use, such as 1, 15.82, −0.1, 3/4, etc, are called real numbers. Positive or negative, large or small, whole numbers or decimal numbers are all real numbers, so called because they are not imaginary numbers.

The symbol for the set of all real numbers is shown

An imaginary number is one that when squared gives a negative result. When we square a real number we always get a positive, or zero, result. For example, 2×2=4, and (-2)×(-2)=4 as well. So how can we square a number and get a negative result? Because we "imagine" that we can. The "unit" for imaginary numbers (the same as "1" for real numbers) is √(-1), and its symbol is i or j.

Basics of number

SLIDE NUMBER 6

May 2019

Key vocabulary

A rational number is any number that can be expressed as the quotient or fraction of two integers, with a numerator and a non-zero denominator .

Since may be equal to 1, every integer is a rational number.

The set of all rational numbers is shown

(originating from the Italian word quoziente)

Irrational numbers are all the real numbers which are not rational numbers.

The most famous irrational number is , sometimes called Pythagoras’ constant. Other examples include , , , etc

Basics of number

SLIDE NUMBER 7

May 2019

Positive and Negative Numbers

In mathematics, direction from zero is indicated by use of positive and negative signs.

Basics of number

SLIDE NUMBER 8

May 2019

-4 -3 -2 -1 0 1 2 3 4

Negative numbers

Positive numbers

Origin

To denote that a number is negative we use a minus sign in front of

the number, and, on occasion, a plus sign is used to denote a positive number but this is commonly assumed.

-9 denotes “negative nine”; +9 or 9 denotes “positive nine”

Greater than, less than, equal to

In mathematics, arrows are used to denote whether a number is greater than, less than or equal to another number.

6 > 3 means that “6 is greater than 3”

-6 < 3 means that “negative 6 is less than 3”

a < b means that “a is less than or equal to b”

b > a means that “b is greater than or equal to a”

It is important to remember that

“ = ” means “equal to”

and not “here’s my next step” or “ the answer is”

Basics of number

SLIDE NUMBER 9

May 2019

Review of main ideas from above:

We will use real numbers, rational and irrational numbers but imaginary numbers are only introduced in the Further Mathematics A-level specification, but here’s something to consider

Consider this incorrect use of the = symbol

(3 + 4) x 6 + 7 = 7 = 42 = 49

So now pause the recording to consider, once you have done this play the recording.

SLIDE NUMBER 10

May 2019

CONSIDER…

Review of main ideas from above:

Could be interpreted mathematically as

“i eight all of pi” or “I ate all of the pie”

(3 + 4) x 6 + 7 = 7 = 42 = 49

The beginning and end of this are true, (3 + 4) x 6 + 7 = 49

But, 7≠ 42 ≠ 49 We will look at the correct order of operations in the next part

SLIDE NUMBER 11

May 2019

CONSIDER…

Session Objectives

The purpose of the session is to:

Understand notation, vocabulary, positive and negative integers and symbols

Calculate using formal written methods of the four operations including brackets, powers, roots and reciprocals

Recognise and apply positive integer powers and associated real roots (square, cube and higher) powers of 2, 3, 4, 5

Describe estimation of calculations and apply the concept to round numbers and measures

Define factors and multiples and use Prime Factor Decomposition to identify HCF and LCM

Define the product rule for counting (combinations)

Perform a range of calculations using fractions, decimals and percentages

SLIDE NUMBER 12

May 2019

Order of operations

Calculations should be completed in a specific order remembered by the acronym BODMAS

Basics of number

SLIDE NUMBER 13

May 2019

rackets

rder (power)

ivision

ultiplication

ddition

ubtraction

These operations are interchangeable

BODMAS

First, let’s consider the calculation from earlier

(3 + 4) x 6 + 7

Basics of number

SLIDE NUMBER 14

May 2019

= 7 x 6 + 7

= 42 + 7

= 49

If we incorporate some more elements of BODMAS, 13 + 12 ÷ (9 - 7)2

= 13 + 12 ÷ (2)2

= 13 + 12 ÷ 4

= 16

= 13 + 3

Columnar Addition

If we need to evaluate 2746 + 578, we use the columnar addition method

Basics of number

SLIDE NUMBER 15

May 2019

2746

578

It is important to ensure that the digits are kept in clear place value columns and the addition sign is included.

Answer: 3324

Columnar Subtraction

If we need to evaluate 2746 - 578, we use the columnar subtraction method

Basics of number

SLIDE NUMBER 16

May 2019

2746

578

It is important to ensure that the digits are kept in clear place value columns and the subtraction sign is included.

Answer: 2168

Long Multiplication

If we need to evaluate 274 x 57, we use the columnar multiplication method

Basics of number

SLIDE NUMBER 17

May 2019

274

Answer: 15618

It is important to ensure that the digits are kept in clear place value columns, the multiplication sign is included and you appreciate that in the second line you are multiplying by a multiple of 10, hence the zero.

Short division

If we need to evaluate 284 ÷ 5, we use the short division method

Basics of number

SLIDE NUMBER 18

May 2019

284

If you have a remainder at the end you can introduce a decimal point and as many zeros as are needed. Alternatively you could give the answer as a mixed number by putting the remainder as the numerator and the divisor as the denominator,

Answer: 56.8

i.e.

Long division

If we need to evaluate 585 ÷ 15, we use the long division method

Basics of number

SLIDE NUMBER 19

May 2019

585

In long division, the remainder is found and shown underneath as shown

Answer: 39

135

Review of main ideas from above:

Consider these questions:

17 + (3 x 6) ÷ 32

672 + 384

965 – 739

361 x 39

391 ÷ 17

So now pause the recording to try these calculations , once you have done this play the recording

SLIDE NUMBER 20

May 2019

CONSIDER…

Review of main ideas from above:

Consider these questions:

17 + (3 x 6) ÷ 32 19

672 + 384 1056

965 – 739 226

361 x 39 14079

391 ÷ 17 23

SLIDE NUMBER 21

May 2019

CONSIDER…

Session Objectives

The purpose of the session is to:

Understand notation, vocabulary, positive and negative integers and symbols

Calculate using formal written methods of the four operations including brackets, powers, roots and reciprocals

Recognise and apply positive integer powers and associated real roots (square, cube and higher) powers of 2, 3, 4, 5

Describe estimation of calculations and apply the concept to round numbers and measures

Define factors and multiples and use Prime Factor Decomposition to identify HCF and LCM

Define the product rule for counting (combinations)

Perform a range of calculations using fractions, decimals and percentages

SLIDE NUMBER 22

May 2019

Powers

Powers are used when a number is multiplied by itself a number of times.

For example,

3 x 3 x 3 x 3 = 34 = 81

They can also be used for reciprocals and roots.

and

The rules of indices (or powers) will be covered in more detail in Session 2 – Indices, roots and surds

Basics of number

SLIDE NUMBER 23

May 2019

Common Powers and Roots

Basics of number

SLIDE NUMBER 24

May 2019

Review of main ideas from above:

Powers are very powerful in mathematics

Modern scientific calculators can find any power of any number – do you know how to do this on yours? https://www.youtube.com/results?search_query=how+to+use+a+scientific+calculator+for+powers

So now pause the recording to consider the use of your calculator, once you have done this play the recording

SLIDE NUMBER 25

May 2019

CONSIDER…

Session Objectives

The purpose of the session is to:

Understand notation, vocabulary, positive and negative integers and symbols

Calculate using formal written methods of the four operations including brackets, powers, roots and reciprocals

Recognise and apply positive integer powers and associated real roots (square, cube and higher) powers of 2, 3, 4, 5

Describe estimation of calculations and apply the concept to round numbers and measures

Define factors and multiples and use Prime Factor Decomposition to identify HCF and LCM

Define the product rule for counting (combinations)

Perform a range of calculations using fractions, decimals and percentages

SLIDE NUMBER 26

May 2019

Rounding

Numbers can be rounded to a given number of decimal places or to a given number of significant figures.

The basic rules are the same for both:

Find the digit that will be the new final digit

Look at the digit that follows it

If it is less than 5, the final digit remains unchanged

If it is 5 or more, then the final digit must be increased by one. If the final digit is 9, then the number before is increased by one, and so on.

See the next slide for examples

Basics of number

SLIDE NUMBER 27

May 2019

Rounding to a given number of decimal places

Rounding 34.6456 to one decimal place (1dp) will give 34.6

Rounding 67.6563 to 2dp will give 67.66

Rounding 8.0998 to 3dp will give 8.100

Note on example 3 that, despite the zeros having no value, they are required to meet the degree of accuracy

Basics of number

SLIDE NUMBER 28

May 2019

Rounding to a given number of significant figures

Rounding 34.6456 to one significant figure (1sf) will give 30

Rounding 67.6563 to 2sf will give 68

Rounding 8.0998 to 3sf will give 8.10

Rounding 0.030456 to 3sf will give 0.0305

Notes

Example 1 – the zero is included as a place value holder

Example 3 – the zero is include to meet the degree of accuracy

Example 4 – the first significant figure is the 3, preceding zeros are not counted but subsequent zeros are

Basics of number

SLIDE NUMBER 29

May 2019

Estimation

It is useful to be able to estimate the value of a calculation, even if you have access to a calculator.

Consider the following, quite complex calculation,

Basics of number

SLIDE NUMBER 30

May 2019

The exact answer is 20.2 (3sf)

If you were to accidentally type 193 instead of 19.3,

the calculator would give the answer 2020 (3sf) which you would know was wrong if you had estimated first

Review of main ideas from above:

Round the following numbers to the degree of accuracy given in brackets:

6.7083 (2dp)

3091.87 (3sf)

0.004001 (2sf)

For the following calculation, first find an estimate then use your calculator to find the exact answer to 3sf

So now pause the recording to consider the questions, once you have done this play the recording

SLIDE NUMBER 31

May 2019

CONSIDER…

Review of main ideas from above:

6.7083 (2dp) = 6.71

3091.87 (3sf) = 3090

0.004001 (2sf) = 0.0040

Exact answer = 1.65 (3sf)

SLIDE NUMBER 32

May 2019

CONSIDER…

Session Objectives

The purpose of the session is to:

Understand notation, vocabulary, positive and negative integers and symbols

Calculate using formal written methods of the four operations including brackets, powers, roots and reciprocals

Recognise and apply positive integer powers and associated real roots (square, cube and higher) powers of 2, 3, 4, 5

Describe estimation of calculations and apply the concept to round numbers and measures

Define factors and multiples and use Prime Factor Decomposition to identify HCF and LCM

Define the product rule for counting (combinations)

Perform a range of calculations using fractions, decimals and percentages

SLIDE NUMBER 33

May 2019

Factors and Multiples

Factors are numbers which can be multiplied together to give the original number. They are usually found in pairs, except in the case of square numbers which always have an odd number of factors.

For example,

Factors of 24 are 1 and 24, 2 and 12, 3 and 8, 4 and 6

Factors of 16 are 1 and 16, 2 and 8, 4 (and 4)

Multiples are numbers that are generated by multiplying by integers. Basically they are the numbers found in the times tables.

For example,

The first 10 multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18 and 20

The first 5 multiples of 100 are 100, 200, 300, 400 and 500

Basics of number

SLIDE NUMBER 34

May 2019

Prime numbers

Prime numbers are special numbers which have exactly two factors – itself and one.

A common misconception is that 1 is a prime number, but it only has one factor.

The Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to any given limit.

It is completed by marking the multiples

of each prime, starting with the

first prime number, 2, on a number grid.

The numbers left unmarked are

the prime numbers.

Basics of number

SLIDE NUMBER 35

May 2019

Prime Factor Decomposition

Any integer can be written as a product of its prime factors which can be found by Prime Factor Decomposition

Let’s write 420 as a product of its prime factors,

Basics of number

SLIDE NUMBER 36

May 2019

420

210

105

Answer: 2 x 2 x 3 x 5 x 7

or 22 x 3 x 5 x 7

Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

If we need to find the HCF or LCM of two or more integers we can use prime factor decomposition.

Let’s find the HCF and LCM of 420 and 66

Basics of number

SLIDE NUMBER 37

May 2019

Answer: 2 x 3 x 11

If we now write both numbers as their products one above the other,

420 = 2 x 2 x 3 x 5 x 7

66 = 2 x 3 x 11

HCF = 2 x 3 = 6

LCM = 2 x 2 x 3 x 5 x 7 x 11 = 4620

Review of main ideas from above:

This youtube video is a quirky explanation of the Sieve of Eratosthenes https://www.youtube.com/watch?v=V08g_lkKj6Q

Find the HCF and LCM of these pairs of numbers using prime factor decomposition:

1. 24 and 60

2. 48 and 72

So now pause the recording to watch the youtube video and to try the technique, once you have done this play the recording

SLIDE NUMBER 38

May 2019

CONSIDER…

Review of main ideas from above:

Find the HCF and LCM of these pairs of numbers using prime factor decomposition:

1. 24 and 60

24 = 2 x 2 x 2 x 3

60 = 2 x 2 x 3 x 5

HCF = 2 x 2 x 3 = 12; LCM = 2 x 2 x 2 x 3 x 5 = 120

2. 48 and 72

48 = 2 x 2 x 2 x 2 x 3

72 = 2 x 2 x 2 x 3 x 3

HCF = 2 x 2 x 2 x 3 = 24; LCM = 2 x 2 x 2 x 2 x 3 x 3 = 144

SLIDE NUMBER 39

May 2019

CONSIDER…

Session Objectives

The purpose of the session is to:

Understand notation, vocabulary, positive and negative integers and symbols

Calculate using formal written methods of the four operations including brackets, powers, roots and reciprocals

Recognise and apply positive integer powers and associated real roots (square, cube and higher) powers of 2, 3, 4, 5

Describe estimation of calculations and apply the concept to round numbers and measures

Define factors and multiples and use Prime Factor Decomposition to identify HCF and LCM

Define the product rule for counting (combinations)

Perform a range of calculations using fractions, decimals and percentages

SLIDE NUMBER 40

May 2019

The Product Rule for Counting

To find the total number of outcomes for two or more events, we can use the product rule for counting in which we multiply the number of outcomes for each event together.

For example, if we wanted to know the total number of combinations on a menu with 5 starters, 3 main courses and 4 desserts we could use the product rule,

5 x 3 x 4 = 60 different meal combinations

Basics of number

SLIDE NUMBER 41

May 2019

This will be explained again in more detail in Session 18 – Probability

Session Objectives

The purpose of the session is to:

Understand notation, vocabulary, positive and negative integers and symbols

Calculate using formal written methods of the four operations including brackets, powers, roots and reciprocals

Recognise and apply positive integer powers and associated real roots (square, cube and higher) powers of 2, 3, 4, 5

Describe estimation of calculations and apply the concept to round numbers and measures

Define factors and multiples and use Prime Factor Decomposition to identify HCF and LCM

Define the product rule for counting (combinations)

Perform a range of calculations using fractions, decimals and percentages

SLIDE NUMBER 42

May 2019

Simplifying Fractions

To simplify a fraction you need to find the HCF of both the numerator and denominator and divide both by this factor. Once a fraction has been simplified in this way it is said to be in its lowest or simplest terms.

For example, to simplify ,we find that the HCF of the numerator and denominator is 9, so dividing both by 9 will give

Basics of number

SLIDE NUMBER 43

May 2019

Adding Fractions

To add a pair of fractions with different denominators you need to:

Find the LCM of both the two denominators

Multiply the numerator by the same factor as the denominator would be multiplied by to make the LCM; do this to both fractions

Add the numerators of the two fractions, leaving the denominator unchanged as the LCM

Simplify the new fraction if possible, or convert to a mixed number if the fraction is improper.

For example,

Basics of number

SLIDE NUMBER 44

May 2019

Subtracting Fractions

To subtract a fraction from another with a different denominator you need to:

Find the LCM of both the two denominators

Multiply the numerator by the same factor as the denominator would be multiplied by to make the LCM; do this to both fractions

Subtract the second numerator from the first, leaving the denominator unchanged as the LCM

Simplify the new fraction if possible, or convert to a mixed number if the fraction is improper.

For example,

Basics of number

SLIDE NUMBER 45

May 2019

Adding or Subtracting Mixed Numbers

Convert the mixed numbers to improper fractions and then follow the process as previously described.

For example,

Basics of number

SLIDE NUMBER 46

May 2019

The same method is used when subtracting mixed numbers

Multiplying Fractions and Mixed Numbers

To multiply a pair of fractions you need to:

Multiply the two numerators to give the new numerator

Multiply the two denominators to give the new denominator

Simplify the new fraction if possible, or convert to a mixed number if the fraction is improper.

For example,

If you need to multiply a pair of mixed numbers you need to first convert to improper fractions, then follow the method above.

Basics of number

SLIDE NUMBER 47

May 2019

Dividing Fractions and Mixed Numbers

To divide a pair of fractions you need to:

Convert the fractions so that they have common denominators

Divide the first numerator by the second numerator

Simplify the new fraction if possible, or convert to a mixed number if the fraction is improper.

For example,

If you need to divide a pair of mixed numbers you need to first convert to improper fractions, then follow the method above.

Basics of number

SLIDE NUMBER 48

May 2019

Converting Fractions to Decimals,

and vice versa

A fraction is another way of representing a division,

so means 3 ÷ 4.

Hence, to convert a fraction to a decimal, simply divide the numerator by the denominator, meaning that

If you need to write a decimal as a fraction, you need to consider the meaning of the decimal. For instance, 0.45 means 4 tenths and 5 hundredths, or 45 hundredths,

therefore

Basics of number

SLIDE NUMBER 49

May 2019

Writing Recurring Decimals as Fractions

A recurring decimal is one that is non-terminating (unlike 0.45) and has a sequence of digits that repeat infinitely.

Recurring decimals are indicated using small dots over the start and finish of the repeating sequence, for example,

To convert to a fraction,

Let

and

Hence

Basics of number

SLIDE NUMBER 50

May 2019

Converting Percentages to Fractions to Decimals, and vice versa

A percentage is simply a proportion given as an amount out of 100

Hence, to convert a percentage to a fraction, simply write the percentage value as the numerator with 100 as the denominator, and then to write as a decimal simply divide.

For example, and

If you need to write a decimal as a percentage, firstly consider how many hundredths it is then write as a fraction and then a percentage.

For example, and

Basics of number

SLIDE NUMBER 51

May 2019

Converting Percentages to Fractions, and vice versa

Write the percentage value as the numerator with 100 as the denominator, and then simplify if possible.

For example, and

If you need to write a fraction as a percentage,

divide the numerator by the denominator to give the fraction as a decimal

then convert the decimal as previously discussed

For example,

and

Basics of number

SLIDE NUMBER 52

May 2019

Finding Fractions and Percentages of amounts

To find a fraction of an amount ,divide by the denominator then multiply the result by the numerator.

For example, to find of 64,

first find of 64 = 64 ÷ 8 = 8, then multiply by 7, 8 x 7 = 56

Hence of 64 = 56

If you need to find a percentage of an amount, firstly consider the percentage as a fraction, then follow the same method

For example, to find 6% of £500, ( of 500)

first find of 500 = 500 ÷ 100 = 5, then multiply by 6, 5 x 6 = 30

Hence 6% of £500 = £30

Basics of number

SLIDE NUMBER 53

May 2019

Review of main ideas from above:

Have a go at the following questions:

1. 2. 3. 4.

5. Convert 0.76 to a fraction in its simplest terms

6. Convert to a percentage

7. Write the recurring decimal as a fraction

8. Find 4% of £250

So now pause the recording to consider the questions, once you have done this play the recording

SLIDE NUMBER 54

May 2019

CONSIDER…

Review of main ideas from above:

Have a go at the following questions:

1. 2.

3. 4.

5. Convert 0.76 to a fraction in its simplest terms

6. Convert to a percentage = 62.5%

7. Write the recurring decimal as a fraction

8. Find 4% of £250 = £10

SLIDE NUMBER 55

May 2019

CONSIDER…

Session Objectives

The purpose of the session is to:

Understand notation, vocabulary, positive and negative integers and symbols

Calculate using formal written methods of the four operations including brackets, powers, roots and reciprocals

Recognise and apply positive integer powers and associated real roots (square, cube and higher) powers of 2, 3, 4, 5

Describe estimation of calculations and apply the concept to round numbers and measures

Define factors and multiples and use Prime Factor Decomposition to identify HCF and LCM

Define the product rule for counting (combinations)

Perform a range of calculations using fractions, decimals and percentages

SLIDE NUMBER 56

May 2019

SLIDE NUMBER 57

May 2019

CONSIDER…

End of Presentation

At this point it would be advisable to go back over the presentation. Ensure that you are fully able to deal accurately and effectively with each session objective.

You should supplement the content of this session with suitable reading, research and discussion with others.

Claire Roberts

SLIDE NUMBER 58

May 2019

GCSE Mathematics – Basics of number

End of presentation