Download 1.1 Understanding prime factors, LCM and HCF

Chapter 1 Number
Specification
GCSE 2010
N c Use the concepts and vocabulary of
factor (divisor), multiple, common factor,
Highest Common Factor (HCF), Least
Common Multiple (LCM), prime number
and prime factor decomposition
FS Process skills
Select the mathematical information
to use
1.1 Understanding prime factors, LCM
and HCF
Concepts and skills
•
•
•
•
Identify factros, multiples and prime numbers from a list of numbers.
Find the prime factor decomposition of positive integers.
Find common factors and common mulitples of two or three numbers.
Find the HCF and LCM of two or three numbers.
Functional skills
•
FS Performance
Level 1 Select mathematics in an
organised way to find solutions
L1 … multiply and divide whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
Students should already know their
• multiplication tables up to 10 × 10.
• be able to find factors, multiples and prime numbers (NC).
Starter
Resources
•
Links
http://www.bbc.co.uk/education/
mathsfile/shockwave/games/gridgame.
html
•
ActiveTeach resources
Multiples and factors quiz
Ladder method interactive
HCF and LCM interactive
•
Check that students understand the terms prime number, factor and multiple.
List the factors of 12. (1, 2, 3, 4, 6, 12) List the multiples of 6 between 10 and 40. (12, 18,
24, 30, 36) List the first ten prime numbers. (2, 3, 5, 7, 11, 13, 17, 19, 23, 29)
Introduce the word ‘common’ into some questions.
Find two common factors of 12 and 18. (1, 2, 3, 6) Find two common multiples of 3 and 4.
(12, 24, 36 etc)
Main teaching and learning
•
•
•
•
•
•
•
Tell students that they are going to find out how to write any positive whole number as
a product of its prime factors. Check that students understand the meaning of the word
product.
Explain that this can be done by using a factor tree (or repeated division). Draw a factor
tree to show how 120 can be broken down into its prime factors (see Example 2).
Discuss the fact that you can start with any two numbers that multiply to give 120. Draw
a second factor tree for 120 starting with a different factor pair to show that the same
result is reached.
Tell students that they are going to find the HCF and LCM of two numbers.
Explain that there are different methods that can be used to do this depending on the
size of the numbers involved.
Discuss the best method for finding the HCF and LCM for two small numbers (e.g. 4 and
6). Show students how these can be found by making a list of the factors and first few
multiples of 4 and 6.
Discuss why this method would not be appropriate for large numbers (e.g. 240 and 280).
Explain how writing large numbers as the product of prime factors can be used to find
the LCM and HCF.
Common misconceptions
•
Remind students to include the multiplication signs when writing a number as a product
of its prime factors. (These are often incorrectly replaced by addition signs or commas.)
Enrichment
•
•
Suggest that students use the Venn diagram method to find the HCF and LCM of three
large numbers (e.g. 240, 300 and 420).
Students might like to know that the HCF of two numbers must be a factor of the
difference between them. So the HCF of 210 and 250 must be a factor of 40. They may
like to explore this and consider why this is the case.
Plenary
•
•
2
common factor
M01_MSAH_TG_GCSE_0822_C01.indd 2
Ask for the HCF of pairs of small numbers e.g. 2 and 6 (2), 4 and 10 (2), 6 and 12 (6).
Ask for the LCM of pairs of small numbers e.g. 2 and 6 (6), 4 and 10 (20), 6 and 12 (12).
common multiple
factor tree
highest common factor (HCF)
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M01_MSAH_TG_GCSE_0822_C01.indd 3
lowest common multiple
prime factor
3A
7
3
3
42
21
b
9
70
3
10
c
84
21
.....................................................................................................
.....................................................................................................
.....................................................................................................
iii 20 ......................................................................
i 10 ......................................................................
b Write out the first six multiples of
i 4 ......................................................................
a Write out the first ten multiples of
ii 16 ......................................................................
ii 6 ......................................................................
iv 16 and 20 ......................................................................
iii 4 and 20 ......................................................................
v 10 and 16 ......................................................................
ii 6 and 16 ......................................................................
i 4 and 10 ......................................................................
b Use part a to help you write down the highest common factor (HCF) of
Factors are numbers that go
exactly into the given number.
iv 16 ......................................................................
iii 10 ......................................................................
v 20 ......................................................................
ii 6 ......................................................................
i 4 ......................................................................
a Write out all the factors of
Remember to draw factor trees first.
d 72 ......................................................................
c 50 ......................................................................
e 100 ......................................................................
b 40 ......................................................................
a 24 ......................................................................
Write each of the following numbers as the product of its prime factors. How can factor trees be drawn
before working through to find out how many factors there are?
a
Complete the following factor trees.
2
6
54
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C
C
i 42 ............................................................................................................................................................................................................................
...............................................................
...............................................................
...............................................................
d 84 and 96 ...............................................................
...............................................................
...............................................................
...............................................................
b 70 and 105 ...............................................................
c 72 and 96 ...............................................................
...............................................................
...............................................................
a 60 and 84 ...............................................................
Find the HCF and LCM of the following pairs of numbers.
.................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................
c Find the HCF and LCM of 70 and 84.
.................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................
b Find the HCF and LCM of 42 and 70.
iii 84 ...........................................................................................................................................................................................................................
ii 70 ............................................................................................................................................................................................................................
M01A_MSAH_TG_GCSE_0822_CDC01.indd 3
8
iv 16 and 20 ......................................................................
ii 6 and 10 ......................................................................
a Use your answers to question 3 to write 42, 70 and 84 as products of their prime factors.
v 6 and 20 ......................................................................
iii 4 and 16 ......................................................................
d 40 ......................................................................
c 36 ......................................................................
Use this factor tree to write 54 as a product of its prime factors.
i 4 and 6 ......................................................................
D
Multiples are the numbers in the times table of the given number.
3B
C
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Section 1.1 Understanding prime factors, LCM and HCF
c Use your answers to parts a and b to help you write down the lowest common multiple (LCM) of
Guided practice worksheet
b 30 ......................................................................
Write down the
factors in pairs.
a 12 ......................................................................
Write down all the factors of each of the following numbers.
M01A_MSAH_TG_GCSE_0822_CDC01.indd 2
6
5
4
3
2
1
Guided practice worksheet
Section 1.1 Understanding prime factors, LCM and HCF
Section 1.1 Understanding prime factors, LCM and HCF
3
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Chapter 1 Number
Specification
GCSE 2010
N d (part) Use the terms square,
positive… square root, cube and
cube root
N e (part) Use index notation for
squares, cubes …
FS Process skills
Use appropriate mathematical
procedures
FS Performance
Level 1 Use appropriate checking
procedures at each stage
1.2 Understanding squares and cubes
Concepts and skills
•
•
•
Recall integer squares from 2 × 2 up to 15 × 15 and the corresponding square roots.
Recall the cubes of 2, 3, 4, 5 and 10.
Use index notation for squares and cubes.
Functional skills
•
L1 … multiply … whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
•
Students should already know how to multiply and divide positive and negative
integers.
Starter
•
Ask students to work out the value of 1 × 1, 2 × 2, 3 × 3 up to 10 × 10 (1, 4, 9, 16, 25, 36, 49,
64, 81, 100) and then 1 × 1 × 1, 2 × 2 × 2 up to 5 × 5 × 5 (1, 8, 27, 64, 125). Identify these as
the square numbers and cube numbers respectively.
Main teaching and learning
•
Explain that 102 is a shorter way of writing 10 × 10 and that (–53) is a shorter way of
writing –5 × –5 × –5.
•
Discuss how square root is the inverse (opposite) of square, therefore 100 = 10
because 102 = 100. Likewise 3 –125 = –5 because (–53) = –125.
•
Ask students if it is possible for them to tell you the square root of any number. Which
numbers can you write down the square root for without a calculator? (The square
numbers.)
•
•
Discuss the fact that each positive number has both a positive and negative square root.
Explain why this is the case, e.g. 5 × 5 = 25 and –5 × –5 = 25.
Common misconceptions
•
•
Remind students that squaring a negative number always gives a positive number.
Warn students of the very common error: 32 = 6.
Enrichment
•
•
Students could investigate the patterns formed from 112, 1112 etc.
•
How many squares are there on a standard chess board? (204 squares)
Some numbers can be expressed as the difference of two squares, for example
42 – 32 = 7, 32 – 12 = 8. Which numbers cannot be expressed as the difference of two
squares? (2, 6, 10, 14, 18, …)
Plenary
•
4
cube
cube number
M01_MSAH_TG_GCSE_0822_C01.indd 4
Ask students to give the values of, for example, 62 (36), 53 (125), 64 (8), 3 64 (4).
cube root
square
square number
square root
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Section 1.2 Understanding squares and cubes
Section 1.2 Understanding squares and cubes
Section 1.2 Understanding squares and cubes
Guided practice worksheet
Guided practice worksheet
1
2
Work out
a 32 .........................................................
b 42 .........................................................
d 72 .........................................................
e 102 .........................................................
c 52 .........................................................
Write down
a 3 1 ................................................................................
Work out
a 23 .........................................................
b 33 .........................................................
d 53 .........................................................
e 103 .........................................................
8
12
16
25
64
b 3 −64
................................................................................
c 43 .........................................................
Here is a list of 8 numbers
4
................................................................................
................................................................................
7
.....................................................................................................................................................................................................................................
1
................................................................................
................................................................................
................................................................................
.....................................................................................................................................................................................................................................
4
d 49 ................................................................................
................................................................................
e 9 ................................................................................
Lizzy says that to work out 62 you do 6 × 2 so the answer is 12. Lizzy is wrong. Explain why.
.....................................................................................................................................................................................................................................
3
c 4 ................................................................................
100
From the numbers in the list write down
a all the numbers that are square numbers
................................................................................
................................................................................
................................................................................
................................................................................
c 3 27 ................................................................................
d 3 −1000 ................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
3
e −8 ................................................................................
.................................................................................................................................................................................................................................
................................................................................
b all the numbers that are cube numbers.
................................................................................
.................................................................................................................................................................................................................................
5
8
Work out
a (–2)2 ................................................................................
Work out
a 52 + 22
b (–2)3 ................................................................................
................................................................................
b 82 – 32
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
c (–6) ................................................................................
d (–8) ................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
2
................................................................................
................................................................................
c 4 × 10
2
................................................................................
................................................................................
d 9 × 16
................................................................................
................................................................................
................................................................................
e 100 × 32 ................................................................................
e (–3) ................................................................................
3
................................................................................
................................................................................
................................................................................
f 92 + 16
................................................................................
................................................................................
................................................................................
................................................................................
6
Write down
a 25 ...............................................................................
b 100 ................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
g 25 × 36 ................................................................................
h 81 + (–1)2 ................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
5A
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5B
M01A_MSAH_TG_GCSE_0822_CDC01.indd 5
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Section 1.2 Understanding squares and cubes
Guided practice worksheet
9
i 72 + 49 ................................................................................
j 42 – (–6)2 ...............................................................................
................................................................................
................................................................................
................................................................................
................................................................................
Work out
3
a 23 × 1
................................................................................
3
b 8 × 52
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
c 42 + 3 −8 ................................................................................
d 3 −125 + 5 ................................................................................
................................................................................
................................................................................
................................................................................
e 53 – 13
................................................................................
................................................................................
f 43 ÷ 82
................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
g 3 1000 × 23 ................................................................................
h (–2)3 ÷ 4 ................................................................................
................................................................................
................................................................................
................................................................................
i (–5)3 × 4 ................................................................................
................................................................................
j
3
−64 × 3 27 ................................................................................
................................................................................
................................................................................
................................................................................
................................................................................
5C
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5
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Chapter 1 Number
Specification
GCSE 2010
N q (part) Understand and use number
operations and the relationships
between them, including … hierarchy of
operations
FS Process skills
Use appropriate mathematical
procedures
1.3 Understanding order of operations
Concepts and skills
•
Multiply and divide numbers using he commutative, associative, and distributive laws
and factorisation where possible, or place value adjustments.
•
Use brackets and the hierarchy of operations.
Functional skills
•
FS Performance
Level 1 Apply mathematics in an
organised way to find solutions…
L1 Add, subtract, multiply and divide whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
Students should already know how to
• add, subtract, multiply and divide positive and negative integers (N a).
•
Understand and use positive number and negative integers, both as positions and
translations on a number line (N b).
Starter
Resources
Resources
Questions for plenary
Variety of calculators
ActiveTeach resources
Squaring quiz
BIDMAS animation
The audience video
•
Give out a variety of different calculators. (The calculator function on less sophisticated
mobile phones is useful here.)
•
Ask students to work out 3 + 5 × 2 on the calculator they have been given. Ask for
the answers from the calculators. You should get the answers 16 (incorrect) and 13
(correct).
•
Discuss why the calculators (which are always correct!) are giving two different
answers.
•
Try some other calculations, e.g. 20 – 14 ÷ 2 (13), 2 × 3 + 4 × 2 (14)
Main teaching and learning
•
Tell students that they are going to find out about the order in which arithmetic
operations should be carried out.
•
Explain to students why it is important that there is a standard order of operations. (So
that we all arrive at the same answer.)
•
Discuss the meaning of the letters in BIDMAS.
Common misconceptions
•
When working out calculations such as 2 × 32 remember to use BIDMAS; this must be
worked out as 2 × 9 = 18.
•
When left with just addition and subtraction then you must work from left to right, e.g.
6 – 10 + 2 = –4 + 2 = –2 (6 – 10 + 2 cannot be worked out as 6 – 12).
Enrichment
•
Using just four 4s and any arithmetic operations, how many of the positive integers can
you make? For example, 4 × 4 + 4 + 4 = 24; 4 ÷ 4 + 4 ÷ 4 = 2.
Plenary
•
6
BIDMAS
M01_MSAH_TG_GCSE_0822_C01.indd 6
operation
Have some pre-prepared questions on the board and ask students to work these out.
For example, 7 + 4 × 2 (15), 24 – (8 × 2) (8).
power, powers
value
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M01_MSAH_TG_GCSE_0822_C01.indd 7
c 3 × 42
h (30 – 12) ÷ 2
.........................................................................
.........................................................................
.........................................................................
d 100 – 82
b (5 + 2)2
.........................................................................
.........................................................................
.........................................................................
.........................................................................
..................................................................
..................................................................
..................................................................
..................................................................
..................................................................
..................................................................
..................................................................
..................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
f 20 + 8 × 2 – 24 ÷ 4
d 45 ÷ (8 – 3)
b (7 + 2) × (16 – 9)
d (29 – 5) ÷ (4 + 2)
b (18 – 13) × 6
In c remember to square the 4 before you multiply by 3.
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
a 52 + 22 .........................................................................
Work out
g 8×7+3×3
e 28 ÷ 4 × 3
c 7×5+3×8
a 37 – (6 × 3)
Work out
c (9 – 3) × (8 + 2)
a 5 × (6 + 3)
.........................................................................
...........................................................................
................................................................................
Work out
d 8 × 10 – 24 ÷ 4 ...........................................................................
....................................................
...................................................
Remember
Brackets
Indices
Divide
Multiply
Add
Subtract
c 9 × 4 + 3 × 7 ................................................................................
................................................................................
a 5 × 6 + 2 × 8 ................................................................................
b 30 ÷ 3 – 2 × 4
d 6 + 18 ÷ 3 .............................................................
c 20 – 4 × 3 ................................................................................
Work out
b 3 × 4 + 2 ...............................................................
a 5 + 2 × 6 ................................................................................
Work out
M01A_MSAH_TG_GCSE_0822_CDC01.indd 7
5
4
3
2
1
Guided practice worksheet
7A
D
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Section 1.3 Understanding order of operations
g
12 – 5 × ( –3)
3
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
h
f
.........................................................................
.........................................................................
.........................................................................
–14 – 6
+2
2
.........................................................................
(15 – 19)
8
.........................................................................
2
.........................................................................
.........................................................................
6 – 20
7 .........................................................................
3
.........................................................................
.........................................................................
.........................................................................
.........................................................................
b 12 –
h 42 – 8 × –3
f (13 – 9) ÷ 2
2
d 5 × 2 + 12
2
b 6 × 4 + 32 × 2
–3 – 15
d 3+ 6
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
4 – 6 23 – 14
e 2 + 3
.........................................................................
2 − 20
c 7+
3
5–9
a 7+ 2
Work out
g 3 × –4 + 5 × 6
2
e 4×2 +3 ×5
2
c 100 + 7 × 5
a 5 × (10 – 7)2
Work out
M01A_MSAH_TG_GCSE_0822_CDC01.indd 8
7
6
Guided practice worksheet
7B
D
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Section 1.3 Understanding order of operations
Section 1.3 Understanding order of operations
7
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Chapter 1 Number
Specification
GCSE 2010
N q Understand and use number
operations and the relationships
between them, including inverse
operations and hierarchy of operations.
N v (part) Use calculators effectively and
efficiently….
FS Process skills
Decide on the methods, operations and
tools, including ICT, to use in a situation
Use appropriate mathematical
procedures
FS Performance
Level 2 Use appropriate checking
procedures and evaluate their
effectiveness at each stage
1.4 Using a calculator
Concepts and skills
•
Understand ‘reciprocal’ as multiplicative inverse, knowing that any non-zero number
multiplied by its reciprocal is 1 (and that zero has no reciprocal because division by
zero is not defined.)
•
•
Find reciprocals.
•
•
•
•
•
Resources
Calculators
Understand and use unit fractions as multiplicative inverses.
Use a calculator effectively and efficiently.
Know how to enter complex calculations.
Understand, and interpret, the calculator display.
Understand that premature rounding can cause problems when undertaking
calculations with more than one step.
Functional skills
•
Resources
Understand that the inverse operation of raising a positive number to a power n is
raising the result of this operation tht power n1 .
L2 Carry out calculations with numbers of any size … to a given number of decimal
places.
Prior key knowledge, skills and concepts
•
Students should be able to use a calculator to enter numbers and carry out the four
arithmetical operations.
Starter
•
Ask students to use their calculators to work out the following
2.452 (6.0025), 3.673 (49.430 863).
•
Ask students how they used their calculators to work out these sums.
Main teaching and learning
•
Tell students that they are going to learn how to use some of the buttons on their
scientific calculators.
•
Ask students to work out various calculations using their calculators and discuss the
key sequences.
3.4 + 7.9
Discuss the different ways that calculations such as
(1.79365…) can be
2.1 × 3
evaluated correctly.
•
Common misconceptions
•
The calculator does not always give the correct answer – it depends what was entered
and how it was entered. Encourage students to work out approximations to calculations
before using their calculators.
•
Remind students to take care when using a calculator to work out a fraction that has a
calculation in the numerator and/or denominator.
Plenary
•
8
inverse operations
M01_MSAH_TG_GCSE_0822_C01.indd 8
Give students various sums to be evaluated using their calculators.
reciprocal
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M01_MSAH_TG_GCSE_0822_C01.indd 9
.........................................................................
.........................................................................
a 1.73 × 7.2
.........................................................................
.........................................................................
.........................................................................
b 56 + 1.23
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
d 3.13 + 92
.........................................................................
.........................................................................
.........................................................................
b 2.32 + 1.52 .........................................................................
Work out, giving your answers correct to one decimal place
c (19.4 – 7.8)2
a (2.2 + 8.9) × 3.5
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
Work out
e 2601
.........................................................................
.........................................................................
.........................................................................
c 3 9261 .........................................................................
d 3 29.791
.........................................................................
.........................................................................
.........................................................................
b 36.69
.........................................................................
.........................................................................
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.........................................................................
.........................................................................
.........................................................................
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.........................................................................
a 529
Work out
e 1.54
.........................................................................
.........................................................................
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.........................................................................
d 35
.........................................................................
.........................................................................
c (–5.1)2 .........................................................................
b 1.2
.........................................................................
.........................................................................
3
Remember, to work out powers on your calculator, use the y x or x or ^ key.
.........................................................................
a 32
2
Work out
M01A_MSAH_TG_GCSE_0822_CDC01.indd 9
4
3
2
1
Guided practice worksheet
9A
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Section 1.4 Using a calculator
465 –
c
a
.........................................................................
.........................................................................
.........................................................................
8.42 + 1.62 .........................................................................
153
d
.........................................................................
.........................................................................
.........................................................................
.........................................................................
722 + 96 .........................................................................
( 42.6 + 9.3)2
.........................................................................
.........................................................................
.........................................................................
6.2 × 1.7
7.5 + 3.4
.........................................................................
b
.........................................................................
43 + 6.22
38.4 – 13.6 .........................................................................
.........................................................................
.........................................................................
Work out, giving your answers correct to three significant figures
.........................................................................
.........................................................................
.........................................................................
2
.........................................................................
.........................................................................
.........................................................................
8.74 4.5
+
1.7 0.6
.........................................................................
b
.........................................................................
 67

+ 453
d 
 .........................................................................
0.23
.........................................................................
.........................................................................
68.2
7.6 .........................................................................
783
c
5.6 − 4.87
a
Work out, giving your answers correct to three significant figures
.........................................................................
.........................................................................
.........................................................................
.........................................................................
3.27 × 6.3
9.1 × 35.4 .........................................................................
.........................................................................
b
8.12
d 7.22 – 3.62 .........................................................................
.........................................................................
.........................................................................
.........................................................................
65.4 + 19.5
c
8.2 − 3.46 .........................................................................
7.21
a 8.3 − 2.76
Take care when working out sums that involve fractions. Work out
the top and bottom of the fraction separately or use brackets.
.........................................................................
Work out, giving your answers
correct to three significant figures
.........................................................................
.........................................................................
.........................................................................
d 83 – 230
.........................................................................
.........................................................................
M01A_MSAH_TG_GCSE_0822_CDC01.indd 10
7
6
5
c 8.42 ÷ 12
Guided practice worksheet
9B
C
D
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Section 1.4 Using a calculator
Section 1.4 Using a calculator
9
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Chapter 1 Number
Specification
GCSE 2010
N f (part) Use index laws for
multiplication and division of integer…
powers
FS Process skills
Use appropriate mathematical
procedures
1.5 Understanding the index laws
Concepts and skills
•
Use index laws to simplify and calculate the value of numerical expressions involving
multiplication and division of integer… powers.
Functional skills
•
L1 … multiply and divide whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
FS Performance
Level 1 Use appropriate checking
procedures at each stage
Students should
• be able to work out the square of a number and the cube of a number.
•
understand index notation e.g. know that 34 = 3 × 3 × 3 × 3.
Starter
Resources
ActiveTeach resources
RP Number knowledge check
RP Multiples problem solving
Follow up
25.1 Using zero and negative powers
25.3 Working with fractional indices
•
•
•
Write 2 3 on the board and ask students what this means (2 × 2 × 2).
Ask them for another way to write 6 × 6 × 6 × 6 (64).
Have other similar examples written ready on the board to use for practice.
Main teaching and learning
•
Tell students that they are going to learn the index laws, which will enable them to
simplify expressions such as 32 × 34 and 65 ÷ 62.
•
Ask students to write out 32 × 34 in full (3 × 3 × 3 × 3 × 3 × 3) and explain that this could be
written as 36.
•
Discuss other similar examples and encourage students to give a general rule for
combining the powers to give a single power (add the powers).
•
Ask students to investigate examples involving division and ask them to come up with a
rule this time (subtract the powers).
•
Ask students to explain the meaning of (72)3 (72 × 72 × 72). Discuss how this can be
simplified to 72+2+2 which can be written as 72×3 or 76. Encourage students to give a
general rule for expressions of the form (am)n (multiply the powers).
Common misconceptions
•
•
A common error is to work out 34 as 3 × 4 rather than 3 × 3 × 3 × 3.
Questions that ask students to ‘Simplify 46 × 43’ or ‘Write 46 × 43 as a power of 4’ want the
answer to be given as 49; they are not asking the student to work out 49. Students should
be encouraged to show their working i.e. writing 46+3 before the final answer to show
their method.
•
Questions that use the word ‘evaluate’ or the phrase ‘work out’ do require an answer
that is not in index form. For example, the answer to ‘Evaluate 22 × 23’ is 32.
•
Students often multiply rather than add the powers in 26 × 25 and divide rather than
subtract the powers in 28 ÷ 24.
•
Students often multiply the numbers as well as adding the powers, i.e. to give the
answer to 34 × 37 as 911 instead of 311.
•
Students often think that 4 is the same as 40 rather than 41.
Plenary
•
10
index number
M01_MSAH_TG_GCSE_0822_C01.indd 10
Ask students to simplify expressions such as 78 × 73 (711), 812 ÷ 82 (810), (64)2 (68)
laws of indices
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Section 1.5 Understanding the index laws
Section 1.5 Understanding the index laws
Section 1.5 Understanding the index laws
Guided practice worksheet
1
Guided practice worksheet
Work out
6
a 43 .........................................................................
b 92 .........................................................................
a (62)4 .........................................................................
c 26 .........................................................................
d 104 .........................................................................
e 35 .........................................................................
2
3
b 6 × 6 × 6 .........................................................................
c 8 × 8 × 8 × 8 × 8 × 8 × 8 ..........................................................
d 17 × 17 .........................................................................
.........................................................................
.........................................................................
d (69)3 .........................................................................
.........................................................................
.........................................................................
a 75 × 73
c 73 × 7
e (63)2 .........................................................................
.........................................................................
C
a m × a n = a m +n
Write as a power of 7
.........................................................................
b 72 × 78 .........................................................................
.........................................................................
.........................................................................
.........................................................................
d 76 × 76 .........................................................................
.........................................................................
.........................................................................
7
Write as a power of a single number
a 73 × 75
.........................................................................
c 84 × 82
a m ÷ a n = a m–n
8
.........................................................................
.........................................................................
d 57 ÷ 5 .........................................................................
.........................................................................
d 37 × 36 ÷ 35 .........................................................................
.........................................................................
.........................................................................
B
Write as a power of 5
b
5 × 58
54 .........................................................................
.........................................................................
.........................................................................
c 52 × 56 .........................................................................
d 56 × 5 .........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
516
3
.........................................................................
.........................................................................
.........................................................................
e 5 ÷ 5 .........................................................................
.........................................................................
5
.........................................................................
.........................................................................
52 × 55
a 53 .........................................................................
b 59 ÷ 56 .........................................................................
c 512 ÷ 54 .........................................................................
5
.........................................................................
.........................................................................
.........................................................................
Write as a power of 5
a 56 ÷ 52 .........................................................................
b (94)2
.........................................................................
e 5 × (54)2 ÷ 52 .........................................................................
e 74 × 7 × 73 .........................................................................
4
b (64)3 .........................................................................
c (68)5 .........................................................................
Write in index form
a 2 × 2 × 2 × 2 × 2 .........................................................................
C
(a m ) n = a mn
Write as a power of 6
53 × 55
Write as a power of 3
a 3 ×3
9
2
c 310 ÷ 34
.........................................................................
b 3 ÷ 3 ........................................................................
.........................................................................
.........................................................................
.........................................................................
d 3 × 34 .........................................................................
.........................................................................
.........................................................................
7
3
5 × 512
e 54 × 56 .........................................................................
.........................................................................
.........................................................................
e 37 × 32 ÷ 36 .........................................................................
.........................................................................
11A
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11B
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Section 1.5 Understanding the index laws
Guided practice worksheet
9
B
Work out
a
3 4 × 32
35 .........................................................................
b
.........................................................................
49
.........................................................................
.........................................................................
.........................................................................
c 4 3 × 4 4 .........................................................................
2 × 28
26
.........................................................................
d
102 × 105
104
.........................................................................
.........................................................................
.........................................................................
.........................................................................
.........................................................................
73 × 75
e 7 × 76 .........................................................................
.........................................................................
.........................................................................
11C
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11
M01_MSAH_TG_GCSE_0822_C01.indd 11
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