Download Square roots - Pearson Schools and FE Colleges

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Key words
Calculations with integers
positive
negative
integer
product
Know how to add and subtract positive and negative numbers
Know how to multiply and divide positive and negative numbers
Know how to use the sign change key on a calculator
The set of positive and negative whole numbers, including zero, are called integers .
To add two integers, use a number line to help. Start on the line at the first number.
To add a positive number count right; to add a negative number count left.
To subtract one integer from another, convert the subtraction into an addition of its
inverse, then use the number line to model the addition.
For example: for 4 2, first convert it into 4 2, because the inverse of 2 is 2.
To multiply together two integers, first multiply together the whole number parts, and
then decide on the correct sign.
To divide one integer by another, first divide the whole number parts, and then decide
on the correct sign.
Rules of signs for multiplication and division:
if both signs are the same, then the result is positive;
if the signs are different, then the result is negative.
The result of a multiplication is called a product .
Example 1
Copy and complete these calculations:
a) 4 6 b) 3 2 4 c)
2 5 Use a number
line to help.
10 9 8 7 6 5 4 3 2 1
a)
4 6 2
b)
3 2 4 9
c)
2 5 2 5 3
0
1 2 3 4 5 6 7 8 9 10
Add a positive,
count right.
Add a negative,
count left.
For subtraction,
add the inverse.
Example 2
2
Copy and complete these calculations:
a) 3 4 5
b) 5 5
c) 16 8
d) 24 3
a)
3 4 5 (3 4) 5 12 5 60
b)
5 5 25
Multiplying same signs positive.
c)
16 8 2
Dividing different signs negative.
d)
24 3 8
Maths Connect 2R
Dividing same signs positive.
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Exercise 1.1 .............................................................................................
Copy and complete these calculations:
a)
d)
g)
3 7 8 3 9 4 6 b)
e)
h)
8 2 7 4 17 6 2 4 c)
f)
i)
Copy and complete these calculations:
a)
d)
g)
8 2 16 4 4 3 20 b) 5 7 e) (9)2 h) (8)2 (3)2 6 2 4 13 8 6 12 36 5 c)
f)
i)
3 3 2 10 2 4 2 5 10 Copy and complete these calculations:
a) 6 (2 3) c) (2 3) (6 2) 3 2 9
e) 3
4(5 3) 4 2 3 b)
d)
f)
Multiply the value outside
the brackets by every
value inside the brackets.
4 1 2(5 7) Copy and complete this subtraction pyramid using only negative
4
numbers. Subtract the number in the left tile from the number in
the right tile to make the number in the tile above.
Copy and complete this table of products, using the sign change key on your calculator.
3
2
5
4
On your calculator press:
6
2
5
4
The product of two integers is 24. Their sum is 2 and the difference between them is
10. What are the two numbers?
Both the product and the sum of three consecutive integers is 6.
a) What are the three integers?
b) Can you find three other consecutive integers that have a product equal to their sum?
A magic square is a square of numbers in which all the rows, columns
and diagonals add up to the same total.
Use the following number cards to make a three-by-three magic square.
3
1, 1, 9, 5, 5, 7, 3, 11, 3
The central number has been placed for you.
Investigation
Using the digit 4 four times, try to make all the numbers from 9 to 9.
You must use four 4’s each time. For example: (4 4) 4 4 8
Calculations with integers
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Key words
Powers and roots
Use index notation to write squares and cubes
Use a calculator to find squares, cubes, square roots and cube roots
Know the cubes of 1, 2, 3, 4, 5 and 10 and the corresponding cube roots
Know the powers of 10
square
index
power
square root
cube
cube root
Some square numbers are:
12 1 1 1
22 2 2 4
32 3 3 9
82 8 8 64
The squares 12, 22, 32, . . . are written in index notation. The
index, or power , is the raised number 2. It tells us how many
times the number is multiplied by itself.
The inverse of squaring is finding the square root .
64
8 ‘the square root of sixty four is eight’. Also, (8)2 64. So 64
8 or 8.
By convention the square root sign ( ) is used to mean the positive square root.
Some cube numbers are:
13 1 1 1 1
23 2 2 2 8
33 3 3 3 27
Here the index or power is 3.
‘Two cubed is eight’ or ‘two to the power of three is eight.’
3
The inverse of cubing is finding the cube root . 8 2
‘the cube root of eight is two’.
We can use index notation to write powers higher than 2 or 3. Some powers of 10 are:
101 10
This is a square number.
2
10 10 10 100
This is a cube number.
103 10 10 10 1000
106 10 10 10 10 10 10 1 000 000 (one million)
109 1 000 000 000 (one billion)
Example 1
Write the value of:
a) 54 625
3
5
b) 125
Example 2
3
b) 125
5 5 5 5 625
5 5 5 125
Write as powers of 10, the number of:
a) millilitres in a litre
b) millimetres in a kilometre.
a) 1 1000 m 103 m
b) 1 km 1 000 000 mm 106 mm
4
a) 54
Maths Connect 2R
1000 mm 1 m and 1 m 1000 1 km
1000 1000 1 000 000
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Exercise 1.2 .............................................................................................
Write the value of:
a)
d)
g)
j)
m)
82
0.5 squared
3
8
10
000
3
1000
Write the value of:
a) 196
e) 0.23
b)
e)
h)
k)
n)
43
0.1 cubed
the square root of 81
2
7
109
3
b) 15
625
3
f) 74.088
c)
f)
i)
l)
o)
105
16
the cube root of 27
3
2
3
1
c) 13.52
g) 246.49
d) 9.23
c) d2
d3
g) ab
d) (2c)3
If a 3, b 1, c 5, d 4, write the value of:
a) a3
2
c
e) a 2b
b) b
3
f) (a
c)
Write as powers of 10, the number of:
a)
c)
e)
g)
centimetres in a metre
millimetres in a metre
centilitres in a litre
centimetres in a kilometre
b)
d)
f)
h)
grams in a kilogram
metres in a kilometre
millilitres in a centilitre
millimetres in a metre
Who am I?
a)
b)
c)
d)
e)
I am six to the power of two, halved.
One tenth of me is ten to the power of two.
I am the difference between two to the power of four and three squared.
I am the square root of four squared.
I am the cube root of two to the power of six.
State which is greater: 82
or 32. Give a reason for your answer.
Work out the value of:
a) (2)2
b) (2)3
c) (2)4
d) (2)5
f) Make a comment about what you notice.
g) Would (2)13 be a negative or positive number?
e) (2)6
Investigation
a) List the first six prime numbers.
b) List the first six powers of 2.
c) Choose a number, say 17. Check to see if it can be written as the total of a prime
number and a power of 2, for example: 17 13 4.
d) Make a list of all the numbers between 4 and 20 that cannot be written like this.
e) Can you make a guess about the type of numbers between 20 and 30 you will not
be able to make this way?
f) Test your hypothesis – is your guess correct?
g) Can you work out why you can’t make these numbers?
Powers and roots
5
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Key words
Indices
Use index (power) notation
Know some simple laws of indices (powers)
The powers of 2 are:
21 2
22 4
23 8
24 16
25 32
26 64
index
power
indices
The powers of 3 are:
31 3
32 9
33 27
34 81
35 243
36 729
Multiplying powers of the same number:
For example: 23 24 27
8 16 128
The result is another power of the same number, given by the total of the powers.
na nb na b
Dividing powers of the same number:
For example: 35 32 33
243 9 27
The result is another power of the same number, given by the subtraction of the
powers.
na nb na b
Example 1
Work out 43 42, giving your answer using index notation.
43 42 4 32 45
Using the laws of indices, we
add the index numbers.
We could write it out in full to
find the answer.
4 4 4 4 4 45
Example 2
Work out 45 43, giving your answer as a single power of 4.
45 43 4 53 42
Division is done by subtracting
the powers.
We could write it out in full to
find the answer.
44444
444
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Exercise 1.3 .............................................................................................
Write the answers to the following using index notation:
a) 32 35 e) 75 71 b) 42 43 f) 32 33 31 c) 31 34 g) 25 23 24 d) 82 82 h) 34 32 Write the answers to the following as a single power:
a) 64 62 b) 54 51 c) 87 83 d) 35 34 5
3
5
3
6 6
2 2
f) g) e) 24 24 4
6
27
Write the solution to these questions as simply as you can by finding a single power
where possible:
a) 34 37 b) 45 43 52 52 52 52
72 75 73 74
d)
c) 53 52
73 78
Calculate the values of Q1, 2 and 3, parts a) and b) only.
John thinks that 32 52 84 4096.
Ali has the answer book and says the answer is 34.
a) Why is John wrong?
b) How do you get the right answer?
By substituting a 3 and b 2 into both sides of the equations, check that the following
are true:
a) a2 a3 a5
b) b4 b5 b9
b5
d) 2 b3
b
c) a4 a2 a2
Find a value for x where this is true:
2x x2
There may be more than one value.
Make sets of three matching cards for a game of Snap. You need six sets in total.
a5 a2
a3
a2 a1
Here is an example.
Now shuffle the cards and deal them between you and a partner. Play the game.
Investigation
Copy and complete these patterns to see if you can find the value of each number to
the power of 0.
2
2
2
23 2 2 2 8
22 2 2
4
21 2
20 2
2
2
3
3
3
33 3 3 3 32 3 3
31 30 3
3
3
Indices
7
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Key words
Square roots
trial and
improvement
estimate
square root
Find an upper and lower bound for a square root
Use a calculator to estimate square roots
Use trial and improvement methods to solve problems
’ Trial and improvement ’ is the name given to the method of estimating an answer,
comparing it with the target answer, then making an improved estimate , comparing
it again, and so on, eventually getting closer and closer to the target.
A table of familiar squares and square roots , helps to
provide the initial estimate.
Number
Square
Number
Square
1
1
10
100
2
4
20
400
3
9
30
900
4
16
40
1 600
5
25
50
2 500
6
36
60
3 600
7
49
70
4 900
8
64
80
6 400
9
81
90
8 100
10
100
100
10 000
To find the square root of 72
, the table can be used to make a first estimate.
8 72
9
72 is between 64 and 81 so the square root must be between 8 and 9.
The number on the left (8) is called the ‘lower bound’, and the number on the right (9)
is called the ‘upper bound’ for the square root.
Example 1
The square root of 153 lies between which two consecutive whole numbers?
122 144 too small.
132 169 too big.
12 153
13
Example 2
By ‘trial and improvement’, find the square root of 78 to two decimal places.
82 is 64 and 92 is 81 so the square root is between 8 and 9.
Try
Value
Result
Limits
8.5
72.25
Too small
8.5 78
9
8.7
75.69
Too small
8.7 78
9
8.9
79.21
Too big
8.7 78
8.9
8.8
77.44
Too small
8.8 78
8.9
8.85
78.3225
Too big
8.8 78
8.85
8.83
77.9689
Too small
8.83 78
8.85
8.84
78.1456
Too big
8.83 78
8.84
8.83 to
77.9689 is closer to 78 than 78.1456 so 78
two decimal places.
8
Maths Connect 2R
It helps to set it out in a table.
It is between 8.5 and 9.
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Exercise 1.4 .............................................................................................
The square root of these numbers lies between which two consecutive whole numbers?
a) 94
b) 170
c) 15.3
d) 8200
e) 0.5
Use ‘trial and improvement’ to find the square roots of these correct to one decimal place:
a) 19
b) 115
c) 5.6
Continue the ‘trial and improvement’ for Q2 to find the square roots correct to two
decimal places.
A square lawn has an area of 45 m2. How long is each side of the lawn correct to
one decimal place?
Daniel says that the square root of 39 is 5.983 27. Jayne thinks he is wrong, but her
calculator is broken. How can she use estimation to check Daniel’s answer?
Use a ‘trial and improvement’ method to find the solution of these equations correct to
one decimal place.
a) x2 71
b) x2 142 0
c) 4x2 82
a) Copy the table below.
b) Estimate how big the square root of each number will be, i.e. units, tens, hundreds,
thousands.
c) Now use your calculator to find the answer to three decimal places.
d) Was your estimate in the second column correct?
Square
root
Estimate
Square root to
3 decimal places
Was the
estimate correct?
8
Units
2.828
Yes
63
247
3 000
70 000
800 000
Investigations
To cover a cube in wrapping paper, with no overlap, requires
a piece of paper with an area of 72 cm2.
a) Find the area of one face of the cube.
b) Find the length of one edge of the cube to one decimal
place.
Use a ‘trial and improvement’ method to find, to two decimal places, the value of x
in this equation: x3 17
Square roots
9
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Key words
Sequences
sequence
term
term-to-term
general term
Generate a sequence given the general term
Find the general term of a sequence
Remember:
We can describe a sequence in two ways:
1. Giving the first term and a term-to-term rule.
2. Giving a general term .
We can express the general term like this: T(n) 2n 1.
To find terms in the sequence we replace the n with the term number:
T(1) 2 1 1 1
T(2) 2 2 1 3
T(3) 2 3 1 5 …
It is easy to find any term in the sequence, for example, the 112th term:
T(112) 2 112 1 223.
Example 1
The general term of a sequence is T(n) 5 3n.
Find the first three terms of the sequence.
T(1) 5 3 1 8
T(2) 5 3 2 11
We substitute the values
n 1, n 2, and so on
into the general term.
T(3) 5 3 3 14
Example 2
The general term of a sequence is T(n) 3n 5.
a) Is the sequence ascending or descending?
b) Describe the sequence by giving the first term and the term-to-term rule.
a) The sequence is descending.
b) T(1) 3 1 5
8
First term 8.
Term-to-term rule = subtract 3
Since the number in front of the n is negative we
know the difference between consecutive terms is
3, so the sequence will be descending.
To find the first term substitute n 1 into the
general term.
The number in front of the n is the value you add
(or subtract) to find the next term in the sequence.
Exercise 1.5 ..........................................................................................
Write down the first five terms of each of the following sequences:
a) T(n) n 4
c) T(n) 9 3n
b) T(n) 7 n
d) T(n) 4n 12
Decide whether each of these sequences is ascending or descending.
a) T(n) n
c) T(n) 5 3n
10
Maths Connect 2R
b) T(n) n 5
d) T(n) 1 n
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a) Write down the first five terms of each of the sequences in Q2.
b) Describe each of the sequences in Q2 by giving the first term and term-to-term rule.
Find the general term of each of the following and write it in the form T(n) :
a)
3, 0, 3, 6, 9, …
b) 12, 8, 4, 0, 4, …
c)
7, 6, 5, 4, …
d) 2, 8, 14, 20
James uses the following spreadsheet to generate the sequence: T(n) 10 3n.
A
1
2
3
4
5
B
7
…
…
…
…
1
2
3
4
5
Column A gives the term number (n).
Column B gives the terms (first term is 7).
a) Write down the values in cells B2 to B5.
b) James generated the sequence by entering formulae in column B that referred to
column A. What formula did he enter in cell B1?
c) Write down the first term and the term-to-term rule for the sequence.
For each sequence, write down the general term in the form T(n) n :
a)
b)
c)
d)
Multiples of 5.
Even numbers.
Odd numbers.
Descending negative integers.
5, 10, 15, 20, . . .
1, 2, 3, 4, 5, . . .
The general term of a sequence is T(n) 3n 1.
a) Copy and complete the table below:
n
1
2
3
4
5
T(n)
4
7
…
…
…
b) We could describe this sequence by: First term 4
Copy and complete the next row of the table:
T(n)
13
133
1333
Term-to-term rule ‘add 3’
1…
1…
c) We can write 3 3 3 3 as 4 3.
Using this notation, copy and complete the next row of the table.
T(n)
1 (1 3) 1 (2 3) 1 (3 3)
1…
1…
d) To find the 5th term of the sequence, how many times do we need to add 3?
e) To find the nth term of the sequence, how many times would we need to add 3?
f) How could we express this algebraically?
Sequences 11
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Key words
Special sequences
Find the next few terms in sequences where the difference between
consecutive terms is not always constant
Explain how a general term relates to a pattern
consecutive
terms
term-to-term
general term
constant
The sequence 5, 8, 11, 14, 17, … is a linear sequence. In a linear sequence the gaps
between consecutive terms are always the same.
The term-to-term rule is ‘add 3’ since the difference between consecutive terms is 3.
Each term in the sequence is two more than the multiples of 3:
1st term 1 3 2 5
2nd term 2 3 2 8
3rd term 3 3 2 11
General term term number 3 2, which we can write as T(n) 3n 2.
We have found the general term of a sequence where the differences between
consecutive terms are constant .
Consider the following sequence. 1, 2, 4, 8, … . The term-to-term rule is ‘multiply by 2’.
Look at the differences between consecutive terms:
Sequence
1
2
1
4
2
8
4
16
8
32
16
We can see that the sequence of differences is the same as the original sequence.
Looking at the pattern of differences, we need to add 32 to find the next term.
Example 1
Write down the first four terms of the sequence given that the first term is 1
and the term-to-term rule is ‘multiply the previous term by 3’.
T(1) 1
T(2) 1 3 3
To find the second term we
multiply the first term by 3.
T(3) 3 3 9
T(4) 9 3 27
Example 2
To find the third term we
multiply the second term by 3.
Look at this sequence:
Write down a general term for the number of squares in the nth pattern and
explain why it works.
1 square is in the first pattern, 4 squares are in the second pattern
and 9 squares are in the third pattern so the sequence is 1, 4, 9 …
This is the sequence of square numbers so the general term is n 2.
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Exercise 1.6 .............................................................................................
Write down the first five terms of each of the following sequences:
a)
b)
c)
d)
First term 2
First term 1
First term 64
First term 1000
Term-to-term rule is ‘multiply by 2’
Term-to-term rule is ‘multiply by 10’
Term-to-term rule is ‘divide by 2’
Term-to-term rule is ‘divide by 10’
a) Copy and complete the diagram for the sequence: 1, 2, 4, 7, 11, 16, …
Sequence
1
2
1
Differences
4
2
…
…
…
b) Use the diagram to help you find the 7th, 8th and 9th terms in the sequence.
Using the method from Q2, find the next three terms for each of these sequences:
a) 3, 5, 8, 12, 17, …
c) 0, 5, 15, 30, 50, …
b) 0, 2, 6, 12, 20, …
d) 1, 1.5, 2.5, 4, 6, …
a) Copy and complete the diagram for the sequence: 100, 98, 94, 88, 80, 70, …
Sequence
Differences
100
98
2
94
4
…
…
…
b) Use the diagram to help you find the 7th, 8th and 9th terms in the sequence.
Using the method from Q4, find the next three terms for each of these sequences:
a) 50, 49, 47, 44, 40, …
c) 100, 90, 81, 73, 66, 60, …
b) 1000, 990, 970, 940, 900, …
d) 50, 40, 20, 10, …
For each of these sequences of patterns, find the general term for the number of squares
and explain why it works.
a)
b)
c)
d)
a) Write down the first five terms of each of the following sequences:
i) T(n) n2 1
ii) T(n) n2 4
iii) T(n) n2 12
2
2
v) T(n) n 0.5
vi) T(n) n2 15
iv) T(n) n 100
b) Look at the difference between consecutive terms in each of the sequences. Describe
any patterns you notice.
c) Explain why you think this pattern occurs.
Special sequences 13