Download +2 - Gore High School

Section A:
Pattern
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Words to know
A list of numbers in order is called a sequence.
Each number in a sequence is called a term.
4, 8, 12, 16, 20, 24, 28, 32, ...
1st term
6th term
Substitute: to put a number in place of a letter
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Name the following sequences:
2, 4, 6, 8, 10, ...
Even Numbers (or multiples of 2)
1, 3, 5, 7, 9, ...
Odd numbers
3, 6, 9, 12, 15, ...
Multiples of 3
5, 10, 15, 20, 25, ...
Multiples of 5
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If n = 5 find
Substitution
3n2 + 4n – 8
Rewrite and then replace letters with numbers
3 x (n
5 )2 + 2 x ( n5 ) – 8
Apply BEDMAS and ‘do’ exponents first
3 x (25) + 10 – 8
75 + 10 – 8
= 77
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Your Turn: If b = 5 calculate 3b + 4:
3 x 5b + 4
15 + 4 = 19
Your Turn: Find the value of 2p – 5 when
P=6
2 x 6p – 5 =
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12 – 5 = 7
Your Turn:
Complete This Table
n
5
10
21
32
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n-3
2
7
18
29
Your Turn:
Complete This Table
x
5
10
21
32
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x+6
11
16
27
38
SLO
Continue a given LINEAR pattern (sequence)
Continue a given pattern using differences
http://www.youtube.com/watch?v=Zj-a_9cd5jc
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Words to know
Linear: straight line graph, even steps
Difference: take away
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Difference tables
A Difference table is made by finding the difference between
two consecutive terms.
Example 1:
2,
7,
+5
12,
+5
17,
+5
22,
+5
27,
+5
32,
+5
37, ...
+5
7 – 2 = 5 (common difference)
Example 2:
24, 17,
–7
10,
–7
–4,
3,
–7
–7
–11, –18, –25, ...
–7
–7
–7
All linear sequences increase or decrease in equal steps.
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Your Turn:
What are the next two terms in the following sequence,
102, 95, 88, 81, 74 . . . ?
Look at the difference between each consecutive term.
102
95
–7
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88
–7
81
–7
74
–7
67
–7
60
–7
Your Turn:
Find the next two terms in the following sequences
First 5 terms
1, 3, 5, 7, 9
5, 8, 11, 14, 17
8, 13, 18, 23, 28
7, 11, 15, 19, 23
2, 12, 22, 32, 42
7, 14, 21, 28, 35
50, 48, 46, 44, 42
19, 16, 13, 10, 7
6, 2, -2, -6, -10
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Next two terms?
11, 13
20, 23
33, 38
27, 31
52, 62
42, 49
40, 38
4, 1
-14, -18
SLO
Finding a formula for a liner pattern
(finding the nth term)
Find the formula for a given number pattern
http://www.youtube.com/watch?v=PhFhnUJIhw0
http://www.youtube.com/watch?v=aDwaTi0UAKU&feature=related
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Matchstick Squares
1
2
How many matches do I need
for pattern number 100?
Filling in this table would
take a long time!!
If we could find a rule from
pattern number to number
of matches it would be easy
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3
Pattern number
Matches
1
4
2
7
3
10
4
13
5
16
100
?
Finding the rule
To find the rule
we first find how many
matches we add each time. 3
Pattern number
Can you see what 3 has to do
with finding the connection
between pattern number and
how many matches you need?
Your rule must work every time!
1
2
3
4
5
3 x Pattern number then add 1
OR (using letters only)
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M = 3n + 1
Matches
4
7
10
13
16
100
How many matchsticks are needed for pattern
number 100?
Pattern number
Using 3 x 100
n +1
= 300 + 1
= 301
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1
2
3
4
5
100
Matches
4
7
10
13
16
?
Match Stick Triangles
Can you find the rule to find how many matches are
needed for each pattern?
Pattern number
1) Draw a table
2) See how much the
numbers are going up by.
3) Use 2 to find the rule
2n + 1
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2
1
2
3
4
5
Matches
3
5
7
9
11
Sequence generator
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Your Turn: Find the Rule
For the two sets of numbers below, a matchstick pattern has
been built and the student has filled in the table. Your
challenge is to find the rule.
Pattern number
1
2
3
4
5
Matches
5
7
9
11
13
2n + 3
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Pattern number
Matches
1
2
3
4
5
4n - 2
2
6
10
14
18
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Finding a formula for a linear pattern
(finding the nth term)
E.g. Find the formula for the following sequence
n 0
1
2
3
4
5
9,
13,
17,
21,
25
5
+4
+4
+4
+4
+4
Step 1: Find the common difference
4
Step 2: use the common difference as the number in front
of n i.e. 4n
Step 3: Find the 0th term i.e. 5
Step 4: add 0th term to step 2 i.e. 4n + 5
Step 5: check that it works
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Your Turn:
Find the nth term formula for the following sequences
First 5 terms
1, 3, 5, 7, 9
5, 8, 11, 14, 17
8, 13, 18, 23, 28
7, 11, 15, 19, 23
2, 12, 22, 32, 42
7, 14, 21, 28, 35
50, 48, 46, 44, 42
19, 16, 13, 10, 7
6, 2, -2, -6, -10
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Formula?
2n – 1
3n + 2
5n + 3
4n + 3
10n – 8
7n
-2n + 52
-3n + 22
-4n + 10
Finding the nth term of a linear sequence
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Finding linear rules
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Paving slabs: find the nth term for each pattern
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Dotty pattern
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SLO
Justifying how linear patterns work for practical problems
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Find patterns and solve “practical” problems with linear patterns.
http://www.youtube.com/watch?v=fNE_I9FJc20&feature=related
(take care, uses Llbs)
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Justifying formulae
When patterns refer to practical problems the formula will
be related to the formula e.g. the formula for the total
number of circles below is 3n + 1
n= 1
2
3
4
5
4
7
10
13
16
Can you explain what 3n+1 has to do with the above
pattern?
3 times n is because each time we advance to the next
term an extra row of 3 are added. The plus 1 is due to
the red circle on the left hand side
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Sequences from practical contexts
The following sequence of patterns is made from L-shaped tiles:
Number of
Tiles
4
8
12
16
The number of tiles in each pattern forms a sequence.
How many tiles will be needed for the next pattern?
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16
Sequences from practical contexts
1 lot of 4
2 lots of 4
Find the nth term
3 lots of 4
4 lots of 4
4n.
Justify the formula in relation to the tiles
The formula is 4n as there are 4 arms and each time 1 is
added to each arm.
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Sequences from practical contexts
Number of
Blocks
4
7
10
13
How many blocks will there be in the next shape?
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13
Sequences from practical contexts
Look at the pattern again:
1st pattern
2nd pattern
3rd pattern
4th pattern
Find a rule for the nth term.
The nth pattern has 3n + 1 blocks in it.
Justify the formula in relation to the tiles
The patterns have 3 „arms‟ each increasing by one block each
time. So the nth pattern has 3n blocks in the arms, plus one
more in the centre.
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Paving slabs
The number of blue tiles form the sequence 8, 13, 18, 23, ...
Pattern
number
1
2
3
Number of
blue tiles
8
13
18
Find the rule for the nth term of this sequence
5n + 3
Justify the formula in relation to the tiles
Each time we add another yellow tile we add 5 blue tiles.
The +3 comes from the 3 tiles at the start of each pattern.
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Tiling patterns
The following patterns are made from tiles:
Pattern 1
5 tiles
Pattern 2
9 tiles
Find the formula for the nth pattern.
Pattern 3
13 tiles
4n + 1
Explain why the formula works in relation to the tiling pattern.
Times by 4 due to 4 arms and add 1 due to centre tile
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Questions to do from the books
Achieve
Merit
Gamma
P30 EX. 3.02
EAS
P4 Q1 – 8
P30 Q124-127
P29 Q116 - 123
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Excellence
SLO
Continue a given Quadratic sequence
Continue a given pattern for a quadratic
http://www.youtube.com/watch?v=ShfgwVSYK6o&feature=related
Complete a 2nd order difference table
http://www.youtube.com/watch?v=p0CEGUg8PXU
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Words to know
Quadratic: a ‘U’ shape graph, formula with
highest power is squared.
E.g. y = 8x2 – 6x + 1
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Sequences that increase in increasing steps
Some sequences increase or decrease in unequal steps.
For example, look at the differences between terms in this
sequence:
5,
6,
+1
8,
+2
11,
+3
15,
+4
20,
+5
26,
+6
33, ...
+7
This sequence starts with 5 and increases by 1, 2, 3, 4, …
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Using a second row of differences
Can you work out the next three terms in this sequence?
1,
3,
+2
8,
+5
+3
16,
+8
+3
27,
+11
+3
41,
+14
+3
58,
+17
+3
78, ...
+20
+3
Look at the differences between terms (click for a hint).
A sequence is formed by the differences so we look at the
second row of differences.
This shows that the differences increase by 3 each time.
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Your Turn:
Find the next two terms (click for hint):
1
2
+1
4
7
+2
+1
+3
+1
11
+4
16
+5
+6
+1
+1
+1
22
Find the next two terms (click for hint):
100,
–1
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99,
97,
–2
94,
–3
90,
–4
85,
–5
79,
–6
72
–7
Your Turn:
Find the next two terms in the following sequences
First 5 terms
1, 3, 6, 10, 15
5, 8, 12, 17, 23
8, 13, 19, 26, 34
7, 11, 16, 22, 29
2, 12, 23, 35, 48
7, 14, 22, 31, 41
50, 48, 45, 41, 36
19, 16, 12, 7, 1
6, 2, -3, -9, -16
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Next two terms?
21, 28
30, 38
43, 53
37, 46
62, 77
52, 64
30, 23
-6, -14
-24, -33
Use to fill in 1st row of table (ignore rest of difference table)
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SLO
Finding a formula for a Quadratic pattern
(finding the nth term)
Find the formula for a quadratic pattern
http://www.youtube.com/watch?v=OUhbsQgheX0
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Finding quadratic formulae
Find the formula for the nth term:
6
9
+3
14
+5
+2
21
+7
+2
30
+9
+2
41
+11
+2
54
+13
+2
First difference
Second difference
As this difference table has gone to a second difference to
find a constant, the formula is a quadratic.
If the second difference is a 2 then the formula contains n 2
If the second difference is a 4 then the formula contains 2n 2
If the second difference is a 6 then the formula contains 3n 2
Etc.
(this is probably a little too much detail for this standard)
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Your Turn: Find the quadratic formulae
Find the formula for the nth term:
6
5
9
+3
+1
+2
14
+5
+2
21
+7
+2
30
+9
+2
41
+11
+2
54
+13
+2
The 0th term is 5 so the formula is n2 + 5 (click to add to
diagram)
If the examples in the Gamma book get more complicated than
this, leave them out
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Use to show how to find 0th term
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Your Turn:
Find the nth term formula for the following sequences
First 5 terms
1, 4, 9, 16, 25
3, 6, 11, 18, 27
0, 3, 8, 15, 24
7, 10, 15, 22, 31
-9, -6, -1, 6, 15
2, 8, 18, 32, 50
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Formula?
n2
n2 + 2
n2 – 1
n2 + 6
n2 – 10
2n2
Finding quadratic rules
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Your Turn:
Revision mix of linear and quadratic sequences
Find the rules of these sequences And these sequences





1, 3, 5, 7, 9,… 2n
6, 8, 10, 12,……. 2n
3, 8, 13, 18,…… 5n
20,26,32,38,……… 6n
7, 14, 21,28,…… 7n
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–1
+4
–2
+ 14





1, 4, 9, 16, 25,… n2
3, 6,11,18,27……. n2 + 2
20, 18, 16, 14,… -2n + 22
40,37,34,31,……… -3n + 43
6, 26,46,66,…… 20n - 14
SLO
Continue cubic patterns
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Words to know
Cubic: a
shape graph, formula with highest
power is cubed.
e.g. y = 5x3 + 6x – 8
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Cubic patterns
1
8
+7
27
+19
+12
64
37
+18
+6
+61
+6
+91
+6
+127
+36
+30
+24
343
216
125
+6
Second difference
Third difference
A cubic has a constant as a third difference
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First difference
Your Turn:
Fill in the following table
n
1
2
3
4
5
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n3 + 4n + 2
7
18
41
82
147
Your Turn:
Fill in the following table
n
1
2
3
4
5
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5n3 - n2 + 3n + 6
13
48
141
322
621
Your Turn:
Fill in the following table
n
1
2
3
4
5
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2n3 + 3n2 + 2n - 7
0
25
80
177
328
Questions to do from the books
(These questions are quadratic only)
Achieve
Merit
P35 EX. 3.03 Q1, 2, 6ab,
Gamma
7ab
P35 Ex 3.03
EAS
P63 Q197-201
P64 Q202-203
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P36 Q130–137
Excellence
SLO
Continue a given exponential sequence
Continue a given pattern [substitute into a formula]
http://www.youtube.com/watch?v=6WMZ7J0wwMI
Find the formula for an exponential pattern
http://www.youtube.com/watch?v=eqPqx5yrFsg
(using y = a.b^x, notice „.‟ = times)
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Some sequences do not have a common difference, however
many times you continue the pattern e.g. find the pattern
below.
2,
4,
+2
8,
+4
+2
16,
+8
+4
+2
+16
+8
+4
+2
32,
+32
+16
+8
+4
+2
+32
+16
+4
+128
+64
+32
+16
+8
+4
+2
128, 256, ...
+64
+8
+2
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64,
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Exponential Equations
If a sequence has no common difference, see if the first
difference is being multiplied to get to the next difference e.g.
3,
5,
+2
9,
+4
x2
17,
+8
x2
+16
x2
33,
+32
x2
65,
+64
x2
129, 257,
+128
x2
523
+512
+256
x2
1025
x2
Use this pattern to predict the next two terms (click for hint)
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Exponential Equations
3,
5,
+2
9,
+4
x2
17,
+8
x2
+16
x2
33,
+32
x2
65,
+64
x2
129, 257, ...
+128
x2
As the first difference is being multiplied by 2, all of the
terms in this sequence are powers of 2 i.e.
2,
4,
8,
16, 32, 64, 128, 256, ...
21
22
23
24
25
26
27
28
Therefore the formula is:
2n + 1
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Your Turn:
Find a formula for the nth term:
5
11
+6
29
x3
245
+162
+54
+18
x3
83
x3
731
+486
x3
Each term in this sequence is three times the term before it.
Therefore the formula has 3n as part of it.
31
32
33
34
35
By inspection, the final expression is 3n + 2
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36
Your Turn:
Fill in the following table
First 5 terms
1, 3, 7, 15, 31
4, 6, 10, 18, 34
0, 6, 24, 78, 240
10, 16, 34, 88, 350
10, 30, 130, 630, 3150
2, 14, 62, 254, 1022
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Next 2 terms
63, 127
66, 130
Formula?
2n – 1
2n + 2
726, 2184
3n – 3
736, 2194
3n + 7
15630, 78130
5n + 5
4n – 2
4094, 16382
Questions to do from the books
Achieve
Merit
P71 Q211–218
P74 Q219-222
P74 Q223-224
Gamma
EAS
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Excellence