Download Patterns and relationships

ALGEBRA
PATTERNS AND RELATIONSHIPS
What are the 2 missing shapes?
What are the 2 missing letters?
Starter
=
=
=
=
=
=
12x
10x
ac
3p
d
14x
= ab
= 8cd
= 6fg
= 2ap
= 8ab
= 6pqr
= 10def
= 6pqr
Note 3:
Patterns
To complete a pattern, look for a rule to get from
one term to another.
Examples: Find the rule for these sequences and
write down the next term:
3,7,11,15,……
The rule is ‘start with 3 and add 4 each time’.
The next term will be 15 + 4 = 19
1,3,9,27,……
The rule is ‘start with 1 and multiply by 3 each time’.
The next term will be 27 x 3= 81
Note 3:
Patterns
Sometimes it helps to find the rule if we look at
the differences between each term.
Examples: Find the next three terms in the following sequences
1,
3,
7,
+2 +4
1,
3,
+2
+4
13,
+6
7,
+8
15,
+8
+16
21,
31,, 43,,
+10 +12
31,
+32
57,,
+14
63, 127, 255,
+64
+128
Note 3:
Patterns
A car salesman has said he wants to sell 10 cars in his
first month of business. Every month after that, he
aims to sell two more cars than the previous month.
Calculate how many cars he aims to sell each month
for the first 5 months of business.
month = 10
2nd month = 10 + 2
1st
= 12
3rd month = 12 + 2
= 14
4th month = 14 + 2
= 16
5th month = 16 + 2
= 18
Alpha IWB
Ex 12.03 pg 312
Note 4:
Finding terms from patterns
Using the rule for the pattern, substitute values in for n
using 1, 2, 3, 4,…..etc until we have the required terms
e.g. Find the first four terms in each of the following rules
2n
n = 1 (term 1)
n = 2 (term 2)
n = 3 (term 3)
n = 4 (term 4)
2
2
2
2
x
x
x
x
(1)
(2)
(3)
(4)
Terms are:
2, 4, 6, 8, ……..
=
=
=
=
2
4
6
8
Note 4:
Finding terms from patterns
Using the rule for the pattern, substitute values in for n
using 1, 2, 3, 4,…..etc until we have the required terms
e.g. Find the first four terms in each of the following rules
n + 2 n = 1 (term 1)
n = 2 (term 2)
n = 3 (term 3)
n = 4 (term 4)
1
2
3
4
+
+
+
+
2
2
2
2
Terms are:
3, 4, 5, 6, ……..
=
=
=
=
3
4
5
6
Note 4:
Finding terms from patterns
Using the rule for the pattern, substitute values in for n
using 1, 2, 3, 4,…..etc until we have the required terms
e.g. Find the first four terms in each of the following rules
2(n + 2) n = 1 (term 1)
n = 2 (term 2)
n = 3 (term 3)
n = 4 (term 4)
2(1
2(2
2(3
2(4
+
+
+
+
2)
2)
2)
2)
Terms are:
6, 8, 10, 12 ……..
=
=
=
=
6
8
10
12
Note 4:
Finding terms from patterns
Using the rule for the pattern, substitute values in for n
using 1, 2, 3, 4,…..etc until we have the required terms
e.g. Find the first four terms in each of the following rules
2(n + 2) n = 1 (term 1)
n = 2 (term 2)
n = 3 (term 3)
n = 4 (term 4)
2(1
2(2
2(3
2(4
+
+
+
+
2)
2)
2)
2)
Terms are:
6, 8, 10, 12 ……..
=
=
=
=
6
8
10
12
Note 4:
Finding terms from patterns
Using the rule for the pattern, substitute values in for n
using 1, 2, 3, 4,…..etc until we have the required terms
e.g. Find the first four terms in each of the following rules
3n - 1 n = 1 (term 1)
n = 2 (term 2)
n = 3 (term 3)
n = 4 (term 4)
3(1)
3(2)
3(3)
3(4)
−
−
−
−
1
1
1
1
Terms are:
2, 5, 8, 11, ……..
= 2
= 5
= 8
= 11
Task: Find the first four terms in each of the following rules
by substituting 1, 2, 3 & 4 in the following rules.
3n 3, 6, 9, 12, …
n +5
6, 7, 8, 9, …
n - 1 0, 1, 2, 3, …
4n + 1 5, 9, 13, 17, …
-2 + 7n 5, 3, 1, −1, …
5(n+2)
n2
n2 + 4
n2 +n
15, 20, 25, 30, …
1, 4, 9, 16, …
5, 8, 13, 20, …
2, 6, 12, 20, …
Alpha IWB
Ex 12.03 pg 313-314
PUZZLE pg 315
Ex 12.04 pg 319-323
Finding General Rules
for number patterns
Think of an integer, multiply it by 3, and subtract 2.
You may have thought of many different numbers:
e.g. 3 × 4 – 2 = 10
3 × 2 – 2 = 4
3 × −7 − 2 = −23
3 × 9 − 2 = 25
What is the same about each sentence?
What is the different about each sentence?
How could you write a sentence that describes all possible sentences?
How could you write that sentence in maths language?
3×n−2
3n − 2
Note 5:
Finding General Rules for
number patterns
A variable is a letter or symbol that is used to describe
infinitely many numbers.
We write rules in the form of y = dn ± x
d and x are numbers that we calculate
To do this:
1.) Find the common difference between the terms.
This number is d.
2.) Subtract the first term from the common difference.
This number is x.
e.g. What is the rule for 3, 5, 7, 9, ….
Note 5:
y = dn ± x
Finding General Rules for
number patterns
d = common difference
x = Term 1 − common difference
e.g. What is the rule for 3, 5, 7, 9, ….
Term number
Term (y)
Common Difference
1
3
2
5
2
x = Term 1 − Common Difference
= 3 − 2
=1
3
7
2
4
9
2
d=2
y = dn ± x
The equation is y = 2n + 1
Note 5:
Finding General Rules for
number patterns
y = dn ± x
d = common difference
x = Term 1 − common difference
e.g. What is the rule for 10, 14, 18, 22 ….
Term number
Term (y)
1
10
Common Difference
2
14
4
x = Term 1 − Common Difference
= 10 − 4
=6
3
18
4
4
22
4
d=4
y = dn ± x
The equation is y = 4n + 6
Note 5:
y = dn ± x
Finding General Rules for
number patterns
d = common difference
x = Term 1 − common difference
e.g. What is the rule for 35, 30, 25, 20 ….
Term number
Term (y)
1
35
Common Difference
2
30
-5
x = Term 1 − Common Difference
=35 −−5
= 35 + 5
= 40
3
25
-5
4
20
-5
d = -5
y = dn ± x
The equation is y = -5n+40
Note 5:
y = dn ± x
Finding General Rules for
number patterns
d = common difference
x = Term 1 − common difference
e.g. What is the rule for 4, 1, -2, -5, ….
Term number
Term (y)
Common Difference
1
4
2
1
-3
x = Term 1 − Common Difference
=4 −−3
=4 + 3
=7
3
-2
-3
4
-5
-3
d = -3
y = dn ± x
The equation is y = -3n+7
Now its your turn!
Find d
and x first
Write rules in the form of y = dn ± x
e.g. What is the rule for :
2, 4, 6, 8, …
d = 2, x = 0
y = 2n + 0
17, 12, 7, 2, …
d = −5, x = 22
y = −5n + 22
18, 27, 36, 45, … d = 9, x = 9
y = 9n + 9
37, 29, 21, 13, … d = −8, x = 45
y = -8n + 45
Alpha IWB
Ex 12.04 pg 319-323
Spatial Patterns
Spatial Patterns
4
7
10
How many matches are there in each pattern?
How many matches will there be in a pattern with 4 squares?
10 + 3 = 13
Spatial Patterns
7
4
Number of Squares (s)
Number of matches
10
1
4
s = Term 1 − Common Difference
= 4 − 3
=1
13
2
7
3
10
4
13
d=3
y = dq ± s
The equation is y = 3q+ 1
Spatial Patterns
3
9
6
Draw the next 2 shapes in this pattern
Number of Triangles (t)
Number of matches
s = Term 1 − Common Difference
= 3 − 3
=0
1
3
2
6
3
9
4
12
d=3
y = dt ± s
The equation is y = 3t + 0
Spatial Patterns
6
11
y = dh ± x
16
Draw the next 2 shapes in this pattern
Number of Houses (h)
Number of matches
x = Term 1 − Common Difference
= 6 − 5
=1
1
6
2
11
3
16
4
21
d=5
y = dh ± x
The equation is y = 5h + 1
Note 6: Using rules, tables & graphs
Sarah babysits on the weekend. She is paid $5.00 each time, plus
$6 per hour that she is there.
Complete the table to show her earnings for 1-5 hours at a time.
1
2
3
4
5
Pay
$11
$17
$23
$29
$35
Plot the coordinates
from the table to a
graph
Rule: 5 + 6h
Wages earned $
Hours worked
40
30
20
10
0
0
2
4
Hours worked
6