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Algebra Patterning and Graphs. Exploring Patterns. e.g. Write the next two numbers in the following patterns and describe the pattern. a) 2, 5, 8, 11, 14,. 17, 20. b) 40, 34, 28, 22, 16,. 10, 4. Add 3. Subtract 6. c) 1, 3, 9, 27, 81,. 243, 729. d) 1, 4, 9, 16, 25,. - PowerPoint PPT Presentation
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Algebra Patterning and
Graphs
Exploring Patternse.g. Write the next two numbers in the following patterns and describe the pattern.a) 2, 5, 8, 11, 14, b) 40, 34, 28, 22,
16,c) 1, 3, 9, 27, 81, d) 1, 4, 9, 16, 25,
17, 20 10, 4
243, 729 36, 49
e.g. Draw the next shape in the pattern and add the number below
10631
Add 3 Subtract 6
Multiply by 3
Square Numbers
Finding a Rule for Linear Patterns- Linear number patterns are sequences of numbers where
the difference between terms is always the same (constant) - Rule generating a linear pattern is: Difference × n ± a constante.g. Write a rule (using n) to describe the following number patterns. n Number
of Squares
(s)
Number of Dots
(d)
1 1 52 4 93 7 134 10 175 13 21
+ 3+ 3+ 3+ 3
+ 4
+ 4
+ 4
+ 4
Rule: s =
Rule: d =3×n 4×n
3×1= 33 = 1- 2
- 2
4×1= 44 = 5+ 1
+ 11. Find the difference between terms and if the same multiply by n
2. Substitute to find constant3. Check if rule works
3×4 – 2
4×4 + 1
e.g. To make these squares, the amount of matches below are needed.
4 7 10 13These results are shown on the table belowSquares (n)
Matches (M)
1 42 73 10410150
a) Draw the next set of squaresb) Create a rule for the pattern and use it to help fill in the gaps in the table
+ 3
+ 3
M = 3×n
3×1 = 33 = 4+ 1
+ 13×10 + 1
13
3×150 + 1
31451
c) Write the rule in wordsThe number of matches equals three times the number of squares plus one.
3×4 + 1 = 13
Simple Quadratic Patterns- Quadratic number patterns are sequences of numbers where the difference between terms is not the same - You need to look at the differences of the differences. If it is a ‘2’, then the rule contains ‘n2’e.g. Write a rule for the following pattern
n Term (T)1 42 73 124 195 28
1. Find the difference between terms
+ 5
+ 3+ 7+ 9
2. If difference is not the same, find the difference of the differences!
+ 2+ 2+ 2
Rule: T =
3. If the 2nd difference is a ‘2’, the rule contains n2
n2
4. Substitute to find constant
12 = 11 = 4+ 342 + 3 = 19
5. Check if rule works
+ 3
Co-ordinates- Are two references used to identify places of interest- The horizontal reference is written first with the vertical second e.g. MapsMathematical Co-ordinates- Are used to describe positions of points
e.g. Plot the following points: A = (1, 3), B = (4, 2), C = (3, -4), D = (-5, 1)
1. Add in axes (if needed)2. Label and number axes x = horizontal, y = vertical
x
y
1234
1 2 3 4 5-3 -2 -1-4-5 -1-2-3-4
3. Plot points using first number as the x co-ordinate and the second the y co-ordinate.4. Label points
AB
C
D
Co-ordinate Patterns- Involves plotting rules that link the x co-ordinate to the y co-ordinatee.g. Complete the tables below and plot the following rulesa) y = 2x b) y = ½x – 1 c) y = -3x + 2
x y = 2x y = ½x – 1
-2
-1
0
1
2x y = -3x +
2 -2
-1
0
1
2
x-5 -4 -3 -2 -1 1 2 3 4 5
y
-5-4-3-2-1
12345
2 x -1 2 x -2
02
-4
4
-2 ½ x -1 – 1
½ x -2 – 1-1-½
-2
0
-1 ½
-3 x -1 + 2
-3 x -2 + 22-1
8
-4
5
Scatterplots- Show relationships between two quantities- Has two axes, each showing a different quantity with its own scale
Height (m)
Age (years)
BobJane
Tom
Mary
e.g.
a) Who is the tallest?
b) Who is the same age?c) Who is the oldest?
Tom
Bob and Mary
Jane
Line Graphs- Show how one quantity changes as another one does
0
20
40
60
80
100
20 40 60 80 100
Sweetcorn boiled then left to cool
Tem
pera
ture
Time (mins)
e.g.a) What is the room
temperature?
b) How long does it take for the sweetcorn to cool down to room temperature?
20ºC
80 – 15 = 65 mins
e.g. Draw a graph to show how the water level may change in the following situation.- The sink is filled ¾ full of water
- All of the dishes are put into the sink at once- The dishes are washed and removed separately- The water is then drained
Wat
er le
vel i
n si
nk
empty
full
time
Doing the dishes
Distance/Time Graphs- Are line graphs with time on horizontal and distance on vertical axis.- If the line is horizontal the object is not moving- The steeper the line, the faster the movement
9 am 11 am 1 pm 3 pmtime of day
dist
ance
from
har
bour
(km
)
10
8
6
4
2
0
Distance of yacht from harboure.g.
a) How far out from the harbour did the yacht travel?b) What happened while the graph was horizontal?c) Which part of the journey was quickest?
5 kmThe yacht was stationary
Steepest
The return journey
Applications- When using graphs to read off values and explain rulese.g. A student lends out his scooter for a fee of $3 and $2 for every km travelled. Complete the table and plot on graph below.Length of
Journey (km)
Charge ($)
12345
0
2
4
6
8
10
12
14
1 2 3 4 5Journey (km)
Cha
rge
($)
Scooter fees
a) What will be the charge for a journey of 2 ½ km?
1 × 2 + 352 × 2 + 37
91113
$8