2. Coloured Blocks Diagram
Is there a pattern to these colours?
Can you use the pattern to predict what
colour will be at a particular point?
How can we investigate if there is a pattern?
3. Table
Block
Colour Black
Position
1 Block Position
2 1 2
3 2 4
4 3 6
5 4 8
6 5 10
Black block is always even.
4. Words: You double the number of the black block
Black Red
Block Position Block Position
1 2 1 1
2 4 2 3
3 6 3 5
4 8 4 7
5 10 5 9
n 2n n 2n – 1
5. Key Outcomes and
Words:
identify patterns and describe
different situations using tables,
graphs, words and formulae
predict
generalise in words and symbols
justify
relationship
7. Tables
Yellow Black Green
Block Position Block Position Block Position
1 1 1 2 1 3
2 4 2 5 2 6
3 7 3 8 3 9
4 10 4 11 4 12
5 13 5 14 5 15
n 3n – 2 n 3n – 1 n 3n
8.
9. Money Box Problem
Mary Money Box
Start €0
Growth per day €2
We want to investigate the total amount
of money in the money box over time.
Is the growth of the money a pattern?
Can we predict how much money will be in
the box on day 10?
10. Money Box Problem
John Money Box
Start €3
Growth per day €2
Is there a pattern to the growth of this money?
Can we use this pattern to predict how much money
will be in the box at some future time?
How can we investigate if a pattern exists?
11. Table for John’s Money Box
Time/days Money in Box/€
0 3
1 5
2 7
3 9
4 11
5 13
Is there a pattern?
Where is the start value in € and growth per day
in this as seen in the table?
What do you notice about successive outputs ?
12. Money Box Problem
Bernie Money Box
Start €4
€2 on week days
Growth per day
€5 on Weekend days
Is there a constant rate of change here?
14. Identifying variables and constants
Money Box Varying Constant
John
Mary
Bernie
What is varying each Day?
What is constant?
Can you put this into words?
15. Mary John Bernie
Time on Horizontal Axis
Total Money on Vertical Axis
* Note: in this example day 0 is a Tuesday.
16. Draw a Graph
22
21
20 Mary
19
John
18
17 Bernie
Amount of Money Spent
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
Days
18. Now, I want you to observe the
pattern.
Explain in words & numbers, how to find
the total amount of money in John’s box
after 15 days.
Time/days Money/€ Change
0 3
+2
1 5
+2
2 7
+2 Do this on your
3 9
white board
+2
4 11
+2
5 13
+2
6 15
19. Now, I want you to generalise.
Explain in words & symbols, how to find
the total amount of money in John’s box
after any given day.
Time/days Money/€ Change
0 3
+2
1 5
+2
2 7
+2 Do this on your
3 9
white board
+2
4 11
+2
5 13
+2
6 15
20. Table for Mary’s Money Box
Money in
Time/days
Box € What is the general
0 0 formula for Mary?
1 2 A = 0 + 2D
2 4
3 6
4 8
5 10
21. John : A = 3 + 2D
Mary : A = 2D + 0
Only seeing the formula :
Can you read the John's start amount?
Can you read Mary's rate of change?
What are we building up to?
y = c + mx or y = mx + c
22. Sunflower growth
Sunflower a b c d
Start height/cm 3 6 6 8
Growth per day/cm 2 2 3 2
Is there a pattern to the growth of these sunflowers?
Can we use this pattern to predict height at some future time?
How can we investigate if a pattern exists?
23. Table for Each Sunflower
Time/days Height/cm Change
0
1
2
3
4
5
25. 22
A and B
21
20
19
18
17
16 Sunflower A
15
14
Sunflower B
Height/cm
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
Time/Days
26. 22
B and C
21
20
19
18
17
16 Sunflower B
15
14
Sunflower C
Height/cm
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
Time/Days
27. 22
C and D
21
20
19
18
17
16 Sunflower C
15
14
Sunflower D
Height/cm
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
Time/Days
28. Formula Representation
• Describe in words the height of the sunflower a,
on any day.
• Describe in symbols the height of the sunflower
a, on any day.
• Identify the variables and constants in this
formula.
• Where do the y – intercept and the slope of the
graph appear in the formula?
29. Pattern of Growth for 4 Different Sunflowers
A B C D
T H T H T H T H
Pattern Pattern Pattern Pattern
days cm days cm days cm days cm
0 3 3 0 6 6 0 6 6 0 8 8
1 5 3+2 1 8 6+2 1 9 6+3 1 10 8+2
2 7 3+2+2 2 10 6+2+2 2 12 6+3+3 2 12 8+2+2
3 9 3+2+2+2 3 12 6+2+2+2 3 15 6+3+3+3 3 14 8+2+2+2
4 11 3+2+2+2+2 4 14 6+2+2+2+2 4 18 6+3+3+3+3 4 16 8+2+2+2+2
5 13 3+2+2+2+2+… 5 16 6+2+2+2+2+… 5 21 6+3+3+3+3+… 5 18 8+2+2+2+2+…
6 15 3+2+2+2+2+… 6 18 6+2+2+2+2+… 6 24 6+3+3+3+3+… 6 20 8+2+2+2+2+…
Describe in words the height of the sunflower a, on any day.
Describe in symbols the height of the sunflower a, on any day.
Identify the variables and constants in this formula.
Where do the y – intercept and the slope of the graph appear in the formula?
30. Pattern of Growth for 4 Different Sunflowers
A B C D
T H T H T H T H
Pattern Pattern Pattern Pattern
days cm days cm days cm days cm
0 3 3 0 6 6 0 6 6 0 8 8
1 5 3+2 1 8 6+2 1 9 6+3 1 10 8+2
2 7 3+2+2 2 10 6+2+2 2 12 6+3+3 2 12 8+2+2
3 9 3+2+2+2 3 12 6+2+2+2 3 15 6+3+3+3 3 14 8+2+2+2
4 11 3+2+2+2+2 4 14 6+2+2+2+2 4 18 6+3+3+3+3 4 16 8+2+2+2+2
5 13 3+2+2+2+2+… 5 16 6+2+2+2+2+… 5 21 6+3+3+3+3+… 5 18 8+2+2+2+2+…
6 15 3+2+2+2+2+… 6 18 6+2+2+2+2+… 6 24 6+3+3+3+3+… 6 20 8+2+2+2+2+…
h = 3 + 2t h = 6 + 2t h = 6 + 3t h = 8 + 2t
y − int ercept = 3 y − int ercept = 6 y − int ercept = 6 y − int ercept = 8
slope = 2 slope = 2 slope = 3 slope = 2
31. Multi – Representation
Table t/d h/cm
0 3
1 5
2 7
3 9
Graph 4 11
5 13
6 15
Words Height = 3 + 2 times
the number of days
h = 3 +2d Symbols
32. Over to you on your White Boards
1. Draw a rough sketch of the graph: y = 2x + 1
33. Over to you on your White Boards
2. I start off with 6 euro in my money box and put in 3 euro each day.
Draw a rough sketch of the graph.
34. Over to you on your White Boards
3. The initial speed of a car is 10 m/s and the rate at which it increases its
speed every second is 2 m/s 2 .
Write down a linear law for the speed of the car after t seconds.
Draw a graph of the law.
30
28 Solution :
26 Speed = 10 + 2(number of seconds travelling)
24 v = 10 + 2t
22 or v = 2t + 10
20
18
16
14
12
10
8
6
4
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
36. All the graphs you a have drawn on your white
boards have been increasing functions. Do this on your
white board
Assess the learning:
Story :
Isabelle has a money box with 20 euro in it. She takes 2
euro out each day to buy sweets in the shop.
Draw a rough graph of how this might look.
Investigate if your graph is close by doing a table.
From your table, what is your rate of change?
Conclusion: Decreasing graph has a negative slope
39. Growing
Squares
Write down a relationship
which defines how many red
squares are required for each
white square.
Hint: You may need more than one representation to help you!
40. Characteristics :
• first change constant
• n term
• linear graph
Are the characteristics of…..
A Linear Relationship
41. Growing Rectangles
Complete the next two rectangles in the above pattern.
There is squared paper in your handbooks.
Is there a relationship between the:
height of the rectangle
length of the rectangle
number of tiles/area of the rectangle?
42. Table
Number of
tiles n in Change of
Height, h Length, l Area = h x l Change
each change
rectangle
1 2 2 2
+4
2 3 6 6 +2
+6
3 4 12 12 +2
+8
4 5 20 20 +2
+10
5 6 30 30 +2
+12
6 7 42 42 +2
+14
7 8 56 56
Investigate the first change and the second change.
What do we notice?
If we let n be the height, write a formula for the area in terms of n,
on your white board.
43. Draw a graph of the table
Number of
tiles n in Change of
Height, h Length, l Area = h x l
each Change change
rectangle
1 2 2 2
2 3 6 6
3 4 12 12
4 5 20 20
5 6 30 30
6 7 42 42
7 8 56 56
What do you observe about the shape of your graph?
44. Characteristics :
• first change varies
• second change constant
• n term
2
• curved graph
Are the characteristics of…..
A Quadratic Relationship
45. Story: How to ask for Pocket Money
“I only want you to give me pocket money for the month of July.
All I want is for you to give me 2 c on the first day of the month,
double that for the second day, and double that again for the 3rd
day... and so on.
On the first day I will get 2 c, on the 2nd day 4 c, on the 3rd day
8c and so on until the end of the month. That is all I want.”
Is this a good deal for my parents
or is it a good deal for me?
46. Investigate using a
Day Table
Money in cent Do this on your
white board
1 2
2 2x2
3 2x2x2
4
5
6
7
8
9
10
If we let n be the number of days, can we write a formula
for the Amount of Pocket Money?
49. What if your Dad trebled the amount of
money each day? Trebling
Doubling
Money/Cents
Time/Days
The money would grow even quicker.
50. Lets look at the Changes in a Table
Money in Change of
Days Change
cent change
1 2
+2
2 4 +2
+4
3 8 +4
+8
4 16 +8
+16
5 32 +16
+32
6 64 +32
+64
7 128 +64
+128
8 256 +128
+256
9 512 +256
+512
10 1024
What do you notice about the Change columns……
They develop in a ratio.
51. Characteristics :
• change develops in a ratio
• Formula: 2 n or 3 n
• Words: Doubling or Trebling
• curved graph that grows very quickly
Are the characteristics of…..
An Exponential Relationship
52. F = P( 1 + i ) t
Ignoring the principal, the interest rate, and the number of years by setting all these variables equal to "1", and
looking only at the influence of the number of compoundings, we get:
yearly ( 1 + 1) 1 = 2
2
1
semi − annually 1 + ÷ = 2.25
2
4
1
quarterly 1 + ÷ = 2.44140625
4
12
1
monthly 1 + ÷ = 2.61303529022...
12
52
1
weekly 1 + ÷ = 2.692596954444...
52
365
1
daily 1 + ÷ = 2.71456748202...
365
8760
1
hourly 1 + ÷ = 2.71892792154...
8760
525600
1
every minute 1 + ÷ = 2.7182792154...
525600
31536000
1
every second 1 + ÷ = 2.71828247254...
31536000
Can you see a pattern in the table? What is the key to this pattern? Students link between words and table. What is the key to this pattern? Can they see that the unit is 3, it repeats every 3. Green is 3 6 9 12 The first green is 3 The second green is 6 The third green is 9 Can they begin to see 3 x 1 st green block is the 3rd position Can they see that 3 x 2 nd green block is the 6th position Can they see that 3x 3 rd black block is the 9 th position 3 is the key Three times every green block gives the position Can they arrive at.....3n If you know where the green blocks are you can get the one before it to be black every time. Can students move from these words to: The one before green is black Can they arrive at 3n – 1. And then to 3n - 2
Identify patterns table words graph formulae Dependent (rate of change) independent variable Start amount in the graph / table/ formulae y intercept Rate of change of dependent variable in the table graph (slope) formula Linear graphs constant first differences between successive y values (outputs) Parallel lines have the same slope that is the rate of change of y with respect to x Increasing functions with positive slope
Adds 2 every day is the pattern Start is €0 The growth of money is seen is seen in the right hand column Each successive output increases by 2 Dependent (rate of change) independent variable Start amount in the graph / table/ formulae y intercept Rate of change of dependent variable in the table graph (slope) formula Linear graphs constant first differences between successive y values (outputs) Parallel lines have the same slope that is the rate of change of y with respect to x Increasing functions with positive slope
What is the difference between Mary & John?
Adds 2 every day is the pattern Start is €4 The growth is seen in the right hand column Each successive output increases by 2
What is the difference between Mary & John?
Adds 2 every day is the pattern Start is €4 The growth is seen in the right hand column Each successive output increases by 2
Constant rate of change What varies is the variable. Here we have 2 variables days and amount of money Which variable depends on which? We call one variable the dependent variable and we call the other variable the independent variable. Which is which above? The values which stay the same are constants....constant rate of change Is there a pattern? Where is the start amount of money in the table? Where is the “rate of change of money per day” in the table
Draw a graph for Mary and John Which variable should go where? Days Amount of Money Independent on X-axis, Dependent on the y-axis X values as inputs and y values as outputs What observations can you make? How is each observation seen in the graph How is each observation seen in the table Are the amounts ever the same. Explain
Where do you see starting values and growth rates in the tables ? (day 0 values and first differences) W hat do you notice about all the first differences? 2 Can students put this into words what they see in the table that is John starts with €4 and each day he gets another 2 How would you get what John has on the ??? Day Only words at ordinary Total = Starts + 2 (Number of Days) T = s + 2 d T = 4 + 2d
Adds 2 every day is the pattern Start is €0 The growth of money is seen is seen in the right hand column Each successive output increases by 2 Dependent (rate of change) independent variable Start amount in the graph / table/ formulae y intercept Rate of change of dependent variable in the table graph (slope) formula Linear graphs constant first differences between successive y values (outputs) Parallel lines have the same slope that is the rate of change of y with respect to x Increasing functions with positive slope
Where is the ‘start height’ for each plant seen in the tables ? Where is the amount the plant grows by each day seen in the tables? What do you notice about the first differences between successive outputs for all the tables? For each of the situations and tables (a), (b), (c), and (d) identify 2 values which are staying the same and 2 values which are varying. Situation and Table Varying Staying the same We call the values which vary ‘ variables ’ and the values which stay the same ‘ constants ’ . We have identified 2 variables. Which variable depends on which? We call one variable the dependent variable and we call the other variable the independent variable.
Where do you see starting values and growth rates in the tables ? (day 0 values and first differences) What do you notice about all the first differences?
Where do you see starting values and growth rates in the tables ? (day 0 values and first differences) What do you notice about all the first differences?
Where do you see starting values and growth rates in the tables ? (day 0 values and first differences) What do you notice about all the first differences?
Graphs are straight lines with constant first differences They have a constant rate of change. You can see this in the table – constant first differences between successive outputs for consecutive inputs When the left hand column goes up by a constant amount the right hand column goes up by a constant ( not necessarily the same as for the left hand column) amount. The slope is the difference between 2 outputs divided by difference between the corresponding inputs. The bigger the slope – the steeper the graph The formula has an amount you multiply the independent variable by and an amount you add - the slope and y – intercept respectively. If the graph goes through (0,0) the amount you add is 0
Various Slopes
Line with constant slope Monica decided to plant a plastic sunflower in the garden whose height was 30 cm. Draw up a table and plot a graph of the height of the sunflower from day 0 to day 5. What shape is the graph? Why is the graph this shape? As the number of days increased what happened to the height of the sunflower? What is the slope of the graph? What formula would describe the height of the sunflower? Line with a negative slope: You have €40 in your money box on Sunday. You spend €5 on your lunch each day for 5 consecutive days (take Monday as day 1). Draw up a table, plot the graph, calculate the slope and write the formula for the situation. If this pattern continues will you ever have €0 in your money box? If so, when? Is sunflower growth a realistic situation for negative slope? As the variable on the x -axis increases what happens to the variable on the y -axis? Can you think of other real life situations which would give rise to linear graphs with negative slopes? (e.g. volume of heating oil in a tank against time in days, assuming constant rate of usage per day, paying off a loan, use of credit on a mobile phone)
