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Welcome to the VULA Mathematical Literacy Holiday Workshop

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Welcome to the VULA Mathematical Literacy Holiday Workshop (April 2009) A Proposed Method for Teaching Ratios i.e. Unit Values How many bags of Surebuild cement ... – PowerPoint PPT presentation

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Title: Welcome to the VULA Mathematical Literacy Holiday Workshop


1
Welcome to the VULA Mathematical Literacy
Holiday Workshop (April 2009)
2
Using a Bicycle to Teach Mathematics
3
The History of the Bicycle
4
In 1870 James Starley made a bicycle with a large
front wheel and a small back wheel. It was called
the "penny-farthing" bicycle
5
  • The average bicycle weighed 20 kg.
  • The problem with the penny-farthing (high
    bicycle) was that it was very dangerous because
    if the rider were to stop the rider could fall
    backwards. Also it was dangerous going down a
    hill.
  • Riders needed a ladder to get on the bicycle.

6
The first bicycle with a chain was made in 1874
by H.J. Lawson.
7
Modern bicycles have come a long way ...
8
Overview of the Bicycle
9
Handle Bars
Brakes
Seat
Front Wheel
Back Wheel
Chain ring
Shock absorber
Spokes
Pedals
Frame
Deralier
Chain
10
The Mathematics of Gears
11
Ratios
12
Ratio defined
  • For example
  • comparison of the number of boys and girls in a
    class
  • comparison of the number of wheelbarrows of
    cement, sand and stone needed for concrete
  • comparison of the distance measured on a map and
    the actual real world distance.

13
We do not include units in a ratio! This is
because the relationship in size between the
values is the same irrespective of the units
being used. For example If the scale of a map
is 1 50 then 1cm on the map 50 cm in actual
distance or 1 m on the map 1 m in actual
distance or 1 km on the map 1 km in actual
distance
14
Another example A concentrate juice must be
mixed in the ratio 1 part concentrate to 3 parts
water. ? 1 litre concentrate will need 3 litres
of water ? 1 ml concentrate will need 3 ml of
water ? 1 jug concentrate will need 3 of the
same jugs of water ? cup concentrate will need 3
of the same cups of water
15
Something to note The scale of a map is 1
500. This means that 1 cm on the map 500 cm
in actual distance
(5 m in actual distance)
(0,005 km in actual distance)
16
The Mathematics of Gears
17
Rear sprocket
Deralier
Front sprocket
Chain Drive
18
Each sprocket is made up of chain rings.
There are most commonly
3 chain rings on the front sprocket
8 at the back
This means there are ???? possible gear
combinations.
19
Front Sprocket
48 teeth
38 teeth
28 teeth
20
Question Will it be harder to pedal in the
biggest chain ring (48 teeth) or the smallest
chain ring (28 teeth)?
48 Teeth
28 Teeth
Why?
21
Back Sprocket
30 teeth
24 teeth
21 teeth
15 teeth
19 teeth
17 teeth
11 teeth
13 teeth
22
Scenario 1 Consider pedalling in the big chain
ring in the front (48 teeth) and the 2nd biggest
chain ring at the back (24 teeth)
24 teeth
48 teeth
23
48 Teeth
24 Teeth
Questions 1. How many times will the wheels turn
for every 1 turn of the pedals? 2. What is the
gear ratio when the chain is on these two chain
rings?
24
Scenario 2 (a) Consider pedalling in the small
chain ring in the front (28 teeth) and the small
chain ring at the back (11 teeth)?
28 teeth
11 teeth
25
11 Teeth
28 Teeth
Questions 1. How many times will the wheels turn
for every 1 turn of the pedals? 2. What is the
gear ratio when the chain is on these two chain
rings?
26
11 Teeth
28 Teeth
Gear Ratio 28 11 ? 2,545 1
27
Scenario 2 (b) Consider pedalling in the small
chain ring in the front (28 teeth) and the 3rd
smallest chain ring at the back (15 teeth)?
28 teeth
15 teeth
28
15 Teeth
28 Teeth
Questions 1. How many times will the wheels turn
for every 1 turn of the pedals? 2. What is the
gear ratio when the chain is on these two chain
rings?
29
15 Teeth
28 Teeth
Gear Ratio 28 15 ? 1 1,8667
30
Scenario 2 (c) Consider pedalling in the small
chain ring in the front (28 teeth) and the 2nd
biggest chain ring at the back (24 teeth)?
28 teeth
24 teeth
31
24 Teeth
28 Teeth
Questions 1. How many times will the wheels turn
for every 1 turn of the pedals? 2. What is the
gear ratio when the chain is on these two chain
rings?
32
28 Teeth
24 Teeth
Gear Ratio 28 24 ? 1,1667 1
33
Scenario 2 (d) What about if you pedal in the
small chain ring in the front (28 teeth) and the
big chain ring at the back (30 teeth)?
28 teeth
30 teeth
34
Questions 1. How many times will the wheels turn
for every 1 turn of the pedals? 2. What is the
gear ratio when the chain is on these two chain
rings? 3. Will this be harder to pedal than
small (48) in front and small (11) at back
(Scenario 2)? 4. Will this be harder to pedal
than big (48) in front and 2nd biggest (24) at
back? (Scenario 1)?
35
Gear Ratio 28 30 ? 0,9333 1
36
Question Does it become harder or easier to
pedal when you change the chain on the back chain
ring to a bigger ring?
37
(No Transcript)
38
Source http//www.explainthatstuff.com/gears.html
39
Tea Time! ?
40
Activity Time
41
How many bags of Surebuild cement are needed to
build a 135 m2 wall?
27 bags
42
What percentage of the total expenditure was
attributed to Goods and Service?
13,5
43
How many British Pounds will you receive if you
exchange R5 000 into Pounds?
360
44
Approximately how long (in hours and minutes)
would it take to travel from Rustenburg to
Zeerust if you travelled at the speed limit?
1 h 36 min
45
A Toyota Verso has an average petrol consumption
rate of 12,6 litres per 100 km. If the current
petrol price is R6,82 per litre, determine how
much it will cost in petrol costs to travel a
distance of 562 km from Durban to Johannesburg.
R483,00
46
A recipe asks for 300 g of flour. How many ml of
flour is this?
500 ml
47
Use the bar scale to determine the actual length
of Wall 4 (in metres).
48
If a person were picked at random from this
sample of people, what is the likelihood that the
person would be obese?
1 in every 23 people 4,3
49
Analysing the Activities
50
How many bags of Surebuild cement are needed to
build a 135 m2 wall?
51
How many bags of Surebuild cement are needed to
build a 135 m2 wall?
52
What percentage of the total expenditure was
attributed to Goods and Service?
53
What percentage of the total expenditure was
attributed to Goods and Service?
54
How many British Pounds will you receive if you
exchange R5 000 into Pounds?
55
How many British Pounds will you receive if you
exchange R5 000 into Pounds?
56
Approximately how long (in hours and minutes)
would it take to travel from Rustenburg to
Zeerust if you travelled at the speed limit?
57
Approximately how long (in hours and minutes)
would it take to travel from Rustenburg to
Zeerust if you travelled at the speed limit?
58
A Toyota Verso has an average petrol consumption
rate of 12,6 litres per 100 km. If the current
petrol price is R6,82 per litre, determine how
much it will cost in petrol costs to travel a
distance of 562 km from Durban to Johannesburg.
59
A Toyota Verso has an average petrol consumption
rate of 12,6 litres per 100 km. If the current
petrol price is R6,82 per litre, determine how
much it will cost in petrol costs to travel a
distance of 562 km from Durban to Johannesburg.
60
A recipe asks for 300 g of flour. How many ml of
flour is this?
61
A recipe asks for 300 g of flour. How many ml of
flour is this?
62
Use the bar scale to determine the actual length
of Wall 4 (in metres).
63
Use the bar scale to determine the actual length
of Wall 4 (in metres).
64
If a person were picked at random from this
sample of people, what is the likelihood that the
person would be obese?
65
If a person were picked at random from this
sample of people, what is the likelihood that the
person would be obese?
66
A Proposed Method for Teaching Ratios i.e. Unit
Values
67
How many bags of Surebuild cement are needed to
build a 135 m2 wall?
10 m2 2 bags
10
10
1 m2
0,2 bags

135
135
135 m2
27 bags
? bags

68
What percentage of the total expenditure was
attributed to Goods and Service?
69
Goods Services Expenditure 66 981,1 mil Total
Expenditure 495 247,7 mil
495 247,7 mil
123 811,93 mil
247 623,85 mil
61 905,96 mil
So, 66 981,1 mil is 13
70
G S Total
66 981,1 mil 495 247,7 mil
495 247,7
495 247,7

0,135
1
100
100
13,5
100

?
71
How many British Pounds will you receive if you
exchange R5 000 into Pounds?
R13,9016 1,00
13,9016
13,9016
0,0719
R1,00

5 000
5 000
R5 000
360

?
72
Approximately how long (in hours and minutes)
would it take to travel from Rustenburg to
Zeerust if you travelled at the speed limit?
80 km 1 hour

80 km
60 min
80
80
1 km
0,75 min

128
128
128 km
? minutes
96 minutes

1 h 36 min
73
A Toyota Verso has an average petrol consumption
rate of 12,6 litres per 100 km. If the current
petrol price is R6,82 per litre, determine how
much it will cost in petrol costs to travel a
distance of 562 km from Durban to Johannesburg.
74
12,6 l 100 km
100
100
0,126 l
1 km

562
562
? l

562 km
70,8 l
So the cost of the petrol for the trip will
be Petrol cost R6,82 / litre 70,8 litres
R482,86
75
A recipe asks for 300 g of flour. How many ml of
flour is this?
60 g 100 ml
60
60
1 g
1,667 ml

300
300
300 g
500 ml
? ml

76
Use the bar scale to determine the actual length
of Wall 4 (in metres).
77
If a person were picked at random from this
sample of people, what is the likelihood that the
person would be obese?
78
Obese Total
390 9 154
390
390
1
23,5

OR

390 9 154
9 154
9154
0,0426
1

100
100

100

4,26
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