Hands-on Variation page 71
Hands-on Variation
p74
Lesson 24
Direct linear variation
p75
Set X:
Applications of direct linear variation
p77
Lesson 25
Partial linear variation
p78
Set Y:
Applications of partial linear variation
p80
Lesson 26
Joint linear variation
p82
Set Z:
Applications of joint linear variation
p84
Lesson 27
Inverse variation
p86
Set AA: Applications of inverse variation
p88
Lesson 28
Square and cubic variation
p89
Set BB: Applications of square & cubic variation
p91
Lesson 29
Inverse square variation
p92
Set CC: Applications of Inverse square variation
p93
Set DD: Car safety and variation
This forms the fifth set of resources in a series on Ratio, Proportion and Variation. The other titles are
‘Hands-on Ratio’, Hands-on Scale’, ‘Hands-on Proportion’ and ‘Hands-on Trigonometry’.
These resources comprise lesson plans to guide you to teach the mathematical ideas, and group activity
task cards to enable students to grapple with these ideas in the context of real applications. There is also a
suggested format for the group to produce a report of there investigations, either orally, based on rough
notes, or in writing. There will be software (spreadsheets) for exploration in that mode. And to tie it all
together there are unit plans.
At the start of each unit there is an introductory note to the teacher discussing the mathematical ideas for the
unit, the pedagogy that will make it work in the classroom, student activities and assessment.
The mathematical ideas in this unit
The topic ‘Variation’ provides a link between direct proportion calculations (as in ‘Hands On Proportion and
Percentages’) and more formal algebra and graphs.
Lesson 24 and set X: Direct linear variation
Direct variation covers all types of variation where the graph passes through the origin. In a pure
mathematical sense, ‘Direct linear variation’ is the same as ‘Direct proportion’. The formulas are all of the
type y = mx and the straight-line graphs pass through the origin. However the range of application in this unit
are more complex than in the previous unit.
Lesson 25 and set Y: Partial linear variation
Partial linear variation corresponds to formulas of the type y = mx + c, and the graphs are straight but do not
pass through the origin. Again there are many applications.
Lesson 26 and set Z: Joint linear variation
Joint linear variation is the product of two variables, such as A = LW. There are many practical examples of
this in real life, although we often keep one of the variables constant; for example, price = rate x quantity has
very many applications where the rate is known as the ‘unit price’.
Lessons 27, 28 and 29 and sets AA, BB, CC and DD: Other types of variation
Non-linear variation is approached by extending the linear variation ideas to show that some relationships
are different, and hence their graphs are not linear.
The most common real life alternative is inverse variation, where the idea is a fixed product; the graph is a
hyperbola and the idea of asymptotes is introduced. Square and cubic variation (both direct, i.e. through the
origin) and inverse square variation have an extensive range of real life applications.
Finally the variation unit is summed up in two similar studies involving car safety. They use spreadsheets.
Braking
This uses ideas of direct linear, and square variation as well as inverse and joint variation to compare the
effects of mass (direct), speed (square) and available braking force (inverse) on the stopping distance.
Cornering
This uses ideas of direct linear, and square variation as well as inverse and joint variation to compare the
effects of mass (direct), speed (square) and radius of turn (inverse) on the force required to make a turn.
Hands-on Variation page 72
Teaching style for this unit
It is intended that the class move as a whole, but that there is considerable choice and flexibility within most
lessons to accommodate the considerable range of interest and readiness in most classes. The plan
alternates between class lessons and group activities. Use the software to reinforce the mathematical ideas
either in school times or at home.
Class lessons
The purpose of the class lessons is to reflect upon previous experiences as a group, and to introduce new
ideas. In the class lessons, teacher-led discussions are particularly important. Try to ask questions that
will lead most students to think mathematically and put the mathematical ideas into their own words. As you
go there will be a rich diversity of experiences to draw upon so that the big, common ideas will become clear
to all.
Group activities
The purpose of the group activities is to let students explore the mathematical applications appropriate to the
mathematical ideas being studied. These sessions should allow students to be hands-on with equipment, to
move around the classroom, and where needed around the building. In these sessions the teacher’s role is
to be a guide and a manager. Let them know your expectations of behaviour, and keep firm control on those
that need it. As you move around the groups, keep them on task, ask questions and seek to raise the level of
thinking. You are teaching them to think for themselves and become team problem solvers! This is best done
initially by providing short timelines, clear directions, and high expectations about reporting.
Student investigating and reporting
The best parallel for these mathematical investigations is a practical science investigation. Students should
either choose or negotiate one of the possible activities available on the day. It is quite OK if some activities
are not tackled, and if all groups choose to do the same one that is fine, providing you have the equipment.
For their chosen activity they should write an Aim, their Method, their Results and their Conclusion. You
can determine the level of detail according to your knowledge of the literacy levels of the students. You might
provide a single sheet with these subheadings as a template.
Reporting should sometimes happen to the class, orally. In these cases, you might not expect such a
detailed written report, but choose last least one group on most sessions to share their aims, methods,
results and conclusions with the rest of the class.
Assessment
Clearly there will be plenty of involvement for you to witness, to document (digital camera, notebook, etc),
and there will be some written reports. The emphasis must be on assessment for learning (formative) so
that you note how well different students are learning by observations and conversations, and adjust your
choice of activities for them on the basis of this feedback.
Hands-on Variation page 73
Unit plan (3 weeks)
Linear variation
Class lesson 24: Direct linear variation
Group activities using card set X (Applications of direct linear variation)
Class lesson 25: Partial linear variation
Group activities using card set Y (Applications of partial linear variation)
Class lesson 26: Joint linear variation
Group activities using card set Z (Applications of joint linear variation)
Other variation
Class lesson 27: Inverse variation
Group activities using card set AA (Applications of inverse variation)
Class lesson 28: Square and cubic direct variation
Group activities using card set BB (Applications of square and cubic variation)
Class lesson 29: Other types of variation
Group activities using card set CC (Applications of other types of variation)
Class lesson 30: Car safety and variation
Two explorations of joint variation
Software for this unit
Floating, World records, Types of variation, Families of graphs, Rectangle area, Linearising variation,
Braking, Cornering,
Hands-on Variation page 74
Lesson plans and activities
Class lesson 24: Direct linear variation
Introduction
It is assumed in this lesson that students have previously experienced some of the material from earlier units
in this set, particularly Hands-on Proportion and Percentages. In those lessons and activities students have
been introduced to proportion and straight-line graphs through the origin to express the ‘double-double’
relationship of proportion. The idea that doubling one quantity also doubles the other (and in fact this holds
for any multiplier) is also the basis of ‘direct linear variation’. The basic formula is still y = mx.
So the first section of this unit should be a review of the ideas from previous work, and it would be helpful if
you could make positive links between this material and student’s previous experiences. If students have not
tackled proportion before this unit you will have to build the basic ideas from scratch.
Lesson ideas
Unit prices
Ask the class to list those items for which we do not normally get a reduction in unit price by buying in bulk.
Examples include petrol, transport, postage and gas. If you double the quantity, you double the cost.
Ask for typical unit prices and help students to produce formulas, such as (2008 prices):
•
c = 1.7v where $c is the cost, and v litres is the quantity of petrol.
•
c = 55n, where c cents is the cost of n stamps
Exceptions to this simple rule will be those for which different rates apply for different quantities, such as
electricity and water.
Currency conversions
Discuss the meaning of ‘exchange rates’ and look up some recent ones on the internet (or get them from the
nightly TV news). Look at the conversion from Australian dollars to selected overseas currencies, and from
those currencies back to Australian dollars, showing the difference between buying and selling rates.
Consider a small international trip requiring exchanging Australian dollars to one currency (exchange 1),
then to another currency (exchange 2). Is this cheaper than exchanging direct from Australian dollars to the
second currency or not?
Write some formulas for the exchange rates, using the same formula (y = mx) as above.
Walking:
The formula y = mx and the direct linear variation graph
How fast do you walk? If students have no real idea of their
walking speed, they can run an experiment.
Distance
walked (m)
Mark out a track 10 m or 20 m long. Students walk up and
back on this track at a ‘normal’ speed and see how far they
walk in 10 seconds. They should convert this speed to
metres per second.
The relationship between distance and time includes their
speed as the multiplier: d = st.
They can draw a simple graph with this formula.
Now they can measure distances by walking and
converting the time taken into metres. Note that doubling
the time must double the distance.
Take time to link this activity with the others – to generalise
the idea of direct linear variation.
Time taken (secs)
Hands-on Variation page 75
Activities Set X: Applications of direct linear variation
1 Stretch and load for a spring
Equipment: rubber band or light spring, paper or plastic cup attached firmly to the end of the rubber band (or
spring), masses (preferably in 100 g units).
Set up a spring or rubber band, hanging vertically from a firm support.
Measure its length when full length but unstretched.
Add mass of 100 g. This is the ‘load’. The band will stretch.
Measure its new length and calculate the amount of stretch. Put the results into the table.
Load (g)
0
Stretch (mm)
0
100
200
300
Draw a graph using Excel, to show your results.
Either find a simple formula by yourself or use Excel to find a simple formula for your data.
2 Rolling speed
Equipment: a plank, a ball and a watch that clearly shows seconds.
Use a book or a small box to set the plank at a small angle. You will release the ball from the top of the
plank.
Use a marker of some sort to show how far the ball has rolled after 1 second. (You might need to roll it
several times to be sure where to place the marker.)
Use a marker of some sort to show how far the ball has rolled after 2 seconds.
Repeat for 3 seconds and continue until the ball has rolled off the end of the plank.
Measure the distances and enter them into a table.
Time (s)
0
Distance (mm)
0
Average speed to this marker (mm/s)
0
1
2
3
Use Excel to draw a graph of the relationship between time and the average speed for that distance.
Use some method to find the simple formula to predict the average speed for any time.
3 Gears
Equipment: some gears (for example, from Meccano, bicycles or Spirograph), ruler
For any one set of gears (with the same tooth size e.g. gears on a bike, or in a Spirograph set), find the
relationship between the number of teeth (count them!) and the diameter of the gear.
Use Excel to draw a graph of the relationship and find a formula for it.
4 Bolts
Examine as many different bolts and screws as you can.
For each find the pitch of the thread. (The pitch is the number of millimetres per turn.) As you turn the bolt
round one turn (with a spanner) the bolt will move this number of millimetres forward.
You will probably find that there are several standard pitches.
Does the pitch depend on the diameter of the bolt?
Hands-on Variation page 76
5 Ramps and slopes
Equipment: a long plank for a ramp, and a chair or something to support one end fairly high.
The point of a ramp is that it saves lifting a heavy load and turns it into sliding or rolling it up the ramp.
As you go up the ramp by 10 cm, how much has the load actually been lifted?
How much has it been lifted when you have gone 20 cm up the ramp?
Complete the table to show the relationship.
Distance up the ramp (cm)
0
Distance raised (cm)
0
10
20
30
Draw a graph of this relationship using Excel.
Work out a formula for this relationship.
6 Dish-washing detergent
Equipment: Some detergent, a bowl, some water and a measuring jug.
When you use a concentrated detergent for washing up, how much detergent do you need for how much
water?
Try a typical amount of wash up water. Measure the amount in the jug.
Tip a typical amount of detergent into the jug, and measure it before tipping it into the water.
Work out the ‘concentration’ in mL per L – the amount of detergent (millilitres) per litre of water.
7 Density
Equipment: small pieces of a wide variety of solids, such as styrofoam, foam rubber, foam plastic, woods,
eraser, paper, glass, pottery, brick, tile, steel. Accurate scales for mass (weight). Small measuring jug or
cylinder, water.
Arrange the solids in what you think is the order of their ‘density’. Density is the mass per unit volume.
Now measure the mass of each solid in grams (g).
Measure the volume of each solid. Partly fill a small jug or measuring cylinder. Then lower the solid into a
measuring jug or cylinder, and observe the rise in the water level. Convert the rise in level (in mL) directly
3
into the volume of the solid (cm ).
3
3
For each solid, calculate the density (g/cm ) by dividing the mass (g) by the volume (cm ).
Compare the results with your estimates of the order.
Try the spreadsheet ‘Floating’.
8 Density of liquids (often called Specific gravity)
Equipment: a measuring jug or cylinder, accurate scales for mass (weight), a collection of different liquids,
such as water, kerosene, brine, oil, milk.
For each liquid measure the mass of a carefully measured volume. In this way find the density (the mass in
grams for each unit of volume in mL).
Which would float on top of others?
Try the spreadsheet ‘Floating’.
9 Atomic and molecular masses
•
Find out how ratios are used to ‘balance’ chemical equations.
•
Find out how molecular masses, in grams, are used to find out mass of reagents and products in
chemical reactions.
Hands-on Variation page 77
Class lesson 25: Partial linear variation
The important difference between direct linear variation (y = mx) and partial linear variation (y = mx + c) is
the constant c. It means that the simple ‘double-double’ rule (that holds for direct linear variation) does not
hold between pairs of values in partial linear variation.
Of course partial linear variation is just the general straight-line formula, but the variation approach to it is
much more practical and not theoretical.
The lesson includes several related examples, and it is the task of the teacher to help link the abstract
mathematical ideas so that students understand what partial linear variation is, and how it differs from direct
linear variation.
Taxi fares
Students will be able to provide the current ‘flagfall’ cost and the ‘travel distance’ cost for local taxi fares. If
not, they should research these on the internet.
In 2008 the Melbourne metropolitan rates were as follows:
Flagfall
Distance
Time
$3.20 when meter is started
$1.526 per kilometre
54.7 cents per minute if the speed is below 21 km/h
For the purposes of simpler discussion, it is useful to round these to simpler numbers, such as $3.00 flagfall
and $1.50 per km. At this stage, ignore the time factor.
Students should make a table of the cost to travel various distances, from 0 to 5 km. They should graph the
values, and see that the graph starts at $3 (flagfall is the constant) and increases at a rate of $1.50 per km
(this is the gradient of the graph). Then they should work out a simple formula, c = 3 + 1.5d, where $c is the
cost to travel d km. (The use of the more exact numbers only makes the formula look more complex.)
Extension: slow travel
If the car is stopped, or travelling slowly due to traffic delays, the driver is not getting paid at a high enough
rate. So the Time rate kicks in. What is the rate (dollars per hour) that is used for slow travel?
At the speed of 21 km/h, what would be the earnings from the Distance rate?
Distance-time graphs
Imagine two runners – maybe a typical student and Bolt, the current world record holder for 100 m.
Bolt decides to give himself a handicap. You can have 50 m start! As you reach 50 m, Bolt takes off and runs
100 m in world-beating time. Who gets to the 100 m mark first?
Clearly it depends on your speed, so we need to find that out first.
Mark out a distance of 25 m. A volunteer student runs the
distance and we find his/her average speed.
Distance (m)
Calculate the time taken for the student to reach 50 m.
Assume that the student can keep running at that speed up to
the 100 m mark.
Bolt runs the distance in 9.7 seconds.
•
Who wins?
•
What is the winning margin (in seconds)?
•
What is the winning margin (in metres) as the winner
crosses the line?
To answer this last question it is helpful to draw a distance time
graph of the race. There are two lines, with Bolt starting after a
number of seconds. (The diagram shows Bolt winning, but this
might not happen; it depends on the speed of the student.) The
vertical line shows the distance margin; the horizontal line
shows the time difference.
Help students to work out the formulas for the two lines.
Time taken (secs)
Hands-on Variation page 78
Activities Set Y: Partial linear variation and “y = mx + c”
1 Hiring a car
The cost of hiring a car, even for one day, depends on several factors: the size of the car, the cost of petrol,
the distance you travel, whether or not you take extra insurance and whether or not the company gives you a
number of free kilometres.
•
Explore the websites of several well-known hire car companies to check on these details and learn about
the actual costs and what affects them. You always have to pay for your petrol, and return the car with a
full tank. Do not ignore this charge.
•
Find some typical charges and explain how they depend on the car size.
•
•
A ‘flat rate’ means that the charge is simply an amount per day, with no extra per kilometre.
(This is a constant daily cost: $y = $c.)
•
A ‘per kilometre’ rate means that you pay for the distance you travel.
(This is a rate, m ($ per km), where x is the number of km you travel.
The cost ($y) is the product $m, so y = mx.)
•
Many deals combine these two, and charge a smaller amount flat rate’ and a smaller amount ‘per
kilometre’. (This is the combination rule. The cost y is given by y = mx + c.)
Write some formulas to explain the real hire car prices you find from the internet.
2 Selling tickets
You are organising a pop concert. You have hired the venue, the band and other things that make up the
expenses. Some costs, such as food, will depend on the number of people who buy tickets. So you know
that you must at least raise that much money through ticket sales, to ‘break even’.
•
Invest some realistic prices for the venue, the band and the other items that should be covered as
expenses.
•
Now set ticket prices at an amount that is realistic – not too much so people will not buy, but not so
cheap that you don’t even cover your costs.
•
Your income depends only on ticket sales – so it is direct variation. Some costs, such as food, will
depend on the number of people who come.
•
Create a table of values showing the financial outcome of your pop concert for different numbers of
people buying the tickets. (You might be able to use a spreadsheet for this purpose.) Your table should
include the fixed costs and the variable costs, such as food, that depend on the number of people. In
addition your table will show the money taken from ticket sales, and the profit (or loss).
•
Try a table like this for different ticket prices, to see what difference it could make.
3 Converting to LPG
You will know that LPG is considerably cheaper to buy at the garage than regular (unleaded) petrol, and
cheaper still than diesel. Research some realistic prices, in terms of rates: cost per litre.
Some car owners have paid an up-front expense to have their car converted to using LPG, usually as a
combined option, so you can continue to use petrol if LPG is not available where you are driving.
•
Take a car in which you are interested, and research the one-off cost of LPG conversion.
An important issue is ‘how long would it take for you to save enough in the cost of fuel to cover the cost of
the conversion?’ Once you have reached that point in time, you start to be ‘ahead. So this is the ‘break even’
point.
To find the breakeven point’ there are three steps:
•
firstly find the number of litres you need for each 100 km for each type of fuel
•
secondly, find the number of kilometres that gives a price saving equal to the cost of the conversion,
•
and finally estimate how many months or years it would take for you to travel that far at the ‘normal’ rate
of car use.
An interesting issue is the question of why so few people have actually made the conversion.
Hands-on Variation page 79
4 Relation between height and mass
For adults or teenagers over 16 years, it is suggested that there is a relationship between height (in cm) and
mass (in kg).
•
To learn about this for yourself, you need to find the height and mass of many people over 16 years.
There might be a difference between the results for males and females, so it would be good to keep
those results separate. For example, you could get results for footballers by accessing the statistics of
different team players. Try www.afl.com.au, and look at the player characteristics.
•
To find the relationship it is good to see all the results on a graph called a scatterplot. You could put
mass horizontally, and height vertically.
•
If it seems reasonable to draw an ellipse around most of the points on the graph, then there is a
relationship between height and mass.
•
If you have used a spreadsheet you can also use it to find a Trendline (under Chart, Add Trendline).
Choose Linear, and the line will be drawn. Estimate the rate of change (m) of the Trendline, and also the
value of the Trendline for zero mass (c). Use these numbers to estimate the formula for the Trendline (y
= mx + c, where y is the height (in cm) and x is the mass (in kg)).
5 Athletic records
We know that there is improvement of athletic performances over time. Is this a regular, steady increase? At
what rate is it occurring?
You can access all the world and Olympic records at www.olympic.org/uk/sports/records/results_uk.asp.
They are not easy to use in a spreadsheet without a little help. So on the CD there is a spreadsheet called
‘World records’. This has 100 m, 1500 m and High jump, all for men, from 1912 until the present day.
•
Select one set of results. You will see the scatterplot. Select the chart, and then choose Chart >
Trendline. The Trendline shows how the changes have occurred.
•
Estimate the rate of change (m) of the Trendline, and also the value of the Trendline for the year 1900
(c). Use these numbers to estimate the formula for the Trendline (y = mx + c).
•
For the 100 m records, y is the time (in seconds) and x is year after 1900. Use this to predict the year
when the 100 m time could reach 9.5 seconds. Note that the world record was broken in 2008; is it likely
that it will ever reach 9.5 seconds? Explain.
•
For the 1500 m records, y is the time (in seconds) and x is year after 1900. Use this to predict the year
when the 1500 m time could reach 200 seconds. Note that the world record has not been broken since
1998; is it likely that it will ever reach 200 seconds? Explain.
•
For the high jump records, y is the height (in m) and x is year after 1900. Use this to predict the year
when the 1500 m time could reach 2.6 m. Note that the world record has not been broken since 1993; is
it likely that it will ever reach 2.6 m seconds? Explain.
6 Ball speeds
Any falling object will get faster (i.e. accelerate) and this is due to the effect of gravity. You can measure the
acceleration due to gravity with a simple experiment.
The first requirement is that you have a building, flag pole or other tall object of which we know the height.
•
You will throw the ball upwards, from a height of about 1.5 m (when you let go the ball). It should slow
down gradually (due to gravity), reach the known height of the pole, and stop exactly at that height.
Then, of course, it will reverse its path and accelerate downwards. As it lands you can measure the time
it takes for the total up-and-down trip. Half of this will be for the downwards section.
•
Because you know the height from which it fell, and the time it took, you can find its average speed. The
final speed will be twice this, since it started from a ‘stopped position’ at the top and steadily increased
its speed on the way down.
•
Find the rate at which the speed increased on the way down. This is the acceleration due to gravity. It is
measured in metres per second – per second.
Hands-on Variation page 80
Class lesson 26: Joint linear variation
Joint linear variation involves the product of two linear variables. If one of these variables is kept constant
then the variation reduces to direct linear variation. There is no constant being added.
The best-known example is Area = Length x Width;
However there are very many others and this lesson will start to open up the possibilities through
consideration of wages: pay = rate x time.
Areas of rectangles
Use questioning to make sure students understand these ideas.
•
The diagram below shows an array of rectangles. The area is width x height.
•
If the height is kept constant – along any row – then the area depends only on the width. For
rectangles of the same height the area depends only on the width (Area = constant x width)
•
If the width is kept constant – up and down any column – then the area depends only on the height.
For rectangles of the same width the area depends only on the height (Area = constant x height).
•
Taken together, area = width x height.
Wages
The amount you are paid depends on two quantities that can vary: the time you work, and the rate of pay.
Discuss this table and ask students to complete it.
Hours worked
0
1
2
3
4
Rates: $0 per hour
$20 per hour
$40 per hour
$60 per hour
$80 per hour
•
If the rate is kept constant – along any row – then the pay depends only on the time.
Pay = constant x time
•
If the time is kept constant – up and down any column – then the pay depends only on the rate.
Pay = constant x rate
•
Taken together, Pay = rate x time.
Hands-on Variation page 81
Tables and 3D graph
Students should be able to complete the
multiplication table shown. Make your own
copy on a whiteboard.
0
10
20
30
40
50
60
70
80
90
0
9
18
27
36
45
54
63
72
81
You can use sheets of paper to cover all
except rectangles that include the bottom
left corner. An example is shown.
0
8
16
24
32
40
48
56
64
72
0
7
14
21
28
35
42
49
56
63
The area of the visible rectangle is the
number in the top right corner. Along any
row we have simple direct linear variation,
and up & down any column also direct linear
variation.
0
6
12
18
24
30
36
42
48
54
0
5
10
15
20
25
30
35
40
45
0
4
8
12
16
20
24
28
32
36
0
3
6
9
12
15
18
21
24
27
0
2
4
6
8
10
12
14
16
18
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
Imagine that each number is a column with
height being the number shown. This is then
a 3D graph of the relationship – see below.
The spreadsheet ‘Types of variation’ has a
tab that shows Joint linear variation. It is
illustrated below. Z = xy.
The same spreadsheet has other types of variation so you can compare them:
•
2-variable linear: z = ax + by
•
direct linear: z = ax (no change for different values of b)
•
partial linear: z = ax + b (no change for different values of y) and
•
joint linear: z = ax times by.
In all cases, a and b are constants (which you can change to see the effect on the graph), and x, y and z are
variables.
Hands-on Variation page 82
Activities Set Z: Joint linear variation and “z = kxy”
1 Volumes of boxes
The formula for the volume of a box is: volume = width x height x depth.
•
Explain why this is an example of joint linear variation. Give some examples.
2 Distance, speed and time
The formula for distance is: distance = speed x time.
•
Explain why this is an example of joint linear variation. Give some examples, using many speeds and
times.
3 Simple interest
The formula for simple interest on a loan is: simple interest = principal (loan) x rate x time.
For a particular amount invested, on what does the amount of interest depend?
Explain why this is an example of joint linear variation. Give some examples.
4 Levers
The formula for the turning effect (torque) of a lever is: turning effect = distance x force.
For example, try pushing a door very close to the hinge!
•
Why do you need a smaller force when you push further from the hinge?
•
Explain why this is an example of joint linear variation. Give some examples.
5 Forces, masses and accelerations
The size of a force can be determined from its ability to make a mass increase in speed, that is ‘accelerate’.
So a small mass can be accelerated by a small force (a small engine in a small car), but a larger mass
needs a much larger force to produce the same acceleration (large trucks need bigger engines). The small
force will have only a little effect on a large mass (me pushing a large truck), and a large force acting on a
small mass will make it accelerate a large amount (e.g. slap the puck with a hockey stick)
•
The formula is: force = mass x acceleration. Explain why this is an example of joint linear variation. Give
some examples, using a range of situations.
6 Ohm’s law for electricity
In science you may have learned about Ohm’s law, linking voltage (or electromotive force E) and current
through resistance. E = IR, using E volts, I amps of current, and R ohms of resistance.
•
Explain why E = IR is an example of joint linear variation. How does a fuse protect a home? What is the
minimum resistance needed to give a current of 10 amps with 240 volts? Why do fuses burn out?
Hands-on Variation page 83
7 Electrical power and energy
The power rating of many electrical appliances is given on the appliance itself. Examples will include a
100 W bulb, and a 1700 W electric jug.
•
List the power rating of common appliances in your home. For each, use the formula P = EI, E = 240
volts and its power rating to calculate the current each uses. Then use E = IR (Ohm’s law) to find the
resistance of each appliance.
•
Explain why P = EI is an example of joint linear variation.
The electrical energy (K) used by each appliance depends jointly on its power rating (P Watts) and the time
(t hours) for which it is used: K = Pt. For example a 100 W bulb burning for 10 hours produces 100 Watthours, or 1 kiloWatt-hour (kWh).
•
Find the energy used by running each appliance above for a typical number of hours.
8 Families of graphs
We have seen how you can use a 3D graph to
show a joint variation relationship. However there
are other ways. One is shown in the spreadsheet
Families of graphs.
Open the spreadsheet and use the tab called
‘rectangle area’, shown. You will see a family of
five line graphs, each of which is direct linear
variation.
•
The one chosen depends on the width you
use for your rectangle.
•
The point along that line depends on the
length (vertical height) that you use.
Explore the possibilities.
Write a sentence or two describing how each
family of graphs operates.
9 Types of variation
This investigation asks you to explore the types of
variation that are displayed in the spreadsheet:
Types of variation. The tabs at the bottom show
different kinds. For each type, 3D graphs are drawn.
The same spreadsheet has other types of variation so
you can compare them:
•
2-variable linear: z = ax + by
•
direct linear: z = ax
•
partial linear: z = ax + b and
•
joint linear: z = ax times by.
In all cases, a and b are constants (which you can change to see the effect on the graph), and x, y and z are
variables.
Explore the possibilities.
Write a sentence or two describing the differences between each type of variation.
Hands-on Variation page 84
Class lesson 27: Inverse variation
The basic idea is that as one quantity gets larger the other gets smaller, and vice versa. However there are
two ways in which this could occur – having a constant total (i.e. the two numbers are added) and having a
constant product (i.e. the two numbers are multiplied). Inverse variation occurs when the values of two
related variables have a constant product.
Many joint variation relationships can be considered here; to become examples of inverse variation, the
product stays constant.
Rectangles with constant area
There are many combinations of height and width that multiply to the same area, particularly if you include
fractions! The diagram on the left shows areas of 24: 24x1, 12x2, 8x3, 6x4, 4x6, 3x8, 2x12 and 1x24.
You can produce a similar result for any constant product using the spreadsheet: Rectangle area.
0
9
18
27
36
45
54
63
72
81
0
8
16
24
32
40
48
56
64
72
0
7
14
21
28
35
42
49
56
63
0
6
12
18
24
30
36
42
48
54
0
5
10
15
20
25
30
35
40
45
0
4
8
12
16
20
24
28
32
36
0
3
6
9
12
15
18
21
24
27
0
2
4
6
8
10
12
14
16
18
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
Sharing a cost
Imagine that a number of people are splitting the cost of a pizza, or a number of pizzas. The grid shows (with
circles) one person paying $8, 2 paying $4 each, 4 paying $2 each, and 8 paying $1 each.
It also shows (with squares) the more expensive $18 pizza, shared by 2, 3, 6 and 18 people.
To fill in the gaps between the numbers shown on the grid, the circles or squares are joined with smooth
curves.
Students will easily see the similarity between the two diagrams.
Note that for inverse variation doubling one value halves the other.
The hyperbola and reciprocals
The hyperbola graph has been introduced above.
Discuss the graph of xy = 1, and the equivalent form y =
1
.
x
Discuss reciprocals, and use calculators to discover a few,
checking that they always multiply to 1, such as 0.25 x 4.
Note that the reciprocals of fractions get very large.
Note also that there is NO reciprocal for 0. It is not ‘infinity’,
it is simply not defined – it has no meaning.
You might want to discuss how the graph approaches the
axes – asymptotically (i.e. but never touching).
Hands-on Variation page 85
Transforming inverse variation to linear direct variation
The table of values shows xy = 24, or y =
x
y
24
. Its graph will be a hyperbola.
x
1
24
2
12
3
8
4
6
6
4
8
3
12
2
24
1
Suppose we find the reciprocal of each y-value. This produces this table.
x
1
1
24
y
2
1
12
3
1
8
4
1
6
6
1
4
8
1
3
12
1
2
24
1
Here is the new graph.
y
1
.5
0
2
4
6
8
10
12
14
16
18
20
22
24
x
Demonstrate the spreadsheet: Linearising variation.
Insert the data from the table. If there is not enough data, simply leave them blank.
The data is plotted in five ways:
•
in black, the data itself (x against y)
•
in red, the x-values are squared and x is plotted against y.
•
in purple, the x-values are cubed and x is plotted against y.
•
1
1
in blue, the reciprocals ( ) are used, and is plotted against y.
x
x
•
in green, the squared reciprocals (
2
3
1
1
2 ) are used, and 2 is plotted against y.
x
x
If one of these transformations is linear, then that indicates the type of variation involved.
1
1
In this case, using the reciprocal of x ( ) transforms the hyperbola into a straight line, so y = k( ) is the
x
x
formula. We then put a pair of values into this formula to find the value of k.
Using x = 24 and y = 1 produces y = k(
1
24
). So k = 24. Hence the formula is y =
.
24
x
Hands-on Variation page 86
Activities Set AA: Inverse variation and “xy = c”
1 Balance beam
Use a metre ruler. Balance it at its centre point. On one end place a small load of (say) 100 grams. You can
balance this is 100 g at the opposite end.
•
How far from the balance point can you balance it with 200 g? Where with 300 g? 400 g? 500 g? Create
a table of values.
•
Is inverse variation involved; if one value is doubled, is the other halved?
•
Write the formula from the table of values. Sketch the graph.
2 Speed and time over constant distance
Imagine that you have a variety of ways of travelling 1200 km. They are set out in the table, with typical
average speeds.
Vehicle
Speed
Time
constant
walk
3 km/h
run
10 km/h
bicycle
20 km/h
car
100 km/h
plane
600 km/h
•
For each trip, calculate the time taken in hours.
•
Check whether or not inverse variation is involved; if one value is multiplied by a number, the other
should be divided by the same number.
•
Work out the constant, and write the formula.
3 Gears on a bicycle
You need a bicycle. The chain connects two gears; your bicycle might have more; if so start with one gear
setting only.
•
The best way to measure the size of the gear is to count the number of teeth it has. Do this for both
pedal gear and gear on the rear axle.
•
Now we are interested in the fact that the rear axle gear moves at a different speed to the pedal gear.
You will need to mark a particular gear cog, so you will know when the gear has turned a whole number
of times.
•
Turn the pedal and count both the number of pedal gear turns and at the same time the number of rear
gear turns. (You should use a fairly large number, between 30 and 60).
•
Multiply the number of turns by the number of teeth for each gear. Is this inverse variation?
If you have more gears on your bicycle, try the same experiment with different sized rear wheel gears.
4 Pressure
Pressure is an amount of force per unit area. You can possibly walk on thin ice without cracking the ice, but
ice skates have a much smaller area on which to concentrate your weight, so skating on this ice is
dangerous.
•
Find your weight – use kilograms as the unit.
•
Find the area of one of your shoes that is actually in contact with the floor as you walk. (This is not just
the area of the shoe, because it doesn’t all touch the floor as you walk.)
•
Work out the pressure you exert on the floor as you walk on it.
•
A skateboard touches the floor over a very small area. The force is concentrated on a small area.
Estimate the weight of yourself plus your skateboard. Estimate the tiny area actually in contact with the
floor. Work out the pressure. This is quite likely to damage many floor surfaces.
•
To protect the floor you should walk and carry the skateboard. Work out the pressure with the same
weight (you plus skateboard) and your shoe area found above.
•
Is this inverse variation? Does multiplying the area by a number result in reducing the pressure by the
same number?
Hands-on Variation page 87
5 Electrical fuses
You may know Ohm’s law, E = IR, which is joint variation involving electromotive force (E volts), current (I
amps) and resistance (R ohms). In Australia we use 240 volts of electromotive force, IR is a constant value
of 240.
If the current gets too large, possibly because we are using too many appliances at once, wires can overheat
and cause an ‘electrical’ fire. A fuse is designed to prevent this. It is a small length of special wire that will
melt when the current gets to a particular value. This will stop the current and prevent a fire.
•
Use the formula IR = 240, to find the resistance (in ohms) of a fuse wire that is designed to melt at 10
amps.
An electric jug makes use of this heating ability of a large current. It uses special wires that will heat up when
the power is on.
•
Use P = EI to find the current in one electric jug (2400 W) operating at 240 volts.
Two jugs at once is a risk as the current may overheat the wires carrying the current around the house.
6 Radio and electromagnetic waves
Electromagnetic waves, such as AM and FM radio, have both wavelengths and frequencies.
•
Find out what KHz and MHz mean.
•
For each AM and FM radio station, look up the two values and put them into a table.
•
Discover whether inverse variation is involved. If one value is doubled, is the other halved? Work out the
constant product value – it is the speed of light!
7 Concentrations
Concentration means the amount of a substance, e.g. salt, per litre of the solvent (e.g. water).
Imagine that you have dissolved 5 g of salt in 1 L of warm water. The concentration is 5 g/L.
•
Suppose you now need to make it into a solution with 2 g/L. What does the volume need to become?
•
Check you answer by working out how many grams of salt there will be in each litre.
•
Now you want all of your 2 g/L solution to be changed into a solution with 8 g/L. You must add salt. How
much?
•
Finally add water to make it a solution with concentration of 5 g/L. How much water is needed?
8 Gas laws
Density and volume
Density means the amount of mass contained in a unit volume; for gases it is measured in g/L. For example
air has a density of 1.293 g/L at 0°C at sea level.
•
As a gas warms up its volume increases. Since it is the same mass of gas, its density decreases. This is
inverse variation, relating density (d) and volume (v). Write the formula.
Pressure and volume
A bicycle or car tyre pumped up very tight has a large pressure. (What is the unit for measuring pressure in a
car tyre? What is a typical value?) When the air is released it greatly increases its volume as the air pressure
decreases.
•
The same amount of air in a tyre twice the size would have half the pressure (once they are both at the
same temperature). This is inverse variation, relating pressure (p) and volume (v). Write the formula.
Hands-on Variation page 88
Class lesson 28: Square and cubic direct variation
Square direct variation describes a relationship where one variable is proportional to the square of the other.
A well-known example is the area of circles; the area depends on the square of the radius.
In contrast, cubic direct variation depends on the cube relationship. It is called cubic to show its simplest
example – the volume of a cube.
Growing cubes
Have a large number of cubes available for the demonstration. The idea is to make cubes of different sizes
and work out the surface area and the volume of each. Before long students will be able to give you the
correct numbers and you won’t have to make the cubes!
Side length
Surface area
Volume
SA/Vol
0
1
2
3
4
5
10
Discuss the formulas that describe the relationships and draw the graphs, noting the differences.
•
2
3
Surface area = 6x , (square variation, parabola), Volume = x (cubic variation, cubic) and
6
Surface area/volume = (inverse variation, hyperbola).
x
Linearising square and cubic variation
Side length
(side)2
Surface area
0
0
0
1
1
6
2
4
24
3
9
54
4
16
96
5
25
150
2
The formula for the new linearised relationship is SA = 6 x (side) .
2
If we use the same method we get V = (side) for the volume.
Side length
(side)3
Volume
0
0
0
1
1
1
2
8
8
3
27
27
4
64
64
5
125
125
Demonstrate this with the spreadsheet: Linearising variation, if you are able (e.g. on an interactive
whiteboard.)
Hands-on Variation page 89
Activities Set BB: Applications of square and cubic variation
1 Oranges
How does the area of the peel of an orange depend on its diameter?
How does the volume of an orange depend on its diameter?
You will need at least three oranges of different size.
•
Measure their diameters (see diagram).
EITHER
Partially fill a measuring jug with water and read the water level. Measure the volume of each orange by
lowering it into a measuring jug and reading the increased water level. The increase is the volume of the
orange.
Diameter
Surface area
Volume or Mass
0
OR: Measure the masses of the oranges using a balance.
•
Draw a graph to show the surface area and then another to show the volume, for each diameter.
If you have trouble working out the formulas, try linearising.
Diameter
(Diameter)2
Surface area
Diameter
(Diameter)3
Volume or Mass
0
0
2 Pizza value
Standard pizza sizes were invented in America, so the diameters are given in inches: small inches, medium
inches and large inches.
At any shop you can buy these three sizes in your favourite topping. Choose your topping and find the prices
for all three sizes.
The pizzas are circles. The area depends on the square of their radius. Does the cost also depend on the
square of the radius? Which pizza is the best value for money? Why?
3 Enlargements
•
You know that squares fit together to make larger squares – you can go to size 2, 3, 4, 5, etc.
When you do, how many squares are needed?
•
Make a set of right-angled triangles, all exactly the same. Can you make enlargements of this triangle to
size 2, 3, 4,and 5?
How many triangles are needed?
•
The shapes below can be used to make enlargements, to size 2, 3, 4 and 5. How many would you
need?
Hands-on Variation page 90
4 Pendulum
Using a watch, learn to tap exactly and regularly in half seconds.
Now create a simple pendulum, using a long length of thread and a single small nut or other heavy mass.
Your challenge is to find the lengths of thread that enable the pendulum to gently swing back-and-forth in the
times in the table.
Time (sec)
Length (cm)
0
0.5
1
1.5
2
Then draw a graph, possibly using Excel or a calculator.
To see whether or not it is square or cubic you might need to linearise.
Finally find the formula.
5 Rolling ball
Using a watch, learn to tap exactly and regularly in half seconds.
Now you need to find or create a gentle slope and roll a ball down it.
The challenge is to find the length that the ball will roll in each of the times in the table. The number of
seconds you will be able to use will clearly depend on the angle and the length of the slope.
Time (sec)
Length (cm)
0
0.5
1
1.5
2
Then draw a graph, possibly using Excel or a calculator.
To see whether or not it is square or cubic you might need to linearise.
Finally find the formula.
6 Car power
How does the power of a car depend on the number and radius of its cylinders? Can you write a formula?
Hands-on Variation page 91
Class lesson 29: Inverse square variation
Inverse square variation combines two previously met ideas.
Inverse means that the second quantity gets smaller as the first gets larger.
Square means that it occurs as the square of the first quantity.
Demonstration
Turn the room lights off. Shine a flashlight onto a surface close to you., say 1 metre. The light is bright.
Then move twice as far away, to 2 metres. The light is not half as bright, but a quarter as bright. This is
because the rays are now spread over four times the area.
Move to 3 m away, and the light is one ninth as bright.
This situation applies to many forms of radiation: light, sound, gravity, magnetism, radio intensity.
Photography
The amount of light that is allowed into a lens is determined by the size of the hole that is opened in front of
the lens for the duration of the photograph. This hole is called the ‘aperture’.
A camera uses f-numbers to label the amount of light that gets in. However as the f-number gets larger the
area of the hole gets smaller.
f-number
2
area (mm )
1.4
2
2.8
4
5.6
8
11
16
22
960
480
240
120
60
30
15
7.5
3.75
What is the relationship?
Find the type of variation
It is clearly inverse, but as the f-number doubles (e.g. from 2 to 4) the area drops to a quarter. This looks like
inverse square.
Check by putting the data into the table in the spreadsheet Linearising variation. This will graph not only
1
1
2
3
x against y (i.e. f-number against area), but also x against y, x against y, against y and 2 against y.
x
x
1
2 against y) shows the type of variation involved. In this case,
x
area depends on the inverse square of the f-number.
The graph that is a straight line (in this case,
Find the formula
We can even find the formula exactly.
Choose two values, e.g. x = 8 and y = 30.
1
The formula y = k 2 and k is just a constant of proportion. We can find its value as follows.
x
1
Substitute the two values,. so that 30 = k x
. This means that k = 30 x 64 = 1920.
64
So y = 1920(
1
1920
2 ) or y =
2 . To find the area, divide 1920 by the square of the f-number.
x
x
Have you noticed the pattern in the f-numbers – either on a camera or in the table above?
Hands-on Variation page 92
Activities Set CC: Inverse square variation
1 Heat energy from the Sun
Heat radiation received by the planets (and Pluto) from the Sun depends on their distance from the Sun.
The table gives the distances from the Sun in Astronomical Units. One AU is the average distance of the
Earth from the Sun.
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Distance (AU)
0.387
0.723
1.00
1.52
5.20
9.54
19.2
30
40
2
Heat (kW/m )
1.4
This is an example of inverse square variation.
Find the constant of proportion k in the formula: y = k
1
2 .
x
Use this to write the formula.
Use the formula to calculate the heat energy per square metre for each of the other planets, and Pluto.
2 Gravity and the Sun
The force of gravity, that keeps the planets in their orbits, clearly is weaker for planets further from the Sun.
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Distance (AU)
0.387
0.723
1.00
1.52
5.20
9.54
19.2
30
40
Gravity
1.00
This is an example of inverse square variation.
Find the constant of proportion k in the formula: y = k
1
2 .
x
Use this to write the formula.
Use the formula to calculate the force of gravity on each of the other planets, and Pluto.
Hands-on Variation page 93
Activities Set DD: Car safety and variation
This section involves two examples of joint variation with a lot in common. Both involve cars and safe driving,
but mathematically they are remarkably similar. Because of their similarities they are handled in very similar
ways.
•
There is a spreadsheet for each, with two parts: a simulation of a car moving along a road, and a set of
graphs showing the various relationships.
•
For each, there is a rather parallel set of questions suggested for the use of the spreadsheet.
You could assign these in either order. There are several ways you might use them.
•
You could run each as class lessons, using the questions suggested as the basis of your lesson
plans.
•
You could run one as a class lesson (possibly using an interactive whiteboard) and discuss the
issues, and use the worksheet to set the other as a similar investigation for students to explore in
small groups.
•
You could set both to be explored by small groups using the worksheet questions as guides, as
assignments that could be graded.
•
Using the worksheet questions as guides you could set one for half the groups in your class, and the
other for the other half – and get the groups to prepare reports for the whole class on what they have
learned.
The mathematical formulas used are these:
2
Stopping distance s =
mv
,
2F
where s metres is the distance taken for a car of mass m kg travelling at v km/h to stop by applying a braking
force F Newtons to the tyres from the road.
Students should understand that the force F depends on the conditions of the brakes, road surface and the
tyres. If the brakes cannot slow the wheels enough, the force will not be enough to slow the car. If the road is
wet or the surface is loose, or the tyres are worn, then the available force is reduced. The car will skid out of
control.
In this spreadsheet the speeds are shown in km/h, but for calculations are converted to m/s by dividing by
3.6.
2
Turning force F =
mv
,
1000r
where F units is the force needed for a car of mass m kg travelling at v km/h to turn in a circular arc of radius
r metres.
Students should understand that the actual force that the road to apply to the tyres depends on the road
surface and the condition of the tyres. If the road is wet or the surface is loose, or the tyres are worn, then
the available force is reduced. The car will skid out of control and fail to turn the bend.
In this spreadsheet the speeds are shown in km/h, and this is used for calculations. For this reason the force
units are ‘arbitrary’.
Hands-on Variation page 94
Stopping distance
When you use brakes to slow a car rapidly, the speed is reduced because the road applies a force to the
tyres. The car experiences an acceleration that is negative. It is negative because the speed gets smaller
each second. Eventually the car stops, when the speed is 0.
Explore
Open the spreadsheet ‘Stopping distance’.
1. Type a mass for your car (1000 kg) and a safe speed of 50 km/h.
2. Suddenly you have 200 m in which to stop before hitting a child. Type the 200 m as the braking distance,
the distance away from the child when you hit the brakes.
3. Your brakes are in good condition, and your braking force is 500 units. Use F9 to see the car move, slow
down and eventually stop.
You avoided the accident. Your stopping distance was less than the braking distance. The force needed to
stop in time was less than the force you applied.
Now let’s change some things and see what else might have happened.
4. Firstly let’s use a heavier car – either a bigger model, more passengers, a load in the boot or pulling a
trailer or caravan. Double the mass. What happens?
5. Go back to the original mass. Increase the speed, say from 50 km/h to 60 km/h. What happens?
6. Go back to 50 km/h. Suppose you were not concentrating, and you have only 150 m to stop. What
happens?
7. Finally let’s make the braking less efficient. This might be for several reasons:
•
the brakes need fixing
•
the road is wet
•
the road has loose stones on it.
•
If your brakes are good but the road is slippery, then your good brakes might lock the tyres and you
skid out of control, taking much further to stop.
For any of these reasons the force you can apply could reduce to 400 Newton. What happens?
Understand
•
Now explore the graphs, using the second tab on the spreadsheet.
•
Each graph shows the effect of changing one of the three variables (Mass, Speed and Force) on the
stopping distance. It uses the values of the other two variables as constants.
•
The red marker on the graph shows the values of mass, speed and force used on the ‘Braking’ part of
the spreadsheet. You can change the values there and then look at the effects on the graphs.
One way to explore the graphs is this.
1. Double the mass and see the effect on the stopping distance. It doubles, showing direct linear variation.
The straight line graph supports this.
2. Double the speed and see the effect on the stopping distance. It quadruples, showing direct square
variation. The parabolic graph supports this.
3. Double the force and see the effect on the stopping distance. It halves, showing inverse variation. The
hyperbola graph supports this.
2
Stopping distance s =
mv
2F
When driving, the only variable you can sensibly control is your speed.
You cannot tell in advance what your braking distance to a possible collision will be.
If your car is loaded, with people or other things, remember to drive more slowly.
If your brakes are not good, or the road is wet or has a loose surface, your braking force will not be
good, so slow down.
Hands-on Variation page 95
Cornering
When your car turns a corner it is because the road applies a force to the tyres that pushes the car round the
corner – making it turn in an arc of a circle. If the road is wet or slippery or your tyres are worn, the road
might not be able to apply enough force to make the car take the bend.
Explore
Open the spreadsheet ‘Cornering’.
1. Type a mass for your car (1000 kg) and a safe speed of 60 km/h.
2. You come to a corner and turn it at 60 km/h with a turning radius of 10 m. Type 10 for the radius.
3. Your tyres are in good condition, and your turning force is 400 units. Use F9 to see the car move safely
around the corner. (It will keep repeating.)
Your available turning force was more than the force needed to make the turn.
Now let’s change some things and see what else might have happened.
4. Firstly let’s use a heavier car – either a bigger model, more passengers, a load in the boot or pulling a
trailer or caravan. Double the mass. What happens?
5. Go back to the original mass. Increase the speed, say from 60 km/h to 70 km/h. What happens?
6. Go back to 60 km/h. Suppose you were not concentrating, and you have to turn with a radius of 8 m.
What happens?
7. Finally let’s make the road grip less efficient. This might be for several reasons:
•
the road is wet
•
the road has loose stones on it.
•
If you turn too sharply or at too high a speed, but the road is slippery, then your tyres will not grip the
road and you skid out of control, failing to take the bend.
For any of these reasons the force you can apply could reduce to 300 Newton. What happens?
Understand
•
Now explore the graphs, using the second tab on the spreadsheet.
•
Each graph shows the effect of changing one of the three variables (Mass, Speed and Radius) on the
turning force needed. It uses the values of the other two variables as constants.
•
The red marker on the graph shows the values of mass, speed and radius used on the ‘Cornering’ part
of the spreadsheet. You can change the values there and then look at the effects on the graphs.
One way to explore the graphs is this.
1. Double the mass and see the effect on the turning force needed. It doubles, showing direct linear
variation. The straight line graph supports this.
2. Double the speed and see the effect on the turning force needed. It quadruples, showing direct square
variation. The parabolic graph supports this.
3. Double the radius and see the effect on the turning force needed. It halves, showing inverse variation.
The hyperbola graph supports this.
2
Turning force needed F =
mv
1000r
When driving, the only variable you can sensibly control
is your speed. If the road is wet or has a loose surface,
your turning force will not be good, so you should slow
down for the bends.
If you ‘cut the corner’ with a radius of 20 m or over,
you can corner at a higher speed.
But you are on the wrong side of the road.
If you meet another car you either collide, or swerve to
miss the car and then try to complete the turn with a very
small radius. You will probably skid and go off the road.
