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Year 9 Measurement
Unit Plan
Length Perimeter Area Volume Capacity Project
Length
How Long is a Piece of String???
The Metre (m)
The metre (m) is the base unit for length in the metric system.
Originally the metre was thought to be one ten-millionth of the distance from the north pole to the equator through Paris, France.
The Metre (m)
From the metre, other units of length were devised to measure smaller and larger distances. Millimetres (mm), Centimetres (cm) and Kilometres (km)
Estimations Worksheet
Conversion Rules
Lesson Summary
3 things that I learned today
2 things that I enjoyed
1 thing that I want to learn about
Conversions Worksheet
Perimeter
The Perimeter
We say that a soccer field has a boundary line that is marked out along the playing perimeter.
However, in mathematics we use the word perimeter when we are talking about the distance around a figure.
Find the Perimeter of These…
…And These…
Can you work out the Formula?
Square:
Rectangle
Try These Ones…
Applied Questions
Circumference
Around we go…
Circumference
The circumference of a circle is the perimeter or length around the circle.
Investigation
You will need: Some Cylinders
Soft drink cans, Toilet roll, Piece of pipe…
Investigation
What to do: Step 1
Measure the diameter, of one object as shown. The distance between the two marks is the diameter of your object.
Investigation
What to do: Step 2
Mark a point on the circumference of your object and then roll it for one complete revolution as shown. The distance between the two marks is the circumference of your object.
Investigation
What to do: Step 3
Copy the table below and fill it in for your objects.
Object Circumference
diameter Circumference
diameter
1.
2.
3.
The Life of Pi (π)
It has been known for thousands of years that whenever the circumference is divided by the diameter the answer is always the same.
You should have found this in the previous investigation (allowing for slight inaccuracies in measurements).
The Life of Pi (π)
The actual value for:Circumference diameter
for any circle lies between 3.14 and 3.15
This value is symbolised by the Greek letter π (pi).
The Life of Pi (π)
So, for any circle,Circumference = π diameter
In other words,
Circumference = π × diameter or
C = π × d.
Circumference
Since the diameter is twice the length of the radius, we can also write:
Circumference = π × 2 × radius or C = π × 2 × r
diameter
radius
Pi
The exact value for π cannot be written down because it is a non-terminating (does not stop) and non-recurring (does not repeat) decimal.
The value of π, correct to 36 decimal places is: π = 3.141 592 653 589 793 238 462 643 383 279 502 884 …
Pi
Try These Questions…
Try These Questions
A cylindrical water tower has a base diameter of 7 m. What is the circumference of the base?
A circular flower bed has a radius of 2.5 m. What is the perimeter of the edge of the bed?
Try These Questions
A bicycle wheel has radius 40 cm. Find the circumference of the wheel. How many kilometres would be travelled
if the wheel rotates 10 000 times? How many times does the wheel rotate
if the bike is ridden 10 km?
An Investigation
Which fits better… A round peg in a square hole, or A Square peg in a round hole?
Try These Questions
Find the perimeter of the door:
Try These Questions
Find the perimeter of these shapes:
Practical Uses of Perimeter…
Sunshine Run – how far do you actually run?
The Octagon – how far is it to walk around the Octagon?
Area
Area is the amount of surface within a two-
dimensional shape.
At Home…
Around the home, there are many surfaces such as driveways, paths, floors, ceilings and walls.
All such surfaces have boundaries, that is, they are enclosed within a two-dimensional (2-D) shape.
Information on cans of paint and bags of fertiliser refer to the area they can cover. Similarly, garden sprinklers cover a certain area of lawn.
Some Measures
1 square millimetre (mm2)
1 square centimetre (cm2)1 square metre (m2)
1 hectare (ha)
Try these questions..
What unit would you use?
Another Measure 1 km2 is one kilometre square which is
1000m × 1000m
Population Density is one measure that uses km2.
It measures how densely populated countries are by: Counting the number of people in the country Measuring the land area and then dividing
PopulationLand Area
Population Density
It measures how densely populated countries are by: Counting the number of people in the
country Measuring the land area and then
dividing
PopulationLand Area
Our Octagon…
Can you work out the area inside the Octagon?
How many people can we fit inside?
North Island vs South Island
Which one would you prefer to live in?
Which is more “crowded” or has highest population density?
North Island vs South Island
3 148 400 people live in the North Island It is 113 729 km².
Therefore the number of people per km² or the population density is:
3 148 400 = 27.7 people per km² 113 729
North Island vs South Island
1 008 400 people live in the South Island
It is 151 215 km².
Therefore the number of people per km² or the population density is:
1 008 400 = 6.7 people per km² !!!151 215
Now you try!
Have a go at some area questions
Page 289, questions 3 & 5
Page 293, question 9, 10 & 11
Sheet (ask Mrs Holman or Mr Porter)
Triangles Investigation
Make a box – make sure the corners have 90o angles.
Triangles Investigation
Put a dot anywhere along the top side of your box.
Now draw lines as shown.
Remember this Investigation?
Put a dot anywhere along the top side of your box.
Now draw lines as shown.
Watch This Clip…
A Triangle is always half a rectangle – no matter what type of triangle you have.
Formula:
Area of a Triangle = ½×length×width
Another Formula
The formulae for the areas of triangles, parallelograms and trapezia can be derived from the formula for the area of a rectangle.
Area of triangle = ½×(base×height)
Why?
Find the Area
Of these Triangles…
Find the Area
Of these Triangles…
Parallelogram
The Area of a Parallelogram is:
Area of parallelogram = base × height
Why?
Find the Area
Of these Parallelograms…
Trapezium
The Area of a Trapezium is:
Area of parallelogram = a + b
2x h
Why?
The reason here is because if we add a second trapezium of exactly the same shape we form parallelogram.
Find the Area
Of these Trapezia…
Area of a Circle
We’re going to do an experiment… Start with a circle of
radius (r) What is the
circumference?
Area of a Circle
Now cut the pieces up of the circle
And line them up as shown. What shape does this look like? Can we work out the area of it?
Area of Circle
And finally…
Some Practice
Have a go at a few of these questions…
Starter…
Use your calculator to find the area of the following circles, correct to 2 decimal places:
Starter…
For a circle of diameter 2.6 cm find, correct to 2 decimal places: its perimeter its area
A sprinkler sprays water in a circle of radius 3.4 m. Calculate the area of lawn it waters.
A goat is tethered to a post by a 5.4 m rope. What area can the goat graze?
Starter…
Find the Area of these shapes:
VolumeThe volume of a three-
dimensional object is the amount of space it occupies, and this
space is measured in cubic units.
Common Measures
1 cubic millimetre (mm3) is the volume of a cube with side length 1 mm.
1 cubic centimetre (cm3) is the volume of a cube with side length 1 cm.
1 cubic metre (m3) is the volume of a cube with side length 1 m.
What Unit would you use?
Volume Challenge
I have a box where the: length = 1cm width = 2cm height = 3cm
Draw it. What is it’s Volume?
Consecutive lengths
Volume Challenge
I have a box where the: length = x cm width = (x+1) cm height = (x+2) cm
If it’s Volume = 24cm3, what are the side lengths?
Consecutive lengths
Volume = 24cm3
What other shaped boxes can I make if the volume = 24cm3
Volume Challenge
I have a box where the: length = x cm width = (x+1) cm height = (x+2) cm
If it’s Volume = 336cm3, what are the side lengths?
Consecutive lengths
Summary
Is there a relationship between the areas of any three consecutive numbers?
Complete the following table:Consecutive Numbers Area
1,2,3
2,3,4
3,4,5
4,5,6
5,6,7
6,7,8
Prisms
A prism is a solid figure with a uniform cross-section.
Prisms
A simple prism is the rectangular prism shown alongside.
Check that you agree with the following facts: There are 2 layers. There are 12 (i.e. 4 × 3) cubes in each layer. There are 24 (i.e. 12 × 2) cubes altogether. The volume of this rectangular prism is 24 cm3.
Boxing On…
The Sweet-tooth Company has hired you to design boxes to hold sixty-four sugar cubes. Each cube has edges of 2 cm, just like multilink cubes. The boxes have to be the shape of boxes (cuboids) as there should not be sugar cubes sticking out.
What sizes of boxes could they have? How many different boxes could be
made?
Try These
Find the Volume of the following shapes:
Volume and Capacity
Use any of the packets from the big box and take measurements.
You need to work out what measurements to take to work out the volume and capacity of the packets that you choose.
Further Prisms
For any prism, the volume can be found by multiplying the end area by the length.
Further Prisms
How would you work out the end area of this shape?
Further Prisms
How would you work out the end area of this shape?
Further Prisms
How would you work out the end area of this shape?
Try These…
An Extra Challenge…
Capacity
The capacity of a container is a measure of the largest volume it can hold.
Capacity
We use the term capacity instead of volume when we talk about fluids (i.e., liquids and gases).
For example, the capacity of a cup is the amount of liquid it can hold.
The litre (L) is the basic unit for the measurement of capacity.
Familiar Capacities
Units of Capacity
Units of Capacity
Conversions
Your Turn…
Calculate the capacity of this container given the internal measurements shown:
Your Turn Again…
Find the capacity, in litres, of a fish tank: 1 m by 2 m by 50 cm
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