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