Pi (π)

Pi (π) is a number that is the ratio of a circle's circumference (the distance around it) to its diameter (the distance across it). No matter the size of the circle, this ratio is always the same, approximately equal to (3.14). You can think of it as how many diameters it takes to go all the way around a circle—it's always just a little over three times. 

Draw a circle or get a circular object such as a cup. Wrap a piece of string around the circle and cut it to length. Count how many times the string can be draped across the diameter of the circle. This is π.

Volume and capacity

Volume is the total amount of space an object occupies while capacity is the maximum amount of fluid a hollow container can hold.

Mr Wide and Mr Tall Using a piece of paper, make a cylinder shape in either landscape (i.e., Mr Wide) or portrait (i.e., Mr Tall) orientation. Materials required: Paper, tape, blocks.

We will test which of the two cylinder designs has the greatest capacity using the following steps:

• Each student makes a tall or wide cylinder using a single piece of paper.
• Students do not need to make a bottom for their cylinders as the table will function as the bottom.
• Tops are not required for the cylinders as easy access is required to fill the cylinders with blocks.
• Although overlapping the paper will give the cylinders more rigidity and strength, do not overlap the paper as this will reduce the capacity.
• Use tape to join the two edges of the paper to complete the cylinder.

Teaching tip: There are many ways to test the two designs but the following steps have proven to be effective:

• Find two students with contrasting designs who are willing to be representatives for the other students.
• Ask all of the students with the Mr Tall design to line up behind the Mr Tall representative (and vice versa) ensuring that all students can see the demonstration table.
• Using small construction blocks, fill the Mr Wide cylinder.
• Rather than count these blocks, simply use the same blocks to fill the Mr Tall cylinder after posing the question, "Let's find out if Mr Tall holds the same, less or more?"

Teaching points:

• During the 'Rope activity' in SILO 2.1 'Shapes and objects', we learned that shape is a determining factor when dealing with perimeter and surface area. Today we have seen that shape affects the volume too.
• Mr Wide had the greatest volume because there was an exponent involved.
• Although Mr Tall had the greatest height, height produces a linear increase.
• Mr Wide had the greatest radius and greatest capacity, as radius produces an exponential increase.
• The following formula shows how to calculate the volume of a cylinder.

There is a handy online calculator for cylinders at https://www.omnicalculator.com/math/cylinder-volume.

Exploring the ratio of surface area to volume

The main thing to note here is that the ratio of surface area to capacity is not fixed as shown in the figure below.

There are many examples of this principle in nature. For example:

• Chewing food into smaller pieces aids digestion as the surface area is increased while the volume remains the same.
• Your fingers are the first parts of your body to get cold as they have a high ratio of surface area to volume.
• Elephants have large ears to help them keep cool as their ears also have a high ratio of surface area to volume.

The ratio of surface area to volume can also be seen when building a campfire. It is common to start with kindling because kindling is small which gives it a larger surface area compared to its volume. This is why kindling burns easily.

Exploring solid objects

The following graph shows the surface area (A) against volume (V) of the five Platonic solids and a sphere. Note the following:

• The surface area decreases for rounder shapes.
• The surface-area-to-volume ratio decreases with increasing volume.
• The intercepts with the dashed lines show that when the volume increases 8 (2³) times, the surface area increases 4 (2²) times.

The following image shows some annotations beside a common medicine cup. Notice how each 10 ml has a different height due to the conical shape of the cup.

Nanotechnology and medicine

The following video (2:26) contains two examples of how nanotechnology is being used in the field of medical research to fight cancer. 

Types of scales

Ordinal scales

An ordinal scale has a non specified degree of variation. In other words, the scale is somewhat arbitrary. An example is the Mohs scale of mineral hardness introduced in 1822 by the German geologist and mineralogist Friedrich Mohs.

What do you notice when you compare the 'Mohs hardness' numbers with the 'Absolute hardness' numbers? Is there a direct correlation?

Create your own ordinal scale for hardness using materials in the classroom

The image below is a table to complete this activity. An A4 worksheet for this activity is available here.

Linear scales

A linear scale has a specified degree of variation. In other words, there is a direct mathematical correlation along the scale due to the way that it has been calibrated. An example is the Celsius scale.

Logarithmic scales

In a logarithmic scale, each interval is increased by a factor of the base of the logarithm, which is often a multiple of 10. Examples of common logarithmic scales are pH (to measure acidity), decibels (sound intensity), and the Richter scale (earthquakes).

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Answer to the exponents question about permutations with six character number plates:

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We welcome your feedback and suggestions

The chief investigator for The SILO Project is Associate Professor Brendan Jacobs, Head of Department STEM Education, University of New England. The SILO Project thrives on incremental improvement so constructive feedback is greatly appreciated. Please contact Brendan via email at bjacobs7@une.edu.au to share your thoughts and recommendations.

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