Transformation: Year 9: Planning tool

Expected level of development

Australian Curriculum Mathematics V9: AC9M9SP02

Numeracy Progression: Understanding geometric properties: P7, Proportional thinking: P7

At this level, students can apply the enlargement transformation to shapes by hand and with dynamic geometry software. They will identify aspects of the shape that remain the same after enlargement, and those that change and justify this with appropriate explanations.

The important aspects to consider are a shape’s dimensions, area, and volume. It is critical to define what is meant mathematically by ‘enlargement’ and ‘similarity’.

Explore different pairs of shapes to determine whether they have been enlarged with similarity or slight deformity. It is beneficial to present some pairs with one shape rotated or reflected position. This tests students to think critically. Some students may jump to the conclusion that the shapes are no longer similar – a useful teachable moment.

Afford opportunities to find relationships between the side lengths and the magnitude of the angles. Ask questions: can all the new side lengths be divided by the same number to return to the original dimensions (dilation). This highlights the first aspects of change and continuity and that although the lengths of the sides are changed by the transformation, the ratios between the sides remain unchanged. If this is not the case, the two shapes are not similar.

Give opportunities for students to explore the area of shapes, and the surface area and volume of 3D shapes. Question students on the pattern of continuation that is observed in the increase in size of the side lengths for all these cases. Dynamic software applications are particularly useful here and can visually demonstrate that area of a 2D shape and the surface area of its corresponding solid is increased by multiplying by the square of the scale factor, and that the volume of the solid is increased by multiplying by the cube of the scale factor. Use questioning again to clarify the distinction between the rules for area, surface area and volume. Why does surface area (for solids) follow the same rule as area (plane shapes)?

This topic is the gateway into trigonometrical ratios – AC9M9SP01

Teaching and learning summary:

Revise congruence and similarity of plane shapes and the associated rules.
Revise ratio in the context of scale drawing.
Use examples of enlargement to help explain the meaning of scale factor.
Test for similarity first before making any other conclusions.
Look for patterns in the changes to the area and volume of enlarged shapes.
Draw connections between these patterns and the meaning of similarity, and dimensionality.
Use the properties of similarity to solve problems.
Use dynamic geometry software to perform enlargements and calculate areas and volumes.
Use reasoning to explain similarities and differences between pairs or sets of similar shapes or objects.

Students:

check whether shapes are similar and use this as the basis for drawing additional conclusions
use dynamic geometry software to enlarge shapes and objects
know the relationship and patterns between side lengths, area and volume, the scale factor, and similarity.
can explain why the side lengths, area, and volume change according to the patterns observed.
can explain why the ratio between sides, magnitude of angles, and form, are the only unchanged aspects.
solve problems relating to similarity and enlargement.

Some students may:

confuse similarity and congruence.
not check the scale factor is the same for every side.
assume a diagram is to scale and make assumptions based on this assumption.
not realise that the orientation of an object does not affect its properties.
believe that corresponding sides are ‘matching’ regardless of the orientation of the shape.
assume that area and volume increase by the same factor as the side lengths.
assume that surface area and area are impacted differently by enlargement.

Stimulate curiosity and allow for students to discover patterns to explore a range of enlargements. A convenient way to clarify this to students is through digital applications like PowerPoint. Demonstrate how a shape can be enlarged by dragging one of its corners to become larger; show that by holding the shift key during enlargement the shape maintains uniformity or it risks distortion. Students learn that similarity is defined by all sides increasing by the same scale factor, and the angles remaining constant.

The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.

Learning intention

We are learning how to test whether an enlarged shape has maintained its form.
We are learning how to apply patterns and draw conclusions about similar enlarged shapes.

Why are we learning about this?

Using one piece of information to infer more complex information is an important skill. Area and volume can be time-consuming to compute as we need to know several measurements and then perform a series of calculations. If we can draw conclusions about these based on a shape’s similarity to one another, we can save ourselves a lot of time and tedious calculation. And don’t discount how smart you’ll feel when you can reel off a series of facts about a shape without taking a single measurement! All thanks to the power of patterns.

What to do

Our homes are filled with screens these days. Televisions, phones, tablets, smart home hubs, laptops, gaming devices, and PCs. Which is your favourite viewing experience? Is there any mathematical similarity between these different screens?

Let’s investigate any patterns that exist between the variety of screens around the home.

Below is a table with a list of devices that you might find at home.

Device	Length	Width	Ratio
Living room TV
Video game console
Smart phone
Laptop

Create your own table to record the dimensions of a variety of screens in your home.
Add an additional column for recording the screen ratio. This is calculated by dividing the length by the width.
You may notice the ratio 16 : 9 is frequently used to refer to 'widescreen'. This ratio means that the length is approximately 1.78 times the width.
Do any of your devices match this common ratio?
Are any of your screens similar? That is only true if they have the same ratio!
If any of your screens are similar you should determine the scale factor between them. This is done by finding the number that you can multiply the dimensions of the smaller screen by to produce the dimensions of the larger screen.
How much larger is the area of the larger screen? (Remember – area increases by the square of the scale factor.)
Consider your favourite screen. Which is the most pleasurable to view?
How could you adjust the dimensions of the other screens to make them similar to your favourite?
List a set of new dimensions for the other screens that would most closely match their original size, while taking on the ratio of your favourite.

Success criteria

I can test objects around my home for similarity and apply the patterns of enlargement to them.
I can test whether an enlarged shape has maintained its form.

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