# Euler’s Formula with jubes and toothpicks

Learning Intention: To distinguish between prisms and other three-dimensional shapes and to work out the relationship between vertices, edges and faces (Euler’s Formula).

These two Year 7 students are making three-dimensional models with jubes and toothpicks (or satay sticks) to record vertices, edges and faces. Start with simple shapes, such as triangular pyramids, cubes and square pyramids, distinguishing between shapes that are prisms and those that are not. When students have made at least six or more shapes ask them to see if they can find a relationship between V (vertices); E (edges) and F (faces). I usually give a clue that they only need to use addition and subtraction (not multiplication or division).

# Tessellations

Image source

Learning Intention: “Describe translations, reflections in an axis, and rotations of multiples of 90° on the Cartesian plane using coordinates. Identify line and rotational symmetries.” “Define congruence of plane shapes using transformations.”

Success criteria: Students will create a design that shows various transformations of a 2D geometric shape and be able to describe the translations, reflections and rotations that have taken place to form a tessellation.

Tessellations have been popular decorations for hundreds of years, as this tiled ceiling of the Sheikh Lotfollah Mosque in Iran (1602-1619) shows. Any shape or shapes that can be repeated to fill a 2D plane can be considered tessellations; so, equilateral triangles, squares, rectangles and hexagons are all simple shapes that can be tessellated.

Maurits Cornelis Escher (1898-1972) is one of the world’s most famous graphic artists. His art has been reproduced extensively, especially his ‘impossible structures’ and his tessellations. You may have seen some of his work in your art classes. This YouTube video, Escher’s Tessellations,  showcases some of his work.

We are going to create our own tessellations using one of the following methods.

1. Create a Tessellation using the paper-cut method (YouTube)

2. How to make an Escher-esque Tessellation (YouTube)

3. How to create a Tessellation (YouTube)

This fantastic site, Tessellations.org, has lots more examples of tessellations and describes different methods for creating your own, including a praying mantis design submitted by an Australian student. I would love you to create a design that you are really proud of that you can also submit to the site.

# Geometry – shapes and objects

Image Source

Learning Intention:

• Students will be able to identify and describe 2D shapes and understand the terms translation, reflection, dilation and rotation.
• They will be able to identify and describe, draw, plan and construct 3D objects.

Success Criteria:

• Students will complete a poster that shows various shapes undergoing transformations and construct a tessellation from appropriate 2D shapes.
• They will identify and describe 2D and 3D shapes from photographs.
• They will draw isometric drawings with dot paper and nets and construct 3D shapes.

So far we have learnt about the properties of polygons, especially triangles and quadrilaterals,  in terms of their sides, angles, perimeter and area. Next we will be looking at TRANSFORMATIONS – how 2D shapes are translated (moved), reflected, rotated and dilated. Your task is to create a poster that shows the letters of your name undergoing each of the following transformations – translation, reflection, rotation and dilation. Draw two copies of each of four block letters of your name and then complete a transformation with one of the letters of each pair.

Polygon matching game – identifying 2D shapes.

Sort the shapes – identify and describe polygons

Polygon sorting – regular and irregular polygons.

More interactives from the Maths Zone (2D shapes)

Classifying 2D and 3D shapes – Geometric Figures Game

Drag and drop 3D shapes – Naming 3D shapes

Matching 3D shapes – identifying 3D shapes

More interactives from the Maths Zone (3D objects)

# Composite shapes

Image Source

Learning Intention:  Understand that the area of all composite shapes can be calculated by breaking the shape down into known areas.

Success Criteria: You will be able to summarize the formulae to calculate the area of the following 2 dimensional shapes and break down a composite shape by identifying each of the known shapes and correctly calculating their area.

• squares and rectangles
• triangles
• parallelograms
• trapeziums
• circles and annuli

Some examples of these kinds of problems are in Exercise 8G (page 326) of MathsQuest8. Here are some other activities to learn about the area of composite shapes.

• An L-shaped room has a perimeter of 20m. What might the area of the floor be?
• The difference in areas of two rectangles is 32cm (squared). What might the widths and lengths of the two rectangles be?

Your homework (due Friday) is one of the following:

• Use Google Maps or Google Earth to find an aerial view of an unusual shape, take a screen shot and calculate it’s area using the scale measurements. Copy your screen shot into “Paint”, add the measurements, calculate the area and email it to me. You may choose a large building, carpark or arena (the MCG or Etihad stadium for example) or perhaps your farm or property.