Let me tell you a story about three little pigs. The first pig built a house that was 2 metres high by 3 metres wide by 4 metres long. The second little pig wanted a bigger house, so he doubled the dimensions – his house was 4 metres high, by 6 metres wide by 8 metres long. The third little pig wanted to have the biggest house, so he doubled the dimensions again and built a house that was 8 metres high, 12 metres wide and 16 metres long.
Your task is to calculate the surface area and volume of each of the three houses and work out the ratio of SA:V for each house. Assuming that all the houses were made of the same materials and labour was not included, which house would be the cheapest to build per unit volume?
Learning Intention: Students will understand how to calculate the area of a triangle, using the rule Area = 1/2 x base x height.
Success Criteria: Students will draw and label at least four different triangles of the same area.
How many different triangles can you draw with an area of 120 cm sq? Use graph paper, where 1 cm ~ 10 cm and label the base and the height, showing any right angles.
Games based learning has been a hot topic in the last year or so and one of the most popular games in educational use has been Minecraft. Although there has been some debate about the value of learning outcomes, many students would agree that Minecraft is a fun way to learn about surface area and volume. This YouTube video, “Minecraft Math – Surface Area and Volume” describes how to calculate the surface area of rectangular prisms and challenges the viewer to calculate the surface area and volume of a huge tower of TNT blocks! Another YouTube video, from the same user, demonstrates the “Volume of Prisms and Pyramids” in Minecraft and offers a challenge to calculate the area of a prism with a pyramid on top.
We spent last lesson looking at the volume and surface areas of various patterns of ‘minecraft’ blocks. Our assumption is that each Minecraft block is 1m x 1m x 1m – a cubic metre. Next lesson I would like you to create your name in Minecraft blocks and measure the volume and the surface area of your construction. Start by using the first letter of your first name. It should be a minimum of five blocks high and three blocks wide. Make sure you take a screenshot of your construction and send it to me by email.
Learning Intention: Students will investigate Euler’s rule that describes the relationship between the number of faces, the number of edges and the number of vertices of 3D objects.
Success Criteria: Students will create 3D shapes using toothpicks and jelly-lollies to represent edges and vertices. They will then count and record in a table the faces, edges and vertices of the shapes and investigate Euler’s Rule.
Today we are going to learn more about 3D shapes and investigate a special relationship between the number of vertices, edges and faces of such shapes. Create the following shapes using toothpicks and jubes:
- triangular prism
- square base pyramid
- pentagonal pyramid
- pentagonal prism
- hexagonal pyramid
- hexagonal prism
Now draw a table with six columns that records the name of the polyhedron, the name of the base shape, the number of sides on the base shape, the number of faces, number of vertices and number of edges. Complete the table for each of the 8 shapes listed above. Now see if you can work out any relationship between the values in your table.
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.
- 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.
- 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.
We will then learn more about 3D shapes and how they are drawn and constructed.
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)
Building in Perth (author’s photo)
1. Can you find an acute, right, obtuse, straight and reflex angle in the image above?
2. Fold an A4 sheet of paper into six sections. In each section draw one of the following angles, estimate it’s size and then measure it. Make sure you label each drawing.
(e) complementary angles
(f) supplementary angles
Learning Intention: You will understand that angles are measured in degrees and be able to name and identify different types of angles and triangles. You will be able to use a protractor effectively and be able to estimate angle sizes.
Success Criteria: You will be able to estimate, draw, measure and identify angles measured in degrees. You will be able to identify, name and draw triangles by side length and by angle size.
This term we will start by measuring angles, using a “rotogram” and then a protractor. You will all need your geometry kit this semester – compass, sharp grey lead pencil, protractor and a good ruler.
What are acute, right, obtuse, straight and reflex angles?
What are equilateral, isosceles and scalene triangles?
What are acute, right and obtuse triangles?
Can you have an equilateral right angled triangle or an obtuse equilateral triangle?
Can you have an obtuse isosceles triangle or a right-angled isosceles triangle?
Task 1: Work in a group of three. Each person should draw and cut out six copies of an equilateral, isosceles or scalene triangle, using different coloured paper. Share your shapes so that each person in the group has two of each type of triangle. Now, tear off the corners of one of each type of triangle and match up the corners to demonstrate the sum of the angles in a triangle.
Task 2: Do the same for acute, obtuse and right angled triangles. Do you think the sum of the angles in a triangle will always be the same? Can you explain why? Do you think the same would be true for quadrilaterals, pentagons and hexagons etc?
The Maths Masters have written a very interesting article in The Age about the triangles in Federation Square.
To understand how cartesian coordinates are used on maps and in mathematics to describe locations and linear equations. Also to understand the relationships between co-ordinates that are reflected across the x and y axis.
Students will be able to draw up a cartesian plane, correctly locate and plot coordinates on the plane, as well as be able to identify where points exist using cartesian coordinates. They will complete the games and activities above and identify what they have learnt from each of the interactives.
Maths is Fun – Cartesian Co-ordinates
Shodor Interactives – General Co-ordinates
The links above take you to several activities to help you learn about Cartesian coordinates. Go to the first activity (Interactive cartesian co-ordinates) and plot three points to form an equilateral triangle in one quadrat. Write the co-ordinates down in your book, then reflect the triangle into each of the three other quadrats. List the plotted points in your book. What do you notice about the relationship between the plotted points?
Now. draw a shape of your own choice in one of the quadrats – it could be a simple polygon, a star, heart or more complicated original design. Then write down the co-ordinates of it’s vertices (in order). Reflect the shape into each of the quadrats and write down their co-odinates. Give a partner your list of co-ordinates and see if they can determine the shape you have created using the co-ordinates given.
Find out the answers to the following questions using your netbook for research:
- What is Descarte’s most famous saying?
- What did he see on a placard in the town of Breda, that challenged him?
- What branch of mathematics is named after Descartes?
- What habit did he give up up in the last year of his life?
- Now find a map of the school and create a set of cartesian co-ordinates on it. Work out the co-ordinates of the most important places in the school.