Euler’s Formula with jubes and toothpicks

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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).

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Welcome Back for Term 2!

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This term we will be studying Measurement and Geometry.

Year 7 Maths (JacPlus Chapter 9 – Measurement and Chapter 5 – Geometry) 

By the end of this term I hope you will be able to:

  • Use appropriate units of measurement
  • Calculate the perimeter of 2D shapes
  • Calculate the area of triangles, quadrilaterals and composite shapes.
  • Identify types of polygons (different triangles and quadrilaterals)
  • Estimate, measure and draw angles between 0 and 360 degrees.
  • Identify the properties of parallel and perpendicular lines and the angles that form between them.
  • Calculate the missing angles in polygons, knowing that the internal angles of a triangle add to 180 degrees.
  • Recognise various transformations (translations, reflections, rotations and dilations)

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Year 8 Maths (JacPlus Chapter 7 – Congruence and Chapter 10 – Measurement)

By the end of this term I hope you will be able to:

  • Use and convert units of measurement for perimeter, area and volume
  • Calculate the area of various quadrilaterals.
  • Calculate the area and perimeter of circles.
  • Calculate the volume of various prisms using formulae.
  • Identify congruent shapes
  • Transform various shapes (translate, dilate, rotate and reflect).
  • Solve geometric problems using congruence.
  • Work out problems around different time zones using the 24 hour clock.

Maths with Scratch!

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Although Victorian Education Week is six weeks away (May 17th to 23rd), I am using some of the school holiday break to play with Scratch, so my Year 7 and 8 Maths classes can participate in the “Crack the Code with Maths” challenge. 

Scratch is simple-to-use software, that allows users to create animations using drag-and-drop commands. I hope to use this free program, pre-installed on our government school laptops, as part of our geometry learning this term.

Scratch uses the Cartesian Co-ordinate system to locate ‘sprites’ on a ‘stage’.The screen is a 480 x 360 rectangle, such that: the X position can range from 240 to -240, where 240 is the rightmost a sprite can be and -240 is the leftmost, and the Y position can range from 180 to -180, where 180 is the highest it can be and -180 is the lowest it can be. The centre of the screen, or ‘origin’, is known as (x=0, y=0) or (0,0).

The following links are some examples of what can be achieved with Scratch.

Student tasks:

  • Join the Scratch community, using your school username (eg. gow0049).
  • Explore the links above and other geometry-related Scratch projects.
  • Create your own Scratch project, drawing a different polygon (closed shape with straight sides) in each of four quadrats.
  • Can you create four different triangles? (equilateral acute, isosceles obtuse, scalene right-angled and one other combination of side-length and angle size).
  • Can you create four different quadrilaterals?
  • Can you create a regular pentagon, hexagon, octagon and nonagon?
  • Draw your initials, like these students in 5/6 Clark/Smith Can you translate and reflect your initials so they appear in all four quadrats?

Tessellations

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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

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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.

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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)

Learning Intentions for this unit

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1. Classify triangles according to their side and angle properties and describe quadrilaterals.

2. Establish the formulas for areas of triangles, rectangles and parallelograms and use these in problem solving.

3. Demonstrate that the angle sum of a triangle is 180 degrees and use this to find the angle sum of a quadrilateral.

Measuring Angles, triangles by side length and angle size.

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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.
Read more

Exterior Angles in a Triangle

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Learning Intention: To understand why, in any triangle, the exterior angle is equal to the sum of the two interior angles not adjacent to it.

Success Criteria: To be able to calculate the missing angles in a triangle, given enough information.

 

 

Angles in Polygons

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Building construction in Perth

Learning Intention: Students will understand the relationships between lines, shapes and their angles in triangles, quadrilaterals and other polygons.

Success Criteria: You will be able to calculate the missing angles in various two-dimensional shapes and parallel lines by knowing the sum of the angles of that shape and which angles are equal, complementary or supplementary.

You know that the sum of the angles in a triangle is 180 degrees and in a quadrilateral is 360 degrees, but do you know how to work out the sum of the angles in any polygon? What is an exterior angle? What are complementary and supplementary angles? What are opposite, corresponding, alternate and co-interior angles? These are the questions we will answer in the next unit of work.

Shapes with Symmetry

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Go to the Shape tool and draw, color, paste, slice, rotate, reflect, expand, and contract various shapes. Create a picture that has vertical or horizontal symmetry (eg. mountain reflected in a lake, a rocket ship, UFO, Christmas tree, an animal – use your imagination). Make sure you take a screen shot, convert it to a jpeg file in Irfanview and email as an attachment to me at my gmail address. This is an assessment task, so use at least five different shapes!