Name:Anne Graham ID 15369662Lesson No: 4-7

Year Level: 8Date:April 2007

Duration: 4 x 50 min

Subject: Mathematics

Topic:Geometry

Students explore two and three-dimensional shapes, and develop simple mathematical models.

Relationship to the Curriculum: VELS Level 5

Number:

Students use measuring devices to construct accurate representations of geometric shapes and solids and to measure their dimensions.

Space:

Students construct accurate two-dimensional and three-dimensional shapes with respect to length, angle and adjacency.

Students transform and manipulate two-dimensional and three-dimensional shapes.

Students explain geometric ideas and develop simple mathematical models based on recognition of physical characteristics and patterns.

Students use nets to construct geometric solids, and use perspective to draw three-dimensional objects.

Students use networks to specify relationships, including traversable networks.

Structure:

Students apply simple geometric transformations, including translation, reflection and rotation.

Students use diagrams to make two-dimensional representations of three-dimensional objects.

Students use mathematical terminology to describe the two and three-dimensional shapes they create.

Working Mathematically:

Students use models to investigate practical situations.

Students read and understand problems, plan and implement a strategy to solve, then check their result using a different method.

They develop mathematical models to deal with unfamiliar situations based on their prior knowledge – they develop deductive reasoning.

Students use technology to investigate patterns and relationships in the characteristics of geometric objects, such as number of faces, edges and vertices.

Aims:

Students need to be empowered to use mathematics to solve problems. This unit aims to enable students to investigate relationships and geometric rules involved in two-dimensional shapes and three-dimensional solids.

To extend prior knowledge to investigate polyhedra, nets and networks.

Learning Objectives:

Affective:

Students develop confidence in their mathematical abilities, building on prior knowledge of: angles, triangles and polygons to extend it to polyhedra, nets and solids, and networks. Through a combination of individual and group work they will learn to critically evaluate their own ideas and those of others.

Behavioral:

Students need to think spatially and visualize two and three-dimensional shapes (example: moebius strip, polyhedra). By having the opportunity to prove mathematical theorems for themselves during class exercises in a cooperative and supportive atmosphere, they come to better appreciate the underlying concepts rather than simply memorizing formulae and how to apply them.

Cognitive:

Students recognize relationships based on characteristics such as faces, vertices and edges, and predict outcomes based on these (e.g. Euler’s rule). Students need to understand the lesson content in order to explain their reasoning in solving problems both orally and in writing.

Procedures for Class Work

When working on tasks individually during class, students should limit discussion with peers as much as possible to give themselves the opportunity to recognize or discover the point of the task and possible solutions for it.

Group work (in pairs or small groups) will also be utilized where appropriate, as it provides opportunities to teach and learn from each other, which reinforces student learning. Groups must contain a mix of students, with respect to social groups, gender and learning preferences or abilities, to give everyone the opportunity to participate fully in activities.

The rules for group work:

You (the student) are responsible for your own behavior.
You must be willing to help everyone in your group.
You may not ask the teacher for help unless ALL members of the group are stuck on the same problem.

Due to the highly visual and creative scope of this lesson block, students who do not ordinarily perform well in Maths may find the concepts more interesting and hence easier to master due to their interdisciplinary nature, for example: spatial and drawing skills to define polyhedra, computer literacy to plot and manipulate images, or language skills to clearly explain solutions.

Assessment and Evaluation

Assessment is an integral component of the lessons, and incorporates:

The number of class tasks and bonus tasks completed
Co-operative participation in group activities
Attitude and contribution in class
Unit test and feedback.

Each student receives a Unit Plan at the start of the learning unit so that they can track their progress for self-assessment, and also so that they can take responsibility for their own learning and be actively involved in the process.

Lessons contained in this assignment are highlighted on the Year 8 Geometry Unit Plan below.

Unit Plan

Lessons involve the rediscovery of mathematical relationships.

Textbook: Student Name: ______
Chapter:
Topic: Geometry
Date / Topic / Textbook* reference / Task Box Compulsory Exercise / Teacher / Task Box
Bonus / Teacher
Mon / Exterior angles of a triangle / 9.1
P397 / 1,2,3,5 / 4,6
Prepzone 9
Wed / Angles on Parallel Lines / 9.2
P399 / 1,2,3,4,9,10 / 7(protractor)
11,13
Fri / Geometric Solids / 9.3
P407 / 1,3,4 / 6
Mon / Polyhedra
Mobius strips / 9.4
P410 / Mobius strips
Platonic Solids Q1,2,6 / Q4. Archimedean Solids
Wed / Drawing & Visualizing 3D Shapes / 9.5
P414 / Class worksheet / Design boxes for 20 sugar cubes
Fri / Nets & Solids
Euler’s Law class exercise / 9.6
P417 / Q1,2
(Class sheets, scissors, glue sticks) / Extension activity
Mon / Networks P424
Traversable Networks / 9.7
9.8
P429 / Sprouts game
Map assignment / 9.8 Q6
Wed / Review of topic
Test examples
Fri / Geometry Test
Mon / Test Review

*Textbook – Heinemann Maths Zone 8 VELS Edition, 2005.

Result / No. of Task Boxes Completed / Teacher’s Signature
Low / 6-8
Medium / 9-11
High / 12-13
Excellent / 14-16

Subject: Mathematics - GeometryLesson No: 4

Topic: PolyhedraYear Level: 8

Duration: 50 min

Aims

To introduce the topic of polyhedra and the concept of faces (sides), and to contrast with the previous lesson on geometric solids.

During this lesson students will:

- Develop an appreciation of different types of geometric shapes, similarities in characteristics, and the mathematical relationships between them.

- Complete the Mobius strip exercise in pairs and use their observations to develop mathematical descriptions of the outcomes.

- Independently complete textbook exercises that reinforce the characteristics of polyhedra.

- Work towards understanding of the difference between geometric solids and polyhedra, and will recognise their component polygons.

- Connect this lesson to prior learning.

Assessment and Evaluation

Learning will be assessed by co-operative participation in class activities (work in pairs for Mobius Strips exercise) and discussions, and level of completion of assigned tasks (Basic and Bonus class tasks).

Time / Procedure / Management/ Organisation/
Resources
5 mins
20
mins
10 mins
10 mins
5 mins /

Tuning in

Review of previous lesson (geometric solids, ask for examples: cone, sphere, torus, cube, prism, cylinder)
Procedural steps
1.Mobius Strips
We are going to continue thinking and working in three dimensions by looking at mobius strips.
Read through top section (introduction) of the instruction sheet for the class before allowing students to collect equipment from the front of the room.
Demonstrate making a normal loop and a mobius loop.
Students collect equipment and work through tasks on Mobius Strips class exercise sheet.
Main points:
Different ways to define the relationship between the final appearance of the mobius strip and the number of twists it has, class contributions and discussion.
It pays to look carefully at things – sometimes they are more than they seem.
2. Polyhedra
Now looking at polyhedra – solids whose faces are regular polygons. Contrast to geometric solids to reinforce prior learning.
Regular (Platonic) polyhedra have congruent faces, all the same shape and size. Discovered by the early Greek mathematicians. There are 5 platonic polyhedra:
Tetrahedron – 4 sides/faces
Hexahedron (cube) – 6 faces
Octahedron – 8 faces
Dodecahedron – 10 faces
Icosahedron – 20 faces
Archimedean Polyhedra
13 of these

Exercise 9.4, P410, Questions 1, 2, 6

Heinemann Maths Zone 8
Q1. State whether shapes shown are polyhedra.
Q2. Determine the number of faces on each of the platonic polyhedra.
Q6. How many and which geometric shapes would be needed to make (all of) 4 tetrahedra, 5 hexahedra, 6 octahedra, 5 dodecahedra & 4 icosahedra.
Wrapping up and review
Who knows what a polyhedron is? What is an example of a platonic polyhedron? How do we know it’s a polyhedron and not a geometric solid? What’s another one? What other sort of polyhedra are there? Why are they different to platonic polyhedra? What did we learn from the mobius strip exercise?
Complete as homework if not finished in class.
Get your Unit Plan sheet stamped for tasks completed before you go.
Extension activity/Homework:
Bonus Task, exercise 9.4 question 4
Name the different shapes that make up the faces of a rhombicosidodecahedron.
Draw a two-dimensional representation of this polyhedron. / Class discussion
Textbook
Lesson notes
Mobius strips instruction sheets handed out, students to work in pairs. Equipment: 30cm paper strips, glue sticks, pens, scissors (safety reminder).
Circulate through the class to keep students on task, and to select those who will share their solutions with the class.
Extension activity sheet available for those students who finish quickly
Refer students to summary page (attached)/ textbook.
Overhead of platonic polyhedra or draw diagrams on board.
Refer students to summary page (attached)
Students work individually in their workbooks, quiet discussion of tasks only.
Books closed, class discussion to reinforce lesson content.
Teacher’s resources / Students’ resources
Textbook – Heinemann Maths Zone 8
Student work & summary sheets
Mobius strips: paper strips, glue sticks, scissors
Overhead projector &/or whiteboard / Textbook – Heinemann Maths Zone 8
Student workbook, stationery items

Textbook Exercises

Based on reference: Heinemann Maths Zone 8 VELS Edition, Chapter 9.4, P410-411

MöbiusStrip – Class Exercise

The Recycle logo is an example of a möbius loop:

A Möbius strip is a surface with only one side and only one boundary component. Two German mathematicians, August Ferdinand Möbius and Johann Benedict Listing, discovered it independently in 1858.

A model can be created by taking a paper strip and giving it a half-twist, then joining the ends together to form a loop.

Car fan belts used to be Möbius strips. With an ordinary belt only the inside of the belt was in contact with the wheels, so it would wear out before the outside did. Since a Möbius strip has only one side, the wear and tear on the belt was spread out more evenly and they would last longer. However, modern belts are made from several layers of different materials, with a definite inside and outside, and do not have a twist. This is because they go around many more engine components than the fan and the alternator. Conveyor belts on production lines are another application.

Mathematical Idea

A twisted loop (b) is very different from a normal loop (a):

Materials Needed: Strips of paper, glue stick, scissors and pens or pencils.

Demonstration

Make two loops from the paper strips provided. For loop (a), stick the two ends together. For loop (b), flip one end of the paper over before you stick it down. This should give you a piece of paper with a half-twist in it. This is a Möbius strip.

The Möbius strip has only got one side. Draw a line or colour down the middle of the strip until you get back to your starting point. You will find that you draw on both sides of the paper. The twist in the paper makes you change sides as you draw around.

The Möbius strip has only one edge. In a different colour draw from one point along the edge of the strip. You will find that you get to the opposite point on the edge before you get back to the starting point.

Cut down the middle of both strips. What do you end up with for loops a and b? ______

Using a new Möbius strip cut about a third of the way in from the edge (you will need to go around the loop twice). What do you get? ______

Double Twist

Just like making a Möbius Strip, but this time flip the strip twice before sticking it down. This will give you a strip with two half-twists in it. This strip has two sides and two edges (check). When you cut down the middle of the strip it will you get? ______

Triple Twist

Make a strip with three half-twists. This strip will have only one side and one edge. When you cut down the middle, the strip will become a knotted loop.

Even numbers of half-twists give two loops, odd gives one loop. There is also a pattern to the number of times they twist around each other.

Write a mathematical statement to describe this relationship ______

Further Activities

Shapes 1

You can also stick two strips together to get some interesting effects. Take one ordinary, untwisted loop and a Möbius loop. Stick them together at right angles to each. If you cut down the middle of both strips, what do you get? ______

Shapes 2 - Hearts

Make two Möbius strips. One strip has to be twisted clockwise, the other anticlockwise. Stick them at right angles as before.

When you cut down the middle of both strips, you will get a pair of hearts linked together. (Note: If you get a single heart and a separate, severely twisted heart, you didn't have the strips twisted in opposite directions.)

Reference:

Regular (Platonic) Polyhedra

A polyhedron is a Platonic solid if

all its faces are congruent convex regular polygons,
none of its faces intersect except at their edges, and
the same number of faces meet at each of its vertices.

Polyhedron / Number of
Vertices / Number of
Edges / Number of
Faces
tetrahedron / / 4 / 6 / 4
cube / / 8 / 12 / 6
octahedron / / 6 / 12 / 8
dodecahedron / / 20 / 30 / 12
icosahedron / / 12 / 30 / 20

Due to their aesthetic beauty and symmetry, the Platonic solids have been a favorite subject of geometers for thousands of years. They are named for the ancient Greek philosopherPlato who theorized that they were constructed from the regular solids.

Reference Wikipedia

Archimedean Solids

In geometry an Archimedean solid is a highly symmetric, semi-regular convexpolyhedron composed of two or more types of regular polygons meeting in identical vertices. They are different to the Platonic solids, which are composed of only one type of polygon. Archimedean solids can all be made using nets of regular polygons.

There are 13 Archimedean solids.

Name
/ Solid / Net / Faces / Edges / Vertices
truncated tetrahedron
/ / / 8 / 4 triangles
4 hexagons / 18 / 12
cuboctahedron
/ / / 14 / 8 triangles
6 squares / 24 / 12
truncated cube
or truncated hexahedron
/ / / 14 / 8 triangles
6 octagons / 36 / 24
truncated octahedron
/ / / 14 / 6 squares
8 hexagons / 36 / 24
rhombicuboctahedron
or small rhombicuboctahedron
/ / / 26 / 8 triangles
18 squares / 48 / 24
truncated cuboctahedron
or great rhombicuboctahedron
/ / / 26 / 12 squares
8 hexagons
6 octagons / 72 / 48
snub cube
or snub hexahedron
or snub cuboctahedron
/ / / 38 / 32 triangles
6 squares / 60 / 24
icosidodecahedron
/ / / 32 / 20 triangles
12 pentagons / 60 / 30
truncated dodecahedron
/ / / 32 / 20 triangles
12 decagons / 90 / 60
truncated icosahedron
or buckyball
or football/soccer ball
/ / / 32 / 12 pentagons
20 hexagons / 90 / 60
rhombicosidodecahedron
or small rhombicosidodecahedron
/ / / 62 / 20 triangles
30 squares
12 pentagons / 120 / 60
truncated icosidodecahedron
or great rhombicosidodecahedron
/ / / 62 / 30 squares
20 hexagons
12 decagons / 180 / 120
snub dodecahedron
or snub icosidodecahedron
/ / / 92 / 80 triangles
12 pentagons / 150 / 60

Reference Wikipedia

Subject: Mathematics - GeometryLesson No: 5

Topic: Drawing 3-D ShapesYear Level: 8

Duration: 50 min

Aims

To show students the advantages of drawing three-dimensional solids on isometric dot paper, and the ease of drawing them using appropriate software.

This lesson is booked in the computer room and incorporates ICT. Students use technology tools to enhance learning and promote creativity.

During this lesson students will:

- Explore different media for drawing three-dimensional solids.

- Use a number of different software packages, possibly including: Illuminations (NCTM), Crocodile Mathematics (licence required), Geometers Sketchpad, and ISDDE.

- Recognise the advantages of ICT and isometric dot paper over freehand drawing.

Use different forms of presentation to enhance student understanding.

Assessment and Evaluation

Learning will be assessed by the completion of class tasks.

Homework

Extension task (if not completed in class).

Time / Procedure / Management/ Organisation/
Resources
5-10 mins
5 mins
10 mins
20 mins
5 mins / Tuning in
Last lesson we started to look at polyhedra – probe for differences between polygons and polyhedra.
Today we’re going to explore drawing 3-D shapes both on isometric dot paper and using computer software.
Procedural steps
Draw a couple of shapes on the board:

These can be difficult to draw freehand.
Manual Exercise
Isometric paper makes the task much easier.
Students to work through the worksheet using isometric dot paper.

ICT Exercise

Students instructed to turn computers on &/or log on, and access the appropriate software, e.g.
Geometers Sketchpad
Crocodile Mathematics
Sofweb

Students to complete the exercises on the class worksheet using ICT.
Wrapping up and review
Finish up on the computers, log off and pack up.
Ask students what they though of the exercises and which medium they preferred to work with and why.
Extension/Homework activity
Design three different boxes to hold 20 sugar cubes (assume each cube is 1cm3).
Work out the surface area of each box (hint: the number of faces of cubes on the outside of the shape). / IT Room with internet access or appropriate software loaded. Students instructed that computers are to be left OFF until instructed to turn them on.
Students draw figures from the board in their workbooks.
Circulate through class to ensure students are on-task.
Class worksheet and isometric dot paper handed out
Circulate to ensure that only the specified software is being used, that students are working individually and are on-task, and to check student solutions.
Class discussion
Homework exercise,
Due next lesson.
Teacher’s resources
IT room & resources (hardware & software)
Whiteboard
Isometric dot paper
Student worksheet / Students’ resources
Workbook & stationery
IT resources

Name:______