Maths Enrichment: Space

AAA

ÁÁÁ ,,, @@@

ÀÀÀ

CCC

ÃÃÃ

BBB

ÂÂÂ

AAA

ÁÁÁ ,,, @@@

ÀÀÀ

CCC

ÃÃÃ

BBB

ÂÂÂ

AAA

ÁÁÁ ,,, @@@

ÀÀÀ

CCC

ÃÃÃ ÂÂÂ

BBB

RIC-0009 2.8

http://www.ricgroup.com.au

Published by R.I.C. Publications

- SPACE -

Foreword This three book series has been written for middle to upper primary students as enrichment and extension activities. The three books cover the major areas of mathematics (number, space and measurement) and provide a variety of activities which aim to motivate and challenge young mathematical minds.

The activities in Maths Enrichment - Space are divided into three areas; tessellations, cubes and networks.

'Tessellations' provide students with specific concepts of a tessellating shape and promotes problem solving and lateral thinking skills.

'Cubes' covers a wide range of spacial concepts including area, surface area, volume and doubling using a problem-solving approach.

'Networks' investigates the concepts of regions, paths, junctions and traversability combining these with consolidation of basic number facts.

Contents Section 1 - Tessellations Page 1 What is Tessellating?

Page 2 Tessellating Hexagons Page 3 Irregular Tessellating Shapes Page 4 Making Tessellations Page 5 Curved Tessellations Page 6 Tessellating Face Page 7 Two Shapes Tessellating Page 8 Tessellating Letters

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• Section 2 - Cubes Page 9 Volume and Surface Area Page 10 Cube Calculations Page 11 Volume Verses Surface Area Page 12 Cube Model Views 1 Page 13 Cube Model Views 2 Page 14 Painted Cubes Page 15 Doubling the Size Page 16 Tripling the Size

Section 3 - Networks Page 17 Networks Page 18 Are they the Same?

Page 19 Traversing Networks Page 20 Traversability Rule Page 21 3-D Networks Page 22 Travelling Through Networks Page 23 Network Regions Page 24 Path Puzzle Page 25-26

Answers

EXTENSION MATHS - SPACE

R.I.C. Publications www.ricgroup.com.au

i

,

,,

, , ,

,

,

,,

,

,

,

What is Tessellating?

Tessellating means the same shape repeating itself without leaving spaces.

For example, rectangles will tessellate but circles will not.

1.

In the spaces below draw a tessellating and a non-tessellating pattern.

Tessellating

Non-tessellating

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

, ,

, ,

, ,

, 2.

Tick the shapes below if they tessellate. Test your answers by drawing more of them in the space provided.

Shape 1

Shape 2

Shape 3

Shape 4

EXTENSION MATHS - SPACE

Tessellating

www.ricgroup.com.au

R.I.C. Publications

1

Tessellating Hexagons 1.

2.

,, , ,

Hexagons are tessellating shapes. With hexagons you can make other tessellating shapes such as the one below. Continue the pattern using two colours.

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• Make up another tessellating pattern based on hexagons in the grid below.

EXTENSION MATHS - SPACE

Tessellating/problem solving www.ricgroup.com.au

R.I.C. Publications

2

Irregular Tessellating Shapes We have looked at regular shapes such as rectangles and hexagons to see if they tessellate. Shapes do not have to be regular to tessellate. Look at the tessellating shape below.

1.

Cut out the irregular shapes below and trace around them on a piece of paper to see if they tessellate. Have a guess first.

Shape

1

2

3

4

© R. I .

C.Publ i cat i ons Tessellates •f orr evi ew pur posesonl y• Guess

Doesn't tessellate

Shape 1

Shape 2

Shape 3 Shape 4

EXTENSION MATHS - SPACE

Irregular tessellating shapes www.ricgroup.com.au

R.I.C. Publications

3

Making Tessellations To make a non-regular tessellating shape you must firstly start with a regular tessellating shape like a square, rectangle or hexagon. You subtract certain parts of the shape and add them on to other parts of the shape. Look carefully at the diagrams below.

Draw in the 'subtractions' (taking from one side) and 'additions' (adding to the opposite side) on the rectangle and the hexagon below so that you end up with a tessellating shape.

Cut out your shapes and trace around them to make a tessellating pattern like the one above.

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

EXTENSION MATHS - SPACE

Constructing tessellating shapes www.ricgroup.com.au

R.I.C. Publications

4

Curved Tessellations Tessellating shapes can be made by 'subtracting' and 'adding' parts to a tessellating shape. You can also do this with curved lines. See how the tessellation below was created from a square.

A B A D B C D C

A bird was drawn into this tessellating shape and the result can be seen below. Colour the tessellating bird pattern.

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

EXTENSION MATHS - SPACE

Constructing tessellations www.ricgroup.com.au

R.I.C. Publications

5

Tessellating Face Complete and colour the tessellating pattern of the face below. Make sure the pattern follows the one which is already started.

Use a maximum of four colours.

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

EXTENSION MATHS - SPACE

Tessellating www.ricgroup.com.au

R.I.C. Publications

6

Two Shapes Tessellating Sometimes a combination of two shapes make a tessellating pattern.

Look at the patterns below. What two shapes are used to make each pattern? Colour the patterns.

1.

2.

Shape 1

Shape 1

Shape 2

Shape 2

3.

4.

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

Shape 1

Shape 1

Shape 2

Shape 2

5.

6.

Shape 1

Shape 1

Shape 2

Shape 2

EXTENSION MATHS - SPACE

Tessellating - combining shapes www.ricgroup.com.au

R.I.C. Publications

7

,

,

,

,

,

, , , ,, ,, , , , , , , , ,, , , , , , , ,

,

, , , Tessellating Letters

The letter 'F' has been drawn and then rotated and flipped so that it tessellates. Use the grids below to see if you can draw other letters that tessellate. An easy one to try is the letter 'S'.

1.

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

2.

Try another letter below.

EXTENSION MATHS - SPACE

Tessellating - rotation of shapes www.ricgroup.com.au

R.I.C. Publications

8

Volume and Surface Area The volume of an object is its 'body'.

The surface area of an object is its 'skin'.

Look at the drawing of the cube model.

The number of cubes used to build the model is 5. This is its volume.

The number of squares (faces) that make up the 'skin' is 22. This is known as the surface area.

Look carefully at the drawings below or make the models if you wish to.

Work out the volume and surface area for each.

Volume

Volume

1.

2.

Surface Area

3.

Surface Area

© R. I .

C.Publ i cat i ons Volume •f orr evi ew pur posesonl y • Volume 4.

Surface Area

5.

Surface Area

Build your own 3D shape using cubes. Draw it below. Work out its volume and surface area.

Volume

Surface Area

EXTENSION MATHS - SPACE

Volume and surface area www.ricgroup.com.au

R.I.C. Publications

9

Cube Calculations To work out the volume and surface area of cubes is quite easy. There is no need to build models or count cubes or faces.

W The volume of a cube can be calculated by multiplying the length by the height by the weight.

W W

L

L H

L H

H

, , ,

, ,

, , , , , LxHxW 1x1x1=1

LxHxW 2x2x2=8

LxHxW 3 x 3 x 3 = 27

The surface area of a cube can be calculated by multiplying the number of squares or faces on one side of the cube by six. We multiply by six because a cube has six faces.

©4R. I .

C.Publ i cat i ons • orr e i ew pur posesonl y• 6 f 3v 5

1

2

1x6=6

4 x 6 = 24

9 x 6 = 54

Calculate the volume and surface area of cubes that have the following lengths. (You may need to use a calculator.)

Length

Volume Calculation

Surface Area Calculation

6 10 25 50 100 1 000 EXTENSION MATHS - SPACE

Calculating volume and surface area www.ricgroup.com.au

R.I.C. Publications

10

Volume Verses Surface Area Work out the volume and surface area of the models drawn below.

Record your answers in the table below.

3 1 2

5

4

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

Model

Volume

Surface Area

1.

Does the model with the largest volume have the largest surface area?

1 2 3 4 5

EXTENSION MATHS - SPACE

Comparing surface area and volume www.ricgroup.com.au

R.I.C. Publications

11

Cube Model Views 1 Viewing cube models from the side, top and front gives you a 'plan' of the model. In fact in most cases you can actually build the model if you were just given these views.

Top View Side View Side View

Top View Front View

Front View

1.

Draw the three views of this model.

Top View

Side View © R. I .

C.Publ i cat i ons f orr evi ew pur posesonl y• Front• View

Draw a cube model based on these views.

2.

3.

Side View

Side View Top View Top View

Front View

Front View

EXTENSION MATHS - SPACE

Cubes in perspective - top, side and front views www.ricgroup.com.au

R.I.C. Publications

12

Cube Model Views 2 When we look at views of models they do not always show what the model really looks like. Sometimes two different models can have the same views.

Look carefully at the models and views below.

Side View

Top View

Front View

There is a way where just the top view alone can show exactly what a model should look like. This is done by showing the number of layers.

© R. I .

C.Publ i ca2t i ons layers 1 layer of 2 of cubes Top View cubes •f orr evi ew pu r p o s e s o n l y • 2 1 2 layers of cubes

2 layers of cubes 2 layers of cubes

2 layers of cubes

2

Top View

2

2

Try to draw the cube models from these layered top views.

1.

2.

2

3 2

1

1

1

1

2

2

EXTENSION MATHS - SPACE

Cubes - observing views www.ricgroup.com.au

R.I.C. Publications

13

,

, ,

, , , , ,

, ,

Painted Cubes

Look carefully at the large cube below. It has been made by smaller cubes.

These smaller cubes have been glued into place. The large cube was then painted.

painted

Try to answer these questions about the little cubes that have been used to make the large painted cube.

1.

How many little cubes have:

(a) only one of their faces painted?

© R. I .

C.Publ i cat i ons f oofr r efaces vi e w pur posesonl y• (c) • three their painted? (b) two of their faces painted?

(d) four of their faces painted? (e) no faces painted?

,,

, , ,

,, ,

,

, , , ,

Answer the same questions but use the cube below as your model.

2.

How many little cubes have: (a) one face painted? (b) two faces painted? (c) three faces painted? (d) four faces painted? (e) no faces painted?

EXTENSION MATHS - SPACE

Cubes - identifying faces www.ricgroup.com.au

R.I.C. Publications

14

Doubling the Size Look carefully at the models below. Model 1 is the original model. Model 2 is a double enlargement of Model 1.

Model 2 Model 1

The drawings below are double scale models. Next to each draw the original models.

2.

1.

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• 3.

4.

EXTENSION MATHS - SPACE

Cubes - enlargement in proportion www.ricgroup.com.au

R.I.C. Publications

15

Tripling the Size Look carefully at the models below. Model 1 is the original model. Model 2 is a triple enlargement of Model 1.

Model 2 Model 1 The drawings below are triple scale models. Next to each draw the original models.

2.

1.

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

3.

EXTENSION MATHS - SPACE

Cubes - enlargement in proportion www.ricgroup.com.au

R.I.C. Publications

16

Networks A drawing which contains regions, junctions and paths is called a 'network'. Look carefully at the network below. The regions, junctions and paths are marked.

Paths

Paths:

Paths are the lines between two junctions.

Junctions Regions

Junctions:

Junctions are where three or more lines meet.

This is called the outside region.

Regions:

Regions are the spaces surrounded by paths. This also includes the outside region.

The above diagram contains 3 paths, 2 junctions and 3 regions.

Work out the number of paths, junctions and regions for the networks below.

1.

© R. I .

C.Publ i cat i ons Junctions •f orr evi ew pur posesonl y• Paths

Regions

2.

Paths

Junctions

Junctions

Regions

Regions

4.

EXTENSION MATHS - SPACE

3.

Paths

5.

Paths

Paths

Junctions

Junctions

Regions

Regions

Networks - introduction www.ricgroup.com.au

R.I.C. Publications

17

Are they the Same?

1.

Look carefully at the networks below. Work out how many paths, junctions and regions for each. (c)

(b)

(a)

Paths

Paths

Paths

Junctions

Junctions

Junctions

Regions

Regions

Regions

The three networks above look different from each other but they are identical networks.

2.

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

Look carefully at the networks below. Draw another two networks that have identical numbers of paths, junctions and regions to the networks given.

(a)

(b)

(c)

EXTENSION MATHS - SPACE

Paths, junctions and regions www.ricgroup.com.au

R.I.C. Publications

18

Traversing Networks Networks are traversable if you can go over each line once only and not lift your pencil off the paper. Look at the traced network below.

This network is traversable. However some networks cannot be traversed. Try to traverse the networks below. Tick 'yes' if you can traverse them and 'no' if you cannot.

X Start/Finish

1.

(a)

Network 1

yes

no

(b)

Network 2

yes

no

(c)

Network 3

yes

no

(d)

Network 4

yes

no

(e)

Network 5

yes

no

2

1

yes no © R .

I .

C .

P u b l i c a t i o n s yes no (g) Network 7 •f orr evi ew pur posesonl y• (f)

Network 6

4 3

5 6 7

EXTENSION MATHS - SPACE

Traversability of networks www.ricgroup.com.au

R.I.C. Publications

19

Traversability Rule Try to traverse the networks below and count the number of junctions that are even and the number that are odd.

Odd Junctions:

Are those that have 3, 5, 7 etc. lines in them. For example:

Even Junctions: Are those that have 4, 6, 8 etc. lines in them. For example:

2.

1.

NO THROUGH ROAD

Odd junctions

Odd junctions

Even junctions © R .

I .

C .

P u b l i cat i on s no Traversable yes no Traversable yes •f orr evi ew pur po4.sesonl y• 3.

Even junctions

Odd junctions

Odd junctions

Even junctions

Even junctions

Traversable

yes

no

5.

Traversable 6.

yes

no

From the information you have gained, can you make a rule for the traversability of networks?

Odd junctions Even junctions Traversable

yes

EXTENSION MATHS - SPACE

no Traversability/junction ratio www.ricgroup.com.au

R.I.C. Publications

20

3-D Networks A three-dimensional object like a cube can also be represented by a network.

1.

How many vertices, edges and faces does a cube have?

Vertices Edges

Vertices Faces

Edges Faces

Study the networks of the three-dimensional objects below. Count the vertices, edges and faces of each.

2.

3.

Vertices ©R .

I .

C.Publ i cat i ons Edges •f orr e vi ew pur posesonl y• Faces Vertices Edges Faces

4.

5.

Vertices Edges Faces

Vertices Edges

6.

Vertices

Faces

Edges Faces EXTENSION MATHS - SPACE

Networks - comparing vertices, edges and faces www.ricgroup.com.au

R.I.C. Publications

21

Travelling Through Networks Some networks allow you to travel through them by crossing each line once only in a continuous path. Look at the diagram below.

,

Start/Finish

Try to travel through the networks below. Tick 'yes' if you can travel through them and 'no' if you cannot.

1.

yes nou © R. I .

C.P bl i cat i ons (b) Network 2 yes no 1l • f o r r e v i e w p u r p o s e s o n y• (c) Network 3 yes no (a)

Network 1

(d)

Network 4

yes

no

(e)

Network 5

yes

no

3

2

4 5

EXTENSION MATHS - SPACE

Travelling through networks www.ricgroup.com.au

R.I.C. Publications

22

Network Regions The regions within a network share borders just like countries do on a map.

Some regions have more borders than others depending upon their shape.

Colour the networks below so that no bordering regions have the same colour. Use as few colours as possible.

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

EXTENSION MATHS - SPACE

Networks and regions www.ricgroup.com.au

R.I.C. Publications

23

Path Puzzle Ian, Julia and Nick wanted to go to the shops, the park and the swimming pool. Can you help them get there by drawing in the paths they take. The only problem is that none of the paths must cross.

Ian

Julia

Nick

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y•

EXTENSION MATHS - SPACE

Networks/problem solving www.ricgroup.com.au

R.I.C. Publications

24

ANSWERS Page 1

Page 2

1.

2.

Teacher check Shape 1 - yes Shape 2 - no Shape 3 - no Shape 4 - yes

1.

2.

Teacher check Teacher check

Page 12

1.

side

top

front Shape

1

Guess

Page 3

Page 8

Page 9

Page 10

Tessellates

4

✔

2.

3.

Page 13

1.

2.

Page 14

1.

✔ ✔

✔

Answers will vary Teacher check Teacher check 1.

Octagon, square 2.

Pentagon, parallelogram 3.

Dodecagon, triangle 4.

Hexagon, parallelogram 5.

Square, triangle 6.

Octagon, star

(a) (b) (c) (d) (e)

6 12 8 0 1

2.

(a) (b) (c) (d) (e)

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• 1.

2.

Teacher check Teacher check

1.

2.

3.

4.

5.

Volume = 7 S.A. = 26 Volume = 10 S.A. = 40 Volume = 12 S.A. = 44 Volume = 13 S.A. = 44 Teacher check. Answers will vary

Length

Volume Calculation

6

6 x 6 x 6 = 216

36 x 6 = 216

10

10 x 10 x 10 = 1 000

100 x 6 = 600

25

25 x 25 x 25 = 15 625

625 x 6 = 3 750

50 x 50 x 50 = 125 000

2 500 x 5 = 15 000

1 000

Model

100 x 100 x 100 = 1 000 000 1 000 x 1 000 x 1 000 = 1 000 000 000

Volume

Surface Area

1

27

54

2

24

68

3

22

66

4

24

74

5

32

120

Page 15

1.

2.

3.

4.

16 20 8 0 4

Surface Area Calculation

50 100

Page 11

3

Answers will vary

Doesn't tessellate

Page 4 Page 5 Page 6 Page 7

2

Page 16

1.

2.

10 000 x 6 = 60 000 1 000 000 x 6 = 6 000 000

3.

Yes

EXTENSION MATHS - SPACE

R.I.C. Publications www.ricgroup.com.au

25

ANSWERS Paths

Page 17

Page 18

8

5

5

2.

11

7

6

3.

20

10

12

4.

15

8

9

5.

27

16

13

a, b, c

Page 23

Teacher check

Page 24

There is no solution to this problem as you cannot go through the shops but must go around. This problem has been included to show that there is not always a solution and students should be able to explain the reasons for a non-solution as well as for a solution.

Paths - 3 Junctions - 2 Regions - 3 Answers may vary

2.

Page 19

1.

Junctions Regions

1.

1.

Page 22

1.

(a) (b) (c) (d) (e) (f) (g)

-

(a) (b) (c) (d) (e)

no no yes no no

yes no no yes yes no yes

© R. I .

C.Publ i cat i ons •f orr evi ew pur posesonl y• Page 20

Odd

Even

Traversable

1.

4

0

No

2.

2

3

Yes

3.

2

0

Yes

4.

0

4

Yes

5.

4

0

No

6.

Page 21

A network with more than two odd junctions can not be traversed.

Vertices

Edges

Faces

1.

8

12

6

2.

5

8

5

3.

6

9

5

4.

10

15

7

5.

6

10

6

6.

4

6

4

EXTENSION MATHS - SPACE

R.I.C. Publications www.ricgroup.com.au

26