Book Two

Problem Solving u t

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ent Acti viti hematics Enrichm es r P e r p i m p a f or M i d d l e t o U ry

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Investigation

Pri

m - E d Publishing

by

G e o rge Moore

Foreword Problem Solving Through Investigation is a series of three books of enrichment activities for middle to upper primary. Book 1 caters for 9-10 year olds, Book 2 for 10-11 year olds and Book 3 for 11-12 year olds. Each book contains 25 separate activity sheets that reinforce concepts in number, measurement and space. These activities have been designed to stimulate interest in mathematics, and can be used in the class or for assignments. Many of the exercises are suitable for partner or group work, providing valuable interaction between students, a desirable feature of maths investigation activities.

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Activities in each book are both fundamental in concept and challenging in application. They address interesting and stimulating areas of mathematics that will assist with the development of positive attitudes.

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Contents

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N umber Operations â&#x20AC;&#x201C; Calculator Work .......................................................................................... 1 Number Patterns (1) ......................................................................................................... 2 Number Patterns (2) ......................................................................................................... 3 Number Patterns (3) ......................................................................................................... 4 Number Systems ............................................................................................................... 5 Index Notation â&#x20AC;&#x201C; Calculator Work ................................................................................... 6 Codes and Sequences ........................................................................................................ 7 Codes & Ciphers ............................................................................................................... 8 Sports Probability.............................................................................................................. 9 Carroll Diagrams ............................................................................................................. 10 What Is My Number? ..................................................................................................... 11 How Many? ..................................................................................................................... 12 Maths Terms Crossword ................................................................................................. 13 M easurement

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Decades and Centuries ................................................................................................... 14 Polygons.......................................................................................................................... 15 Map Reading ................................................................................................................... 16 Graphs............................................................................................................................. 17 Hi, Pi! .............................................................................................................................. 18

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Area ................................................................................................................................. 19 Line Segments ................................................................................................................. 20 Coordinates..................................................................................................................... 21 Rigid Shapes .................................................................................................................... 22 Polygon Patterns ............................................................................................................. 23 Tessellations .................................................................................................................... 24 Shapes & Solids Sleuth .................................................................................................... 25 Answers .......................................................................................................................... 26 Prim-Ed Publishing

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Problem Solving Through Investigation Book 2

Operations – Calculator Work

N

Clue

Calculation (8 x 20 x 5) + 7

2. Fish breathe with this

(5787 ÷ 3) x 4

3. A best selling book

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1. A difficult tennis shot

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This exercise will give you practice on your calculator. Work out the sums, then turn the calculator upside down and read the word in the display window. If there is no word, do your calculation again! (Work out the brackets first).

191 x 11 x the product of 2 and 9

4. Used in cold countries 5. A ship’s records are kept in it 6. A fish or part of a shoe

(707 - one century) 3 x 5 x 247

(L x CX) + VII

g

7. You’d be upset by this

485375 - 24000

in

8. It has to be repaired sometimes 300 tens + XLV 9. Used in the garden

35 hundreds and 4 (40 x 200) - 200 - 82

11. To walk as if injured

400 thousand -21 196

12. A marsh or swamp

(12 x L) + VIII

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10. A bird has one

half of 7614

14. Shouldn’t be told

15 score + XVII

15. You’ll find them in a gaggle

(35 x 1000) + 336

Note: Never accept that the calculator answer is always correct. You may have pressed the wrong button! Always estimate the answer first and then compare your estimate with the calculator answer.

Prim-Ed Publishing

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Problem Solving Through Investigation Book 2

Number Patterns (1)

0

36

12 0

27

15

36

6

40

3

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18

48

30

Multiples of three

28

8

24

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In the first circle multiples of three (3 x table) are arranged around the circumference. Start at 0 and then rule a line to 3, then from 3 to 6, then 9 from 6 to 9, and so on. You will end up with a pattern. Colour all of the triangles and quadrilaterals in this pattern. Then do the 21 same with the other circles. Ensure that adjacent shapes with a common side are not the same colour! 12 33 24

N

5

45 20

16

30

20 4 32

g

44

in

Multiples of four 18

30

35

10 40

72

54

0

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50 25

Multiples of five

60

66

Look at each circle and write down the differences between adjacent numbers. Write a comment about these secondary patterns.

48

0

12

55

15

6

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42

60

36 24

Multiples of six

Remember, start at 0, finish at 0! Prim-Ed Publishing

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Problem Solving Through Investigation Book 2

Number Patterns (2)

N

3

6

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Triangular numbers are numbers whose units can be arranged into triangles.

10

15

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Can you see a pattern?

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Diophantus, a Greek mathematician, was interested in patterns in numbers and discovered one pattern described below.

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Use your calculator and multiply each triangular number in the circles by 8 and then add 1. The first has been done for you.

36

45

55

66

78

91

105

x8

x8

x8

x8

x8

x8

x8

x8

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28

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= = = 224 = = = = = +1 +1 +1 +1 +1 +1 +1 +1 = = = 225 = = = = = Now write down one statement about the numbers in the boxes.

Did you find out what Diophantus discovered more than 1500 years ago? Prim-Ed Publishing

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Problem Solving Through Investigation Book 2

Number Patterns (3)

N

Number patterns are found in all areas of maths (tables, calendars, square numbers and so on). Use the â&#x20AC;&#x2DC;100â&#x20AC;&#x2122; squares below to find the next three numbers in each series. First of all, colour in the squares of the given numbers. Then follow the pattern to colour in the next three numbers in the series. Write these numbers in the answer boxes. Use different colours for each series. Example:

1

2

3

4

5

6

7

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9

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99 100

1

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99 100

2, 13, 24, 35, 46,

57

68

79

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Now try these! 1. 72, 83, 74, 85, 76,

2. 88, 77, 68, 57, 48,

in

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3. 20, 19, 29, 28, 38,

4. 11, 2, 13, 4, 15,

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5. 21, 23, 43, 45, 65,

6. 94, 84, 75, 65, 56,

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7. 1, 12, 23, 34, 45,

8. 71, 82, 73, 84, 75,

9. 11, 2, 3, 14, 5, 6, Discuss the patterns with your teacher. Did you find the secondary number patterns (differences between adjacent numbers) to check your answers?

Prim-Ed Publishing

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secondary number pattern

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1

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99 100

Problem Solving Through Investigation Book 2

Number Systems

N

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Throughout history, unlike animals, human beings have been able to count their sheep, cattle, money, measures of wheat, soldiers in their armies, etc. To record their totals they used symbols to represent numbers. These symbols are called numerals and they vary from country to country. To record the number of birds here we would use the Hindu-Arabic numeral 5. The Chinese would use . Ancient Romans would use the Roman numeral V. Listed below are some of the numerals used by ancient peoples. The Egyptians used familiar things for some of their numerals, for example (coiled rope) (lotus flower). and

Ancient Babylonians

1

1

3

3

4

5

1

I

10

9

III

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1000 10000

100000

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The Ancient Egyptians usually put their smaller numbers to the left, so 13 was written as .

IV

10

V

13

15

X

40

XL

50

50

L

62

100

C

1000

M

10

20

10

D

7

12

IX

500

5

7

in

100

3

3

g

5

1

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Ancient Roman

Ancient Egyptian

Central American Mayan

41 23 25 31 The numeral for 1 and 60 was the same, but the 60 numeral was moved to the left (place value).

35 When the â&#x20AC;˘ symbol was high it stood for 20.

Use these lists to find the Hindu-Arabic equivalents of the numbers below: 1. 2.

= =

3. 4. Prim-Ed Publishing

= =

1. MDC =

1.

2. LVII =

2.

3. XCIV =

3.

4. XXV =

4. 5

= = = =

1.

=

2.

=

3.

=

4.

=

Problem Solving Through Investigation Book 2

Index Notation – Calculator Work

N

This exercise will give you practice on your calculator. Work out the sums, then turn the calculator upside down and read the word in the display window. If there is no word, do your calculation again. Work out the brackets first.

Calculation (5 x 102) + 141

2. There’s nothing in it!

(37 x 102) + 41

3. A dog can do this

22% of 29 x 102

4. A hardworking insect 5. A natural protective cover 6. To recede like the tide

(77 x 103) + 300 + XLV 93 + CL + 41

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(7 x M) + 102 + V 142 + 107 + 116

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9. A garden tool

(3⁄5 of D) + (62 + 2)

383 + 306

7. Absolute heaven! 8. The farmer needs it

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1. Belonging to him

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Clue

10. A high sheen

(51⁄2% of 106) + 76

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11. Needed by industrial nations 5% of 142 x 102 (7 x 103) + (33 x 22)

13. To stare at rudely

1

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12. A painful swelling!

⁄2 of 302 x 23 + 160

14. Fluid secreted by the liver

(20% of 18090) + 102

15. We should all have these!

(5 x 1003) + 318 804

You’ll need to know your Roman numerals too! Prim-Ed Publishing

6

Problem Solving Through Investigation Book 2

Codes and Sequences

N

Find out the three missing numbers in each sequence below. Then use the code wheel to find out which letter each number represents. If you have completed the sequence correctly your answer should spell a word. Good hunting! Example: 4, 8, 12, 16, 20, 24 (L, O, W)

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S U PER

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J S O G L 17 18 19 20 2 N 1

L

O

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16

16

22

1 12 13 14 1 T 5

D ER

Z A K W Y 23 24 1 2 3 T

P ER DE

F O S P

1 10

DU

O

4

CO

24 = W

I E B6 7 H W 8 D 5 9

20 =

3, 2, 6, 2, 9, 2,

,

,

(

)

2.

9, 2, 8, 3, 7, 4,

,

,

(

)

(

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(

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(

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(

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(

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(

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1.

1, 1, 2, 4, 7,

4.

4, R, 8, S, 12, T,

,

, , U,

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3.

5.

24, 7, 21, 8, 18, 9,

6.

14, 13, 1, 12, 11, 2, 10, 9, 3,

7.

2, 3, 4, 6, 6, 9, 8,

8.

18, 5, 15, 5, 12, 5,

9.

5, 5, 8, 6, 6, 10, 7, 7, 12, 8,

10.

12, 2, 10, 3, 8, 4,

Prim-Ed Publishing

, V,

,

,

, ,

,

, ,

,

,W

, , , 7

,

Problem Solving Through Investigation Book 2

Codes & Ciphers

N

Codes to send secret messages have been used for thousands of years. The Roman Emperor Julius Caesar used Roman letter symbols in an alphabet substitution code where letters were moved to the right and represented the letter above them. Use this code below and decipher the message. For example: WZY = bed.

Message: V

H

D

Z

X

V

I

T

H

Z

!

J

J

Y

V

K

C

Z

M D

I

B

X

J

Y

Z

T

J

O

P

B

D

Q

Z

g

Y

B

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A B C D E F G H I J K L M N O P Q R S T U V W X Y Z V W X Y Z A B C D E F G H I J K L M N O P Q R S T U 1 3 2

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Hundreds of years ago scientists communicated with each other in simple codes so that no one would learn their secrets. During wars, mathematicians have been used to break enemy codes so that they would know the enemyâ&#x20AC;&#x2122;s plans. Decipher the pigpen code below and see what the enemy plans to do!

Pigpen Code

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A B C

J

D E

F

G H

I

L

N O P

W

Q R S T U V

M

K

Z

X Y

Now choose one of these codes and write a message for your partner. Prim-Ed Publishing

8

Problem Solving Through Investigation Book 2

Sports Probability

N

Below is a Saturday sport fixture list. Complete the lines as shown in the first example.

Eagles versus Bears.

Probability of Eagles win =

1 out of 2

Rovers versus United. Probability of United win =

=

City

=

Souths versus Norths. Probability of Souths win =

50% chance of winning

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versus Swans. Probability of Swans win =

1 2

=

=

The probability of forecasting four correct wins

=

1 2

x

1 2

x

1 2

x

1 2

=

1 16

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= 1 chance in 16

= 16 to 1 chance.

1. Now pick your own names from a sports competition round and fill in the spaces below, following the example above.

. Probability of

in

versus

. Probability of

g

versus

win =

=

win =

=

. Probability of

win =

=

versus

. Probability of

win =

=

. Probability of

win =

=

ew

versus

versus

correct wins =

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The probability of forecasting

=

= =

2. Repeat (1) on some paper, this time using 16 teams. Do you think you could pick all of the winners all of the time? Why?

Prim-Ed Publishing

9

Problem Solving Through Investigation Book 2

Carroll Diagrams

N

in

7 boys 3 girls 15 males 21 adults

5 customers 2 female shoppers 4 males 3 shop assistants

Males

How many people in the shop?

How many females?

How many women in the shop?

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How many people altogether?

Females

In an adult class there are more male basketballers than female. There are:

Females

12 males altogether 5 males who play tennis 13 adults who play basketball 8 females who play sport

Males

Females

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18 girls 20 females 5 teachers 48 students

Males

Students Teachers

At a school camp there are:

Both

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Curved

One-way grid

In a shop there are:

Females

g

Males

Straight

Customers Shop Assistants

In a room are:

SHAPES

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Carroll diagrams are grids used in classification activities. One is shown on the right. They are named after Alice in Wonderland author Lewis Carroll, who was also a mathematician who liked exploring maths problems. Carroll diagrams can also be used in more difficult problem-solving activities. Use the two-way grids below to solve the problems. You canâ&#x20AC;&#x2122;t just add up the numbers given in each problem because some people are included in two categories, for example: boys are also males, adults are males and females. Think carefully and sort the numbers onto the grids. Check your grids agree with the given information when youâ&#x20AC;&#x2122;ve finished!

How many males on camp? How many at the camp altogether?

How many people in the class? How many females play tennis?

Prim-Ed Publishing

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Problem Solving Through Investigation Book 2

What Is My Number? 1.

Terms Used

My number has four digits. (a) My first digit is a square number. (b) My first digit + my fourth digit = 15 (c) My fourth digit is a triangular number. (d) My second digit is 1/2 of my third digit. (e) My third digit is a different square number from the first digit.

My number has four digits. (a) My second digit < my first digit. (b) My first digit is 1/10 of 50. (c) My second digit is a square number. (d) My third digit is an even triangular number. (e) My fourth digit is 1/2 of my second digit.

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2.

Digit Square number Triangular number > and < symbols Multiple

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Answer = ___ ___ ___ ___

N

Answer = ___ ___ ___ ___

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My number has four digits. (a) My third digit is a square number. (b) My second and third digits are consecutive. (c) My second digit is a triangular number. (d) My fourth digit is 1/2 of my third digit. (e) My first digit is a multiple of my third digit, and is > 5.

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3.

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Answer = ___ ___ ___ ___

Prim-Ed Publishing

4.

My number has four digits. (a) My second digit < my third digit. (b) My first digit is an odd triangular number and > my third digit. (c) My fourth digit is a square number, and a multiple of my first digit. (d) My third digit is 1/3 of my first digit. Answer = ___ ___ ___ ___

11

Problem Solving Through Investigation Book 2

How Many?

N

Complete each stage of each problem below. You may use your calculator!

1. The first powered flight by Orville Wright in 1903 was about thirty-seven metres. How many millimetres is this?

=

cm x

=

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2. World War II lasted from 1 September 1939 to the defeat of Japan on 2 September 1945. How many hours was this? (Assume that 1 year = 365 days. Ignore the leap years.)

6 yrs x

mm

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37 m x

=

days x

=

hrs

=

cm

=

mx

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190 km x

in

g

3. The Suez Canal links the Mediterranean Sea to the Red Sea. It is 190 km long. How many centimetres is this?

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4. A cubic metre container holds one kilolitre of water. How many millilitres is this?

1 kL x

=

Lx

=

mL

5. William Shakespeare lived from 1564 to 1616, a total of 52 yrs. How many days did he live?

52 yrs x

Prim-Ed Publishing

=

weeks x

12

=

days

Problem Solving Through Investigation Book 2

Maths Terms Crossword 1

2

N

3

4

5 7

8

9

10

11 13

15

16 18

14 17

19

20 22

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21

12

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If you are not too sure about some of the terms used in the crossword, a dictionary may help!

6

23

25

Clues Across

g

Three score. 17 is trebled. Years in 9 decades. 17 squared (172). Its Roman numeral is L.

8. 5 x

x 3 = 300

8 (3 x 3) = Years in a millennium. √100 4, 5, 8, 10, are some of this number’s factors. 75% of 20. 5 x 103 = The 5th prime number. The quotient for 906 ÷ 3. Its Roman numeral is M. 102 + (3 x 0 x 10 x 5) Prime number between 19 and 24. The 11th composite number.

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9. 10. 13. 15.

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16. 17. 19. 21. 22. 23. 25. 26.

Prim-Ed Publishing

26

Clues Down

in

1. 3. 5. 6. 7.

24

13

1. 2. 3. 4. 5. 6. 7. 10. 11. 12. 13. 14. 18. 20. 21. 23. 24.

82 √900 Its Roman numeral is D. The product of 7 and 26. 30 squared (302). 33 (3 cubed). 2 ⁄3 of 75 A gross. 6 x 1⁄2 x 5 x 0 x 2 103 CLI = 20 cubed (203) (14 x 12) - (13 x 12) MC is its Roman numeral. Runs in a triple century in cricket. A baker’s dozen. The decimal system is based on this number.

Problem Solving Through Investigation Book 2

M

1. How many decades are there between Australian Norman Brooks winning the Men’s Tennis Singles title at Wimbledon and John Newcombe’s first Wimbledon Singles title?

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2. How many centuries are there between the signing of the Magna Carta by King John of England and the defeat of Napoleon at the Battle of Waterloo?

3. How many decades are there from Orville Wright’s first powered flight in a plane to the first ascent of Mount Everest by New Zealander Sir Edmund Hillary?

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4. How many centuries are there between the Spanish Armada’s attempted invasion of England and the Olympic Games at Seoul?

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5. How many decades are there between the arrival of the First Fleet in Sydney with convicts from England and the birth of Admiral Richard Byrd (the first man to fly over the North and South Poles)?

6. How many centuries are there between the Great Fire of London and the change to decimal currency in Australia?

You will need to use an encyclopaedia! Prim-Ed Publishing

14

Problem Solving Through Investigation Book 2

Polygons

M

To prove the three angles of any triangle add up to 180째: 1. Draw any triangle. 2. Label the angles A, B and C and draw in the dotted lines as shown on the left. 3. Cut or tear along the dotted lines. 4. Glue the vertices (corners) labelled A, B, C above the line ST with each vertex (corner) touching point X.

B

A

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C

Now do this yourself, and complete the statement below.

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B

A

S

C T

X

째.

I found that angles A + B + C in my triangle added up to

A

g

To prove the four angles of any quadrilateral add up to 360째:

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in

B

C

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D

1. Draw any quadrilateral shape. Quadrilaterals have four sides. 2. Label the angles A, B, C and D and draw in the dotted lines as shown on the left. 3. Cut or tear along the dotted lines. 4. Glue the vertices (corners) labelled A, B, C and D so that each vertex (corner) touches point Y.

B D

Now do this yourself, and complete the statement below.

Y C

A

I found that angles A + B + C + D in my quadrilateral added up to Prim-Ed Publishing

15

째.

Problem Solving Through Investigation Book 2

M

600km 500km

Sometimes, in road directories and newspapers, concentric circles (different-sized circles with the same centre) are used to make calculations of distances from place to place quicker and easier. In the diagram on the T H left, the first circle is 1 cm from capital city X, the second circle is 2 cm from X, the third circle is 3 cm from X, and so on. As 1 cm equals 100 km on this map, all points on the first circle are 100 km from X.

400km

I

300km

K F

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200km 100km

X

Sa m

E

Z

B

Y

g

Scale: 1 cm â&#x17E;¨ 100 km

in

Now use this map to answer the following questions: 1. Which town is 400 km from the capital?

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2. Which two towns are 100 km apart? 3. Which two towns are the same distance from the capital? 4. Mark the location of a satellite town (S) 300 km from X.

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5. How much greater is the distance from X to B than from X to E? 6. Which two towns are the greatest distance from X? 7. Which town is twice the distance that E is from X? 8. How many towns are there which are more than 300 km from X? 9. During one week a sales consultant made return journeys from X to towns T, I and E. How far did he travel?

10. Which town is approximately 350 km from X? Prim-Ed Publishing

16

Problem Solving Through Investigation Book 2

Graphs

M

ow

Sh

Bar Graph of Shadow Cast by Our Stick

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The stick should be vertical.

Find a straight stick, about 30 cm long, and push it firmly into the ground. Make sure it is in an open space and in full sunlight all day. Now measure the length in millimetres of the shadow that the stick casts every 30 minutes during the day. Perhaps your teacher could send a class member and a partner (to check measurements) out at set times. Record the results on the bar graph below.

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in

g

2. Why does this happen?

1. What happens to the columns on your graph?

Times Measured

Discussion Point

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If you carried out this experiment with the same stick, at the same spot and at the same times, one graph in summer and one in winter, what would you discover about the columns when comparing the two graphs?

Prim-Ed Publishing

17

Problem Solving Through Investigation Book 2

Hi, Pi!

M

Ancient Greek mathematicians noticed a relationship between the diameter of a circle and its circumference.They called this relationship pi and gave it the symbol .

Definitions:

A circumference is the distance around a circle (Its perimeter)

Ď&#x20AC;

B

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A diameter is a line joining two points on a circle and passing through the centre of the circle.

With a partner, measure and record the diameters and circumferences of these circles in the table below and find this relationship. Make all of your measurements A in millimetres.

Sa m

C

Use a calculator to find this!

Diameter (D)

Circumference (C)

A B

D

in

E F

ew

G H

D

g

C

CĂˇD

Vi

Use your own circles for F, G and H. You could measure a frisbee, a bangle, the rim of a round cup for example.

When you and your partner have completed the table above, complete the statement below: We found the circumference was approximately

E

times

the diameter. This means that the diameter is approximately of the circumference. Prim-Ed Publishing

18

Problem Solving Through Investigation Book 2

Area

S

Work out the area of the shaded part for each diagram. There is a short way of doing each one without doing too many long calculations. You may use your calculator.

16 cm

1.

pl e

18 m

16 cm

2.

8 cm

cm2

m2

Area =

Sa m

Area =

18 m

24 cm

3.

15 cm

6 cm

4.

cm2

g

in

14 cm

Area =

28 m

ew

8m

5.

cm2

Area =

Area =

m2

20 m

Vi

6.

20 m

7.

Area of circle is 264 cm2

Area = Prim-Ed Publishing

Area =

m2 19

cm2

Problem Solving Through Investigation Book 2

Line Segments

S

A line segment is a set of points with two end points.

With one point we can draw no line segments.

B A

End point A

End point B

A

B

With two points we can draw one line segment.

Line Segment AB

C With three points we can draw three line segments.

A

B

A

B D

With four points we can draw six line segments.

C

1. Complete the diagrams below and complete part A of the table to the right.

Line segments

Shape outline

1

0

-

2

1

-

3

3

Triangle

4

6

5 6

in

g

5

7 8

6

Vi

ew

B

7

9

Nonagon

10

Decagon

11

Hendecagon

12

Dodecagon

2. Look for the pattern down the second column of the table above. Then without drawing any diagrams complete the second column in Part B of the table.

8

Prim-Ed Publishing

Points

Sa m

A

pl e

A

20

Problem Solving Through Investigation Book 2

Coordinates

29 3

Coordinates are used on maps and in our road directories to locate places by using ordered pairs of numbers as a reference.

B

2

Point A is referenced by the ordered pair (1, 2). Point B is referenced by the ordered pair (3, 3).

A

1 0

1

2

S

Always read along the horizontal axis first.

3

pl e

This system of coordinates was invented by the 17th century mathematician Descartes and is called Cartesian coordinates after him.

N

Sometimes grids are a combination of cardinal points (compass directions) and ordered pairs. For example, the location of the capital city on the map on the right is (4W, 1N).

Sa m

Yacht Club

Light House

4

Airport

Bridge

3 2

Capital City

6

5

4

g

7

3

2

1

River Mouth

1

0

0

1

2

2

3

4

5

Zoo

1

Farm

6

7

E

Lake

3

ew

4 5

Mountain Peak

6

Vi

7

Find the grid locations of the following locations on the island:

S

1. Airport

6. Bridge

2. Mountain peak

7. Zoo

3. Lighthouse

8. Yacht club

4. Farm

9. Golf club

5. River mouth Prim-Ed Publishing

Golf Club

6 5

in

W

7

10. Lake 21

Problem Solving Through Investigation Book 2

Rigid Shapes Card

S Make an equilateral triangle out of three equal strips of thick card. Push the triangle in the direction of the arrow. You will find it will not move because a triangle is a rigid (= strong) shape.

Push

pl e

Drawing Pin Drawing Pin

Now do the same with a rectangle as shown. Push it. Is it rigid?

Diagonal Strut

Push

Sa m

Put in a diagonal strut made of card, as shown in the diagram. Does this make it rigid? Why is this?

Number of sides

of struts to make rigid

of triangles formed

Triangle

3

0

1

Vi

in

g

Make the necessary shapes and complete the final two columns. The lengths of the sides of the shapes need not be equal. Cut out pictures of bridges, buildings, objects and so on, which clearly show the use of the two common rigid shapes Number Number (triangles, circles).

Rectangle

4

1

2

Pentagon

5

Hexagon

6

Heptagon

7

Octagon

8

ew

Shape

Prim-Ed Publishing

22

Problem Solving Through Investigation Book 2

Polygon Patterns

S

A polygon is a two-dimensional shape made up of three or more straight line segments. Select and mark a point P on a vertex (corner) on each polygon shown. Then draw each polygonâ&#x20AC;&#x2122;s diagonals from P. Some of the polygons are regular shapes (all sides and angles are congruent) and some are irregular. Now fill in the table. Remember! One of the vertices (corners) will be point P. Some have been done for you.

P

P

pl e

Triangle

Sides

Diagonals

Triangles Formed

Totals

Triangle

3

0

1

4

4

1

2

7

Pentagon

5

2

3

10

Heptagon

Hendecagon

hexagon

ew

Dodecagon

in

Nonagon Decagon

Pentagon

g

Octagon

Sa m

Polygon

Hexagon

P

Vi

Now look at the number patterns down the four columns and work out the answers for the last three shapes without drawing them! Explain how you found your answers.

heptagon

octagon

nonagon

Prim-Ed Publishing

23

Problem Solving Through Investigation Book 2

Tessellations

S

Shapes which tessellate are shapes which fit together in a pattern without gaps between. Some floor tiles tessellate. The word comes from tesserae (singular: tessera), the name given to small tiling blocks used by the Romans.

Colour in the shapes on the right which you think will tessellate. You may have to test

Rectangle

Trapezium

pl e

some with cut-out shapes.

Circle

Ellipse

Sa m

Equilateral Triangle

Hexagon

Rhombus

Scalene Triangle

Isoceles Triangle Iscosceles Triangle

in

g

A famous 20th century Dutch artist called Mauritz Escher used tessellating shapes in some of his art and mathematicians found his patterns very interesting.

Vi

ew

The diagrams below show how the interlocking patterns are done.

Shapes are cut out of a basic tessellating shape and stuck on the opposite side from where they are cut.

The new shape is then used as a template on different coloured paper and then the cut-out shapes are fitted together.

Now make three tessellating shapes, using graph paper to make the template, and coloured paper for the tiles. Prim-Ed Publishing

24

Problem Solving Through Investigation Book 2

Shapes & Solids Sleuth X

Q

L

E

N

O

C

H

F

E

P

S

N

U

P

L

B

E

P

E

S

C

T

C

B

A

T

L

P

Q

T

P

C

H

L

U

N

D

V

I

Z

U

Y

T

A

E

X

B

T

R

W

P

S

I

F

A

L

X

B

E

R

I

X

S

L

L

M

G

E

A

H

R

A

L

D

T

P

A

L

T

E

T

R

S

Z

E

B

F

I

R

B

U

E

M

pl e

T

E

C

A

G

O

N

G

S

O

B

T

L

N

E

O

P

A

H

E

D

R

O

N

H

S

P

R

I

S

M

F

E

O

C

T

A

G

O

N

W

R

A

N

T

C

L

N

C

O

N

E

L

I

S

O

S

C

E

L

E

S

Sa m

The names of common 2-dimensional shapes and 3-dimensional solids are hidden in this word sleuth. Find these names and then write them next to their correct definition (description) below. Colour in shapes as you find them.

S

A prism with six congruent square faces.

2.

A 2-dimensional shape with seven sides and seven angles.

3.

A type of triangle with two congruent sides.

4.

A simple closed curve similar to an oval shape.

5.

A 2-dimensional shape with four sides.

6.

A 2-dimensional shape with eight sides and eight angles.

ew

in

g

1.

7.

A 3-dimensional shape with four faces.

8.

A 3-dimensional shape with a circular base and one vertex.

9.

A type of triangle with congruent sides and angles. A 2-dimensional shape with six sides and six angles.

11.

A type of triangle with no congruent sides or angles.

12.

A 10-sided 2-dimensional shape with ten angles.

13.

A ball shape with all points on its surface being the

Vi

10.

same distance from its centre. 14.

A quadrilateral with one pair of parallel sides.

15.

A 3-dimensional shape with at least one pair of opposite

s t p

congruent faces which are parallel.

Prim-Ed Publishing

c h i e q o t c e h s d

25

Problem Solving Through Investigation Book 2

Answers P1. Operations – Calculator Work 1. lob 2. gill 3. 4. sleigh 5. log 6. 7. loss 8. shoe 9. 10. bill 11. hobble 12. 13. lobe 14. lie 15.

9. 8, 14, 9 (how) 10. 6, 5, 4 (bed)

bible sole hose bog geese

P8. Codes and Ciphers First message: I am good at deciphering any code you give me.

P2. Number Patterns (1) Answers will vary.

Second message: Sixty tanks will attack from the north at noon.

In a room:

3. 94

ew

3. 23

2. 28

3. 11

4. 25

4. 90

P6. Index Notation – Calculator Work 1. his 2. hole 3. beg 4. bee 5. shell 6. ebb 7. bliss 8. soil 9. hoe 10. gloss 11. oil 12. boil 13. ogle 14. bile 15. hobbies

Vi

Males

Students Teachers

7 3

Females

3 2 30 18

(a) 33 (b) 53

In a shop:

Females

1 2 3 2

(a) 8 (b) 4

Males

Females

5 2 7 6

(a) 20 (b) 2

P11. What is my Number? 1. 9246 2. 5462 3. 8342 4. 3019

P7. Codes and Sequences 1. 12, 2, 15 (pat) 2. 6, 5, 5 (bee) 3. 11, 16, 22 (sly) 4. 16, 20, 24 (low) 5. 15, 10, 12 (top) 6. 8, 7, 4 (hid) 7. 12, 10, 15 (pot) 8. 9, 5, 6 (web) Prim-Ed Publishing

8 13

Males

4. 24

Females

(a) 31 (b) 16

Customers Shop Assistants

2. 57

Babylonians 1. 72 2. 30 Mayan 1. 17

4. 1 024

At a school camp:

3. 11 202

Males

g

Roman 1. 1 600

P10. Carroll Diagrams

in

P5. Number Systems Egyptian 1. 100 011 2. 123

P9. Sports Probability 1. Probability of forecasting 5 correct wins = –12 x –12 x –12 x –12 x –12 = 32 –1 2. Probability of forecasting 8 correct wins 1 =— 256 No, because it’s a chance process.

Sa m

P4. Number Patterns (3) 1. 87, 78, 89 2. 37, 28, 17 3. 37, 47, 46 4. 6, 17, 8 5. 67, 87, 89 6. 46, 37, 27 7. 56, 67, 78 8. 86, 77, 88 9. 17, 8, 9

pl e

P3. Number Patterns (2) 289, 361, 441, 529, 625, 729, 841. The boxed numbers are square numbers.

26

Problem Solving Through Investigation Book 2

(continued) P18. Hi, Pi! The circumference is approximately 3 times the diameter. This means the diameter is approximately –13 of the circumference.

P12. How Many? 1. 37 m x 100 = 3 700 cm x 10 = 37 000 mm 2. 6 yrs x 365 = 2 190 days x 24 = 52 560 hours 3. 190 km x 1 000 = 190 000 m x 100 = 19 000 000 cm 4. 1 kL x 1 000 = 1 000 L x 1 000 = 1 000 000 mL 5. 52 yrs x 52 = 2 704 weeks x 7 = 18 928 days

P19. Area 1. 64 cm2 2. 81 m2 3. 180 cm2 4. 42 cm2 5. 112 m2 6. 200 m2 7. 66 cm2

pl e

P20. Line Segments Missing parts of the table are: 10 pentagon 15 hexagon 21 heptagon 28 octagon 36 45 55 66

P13. Maths Terms Crossword 2

6 0

3

3

5

4

9 0

7

0

7 2

11

1 0 0 0 16

4 0

19

1

8

5 0 0 0

20

1 1

21

0

17

1 5

18

1

14

1 0

15

12

0

13

4

0

0

22

3 0 2

1 0 0 0

23

0

24

1 0 0

25

2 3

1

26

0

2 0

P21. Coordinates 1. 3W, 3N 2. 3E, 5S 5. 3E, 2N 6. 1E, 2N 9. 5E, 7N 10. 1E, 2S

g

0

2 8 9

9

2 0

10

1

6

8

5 0

4

5 1

Sa m

1

in

P14. Decades and Centuries 1. 6 decades (1907 to 1967) 2. 6 centuries (1215 to 1815) 3. 5 decades (1903 to 1953) 4. 4 centuries (1588 to 1988) 5. 10 decades (1788 to 1888) 6. 3 centuries (1666 to 1966)

ew

Vi

P24. Tessellations The circle and the ellipse will not tessellate.

T & H or X & F Answer varies H&Z six K

P25. Shapes & Solids Sleuth 1. cube 2. 3. isosceles 4. 5. quadrilateral 6. 7. tetrahedron 8. 9. equilateral 10. 11. scalene 12. 13. sphere 14. 15. prism

P17. Graphs 1. Shadows grow shorter up to noon and longer after noon. 2. The sun’s position in the sky affects the lengths of shadows. Discussion point Average length of shadows would be longer in winter as sun has a lower trajectory across the sky. Prim-Ed Publishing

P22. Rigid Shapes The strut in the rectangle makes it rigid because it divides the rectangle into two strong triangles. The numbers in columns 3 and 4 follow consecutively. P23. Polygon Patterns Columns 2, 3, 4 increase consecutively. Column 5 goes up in 3’s. Children work out the last three shapes in the table by following the number patterns down the columns.

P15. Polygons No answers required. P16. Map Reading 1. B 2. 3. H & Z 4. 5. 200 k 6. 7. B 8. 9. 2000 km 10.

3. 7E, 5N 4. 5W, 3S 7. 6E, 1S 8. 3W, 6N

27

heptagon ellipse octagon cone hexagon decagon trapezium

Problem Solving Through Investigation Book 2

0065 Problem Solving through Investigation Book 2