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The Maths Zone

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Rainy Day

Maths at Home

A maths activity collection

Activity Guide

Rainy Day Maths at Home © The Maths Zone 2

Contents Introduction

Activities and Games

4 Big Hands

6 Maths Word Anagrams

7 The Tower of Hanoi

12 Paper folding

13 Mancala

15 Pentomino

17 Nim

18 The Game of Pig

Puzzles

19 Matchstick Patterns

20 Roman Matchsticks

21 Coin Patterns

22 Multiply and Multiply Again

23 Cubic Cuts

24 Dock the Clock

25 Making 4 from 3 and 5

26 Congruent Cuts

Projects

27 Sequences

28 Braiding

32 The Bridges of Königsberg

36 Braille

39 Maths Word Anagrams Answers


Introduction Welcome to the Mayor’s Fund for London Rainy Day Maths at Home Activities Book. This is a book full of activities, games, puzzles and projects for you to do if you are stuck indoors and need a change of pace!

There are different types of activities, some are short, some are long. There is no order. Find one that looks interesting and have a go. Then come back to the others. Some things need to be printed and cut out. If you can print on thicker paper or thin cardboard that is better. If counters are needed and you don’t have

enough, use beads or dry beans. With one exception, we haven’t given you the answers to any of the activities. This is quite deliberate! You will be too tempted to look. After you have worked on a problem for a long time, it is interesting to look it up on the internet to find out what other people think and to find more ideas.

Some advice: DON’T give up! If you are truly stuck, leave it and come back to it later. DON’T look the answers up on the internet! DON’T explain the answers to anyone else but DO share the activities. DO work together with someone else or on your own, whichever you prefer.


Big Hands Number your fingers. Both little fingers should say 6 Try to remember the numbers. Don't use a pen unless you are sure it’s OK.

Your left hand

Your right hand

Now you can do all of the difficult times tables on your fingers. This is how it works:

10

9 8

7

6

6

7

8 9

10


Big Hands (continued)

To do 7 8 put the 7 finger next to the 8 finger.

Count these two fingers with the ones below.

You should count 5 fingers.

There are 3 fingers and 2 fingers above.

Multiply the two numbers together

3 2 makes 6.

Put the two digits 5 and 6 together to make 56 7 x 8 = 56 Practise to make sure you can make it work! Try 77, 89, 99.

When you can do that, try these: 1010, 76, 66.

(They are a bit odd, but they still work!)

10 9

8

7

6

6

7

8

9

10 3 × 2 = 6

3 + 2 = 5


Maths Word Anagrams (Answers at the back of the book)

tie rum zap

sub toe

dubico

cry lined

top crud

rag rule

go ax hen

ire of cats

raques

i hog trams

veruc

grail moth

pragh

lie ran

fit acorn

mad lice

me rip

nagel

quote nia

ben rum

oar it


The Tower of Hanoi

The Tower of Hanoi was invented by the French mathematician, Edouard Lucas, in 1883. The idea came from an old story about training young priests in a Hindu temple.

The story says that at the beginning of time the priests in the temple were given a stack of 64 gold discs,

each one a little smaller than the one beneath it.

The priests' job was to move the 64 discs to the pole at the end. There was one important rule: a large disc could never be placed on top of a smaller one.

The priests worked very fast, day and night. But, it is said that it would take so long that when they finished their work the temple would crumble into dust and the world would vanish. So, why does it take so long?


The Tower of Hanoi (continued)

Solve the puzzle with 3 discs

Practice doing the puzzle using only 3 discs.

Keep going until you are quite sure you have found the quickest way.

Copy each move onto the three disc record sheet.

How many pictures did you need?

Time yourself. Write down your best time.

Now solve the puzzle with 4 discs.

Use the Tower of Hanoi record sheet to record your moves. How many pictures did you need for 4 discs? How long did it take?

Now solve the puzzle with 5 discs.

How many pictures will you need for 5 discs? You don't need to draw them all this time. Work it out.

Look how many you needed for 3 discs and for 4 discs.

How long will it take to solve the puzzle with 5 discs.

How long will 64 discs take? Now work out how long it will take to solve the puzzle with the 64 discs that the priests were given. (A calculator will help!)

Remember that your answer will make more sense if you use years rather than seconds.

To find out more and to try the puzzle online go to:

http://nrich.maths.org/ and search for ‘Tower of Hanoi’


Tower of Hanoi 3 Disc Record sheet

Time __________________


Time __________________


Make a Tower of Hanoi Puzzle

Use a pair of compasses to neatly copy the five discs onto thick paper or card. Use scissors to cut them out. Place them on top of Pole 1, in order of size. Put the largest one at the bottom, like in the picture above.

Pole

1

Pole

2

Pole

3


Paper Folding Challenge: Take a piece of paper and fold it in half, then do it again and again until you have folded 10 times. Can you do it? Try again with a bigger piece of paper!

ONLY Read this if you have given up …

Why is it so difficult? Think about how

thick the paper would be after 10 folds.

A sheet of paper is roughly 01 millimetres

thick.

After one fold we multiply by 2.

Use a calculator, press 01 2 =

For the second fold multiply by two again.

Carry on until you have pressed 2, ten times.

Your answer is a big number, but it is millimetres.

There are 10 millimetres in a centimetre. So keep your

answer in the display and press 10 =

This tells you how thick the paper is. Look at the

bottom of the page to check your answer!

Now find out how thick the paper would be if you

folded it 20 times. ( 1000 for metres) What happens

if you fold it 30 times? How many kilometres would

this be?

The paper should be a little more than 10 centimetre thick after 10 folds. Could you fold that?

After 20 folds it would be over 100 metres!

After 30 folds it would be over 100 kilometres!


Mancala This game is played throughout Africa, but every country on the continent, even language groups within each country, has its own version which has slightly different rules. In Tanzania it is called Soro or Bao, the Igbo in Nigeria people call their games Azigo or Okwe and the Ashante in Ghana have Wari or Aware.

Make a Mancala board on paper and 36 counters.

• Place 3 counters in each of the "bowls" and decide who

goes first.

• When it is your turn you pick up all the beads from one of the bowls on your side and drop them one at a time in each bowl as you move anti-clockwise. This includes placing a bead in the special end cups (called the Mancala).

• If your last bead drops into an empty bowl on your side of the board then you capture. You take the bead in your bowl and any beads in the bowl opposite (i.e. on your opponent’s side). You place the captured beads in your end cup (the one on your right).

• If the last bead drops into your Mancala (on your right) then you get another turn and can pick up from any other bowl on your side.

• The game ends when one complete side of the board is empty. When this happens you capture all the counters on your side and put them in your Mancala

• The player with the most counters in their Mancala wins.


Mancala

Put your captured pieces here in the Mancala

bowls Opponent

Player

Pick up from here

Capture these 4 plus your one

Drop 1

Drop 1 Pick up from here

Drop 1

Example of first move (Get a second turn)

Drop last bead here

Capturing

Finishing

The top player captures all the top beads and wins!


Pentomino This is a puzzle game using pentominoes. These are shapes made out of 5 squares placed side to side. This is the complete set:

The next page has a bigger copy you can print and cut out.

Check that you have all 12 pieces. These are all of the different shapes that can be made of 5 squares. (See if you can find another one!) Search for ‘pentomino’ online and you will find images to make your own set of pentominoes. You can also find some really exciting challenges on the Wikipedia page for Pentomino.

When you are an expert, try fitting all of the pieces to make an 8x8 square with a 2x2 square missing from the middle! You know there are 12 different Pentominoes. That’s five squares fitted together side-to-side. Investigate how many different –ominoes there are with different numbers of squares. Di-ominoes; two squares (normally called dominoes) are well known. The

games of Tetris uses Terominoes (4 squares). Can you find a relationship between the number of squares and the number of shapes?


Nim

Play the game with one other person. Play Game 1

first, then Game 2.

Game 1

Place 21 counters on the table.

Each player takes it in turn to remove 1, 2, or 3

counters.

The player who is forced to take the last counter

loses.

Game 2

Place 12 counters in three rows of 3, 4 and 5

counters.

Each player removes counters in turn.

They may remove any number of counters (up to

all of them) from one row.

The player who is forced to take the last counter

loses.

Now think as a mathematician …

For each game:

• Work out a winning strategy.

• Can one of the players always be sure of

winning? How?

• Does it matter if you change the number of

counters or the number of rows or the number

you take away each time?


The Game of Pig

Playing the Game

A game for two players, using two dice.

The winner is the first player to get 100 or more

points. Both players start with zero points.

Players take turns:

Throw the two dice together. Add the total. Throw

again and add the total to what you had before.

Keep going as many times as you like adding the

total at each throw to your turn total. BUT:

• If you throw one 6 your turn ends and nothing

is added to the score for that turn.

• If you throw a double 6 your turn ends and

your score goes back to zero.

Developing a Strategy

• Play the game many times.

• Decide on your strategy.

• Share your strategy with your partners.

• Which strategy seems to be most successful.

• Discuss why.

Change the Game

Try alternative versions of the game. Change your

strategy to be successful with the new rules.

1. A one-die version: Roll just one die. If you roll

a six, your turn ends and your total reverts to

what it was at the start of your turn.

2. A friendlier version: double sixes don't reset

your total score to zero, but one or two sixes

still ends your turn.

3. If you roll a one, your turn ends, if you roll a

double one, your total score is reset to zero.

Sixes are safe.


Matchstick Patterns Make 2 triangles using 5 matches

Move 3 of these matches to Make exactly 5 triangles Remove 4 of these matches to leave exactly 4 triangles

Remove 4 of these matches to leave exactly 9 squares Add eight more matches to divide this shape into 4 equal pieces


Roman Matches All of these puzzles are sums in Roman numerals. Make them correct by moving the matches stated. Move one match to make this correct

Add one match to make this correct

Move two matches to make this correct

Remove three matches to make this correct

Move one match to make this correct


Coins

Move one of these coins to make two rowswith 4 coins in each row

12 coins make four rows with four coins in each row. Use the same 12 coins to make four rows with five coins in each row

Including the diagonals, this H shape has five lines of three coins. Add two more coins to make ten rows with three coins in each line

This square has four lines each with nine coins.

Remove four coins and make four lines still with nine coins in each with the remaining 24 coins

Remove another four coins and make four lines still with nine coins in each with the remaining 20 coins

Remove another four coins and make four lines still with nine coins in each with the remaining 16 coins.


Multiply and Multiply Again

Multiply 12345679 × 8 No calculators!

Then multiply the answer by 9

What do you notice about your final answer?

Try it with other one digit numbers but always multiply by 9 the second time. Does it work every time?

1 2 3 4 5 6 7 9

× 8

× 9

1 2 3 4 5 6 7 9

× 9


Cubic Cuts These cubes have been cut into two symmetrical halves along planes of symmetry.

Find all the other ways of doing this.

Now find different ways of cutting a cube into three congruent shapes. (This is very hard)

Congruent means the same shape and size.


Dock the Clock Using two straight lines, cut the clock face into three parts, each of which contains numbers which add up to 26

How many lines would you need to cut the clock so that each piece added up to 13? Show how you would do it.


Making 4

from 3 and 5

A large cask contains many litres of water. You have a 5 litre jug and a 3 litre jug.

Describe (either with word or diagrams or both) how you would measure out exactly 4 litres of water. You are allowed to pour water back into the cask. Now try these:

1. Measure exactly 4 litres with an 8 litre jug and a 5 litre jug.

2. Measure exactly 8 litres with an 11 litre jug and

a 6 litre jug. 3. Measure exactly 5 litres with an 11 litre jug and



a 11 litre jug.


Congruent Cuts Cut this shape into two congruent pieces. Congruent means the same size and shape. The cut is not a straight line.

Practise on these:

Now cut the shape into three congruent pieces.

And finally, into four congruent pieces


Sequences

The teacher's question has three parts.

• Where to start? Gauss started at 1.

• Where to finish? Gauss finished at 100.

• How much to go up in? Gauss went up in 1’s.

He had to work out 1 + 2 + 3 + 4 + ……… + 99 + 100.

Example

Start at 2, finish at 80, and go up in twos.

Work out 2 + 4 + 6 + 8 + …….. + 78 + 80.

80 + 2 = 82, 78 + 4 = 82, 76 + 6 = 82 ……………

Go up to 80, counting in 2’s.

That means there are 40 numbers.

This makes 20 pairs and each pair makes 82.

So, the answer must be 82 20 = 164 10 = 1640.

Design your own and race your friends.

One day at school, the teacher told Gauss’s class to add up all of the numbers from 1 up to 100. The teacher wanted a rest and thought it would take ages.

The class started to add up; one plus two plus three plus four and on they went. Gauss thought for a moment;

1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101 ………

There are 100 numbers. So, there must be 50 answers of 101. 101 50 = 505 10 = 5050.

Just as the teacher was about to leave the room

Gauss called out: “It’s 5050 miss!” He was right, so the teacher had to stay!


Braiding You need a braiding square made out of stiff cardboard.

The numbered slots should be cut neatly and the hole in the middle should have been cut out. Cutting the braid (wool will be OK)

Cut 4 lengths of colour 'A' and 4 lengths of colour 'B'. (About one arms length or 50 cm is fine) Cut 1 x extra length of either colour roughly 30cm.

Take the bundle of 8 longer pieces and tie them in the middle with the extra piece.

Use the knot shown in the illustration.

This is called a larkshead knot.

Braiding square


Braiding (continued)

Threading the Card

Thread the larkshead knot down through the centre hole in the card so that the 8 cut ends lie neatly over the numbers.

There should be 2 of each colour on each side.

Thread colours 'A' and 'B' under a slight tension (pulling against the knot) into the slot numbers shown above:

Thread colour 'A' into slots 2, 3, 10, 11, 14, 15, 22, 23

Thread colour 'B' into slots 4, 5, 8, 9, 16, 17, 20, 21

The card is now ready to work the braid.

Working the Braid

You are now going to swap over the pairs of braid from opposite slots.

First you move one piece out of the way then you do

the swap. Then you move on to another pair.

Take a numbers sheet to keep track of where you slot the braid. When you have finished, take the braid off the card and tie a not in the end of it to prevent it from unravelling.

6

5

4

3

2

1

13

14

15

16

17

18

7 8 9 10 11 12

24 23 22 21 20 19


Working the Braid (Numbers sheet)

The first swap is slots 3 and 17. Move 3 into 1, then 17 into 3, then 1 into 17. This can be written: 3 → 1, 17 → 3, 1 → 17

The next swap is 5 and 15. We do this: 5 → 6, 15 → 5, 6 → 15

Then swap 8 and 22. We do this: 8 → 7, 22 → 8, 7 → 22





Finally swap 11 and 21. We do this: 11 → 12, 21 → 11, 12 → 21

Notice that when you have done this procedure for a second time, the threads will have returned to their original position.

Reference: Braids: 250 Patterns from Japan, Peru and Beyond, Roderick Owen, Interweave Press, 1995.

Keep repeating this pattern:

3 → 1 17 → 3 1 → 17

5 → 6 15 → 5 6 → 15

8 → 7 22 → 8 7 → 22

10 → 12 20 → 10 12 → 20

2 → 1 16 → 2 1 → 16

4 → 6 14 → 4 6 → 14

9 → 7 23 → 9 7 → 23

11 → 12 21 → 11, 12 → 21


Braiding Squares

Photocopy onto stiff card. (E.g. the back of a pad of paper). Cut out the two squares. Cut out the shaded hole in the middle. Cut carefully into the 6 notches on each side.


The Bridges of Königsberg This is a picture of the town of Königsberg. (It is now called Kalliningrad and is in the Russian Federation. Can you find it on a map?).

There is a famous mathematical problem about the town of Königsberg.

This is the problem:

• Walk over all of the bridges.

• You must cross each one only once.

Can it be done?

Draw your route with a pencil on the practice maps. If it doesn’t work rub it out and try again!

Leonhard Euler was the first to solve the problem mathematically.

His ideas led to the development of a branch of mathematics called topology.


Königsberg Practice Maps


The Bridges of Königsberg Practice Sheet 1 Try these different sets of bridges.

Can you work out why it is possible to walk over some sets of bridges and impossible to walk over others?

Does it make a difference where you start?

Make up one possible and one impossible set of bridges of your own.


The Bridges of Königsberg Practice Sheet 2

Now try these sets of bridges at Königsberg

Make up one possible and one impossible set of

bridges of your own.


Braille Braille is a way of printing books which can be read by people who are blind.

It uses raised dots which can be felt by someone putting their

finger on the paper.

Each letter fits into a space that is 2 dots wide and 3 dots high.

It was invented by a French man called Louis Braille. He was born in 1809 and died in 1852. He became blind himself when he was three years old.

Louise Braille worked as a professor in Paris. His students were blind too. In 1829 he invented his

system to help the blind to read.

It is now possible to use computers to turn normal

typed letters into printed Braille books.


Braille (continued)

Why did Louis Braille decide to use a pattern 2 dots wide and 3 dots high (a total of 6 dots)?

How many different patterns are needed?

There are 26 letters, 10 number characters, punctuation marks and many other symbols.

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 1 2 3 4 5 6 7 8 9 0 . , : ; ? ! £

We need a different pattern for each character.

That makes a total of at least 43 patterns How many different patterns can you make? This pattern has This pattern has 3 raised dots. 4 raised dots.


Braille Recording Sheet

Draw all the arrangements on this sheet. In each pattern, colour in the raised dots. Hints • Start with all the patterns with only one dot raised • Then do patterns with 2 dots raised and so on.


Maths Word Anagrams ANSWERS

tie rum zap trapezium

sub toe obtuse

dubico cuboid

cry lined cylinder

top crud product

rag rule regular

go ax hen hexagon

ire of cats factorise

raques square

i hog trams histogram

veruc curve

grail moth algorithm

pragh graph

lie ran linear

fit acorn fraction

mad lice decimal

me rip prime

nagel angle

quote nia equation

ben rum number

oar it ratio

Rainy Day Maths at Home - mayorsfundforlondon.org.uk · Welcome to the Mayor’s Fund for London...

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