Contributed by:
This book is for mathematics teachers working in higher primary and secondary schools in developing countries. The book will help teachers improve the quality of mathematical education because it deals specifically with some of the challenges which many maths teachers in the developing world face, such as a lack of ready-made teaching aids, possible textbook shortages, and teaching and learning maths in a second language.
1.
THE MATHS TEACHER’S HANDBOOK
JANE PORTMAN
JEREMY RICHARDON
2.
Who is this book for?
This book is for mathematics teachers working in higher primary and
secondary schools in developing countries. The book will help teachers
improve the quality of mathematical education because it deals
specifically with some of the challenges which many maths teachers in
the developing world face, such as a lack of ready-made teaching aids,
possible textbook shortages, and teaching and learning maths in a
second language.
Why has this book been written?
Teachers all over the world have developed different ways to teach maths
successfully in order to raise standards of achievement. Maths teachers
• developed ways of using locally available resources
• adapted mathematics to their own cultural contexts and to the tasks and
problems in their own communities
• introduced local maths-related activities into their classrooms
• improved students’ understanding of English in the maths
classroom.
This book brings together many of these tried and tested ideas from
teachers worldwide, including the extensive experience of VSO maths
teachers and their national colleagues working together in schools
throughout Africa, Asia, the Caribbean and the Pacific.
We hope teachers everywhere will use the ideas in this book to help
students increase their mathematical knowledge and skills.
What are the aims of this book?
This book will help maths teachers:
• find new and successful ways of teaching
maths
• make maths more interesting and more
relevant to their students
• understand some of the language and
cultural issues their students
experience.
Most of all, we hope this book will
contribute to improving the quality of
mathematics education and to raising
standards of achievement.
3.
WHAT ARE THE MAIN THEMES OF THIS BOOK?
There are four main issues in the teaching and learning of
Teaching methods
Students learn best when the teacher uses a wide range of teaching
methods. This book gives examples and ideas for using many different
methods in the classroom,
Resources and teaching aids
Students learn best by doing things: constructing, touching, moving,
investigating. There are many ways of using cheap and available
resources in the classroom so that students can learn by doing. This
book shows how to teach a lot using very few resources such as bottle
tops, string, matchboxes.
The language of the learner
Language is as important as mathematics in the mathematics
classroom. In addition, learning in a second language causes special
difficulties. This book suggests activities to help students use language
to improve their understanding of maths.
The culture of the learner
Students do all sorts of maths at home and in their communities. This
is often very different from the maths they do in school. This book
provides activities which link these two types of rnaths together.
Examples are taken from all over the world. Helping students make this
link will improve their mathematics.
4.
HOW DID WE SELECT THE ACTIVITIES AND TEACHING
IDEAS IN THIS BOOK?
There are over 100 different activities in this book which teachers can use
to help vary their teaching methods and to promote students’
understanding of maths.
The activities have been carefully chosen to show a range of different teaching
methods, which need few teaching aids. The activities cover a wide range
of mathematical topics.
Each activity:
• shows the mathematics to be learned
• contains clear instructions for students
• introduces interesting ways for students to learn actively.
What is mathematics?
Mathematics is a way of organising our experience of the world. It
enriches our understanding and enables us to communicate and make
sense of our experiences. It also gives us enjoyment. By doing
mathematics we can solve a range of practical tasks and real-life
problems. We use it in many areas of our lives.
In mathematics we use ordinary language and the special language of
mathematics. We need to teach students to use both these languages.
We can work on problems within mathematics and we can work on
problems that use mathematics as a tool, like problems in science and
geography. Mathematics can describe and explain but it can also predict
what might happen. That is why mathematics is important.
Learning and teaching mathematics
Learning skills and remembering facts in mathematics are important but
they are only the means to an end. Facts and skills are not important in
themselves. They are important when we need them to solve a problem.
Students will remember facts and skills easily when they use them to
solve real problems.
As well as using mathematics to solve real-life problems, students should
also be taught about the different parts of mathematics, and how they fit
Mathematics can be taught using a step-by-step approach to a topic but it
is important to show that many topics are linked, as shown in the diagram
on the next page.
It is also important to show students that mathematics is done all over the
5.
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Although each country may have a different syllabus, there are many
topics that are taught all over the world. Some of these are:
• number systems and place value
• arithmetic
• algebra
• geometry
• statistics
• trigonometry
• probability
• graphs
• measurement
We can show students how different countries have developed
different maths to deal with these topics.
How to use this book
This book is not simply a collection of teaching ideas and activities. It
describes an approach to teaching and learning mathematics.
This book can be best used as part of an approach to teaching using a
plan or scheme of work to guide your teaching. This book is only one
resource out of several that can be used to help you with ideas for
activities and teaching methods to meet the needs of all pupils and to
raise standards of achievement.
There are three ways of using this book:
Planning a topic
Use your syllabus to decide which topic you are going to teach next, Find
that topic in the index at the back of the book. Turn to the relevant pages
and select activities that are suitable. We suggest that you try the
activities yourself before you use them in the classroom. You might like to
discuss them with a colleague or try out the activity on a small group of
students. Then think about how you can or need to adapt and improve the
activity for students of different abilities and ages.
6.
Improving your own teaching
One way to improve your own teaching is to try new methods and
activities in the classroom and then think about how well the activity
improved students’ learning. Through trying out new activities and
working in different ways, and then reflecting on the lesson and
analysing how well students have learned, you can develop the best
methods for your students.
You can decide to concentrate on one aspect of teaching maths:
language, culture, teaching methods, resources or planning. Find
the relevant chapter and use it.
Working with colleagues
Each chapter can be used as material for a workshop with
colleagues. There is material for workshops on:
• developing different teaching methods
• developing resources and teaching aids
• culture in the maths classroom
• language in the maths classroom
• planning schemes of work.
In the workshops, teachers can try out activities and discuss the
issues raised in the chapter. You can build up a collection of
successful activities and add to it as you make up your own,
individually or with other teachers.
7.
CHAPTER 1
TEACHING METHODS
This chapter is about the different ways you can teach a topic in the
classroom. Young people learn things in many different ways. They
don’t always learn best by sitting and listening to the teacher.
Students can learn by:
• practising skills on their own
• discussing mathematics with each other
• playing mathematical games
• doing puzzles
• doing practical work
• solving problems
• finding things out for themselves.
In the classroom, students need opportunities to use different ways
of learning. Using a range of different ways of learning has the
following benefits:
• it motivates students
• it improves their learning skills
• it provides variety
• it enables them to learn things more quickly.
We will look at the following teaching methods:
1 Presentation and explanation by the teacher
2 Consolidation and practice
3 Games
4 Practical work
5 Problems and puzzles
6 Investigating mathematics
Presentation and explanation
by the teacher
This is a formal teaching method which involves the teacher
presenting and explaining mathematics to the whole class. It can be
difficult because you have to ensure that all students understand.
This can be a very effective way of:
• teaching a new piece of mathematics to a large group of students
• drawing together everyone’s understanding at certain stages of a
topic
• summarising what has been learnt,
Planning content before the lesson:
• Plan the content to be taught. Check up any points you are not
sure of. Decide how much content you will cover in the session.
8.
• Identify the key points and organise them in a logical order. Decide
which points you will present first, second, third and so on.
• Choose examples to illustrate each key point.
• Prepare visual aids in advance.
• Organise your notes in the order you will use them. Cards can be
useful, one for each key point and an example.
Planning and organising time
• Plan carefully how to pace each lesson. How much time will you
give to your presentation and explanation of mathematics? How
much time will you leave for questions and answers by students?
How much time will you allow for students to practise new
mathematics, to do different activities like puzzles, investigations,
problems and so on?
• With careful planning and clear explanations, you will find that you
do not need to talk for too long. This will give students time to do
mathematics themselves, rather than sitting and listening to you
doing the work.
You need to organise time:
• to introduce new ideas
• for students to complete the task set
• for students to ask questions
• to help students understand
• to set and go over homework
• for practical equipment to be set up and put away
• for students to move into and out of groups for different activities.
Organising the classroom
• Organise the classroom so that all students will be able to see you
when you are talking.
• Clean the chalkboard. If necessary, prepare notes on the
chalkboard in advance to save time in the lesson.
• Arrange the teacher’s table so that it does not restrict your
movement at the front of the class. Place the table in a position
which does not create a barrier between you and the students.
• Organise the tables and chairs for students according to the type of
activity:
- facing the chalkboard if the teacher is talking to the whole
group
- in circles for group work.
• Develop a routine for the beginning of each lesson so that all
students know what behaviour is expected of them from the
beginning of the session. For example, begin by going over
homework.
• Create a pleasant physical environment. For example, display
students’ work and teaching resources - create a ‘puzzle corner’.
9.
• It is very important that your voice is clear and loud enough for all
students to hear.
• Vary the pitch and tone of your voice.
• Ask students questions at different stages of the lesson to check they
have understood the content so far. Ask questions which will make
them think and develop their understanding as well as show you that
they heard what you said.
• For new classes, learn the names of students as quickly as
possible.
• Use students’ names when questioning.
• Speak with conviction. If you sound hesitant you may lose
students’ confidence in you.
• When using the chalkboard, plan carefully where you write things. It
helps to divide the board into sections and work through each
section systematically.
• Try not to end a lesson in the middle of a teaching point or
example.
• Plan a clear ending to the session.
Ground rules for classroom behaviour
• Students need to know what behaviour is acceptable and
unacceptable in the classroom.
• Establish a set of ground rules with students. Display the rules in the
classroom.
• Start simply with a small number of rules of acceptable behaviour. For
example, rules about entering and leaving the room and rules about
starting and finishing lessons on time.
• Identify acceptable behaviour in the following situations:
- when students need help
- when students need resources
- when students have forgotten to bring books or homework to the
lesson
- when students find the work too easy or too hard.
10.
Consolidation and practice
It is very important that students have the opportunity to
practise new mathematics and to develop their
understanding by applying new ideas and skills to new
problems and new contexts.
The main source of exercises for consolidation and practice
is the text book.
It is important to check that the examples in the exercises
are graded from easy to difficult and that students don’t start
with the hardest examples. It is also important to ensure that
what is being practised is actually the topic that has been
covered and not new content or a new skill which has not
been taught before.
This is a very common teaching method. You should take
care that you do not use it too often at the expense of other
Select carefully which problems and which examples
students should do from the exercises in the text book.
Students can do and check practice exercises in a variety of
ways. For example:
• Half the class can do all the odd numbers. The other half
can do the even numbers. Then, in groups, students can
check their answers and, if necessary, do corrections. Any
probiems that cannot be solved or agreed on can be given
to another group as a challenge.
• Where classes are very large, teachers can mark a
selection of the exercises, e.g. all odd numbers, or those
examples that are most important for all students to do
correctly.
• To check homework, select a few examples that need to
be checked. Invite a different student to do each example
on the chalkboard and explain it to the class. Make sure
you choose students who did the examples correctly at
home. Over time, try to give as many students as possible
a chance to teach the class.
You can set time limits on students in order to help them work
quickly and increase the pace of their learning.
• When practising new mathematics, students should not
have to do arithmetic that is harder than the new
mathematics. If the arithmetic is harder than the new
mathematics, students will get stuck on the arithmetic and
they will not get to practise the new mathematics.
11.
Both the examples befow ask students to practise finding the
area of a rectangular field. But students will slow down or
get stuck with the arithmetic of the second example.
• Find the area of a rectangular field which is 10 rn long and
6 m wide. (correct way)
• Find the area of a rectangular field which is 7.63 m long
and 4.029 m wide. (wrong way)
• Questions must be easy to understand so that the skill
can be practised quickly.
Both the examples below ask the same question. Students will
understand the first example and practise finding the area of a
circle. In the second exampte they will spend more time
understanding the question than practising finding the area.
• A circular plate has a radius of 10 cm. Find its area. (good)
• Find the area of the circular base of an electrical reading
lamp. The base has a diameter of 30 cm. (bad)
Games
Using games can make mathematics classes very enjoyable, exciting and interesting. Mathematical
games provide opportunities for students to be actively involved in learning. Games allow students
to experience success and satisfaction, thereby building their enthusiasm and self-confidence.
But mathematical games are not simply about fun and confidence building. Games help students to:
• understand mathematical concepts
• develop mathematical skills
• know mathematical facts
• learn the language and vocabulary of mathematics
• develop ability in mental mathematics.
TOPIC Probability
• Probability is a measure of how likely an event is to happen.
• The more often an experiment is repeated, the closer the outcomes get to the theoretical
probability.
Game: Left and right
A game for two players.
Make a board as shown.
You will need:
• a counter e.g. a stone,
a bottle cap.
• two dice
• a board with 7 squares
12.
Place the counter on the middle square. Throw two dice. Work out
the difference between the two scores. If the difference is 0,1 or 2,
move the counter one space to the left. If the difference is 3, 4 or
5, move one space to the right. Take it in turns to throw the dice,
calculate the difference and move the counter. Keep a tally of how
many times you win and how many you lose. Collect the results of
all the games in the class.
• How many times did students win? How many times did students
lose?
• Is the game fair? Why or why not?
• Can you redesign the game to make the chances of winning:
- better than losing?
- worse than losing?
- the same as losing?
TOPIC Multiplying and dividing by decimals
Multiplying by a number between 0 and 1 makes numbers smaller.
Dividing by a number between 0 and 1 makes numbers bigger.
Game: Target 100
A game for two players.
Player 1 chooses a number between 0 and 100. Player 2 has to
multiply it by a number to try and get as close to 100 as possible.
Player 1 then takes the answer and multiplies this by a number to
try and get closer to 100. Take it in turns. The player who gets
nearest to 100 in 10 turns is the winner.
Change the rules and do it with division.
TOPIC Place value
Digits take the value of the position they are in.
The number line is a straight line on which numbers are placed in
order of size. The line is infinitely long with zero at the centre.
Game: Think of a number (1)
A game for two players.
Player 1 thinks of a number and tells Player 2 where on the
number line it lies, for example between 0 and 100, between -10
and -20, 1000 and 2000, etc. Player 2 has to ask questions to find
the number. Player 1 can only answer ‘Yes’ or ‘No’.
Player 2 must ask questions
like: ‘Is it bigger than 50?’
‘Is it smaller than 10?’
Keep a count of the number of questions used to find the number ‘
and give one point for each question.
Repeat the game several times. Each player has a few turns to
choose a number and a few turns to ask questions and find the
number. The player with the fewest points wins.
13.
TOPIC Properties of numbers
• Numbers can be classified and identified by their properties e.g. odd /even, factors,
multiple, prime, rectangular, square, triangular.
Game: Think of a number (2)
A game for two players.
Player 1 thinks of a number between 0 and 100. Player 2 has to find
the number Player 1 is thinking of. Player 2 asks Player 1 questions
about the properties of the number, for example
‘Is it a prime number?’
‘Is it a square number?’
‘Is it a triangular number?’
‘Is it an odd number?’
‘Is it a multiple of 3?’
‘Is it a factor of 10?’
Player 1 can only answer ‘Yes’ or ‘No’.
Player 2 will find it helpful to have a 10 x 10 numbered square to cross off the
numbers as they work.
Each player has a few turns to choose a number and a few turns to
ask questions and find the number.
TOPIC Algebraic functions
• A function is a rule connecting every member of a set of numbers to a unique
number in a different set, for example x -> 3x,
x -> 2x + 1
Game: Discover the function
A game for the whole class.
Think of a simple function, for example x 3
Write a number on the left of the chalkboard. This will be an IN number, though it is important
not to tell students at this stage. Opposite your number, write the OUT number. For example:
10 30
Show two more lines. Choose any numbers and apply the function rule x 3:
5 15
7 21
Now write an IN number only and invite a student to come to the board to write the OUT
11 ?
14.
If they get it right, draw a happy face. If they get it wrong, give them a
sad face then other students can have a chance to find the correct
OUT number. When students show that they know the rule, help them
find the algebraic rule. Write x in the IN column and invite students to
fill in the OUT column;
x ?
The game is best when played in silence!
When students have shown that they know the function, try
another. The board will begin to look like this:
You could extend the game in these ways:
• Try a function with two operations, for example x 2 + 1
• Introduce the functions: square, cube and under-root.
• Challenge pupils to find functions with two operations which
produce the same table of IN and OUT numbers.
• Challenge students to show why the function: x 2 + 2 is the same as
the function: +1 x 2.
In algebra, this is written as 2x + 2 and (r + 1)x2 or 2(r + 1),
• How many other pairs of functions that are the same can they find?
• Challenge students to find functions which don’t change numbers -
when a number goes IN it stays the same. An easy example is x 1!
TOPIC Equivalent fractions, decimals and percentages
• Fractions, decimals and percentages are rational numbers. They can
all be expressed as a ratio of two integers and they lie on the same
number line. All these are equivalent: 1/2= 2/4= 0.5 = 50%.
Game: Snap (1)
A game for two or more players.
You will need to make a pack of at least 40 cards. On each card write
a fraction or a decimal or a percentage. Make sure there are several
cards which carry equivalent fractions, decimals or percentages (you
can use the cards shown on the next page as a model).
15.
Shuffle the cards and deal them out, face down, to the players. The
players take it in turn to place one of their cards face up in the
middle. The first player to see that a card is equivalent to another
card face up in the middle must shout ‘Snap!’, and wins all the cards
in the middle, The game continues until all the cards have been won.
The winner is the player with the most cards.
TOPIC Similarity and congruence of shapes
• Plane shapes are similar when the corresponding sides are
proportional and corresponding angles are equal.
• Plane shapes are similar if they are enlargements or reductions of
each other.
• Plane shapes are congruent when they are exactly the same size
and shape.
Game: Snap (2)
A game for two or more players.
You will need to make a pack of at least 20 cards with a shape on
each card. Make a few pairs of cards with similar shapes and a few
pairs of cards with congruent shapes. The game is played in the
same way as Snap (1) above.
To win the pile of cards, the students must call out ‘Similar’ or
‘Congruent’ when the shapes on the top cards are similar or
congruent.
16.
TOPIC Estimating the size of angles
• Angle is a measure of turn. It is measured in degrees.
• Angles are acute (less than 90°), right angle (90°), obtuse
(more than 90° and less than 180°) or reflex (more than 180°).
Game: Estimating an angle
Game for two players.
Game A
Player 1 chooses an angle e.g. 49°. Player 2 has to draw that angle without using a protractor.
Player 1 measures the angle with a protractor. Player 2 scores the number of points that is the
difference between their angle size and the intended one. For example, Player 2’s angle is
Player 2 tries to draw a 49° angle measured to be 39°. So Player 2 scores 10 points (49°-39°).
without a protractor
Take it in turns. The winner is the player with the lowest score.
Game B
Each player draws 15 angles on a blank sheet of paper. They swap papers and estimate the size of
each angle. Then they measure the angles with a protractor and compare the estimate and the
exact measurement of the angles. Points are scored on the difference of the estimate and the
The angle measures 39°.
Player 2 scores 10 points (49°-39°)
actual size of each angle. The player with the lowest score wins.
Practical work
Practical work means three things:
• Using materials and resources to make things. This involves
using mathematical skills of measuring and estimation and a
knowledge of spatial relationships.
• Making a solid model of a mathematical concept or
relationship.
• Using mathematics in a practical, real-life situation like
in the marketplace, planning a trip, organising an event.
Practical work always involves using resources.
TOPICS Shapes, nets, area, volume, measurement,
scale drawing
Activity: Design a box
A fruit seller wants to sell her fruit to shops in the next large
town. She needs to transport the fruit safely and cheaply. She
needs a box which can hold four pieces of fruit. The fruit must
not roll about otherwise it will get damaged. The box must be
strong enough so that it does not break when lifted.
17.
In pairs, students can design a box which holds four pieces of fruit.
Students need to make scale drawings of their design. Then four
box designs can be compared and students can decide which
design would be best for the fruit seller. Once the best design has
been chosen, students may want to cut and make a few boxes
from one piece of card. They can work from the scale drawing
A box for bananas and test the design they chose.
To choose the best box design, students need to think about:
• Shapes
• the strength of different box shapes
• the shape that uses the least amount of card
• the shape that packs best with other boxes of the same shape
• Nets
• all the different nets for the shape of the box
• where to put the tabs to glue the net together
• how many nets for the box fit on one large piece of card
A net for the banana box
without waste
• Area
• surface area of shapes such as squares, rectangles, cylinders,
triangles
• total surface area of the net (including tabs)
• which box shapes use the smallest amount of card
A box for oranges • Volume
• the volume of boxes of different shapes
• the smallest volume for their box shape so the fruit does not
roll about
circumference of • Measurement
• the size of the fruit in different arrangements
orange box
• the arrangement that uses the least space
• the accurate measurements for their chosen box shape
• Scale drawing
• which scale to use
Net for the box of oranges • scaling down the accurate dimensions of the box, according to
the scale factor
• how to draw an accurate scale drawing of the box and its net
TOPICS Accurate measurement, graphs and relationships
Activity: 10 seconds
You will need: Design a pendulum to measure 10 seconds exactly. The pendulum
• string must complete exactly 10 swings in 10 seconds. Experiment with
• drawing pins different weights and lengths of string until the pendulum
• a ruler completes 10 swings in 10 seconds.
• a watch
• some weights, for • Accurate measurement
example stones Students need to measure the mass of the weights, the time of 10
swings, length of the string etc.
18.
• Graphs and relationships
Students need to decide what affects the length of time for 10
swings and how it affects it. For example, how does increasing
or decreasing the length of string or the weight of the stone
affect the time taken for 10 swings? To discover these
relationships, students can draw graphs of the relationship
between time and length of string or between time and weight.
TOPICS Estimation, area, inverse proportion, scale drawings,
Pythagoras’ Theorem, trigonometry
Activity: Shelter
Give students the following problem.
You and a friend are on a journey. It is nearly night time and you
have nowhere to stay. You have a rectangular piece of cloth
measuring 4 m by 3 m. Design a shelter to protect both of you
from the wind and rain.
Decide:
• how much space you need to lie down
• what shape is best for your shelter
• what you will use to support the shelter - trees, rocks etc?
Help pupils by suggesting that they:
• begin by making scale drawings of possible shelters
• make a model of the shelter they choose
• estimate the heights and lengths of the shelter.
To solve the design problem, students need to:
• Do estimations
• of the height of the people who will use the shelter
• of the floor area of the shelter
• Calculate area
• of the floor of different shelter designs such as
rectangles, squares, regular and irregular polygons,
triangles, circles
• Understand inverse proportion
• for example, if the height of the shelter increases, the floor
area decreases
• Make scale drawings of different possible shelters
• based only on a few certain dimensions like length of one or
two sides, radius
• Use Pythagoras’ Theorem and trigonometry
• to calculate the dimensions of the other parts of the shelter
such as lengths of other sides and angles
19.
TOPIC Probability
• different outcomes may occur when repeating the same
experiment
• relative frequency can be used to estimate probabilities
• the greater the number of times an experiment is repeated, the
closer the relative frequency gets to the theoretical probability.
Activity: Feely bag
Put different coloured beads in a bag, for example 5 red, 3 black
and 1 yellow bead. Invite one student to take out a bead. The
student should show the bead to the class and they should note its
colour. The student then puts the bead back in the bag. Repeat
over and over again, stop when students can say with confidence
how many beads of each colour are in the bag.
Activity: The great race
You will need: Roll two dice and add up the two numbers to get a total. The
• a grid for the race track, as runner whose number is the total can be moved forward one
shown square. For example,
• 2 dice
• a stone for each runner = 9, so runner 9 moves forward one square.
which can be moved along Play the game and see which runner finishes first. Repeat the
the race track game a few times. Does the same runner always win? Is the
game fair? Which runner is most likely to win? Which runner is
least likely to win? Change the rules or board to make it fair.
TOPICS Triangles, quadrilaterals, congruence, vectors.
Activity: Exploring shapes on geoboards
You will need: Make a few geoboards of different shapes and sizes. Students
• nails can wrap string or elastic around the nails to make different
• pieces of wood shapes on the geoboards like triangles, quadrilaterals. They can
• string, coffon or elastic bands investigate the properties and areas of the different shapes.
20.
For example:
• How many different triangles can be found on a 3 x 3 geoboard? Classify the
triangles according to: size of angles, length of sides, lines of symmetry, order
of rotational symmetry. Find the area of the different triangles.
• How many different quadrilaterals can be made on 4 x 4 geoboards?
Classify the quadrilaterals according to: size of angles, length of sides, lines of
symmetry, order of rotational symmetry, diagonals. Find the area of the different
quadrilaterals.
• How many different ways can a 4 x 4 geoboard be split into:
- two congruent parts?
- four congruent parts?
• Can you reach all the points on a 5 x 5 geoboard by using the three vectors
shown? In how many different ways can these points be reached? Always
start from the same point. You can use the three types of movement shown in
the vectors in any order, and repeat them any number of times. Explore on
different sized geoboards.
Problems and puzzles
This teaching method is about encouraging students to learn mathematics
through solving problems and puzzles which have definite answers. The
key point about problem-solving is that students have to work out the
method for themselves.
Puzzles develop students’ thinking skills. They can also be used to introduce
some history of mathematics since there are many famous historical maths
Textbook exercises usually get students to practise skills out of context.
Problem-solving helps students to develop the skills to select the appropriate
method and to apply it to a problem.
TOPIC Basic addition and subtraction
Activity: Magic squares
Put the numbers 1,2,3, 4, 5, 6, 7, 8, 9 into a 3 x 3 square to make a
magic square. In this 3x3 magic square, the numbers in each vertical
row must add up to 15. The numbers in each horizontal row must add
up to 15. The diagonals also add up to 15.15 is called the magic
number.
21.
• How many ways are there to put the numbers 1-9 in a magic 3 x 3
square?
• Can you find solutions with the number 8 in the position shown?
• There are 880 different solutions to the problem of making a 4 x 4 magic
square using the numbers 1 to 16. How many of them can you find
where the magic number is 34?
• What are the values of x, y and 2 in the magic square on the right?
(The magic number is 30.)
TOPIC Multiplication and division of 3-digit numbers
Activity: Digits and squares
The numbers 1 to 9 have been arranged in a square so that the
second row, 384, is twice the top row, 192. The third row, 576, is
three times the first row, 192. Arrange the numbers 1 to 9 in
another way without changing the relationship between the
numbers in the three rows.
TOPIC The four operations on single-digit numbers
Activity: Boxes
Put all the numbers 1 to 9 in the boxes so that all four equations
are
correct.
Fill in the boxes with a different set of numbers so that the
four equations are still correct.
TOPIC Squaring numbers and adding numbers
• To square a number you multiply it by itself.
Activity: Circling the squares
Place a different number in each
empty box so that the sum of the
squares of any two numbers next to
each other equals the sum of the
squares of the two opposite
numbers.
For example: 162 + 22 = 82+ 142
22.
TOPIC Addition, place value
Activity: Circling the sums
Put the numbers 1 to 19 in the boxes so that three
numbers in a line add up to 30.
TOPIC Surface area, volume and common factors
Activity: The cuboid problem
The top of a box has an area of 120 cm2, the side has an area of 96
cm2 and the end has an area of 80 cm2. What is the volume of the
box?
TOPIC Shape and symmetry
Activity: The Greek cross
A Greek cross is made up of five squares, as shown in the diagram.
• Make a square by cutting the cross into five pieces and
rearranging the pieces.
• Make a square by cutting the cross into four pieces and
rearranging them.
A Greek cross • Try with pieces that are all the same size and shape. Try with all the
pieces of different sizes and shapes.
TOPIC Equilateral triangles and area
An equilateral triangle has three sides of equal length and three
angles of equal size.
Activity: Match sticks
• Make four equilateral triangles using six match sticks.
• Take 18 match sticks and arrange them so that:
- they enclose two spaces; one space must have twice the area of
the other
- they enclose two four-sided spaces; one space must have three
times the area of the other
- they enclose two five-sided spaces; one space must have three
times the area of the other
23.
TOPIC Addition, place value
Activity: Decoding
Each letter stands for a digit between 0 and 9. Find the value of each
letter in the sums shown.
TOPIC Forming and solving equations
Activity: Find the number
1. Find two whole numbers which multiply together to make 221.
2. Find two whole numbers which multiply together to make 41.
3. I am half as old as my mother was 20 years ago. She is now 38.
How old am I?
4. Find two numbers whose sum is 20 and the sum of their squares
is 208.
5. Find two numbers whose sum is 10 and the sum of their cubes is
370.
6. Find the number which gives the same result when it is added to
3-3/4 as when it is multiplied by 3-3/4.
TOPIC Percentages
Activity: Percentage problems
1. An amount increases by 20%. By what percentage do I have to
decrease the new amount in order to get back to the original
amount?
2. The length of a rectangle increases by 20% and the width
decreases by 20%, What is the percentage change in the area?
3. The volume of cube A is 20% more than the volume of cube B.
What is the ratio of the cube A’s surface area to cube B’s surface
area?
TOPIC Probability
Activity: Probability problems
• To calculate the theoretical probability of an event, you need to list
all the possible outcomes of the experiment.
• The theoretical probability of an event is the number of ways that
event could happen divided by the number of possible outcomes
of the experiment.
1. I have two dice, I throw them and I calculate the difference. What
is the probability that the difference is 2? How about other
differences between 0 and 6?
2. I write down on individual cards the date of the month on which
everyone in the class was born. I shuffle the cards and choose
two of them. What is the probability that the sum of the two
numbers is even? What is the probability that the sum of the two
numbers is odd? When would these two probabilities be the
same?
24.
3. Toss five coins once. If you have five heads or five tails you have
won. If not, you may toss any number of coins two more times to
get this result. What is the probability that you will get five heads
or five tails within three tosses?
4. You have eight circular discs. On one side of them are the
numbers 1, 2, 4, 8, 16, 32, 64 and 128. On the other side of each
disc is a zero. Toss them and add together the numbers you see.
What is the probability that the sum is at least 70?
5. Throw three dice. What is more likely: the sum of the numbers is
divisible by 3 or the multiple of the numbers is divisible by 4?
Investigating mathematics
Many teachers show students how to do some mathematics and then
ask them to practise it. Another very different approach is possible.
Teachers can set students a challenge which leads them to discover
and practise some new mathematics for themselves. The job for the
teacher is to find the right challenges for students. The challenges need
to be matched to the ability of the pupils.
The key point about investigations is that students are encouraged to
make their own decisions about:
• where to start
• how to deal with the challenge
• what mathematics they need to use
• how they can communicate this mathematics
• how to describe what they have discovered.
We can say that investigations are open because they leave many
choices open to the student. This section looks at some of the
mathematical topics which can be investigated from a simple starting
point. It also gives guidance on how to invent starting points for
investigations,
TOPIC Linear equations and straight line graphs
• An equation can be represented by a graph.
• There is a relationship between the equation and the shape of the
graph.
• A linear equation of the form y = mx + c can be represented by a
straight line graph.
• m determines the gradient of the straight line and c determines
where the graph intercepts the y axis.
Investigation of graphs of linear equations
Write on the board:
The y number is the same as the jt number plus 1.
Ask students to write down three pairs of co-ordinates which follow
this rule. Plot the graph.
25.
Change the rule:
The y number is the same as the x number plus 2.
Ask students to write down three pairs of co-ordinates which follow
this rule. Plot the graph on the same set of axes.
Ask students what they notice about the gradients of the straight line
graphs and the intercepts on the y axis.
Ask students to write the rules on the board as algebraic equations.
Students can then plot the graphs of the following rules:
• The y number = twice the x number
• The y number = three times the x number
• The y number = three times the x number plus 1
Ask students to write the rules as algebraic equations.
Students can work on their own to understand the relationship
between straight line graphs and linear equations. The instructions
below should help them.
Make your own rules for straight line graphs. Plot three co-ordinates
and draw the graphs of these rules.
Make rules with negative numbers and fractions as well as whole
numbers.
Write the equations for each rule and label each straight line graph
with its equation.
Describe any patterns you notice about the gradient of the graphs
and their intercept on the y axis. Do the equations of the graphs tell
you anything about the gradient and the intercept on the y axis?
TOPIC Area and perimeter of shapes
• Area is the amount of space inside a shape.
• Perimeter is the distance around the outside of a shape.
• Area can be found by counting squares or by calculation for regular shapes.
26.
Investigation of area and perimeter
1. A farmer has 12 logs to make a border around a field. Each log is
1 m long. The field must be rectangular.
What is the biggest area of field the farmer can make? What is
the smallest area of field the farmer can make? The farmer now
has 14 logs. Each log is 1 m long. What are the biggest and
smallest fields he can make? Explore for different numbers of
logs.
2. A farmer has 12 logs. Each log is 1 m long. A farmer can make a
field of any shape.
What is the biggest area of field that the farmer can make? What
is the smallest area of field the farmer can make? Explore for
different numbers of logs.
3. You have a piece of string that is 36 m long Find the areas of all
the shapes you can make which have a perimeter of 36 m.
4. A piece of land has an area of 100 mz. How many metres of wire
fencing is needed to enclose it?
TOPIC Volume and surface area of solids
• Volume is the amount of space a solid takes up.
• Volume can be found by counting cubes or by calculation for
regular solids.
• Surface area is the area of the net of a solid.
• Surface area can be found by counting cubes or by calculation for
regular shapes.
Investigation of volume and surface area of solids
1. You may only use 1 sheet of paper. What is the largest volume
cuboid you can make?
2. You are going to make a box which has a volume of 96 cm cubed
or 96 cm3. The box can be any shape. What is the smallest
amount of card you need?
3. You have a square of card. The card is 24 cm x 24 cm. You can
make the card into a box by cutting squares out of the corners
and folding the sides up.
Make the box with the biggest volume. What is the length of the
side of the cut-out squares? Try for other sizes of square card.
Try with rectangular cards.
4. You have a piece of card which is 24 cm x 8 cm. The card is
rectangular What is the biggest volume cylinder you can make?
5. You are going to make a cylinder. The cylinder must have a
volume of 80 cm3. What is the smallest amount of card you
need?
Topic Simultaneous equations
• Simultaneous equations are usually pairs of equations with the
same unknowns in both equations. For example:
x + y = 10
x-y=4
27.
• When simultaneous equations are solved, the unknowns have the
same value for both equations. For example, in both equations
above, x = 7 and y = 3.
One of the simultaneous equations cannot be solved without the
other.
Investigation of simultaneous equations
Simultaneous equations can be solved by trial and improvement, by
using equation laws and/or by substitution.
Write an equation on the top of the board, for example x + y = 10.
Divide the rest of the board into two columns. Ask each student to
do the following:
• Think of one set of values for x and y which makes the equation
on the board true. Do not tell anyone these values.
• Make up another equation in x and y using your values.
Invite students one by one to say the equations they have made up.
If their equation works with the same values as the teacher’s
equation, write it in the left hand column; if it does not work then
write it in the right hand column. Ask students to:
• Work out the values of x and y for each set of equations.
• Discuss the methods they used to solve each set of simultaneous
equations.
Study the two lists of equations on the board:
• Are any pairs the same?
• Can any of the equations be obtained from one or two others?
Topic Tessellations
• A tessellation is a repeating pattern in more than one direction of
one shape without any gaps.
• A semi-regular tessellation is a repeating pattern in more than one
direction of two shapes without any gaps.
• A regular shape will tessellate if the interior angle is a factor of
360°.
• Semi-regular tessellations work if the sum of a combination of the
interior angles of the two shapes is 360°.
Investigation of tessellations
Give students a collection of regular polygons. Ask them to find out:
• Which polygons can be used on their own to cover a surface
without any gaps?
• Which two polygons can be used together to cover the surface
without any gaps?
• Explain why some shapes tessellate on their own and others
tessellate with a second shape.
28.
TOPIC The relationship between the circumference, radius,
diameter and area of circles
• The formula for the circumference of a circle is 2(pi) r
• The formula for the area of a circle
is (pi) r2
• Assume that pi = 3.14 for this
exercise
Investigation of circles
Measure the radius and the diameter of a variety of tins and circular objects.
You will need:
For each circle, work out a way to measure the area and circumference.
• tins
• circular objects, for example List all the results together in a table. Try to work out the relationship
plates, lids, pots between:
• cardboard circles of • radius and diameter
different sizes • radius and circumference
• radius and area
radius diameter circumference area
TOPIC Fractions, decimals and percentages
Investigation of fractions, decimals and percentages
Put 6 pieces of fruit on three tables as shown. Use the same kind of fruit,
such as 6 apples or 6 bananas. Each piece of fruit must be roughly the same
size.
Line up 10 students outside the room. Let them in one at a time. Each
student must choose to sit at the table where they think they will get the
most fruit.
Before the students enter, discuss the following questions with the rest of the
class:
• Where do you think they will all want to sit?
• How much fruit will each student get?
• If students could move to another table, would they?
• Is it best to go first or last?
• Where is the best place to be in the queue?
When all 10 students are seated, ask students to do the following:
• Write down how much fruit each student gets. Write the amount as a
fraction and as a decimal.
• Write down the largest amount of fruit any one student gets. Write this
amount as a percentage of the total amount of fruit on the tables.
29.
Repeat the activity with a different set of students sent outside the
room. Try with a different number of tables or a different number
of pieces of fruit or a different number of students.
TOPIC Line symmetry
• In a symmetrical shape every point has an image point on the
opposite side of the mirror line at the same distance from it.
Investigation of symmetrical shapes
Make three pieces of card like the ones shown.
How many different ways can you put them together to
make a symmetrical shape?
Draw in the line(s) of symmetry of each shape you make.
Now invent 3 simple shapes of your own and make up a
similar puzzle for a friend to solve.
TOPIC Number patterns and arithmetic sequences
• A mathematical pattern has a starting place and one clear
generating rule.
• Every number in a mathematical pattern can be described by the
same algebraic term.
Investigation of number patterns
Fold a large piece of paper to get a grid. Label each box, as shown,
according to its position in the row.
Choose a starting number and put it into the first box in Row 1 .
Choose a generating rule, for example:
• Add 3 to the previous number.
Fill the row with the number pattern.
Choose other starting numbers and generating rules and
create rows of number patterns.
Investigate the link between the label and number in the
box. For example:
Box Number
1 10
2 20
3 30
30.
Which number would go in the 10th box of each number pattern in
your grid? 100th box? nth box?
TOPIC Conducting statistical investigations.- testing
hypotheses, data collection, analysts and
interpretation
Doing a statistical investigation
Hypothesis: Form 4 girls are fitter than Form 4 boys.
Step 1 Use a random sampling method to select 20 girls and 20
boys in Form 4.
Step 2 Decide how you will test fitness, for example:
• number of step-ups in one minute
• number of push-ups in one minute
• number of star jumps in one minute
• time taken to do 10 sit-ups
• pulse rate before any activity, immediately after activity, 1 minute
after activity, 5 minutes after activity, 10 minutes after activity.
Step 3 Design a data collection sheet. Prepare a record sheet for
the girls and a similar one for the boys.
Is there a correlation between any of the activities? Could these be
combined to give an overall fitness rating?
Step 4 Collect necessary
resources like a stop watch.
Find a suitable time and place
to conduct the fitness tests.
Step 5 Collect and record
data. Make sure the tests are
fair. For example, it may be
unfair to test boys in the
midday heat and girls in the
late afternoon. To be fair,
each girl and boy must go
through the same tests, in the
same order, under the same
conditions.
Step 6 Analyse data by comparing the mean, mode, median and
range of number of step-ups for girls and boys. Do the same»for
the number of push-ups, star jumps etc.
Is there a correlation between any of the activities? Could these be
combined to give an overall fitness rating?
Step 7 Select ways of presenting the data in order to compare the
fitness of girls and boys.
Step 8 Interpret the data. What are the differences between boys’
and girls’ performances on each test? Overall?
Step 9 Draw a conclusion.
Is it true that Form 4 girls are fitter than Form 4 boys? Is the
hypothesis true or false?
31.
Other hypotheses to test
Young people eat more sugar than old people. The
bigger the aeroplane, the longer it stays in the air.
Three times around your head is the same as your
height. The bigger the ball, the higher it bounces.
To test any hypothesis, each of the following steps
must be carefully planned:
• Choose your sample.
- How many people/aeroplanes/bails etc. will you include in
your sample?
- How will you select your sample so that your data is not
biased?
• Choose a method of investigation:
- Will you observe incidents in real life?
- Will you need to do research, for example in the library to
find out about the patterns of behaviour you are
investigating?
- Will you need to design a questionnaire or interview
questions to get information from people like how much
sugar they eat per day or per week?
- Will you need to design an experiment such as drop five
balls of different sizes from the same height and count the
number of bounces?
• Decide how to record data in a user-friendly format.
• Make sure the data is collected accurately and without bias.
• Choose the measures to analyse and compare data.
- Will you work with mean, median and/or mode?
- Will range be helpful? Will standard deviation be useful?
• Choose how to present the relevant analysed data.
- Will you use a table, bar chart, pie chart, line graph?
• Interpret the findings of your investigation.
• Draw a conclusion.
- Is the hypothesis true or false? Is the hypothesis
sometimes true?
32.
In this chapter we look at how you can use resources and practical activities to improve
students’ learning. We look at ways in which you can use a few basic resources such as
bottle tops, sticks, matchboxes and string to teach important mathematical ideas and
Why use resources and teaching aids
Spend some time thinking about the question:
What are the advantages and disadvantages of using
resources, practical activities and teaching aids in the
classroom?
Compare your ideas with the list below:
Actively involves students
Motivates students
Makes ideas concrete
Shows maths is in the real world
Allows different approaches to a topic
Gives hands-on experience
Makes groupwork easier
Gives opportunities for language development
Organising the activities
Monitoring work
Planning the work
Storing resources
Noisier classroom
Possible discipline problems
On balance, using resources and activities can greatly improve students’ learning. The
main difficulty from the teacher’s point of view is organising, planning and monitoring the
activities. We shall discuss these problems in Chapter 5.
What resources can be used?
Sticks, corks, bottle tops, cloth, matchboxes, envelopes, shells, string, rubber bands,
drawing pins, beads, pebbles, shoe laces, buttons, old coins, seeds, pots and pans,
washing line, newspaper, old magazines, paper and card, twigs, odd pieces of wood, old
cardboard boxes and cartons, clay, tins, bags, bottles, people and most importantly, the
There are many other things that you will be able to find around the school and local
33.
MAKING RESOURCES
Some resources take a long time to make but can be used again and
again, others take very little time to make and can also be used again
and again. But some resources can only be used once and you need
to think carefully about whether you have the time to make them.
You also need to think about how many of each resource you need.
Are there ways you can reduce the quantity? For example, can you
change the organisation of your classroom so that only a small group
of students use the resource at one time? Other groups can use the
resource later during the week.
Get help with preparing and making resources. Here are some ideas:
• Students can make their own copies.
• Make resources with students in the maths club.
• Run a workshop with colleagues to produce resources. Share the
resources with all maths teachers at the school.
• Invite members of the local community into the school to help
make resources.
• Pace yourself. Make one set of resources a term. Build up a bank
of resources over time.
Find ways of storing resources so that they are accessible and can be
re-used. Perhaps one student can be responsible for making sure the
resources are all there at the beginning and end of the lesson.
On the following pages, we give some mathematical starting points
for using resources which don’t need a great deal of work to
prepare.
Using bottle tops Reflection
TOPIC REFLECTION
• Every point has an image point at the same distance on the
opposite side of the mirror line.
Activity
Place 5 bottle tops on a strip of card as shown.
You will need:
• bottle tops
• small mirrors
• strips of card
Place a mirror on the dotted line. One student sits at each end. Ask
each other: What do you see? What do you think the other student
sees? Move the mirror line. What do you see? What does the
other student see?
Try different arrangements with double rows of bottle tops or
different coloured bottle tops.
34.
TOPIC Estimation
• Any unit of measurement can be compared with another unit of
measurement, for example a metre can be compared with centimetres,
inches, hands, bottletops etc.
Activity
Form two teams for a class quiz on estimation. Each team prepares a set
of questions about estimation. For example:
How many bottle tops would fill a cup? a cooking pot?
a wheelbarrow? a lorry?
How much would a lorry load of bottle tops weigh?
How many bottle tops side by side measure a metre? a kilometre?
the length of the classroom?
Each team prepares the range of acceptable estimations for their set of
questions. The team that makes the best estimations in the quiz
wins.
TOPIC Co-ordinate pairs and transformations
• Co-ordinate pairs give the position of a point on a grid. The point
with co-ordinate pair (2,3) has a horizontal distance of 2 and a
vertical distance of 3 from the origin.
• Transformations are about moving and changing shapes using a
rule. Four ways of transforming shapes are: reflection, rotation,
enlargement and translation.
Activity for
co-ordinates
Draw a large pair of axes on the
ground or on a large piece of card on
the ground. Label they and x axes.
Place 4 bottle tops on the grid as the
vertices (corners) of a quadrilateral.
Record the 4 coordinate pairs. Make
other quadrilaterals and record their
co-ordinate pairs.
Sort the quadrilaterals into the following categories: square, rectangle,
rhombus, parallelogram, kite, trapezium. In each category look for
similarities between the sets of co-ordinate pairs.
Activities for transformations
• Reflection: every point has an image point at the same distance on the
opposite side of the mirror line.
35.
Place 4 bottle tops, top-side up,
to make a quadrilateral. Record
the co-ordinate pairs. Place
another 4 bottle tops, teeth-side
up, to show the mirror image of
the first quadrilateral reflected
in the line y = 0. Record these
coordinate pairs. Compare the
coordinate pairs of the first
quadrilateral and the reflected
quadrilateral.
Show different quadrilaterals
reflected in the y = 0 line. Note
the co-ordinates and investigate
how the sets of co-ordinates are
related.
Make reflections of quadrilaterals in other lines such as x = 0, y = x.
• Rotation; all points move the same angle around the centre of
Place bottle tops, top-side up, to make a shape. Record the co-
ordinates of the corners of the shape. Place another set of bottle
tops, teeth-side up, to show the image of the shape when it has been
rotated 90° clockwise about the origin. Record these new co-
ordinates. Compare the two sets of co-ordinate pairs.
Show different shapes rotated 90° clockwise about the origin. Note
the co-ordinates and investigate how the sets of co-ordinates are
Now try rotations of other angles like 180° clockwise, 90°
• Enlargement: a shape is enlarged by a scale factor which tells you
how many times larger each line of the new shape must be.
Place bottle tops, top-side up, to make a shape. Record the co-
ordinates of the corners of the shape. Place another set of bottle
tops, teeth-side up, to show the image of the shape when it has been
enlarged by a scale factor of 2 from the origin. Record these new
co-ordinates. Compare the two sets of co-ordinate pairs.
Show different shapes enlarged by a scale factor of 2 from the
origin. Note the co-ordinates and investigate how the sets of co-
ordinates are related.
Now try enlargements of other scale factors such as 5, 1/2, -2. Try
enlargements from points other than the origin.
• Translation: all points of a shape slide the same distance and
direction.
Place bottle tops, top-side up, to make a shape. Record the co-
ordinates of the corners of the shape. Place another set of bottletops,
teeth-side up, to show the image of the shape when it has been
translated. Record these new co-ordinates. Compare the two sets of
co-ordinate pairs.
36.
Show different shapes translated. Note the co-ordinates
and investigate how the sets of co-ordinates are related.
Now try different translations and see what happens.
TOPIC Combinations
• All possible outcomes can be listed and counted in a systematic way.
How many ways can you arrange three different bottle tops in a line?
Investigate for different numbers of bottle tops.
TOPIC Growth patterns, arithmetic
progressions and geometric progressions
• A growth pattern is a sequence which increases by a given amount
each time.
• Algebra can be used to describe the amount of increase.
• Arithmetic progressions have the same amount added each time.
• Geometric progressions have a uniformly increasing amount added
each time.
Make Pattern 1 with bottle tops.
How many bottle tops in each pattern? How many bottle tops are added
each time?
Complete the following, filling in the number of bottle tops per term:
Term 1: 1 Term 2:1 + __ Term 3: 1 + _ + _ Term 4:1 +_+_+_
Write the algebraic rule for the nth term.
Make each of the patterns on the next page with bottle tops. For each
pattern, work out:
• the number of bottle tops in each term
• the amount of bottle tops added each time.
Work out the rule for the increase as an algebraic expression.
Write down the number of bottle tops in the 5th term, 8th term, nth term.
Decide if each sequence is a geometric or arithmetic progression.
37.
Make up some growth patterns of your own to investigate.
TOPIC Loci
• A locus is the set of all possible positions of a point, given a rule.
• The rule may be that all points must be the same distance from a
fixed point, a line, 2 lines, a line and a point etc.
Activity
You will need: • Put one bottle top top-side up on the floor. Place the other bottle tops
• a collection of bottle tops teeth-side up so that they are all the same distance from the one that
• chalk is top-side up.
• Draw a line on the floor. Place the bottle tops so that they are all the
same distance from the line.
• Put two bottle tops, top-side up, on the floor. Place the other bottle
tops, teeth-side up, so that they are all the same distance from both
the tops which are top-side up.
• Draw two intersecting straight lines on the floor. Place several
bottle tops so that they are all the same distance from both lines.
• What does the locus of points look like for each of the above rules?
USING STICKS
TOPIC GROWTH PATTERNS
• A growth pattern is a sequence which increases by a given
amount each time.
• Algebra can be used to describe the amount of increase.
• A formula in algebra can be used to describe all terms in a
pattern.
Activity
Use matchsticks or twigs to create this triangle pattern.
Term 1 Term 2 Term 3 Term 4
38.
How many triangles and how many sticks in each term of the pattern?
Figure 2.6 How many sticks are added in each term?
How many triangles will there be in the 5th term? 8th term? 60th term? nth term?
How many sticks will there be in the 5th term? 8th term? nth term?
Investigate the relationship between the number of sticks and the
number of triangles.
Explore the relationship between the number of sticks and the number of squares in
the two patterns below.
Pattern 1
Pattern 2
• Quadratic patterns
How many sticks in a 1 x 1 square? a 2 x 2 square? a 3 x 3
square? an n x n square?
etc
• How many sticks for an n x n x n triangle?
• Is there a number of sticks that will form both a square and a triangle
pattern?
39.
TOPIC Area and perimeter
• Area is the amount of space inside a flat shape.
• Perimeter is the distance around the outside of a flat shape.
Activity
• Use the same number of sticks for the perimeter of each
rectangle. Create two rectangles so that:
- the area of one is twice the area of the other
- the area of one is four times the area of the other.
• Use the same number of sticks to form two quadrilaterals so that
the area of one is three times the area of the other.
TOPIC Standard and non-standard units of
measurement
• We can measure length, area, volume, mass, capacity,
temperature and time.
• Non-standard units of measurement differ from place to place.
• Standard units of measurement are used in many places.
• Most countries use the metric system of units.
Common standard units of measurement:
Length metres, millimetres, kilometres
Area square kilometres, hectares
Volume cubic metres, cubic centimetres
Mass grams, kilograms, tonnes
Capacity litres, millilitres
Temperature degrees Celsius
Time seconds, minutes, hours, days
Activities to explore non-standard units
• In groups of four, think of four different non-standard units to
measure length, for example an exercise book, a local non-
standard unit, a handspan. Estimate and then measure the
length of various things with all four non-standard units. For
example, measure the dimensions of the doors and windows in
the classroom, the height of your friends etc.
• Use four sticks of different lengths. Measure various things with
the different sticks. Which stick is best for which object? Why?
• Find four different non-standard containers like tins, bottles,
cups. Measure different amounts of liquid (such as water) and
solids (such as sand, grain) with the different measures.
• What non-standard units would be useful to measure mass?
• What units are used in local markets and shops?
40.
Activities to explore standard units
• Make sticks of different lengths of standard units such as 1 cm,
5 cm, 100 cm and 1 metre. Use them to estimate and measure
the lengths of various things. Which stick is best for which object?
Activities to compare standard and non-standard measures
• Compare the measurements made using non-standard units with
those measurements made using standard units. For example:
How many cups are equal to one litre?
How many handspans are equal to one metre?
• Are any non-standard units particularly useful? Draw up a table
which shows the relationship between a useful non-standard unit
and a standard unit.
Using Cuisenaire rods
TOPIC Algebraic manipulation
• equivalences: 2 (3a + b} - 6a + 2b = 3a + b + 3a + b = ..., etc
• basic conventions: a + a + a = 3a, and 3b - 2b + 5b = 6b
• collecting like terms and simplifying:
2a + 3b + 4a + c -6a + 3b + c
• The add-subtract law: a + b - c. a = c - b, b = c - a are all
equivalent
• the subtracting bracket laws: a-(b±c) = a-b + c
• commutativity: a + b = b + abuta-b = b-a
• associativity: a + (b + c) = (a + b) + c, a - (b - c}not equal to (a - b) - c
• multiplying out brackets: 3(2a + b) = 6a + 3b
• factorising: 4a + 2b = 2(2a + b)
Cuisenaire rods take a long time to make but can be used for many
activities, last for years and can be shared by everyone in the maths
department.
Choose a lot of sticks that are about the same diameter; bamboo is
ideal. Cut them into lengths and colour them so that you have:
50w rods 1 cm long coloured white
50r rods 2 cm long coloured red
40g rods 1 cm long coloured light green
50p rods 4 cm long coloured pink
40y rods 5 cm long coloured yellow
40d rods 6 cm long coloured dark green
50w rods 1 cm long coloured white
30b rods 7 cm long coloured black
30t rods 8 cm long coloured brown
30B rods 9 cm long coloured blue
20O rods 1 cm long coloured orange
41.
Activitiy 1
• Two or more rods laid end to end make a rod train. The rod train made
from a pink rod and a white rod is the same length as the yellow rod.
Find all the different rod trains equal in length to a yellow rod. List your
answers. Then make trains equal to other colour rods.
Activitiy 2
• In this activity
represents p + r
represents p - r
Answer the following questions using your set of Cuisenaire rods.
For these questions your answer should always be a single rod.
Question 8
Question 21
42.
Activity 3
Test the following to see if they are true or false.
1 r+ g = g + r
2 w+r+g=r+w+g
3 3r = r + 2r
4 y -r = r - y
5. r + (p + y) = (r + p) + y
6 b-(r + w) = b-r-w
7 b-2r = b-r-r
8 (b + y)-p = b + (y-p)
9 (t - p) - w = t - (p - w)
10 3y-2p = (2y - p) + (y - p)
Now make up some of your own to test.
Activity 4
Lay out the red and green Cuisenaire rods end to end as a rod train:
Do this again so you have all 4 rods lying end to end as one rod train:
This is 2 lots of (red + green) or 2 (r + g)
You can lay the rods out in many ways. For instance:
r + r + g + g or 2r + 2g
g + 2r + g
Since these rod trains all use the same rods, you can say that they are
equivalent.
So you can write:
2(r + g) = r + r + g + g
= 2r + 2g
= g + 2r + g
• Write down as many other equivalent forms to 2(r + g) as you can.
• Set up each of the following with rods. For each case, set up and write
down as many equivalent forms as you can.
1 2(g+p) 4 2(3r + 2p)
2 3(g + y) 5 3{g + 2p + 3r)
3 3(2w + g}
Activity 5
You can do something similar when you have subtraction signs. The yellow
minus the red is set up as follows:
r This gap is y - r
y
43.
The total gap is (y - r) + (y - r) or 2(y - r)
If you move a red rod across you can have:
or that gap could be:
y-2r +y
So, since all the gaps are of the same length, you can say that
(y - r) + {y - r}
2(y-r)
2y-2r
y-2r + y are all
equivalent forms.
• Can you find any more equivalent forms to 2(y - r)?
Write them all down if you can.
• Set up each of the following with rods. Write down as many
equivalent forms as you can for each one.
1 2(b - p) 4 3(2y - g)
2 3(y - r) 5 3(4y - 3g)
3 2(2g - r)
Activity 6
You have seen that 2(r + g) = 2r+2g
When you go from 2(r + g) to 2r + 2g it is called multiplying out.
When you go from 2r + 2g to 2(r + g) it is called factorising
These are special equivalent forms. You can use rods for the next
set of questions, or do without them.
Multiply out:
1 3(y + b)
2 2(3p + w)
3 4(2y + B)
4 3(g + w)
5 3(4w-g)
6 5(3p-y)
7 4(3b+ 2g)
8 3(2y + r - g)
9 5(3t - 2b)
10 4(3p + 2w-3g)
44.
Factorise
1. 2g + 2w
2. 3g - 3r
3. 3b - 6w
4. 4g +2w
5. 3t + 9r
6. 4y + 6p
7. 5;y - 5w
8. 6g + 9w
9. 2p + 4g + 6r
10 3y-6g + 3p
Do these without rods. Write down as many equivalent forms as you can.
1. 2(x + y)
2. 3(x + y)
3. 2(3x + y)
4. 3(2x - y)
5. 5(2x + 3y)
6. x + 2y + 3x + 5y
7. 2x + 3y - x - y
8. 3y + 7x - y - 3x
9. x + y + 4x - 2y + 2y + 3y
10. 3x - y + 2x + 6y
Activity 7
Solve the equations
Activity 8
Test the following to see if they are true or false.
1. r + g = g + r 5. r + (y - p) = (r + y) - p
2. (w + p) + g = w + (p + g) 6. O - (y + p) = O - y - p
3. 2(g + w) = 2g + w 7. B - (r + w) = B - r - W
4. y - r = r - y 8. (w + O) - y = w + (O - y)
9. B - 2r = B - r + r
10. (b + y) - p = b + (y - p)
Now try to write 6p - 4y in at least 5 different ways.
45.
U SING MATCHBOXES
• The surface area of a solid is the sum of the
areas of all the faces of the solid.
Activity
Calculate the surface area of a closed matchbox.
How many squares 1 unit would cover the matchbox?
1 unit
How many different nets of the matchbox are there?
Put two matchboxes together. How many different cuboids can you
make? What is the smallest surface area?
Investigate the smallest surface area of a cuboid made from:
• three matchboxes
• four matchboxes
• eight matchboxes
TOPIC Length and area scale factors
• When you increase the lengths of the sides of a shape by a scale factor,
the area of the shape is increased by the square of the scale factor.
Activity
Construct a giant matchbox which is three times the size of an
ordinary matchbox.
What is the area of each side of the giant matchbox? Explore
Pythagoras’ Theorem the lengths and areas of other sized matchboxes.
2 2 2
a =b +c
TOPIC Area of rectangles and Pythagoras’ Theorem
• The area of a rectangle is equal to length x width.
• Pythagoras’ theorem states a2 = b2 + c2 when a is the side opposite
the right angle in a right-angled triangle
Activity
a2 = b2 + c2
• In each picture at the top of page 45, a rectangular piece of stiff card is
placed inside the tray of a matchbox.
Measure the sides of a matchbox tray. Use these dimensions and
Pythagoras’ Theorem to work out the dimensions of the rectangular
pieces of card in each picture.
46.
When you have worked out the length and width of each rectangle, cut the rectangles
to size and see if they fit into a matchbox tray. Did you calculate the sides of the
rectangles correctly?
Calculate the area of each rectangle. Which has the largest area?
• What is the largest triangle that can fit inside a matchbox?
• What is the formula for the area of a triangle?
TOPIC Views and perspectives
• 3-D, or three-dimensional, solids can be looked at from above, the side or the front.
• These views can be drawn in two dimensions as plans and elevations.
• 3-D solids can also be drawn using isometric drawing.
Activity
box standing on end • Here is a top view of a solid shape made from three matchboxes. Make the
structure from three matchboxes. Draw the side and
front views.
box standing on end • Make your own matchbox structure using 4 matchboxes. For each structure, draw
the top view. Give the top view to another student. Ask him/her to make the
structure and draw the side and front views.
View from above
• Make a matchbox structure from three matchboxes so that the top, side and front
views are all the same.
• How many different top views can be made using three matchboxes? Explore
for different numbers of matchboxes.
TOPIC Combinations
• All possible outcomes can be listed and counted in a systematic
way.
Activity
Here are some different ways of arranging three matchboxes.
How many different ways can you find?
Record and count all the different arrangements in a systematic way.
Work with other numbers of matchboxes, for example five. List and count all the
possible different arrangements of the matchboxes. Find ways of recording your
work.
47.
USING STRING
TOPIC Ordering whole numbers, fractions and decimals
• Place value uses the position (place) of a digit to give it its value. For
example:
In 329, the 3 has the value of 300 as it is in the hundreds column. In 0.034,
the 3 has a value of three hundredths as it is in the hundredths column.
A number line made from string
• Tie a piece of string to make a straight line across the classroom.
This represents the number line. Use clothes pegs to peg the
following numbers in the correct place on the line.
10 11 23 15 4 0 25 1
Make five more cards, some with negative numbers. Peg the cards in the correct place on the number line.
Peg the cards 0 and 1
at either end of the number line. Make cards which fit on this number line. Peg them in the correct places.
Where will you peg the cards you made if the ends are labelled 0 and 100? 4 and 4.5? 0.1 and 0.2? 10000 and
1,000,000? 1/2 and 3/4?
Put 1 in the middle. What could be at each end of the number line?
What if .7 is in the middle? 3/8 -23
What could be at the ends of the number line in each case?
• Make sets of cards to show the two times table: 2, 4, 6, 8 up to 24. Put them on the number line with
the correct spacing. Predict what the spacing will be for other times tables. Try them out. What about
the spacing of sets of numbers 1, 2, 4, 8, 16, ...
TOPIC Probability
• Probability is about the likelihood of an event happening.
• To describe the likelihood of an event happening, we use probability words like: very likely, evens,
certain, unlikely, impossible, probable.
Activity
• Tie a piece of string to make a straight line across the classroom. Peg cards 0 and 1 on the
ends of the line.
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