MEI Conference 2012
Short Open-Ended
Investigations for KS3
Kevin Lord
Kevin.lord@mei.org.uk
10 Ideas for Short Investigations
These are some of the investigations that I have used many times with a
variety of groups. Besides providing opportunities for students to apply
mathematics and practise their problem-solving skills, these activities help to
reinforce number and calculation skills, geometry, probability and statistics
ideas.
If sufficient time is allowed most of these investigations can be extended
and will lead to generalisations and algebraic expressions.
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Number Chains
HOTPOs
Magic Total 34
20-line Game
Patterns on a times-table square
Angles in Stars
Euler’s Theorem for Networks
Snakes and Ladders Investigation
Isle of Wight Problem
Cereal Packet Free Gift
MEI is a registered charity, number 1058911 NUMBER CHAINS
Numbers can form surprising chains when very simple rules are applied to
their digits.
Look at the numbers in this chain . . . .
START
16
31
8
40
4
20
?
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 Can you see a rule which might link the numbers together?
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What numbers do you think come next?
Once you have discovered the rule, think about these questions and make
some predictions.
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How many numbers do you think will be in the chain?
What do you think will be the highest number in the chain?
Do you think the chain will go on forever, or stop at some point?
Do you think the numbers will begin to repeat themselves?
Carry on the chain and investigate what happens to numbers
a) Describe the rule for linking the numbers in this chain in your own
words so that someone else would understand.
(Use an example or diagram to help show how the rule works.)
b) Write down how many numbers are in the chain.
c) Write down any numbers which are missing and find out what
happens if you start with one of these.
d) Try a slightly different rule and investigate what happens this time.
MEI is a registered charity, number 1058911 HOTPO
A little number investigation, which practices dividing by 2 and multiplying by
3, known as the “Half or Triple Plus One” - HOTPO
Starting with any number, Un
If Un is EVEN, then divide by 2
Un+1 = Un/2
If Un is ODD, then triple and add 1
Un+1 = 3Un + 1
And continue
For example, Un = 13
13, 40, 20, 10, 5, 16, 8, 4, 2, 1, (4, 2,1
begins to repeat after 1 is reached)
 Where do the sequences end? (Is this true for all numbers?)
 How long is the sequence for different starting numbers?
 What is the “Highest Value” reached in a sequence?
 Could you make a map of the routes of different number?
MEI is a registered charity, number 1058911 MAGIC TOTAL 34
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Above is a 4 by 4 square numbered 1 to 16.
By choosing just one number in each row or column as shown by the
shaded squares you can check the total is 34.
Is this always true?
 What other patterns of 4 squares can you find that also add up to
34?
Try to think about the symmetry of the pattern of squares you choose.
Extension
 What is the magic total for other sizes of square, such as a 5 by 5?
 Do rectangular grids also have a magic total?
How would you explain the rules for rectangles?
MEI is a registered charity, number 1058911 MAGIC TOTAL 34
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MEI is a registered charity, number 1058911 Number Line Game
Start with a number line, numbered 0 to 20.
You can add or subtract 2 numbers on the line so that the answer is also on
the line.
For example
6 + 4 = 10
10 take away 9 makes 1
1…….
Cross out the numbers you use. Circle the numbers you end on. Then cross
these out as you start with them for the next calculation.
 You decide any other rules for the game.
 How will you know who has “won”?
 You could play – Collaboratively – Try to fill in the whole line.
 You could play – Competitively - Stop your partner from going.
 How could this be adapted to be less / more challenging?
 What observations can you make about this game?
 What are good strategies for winning the game?
MEI is a registered charity, number 1058911 Patterns on a Times-table Square
Starting with a 12x12 times-table square investigate the patterns made by
colouring in all the numbers with a particular property.
Example 1 : Multiples of 4
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Example 2: Factors of 24
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MEI is a registered charity, number 1058911 x
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MEI is a registered charity, number 1058911 STARS
If a star has 5 points then you can
draw it using 5 straight lines, without
removing your pencil from the paper.
For stars that have an odd number of
points, you can draw them in this way,
with the number of points equal to the
number of lines.
 What is the sum of the interior angles of the 5 points of a pentagram
(five pointed star)?
 What is the sum of the exterior angles of
the pentagram?
 What is the sum of the interior angles of the
points on a 7 or 9 pointed star?
 Could you write a procedure using Logo to draw any star with an odd
number of points?
MEI is a registered charity, number 1058911 Euler’s Theorem for Networks
Investigate the relationship between the number of regions of
a network and the number of lines and nodes.
For example, here is a simple network diagram.
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There are 4 nodes and 5 lines.
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It divides the paper into 3 regions, 2 inside and one outside.
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a) For the 8 networks below sketch them and shade in the regions.
b) Record your results in a table with the following headings.
Network
1
2
No of Regions
No. of Nodes
No. of Lines
c) Look for a relationship between the number of Regions, R, Lines, L
and Nodes, N.
MEI is a registered charity, number 1058911 SNAKES and LADDERS
The game of Snakes and Ladders has a long history and originated in
Indian where it was a game for teaching children how to live a moral and
good life. Good deeds allowed you to climb the ladders and bad deeds
meant that you slipped back down the snakes to the bottom.
Boards are not all the same, some have more squares, some have more
snakes or more ladders.

How would you compare two different board to decide which one is
harder or easier to reach the top?

Can you calculate the length of each snake and each ladder?

Why is the average length of snake or ladder useful?
MEI is a registered charity, number 1058911 The Isle of Wight
The Problem
When I was at school I remember being told that
“all the people in the world could fit on the Isle of Wight”.
Is this still true?
I expect that the area of the Isle of Wight has not changed very much in 30
years but what about the population of the world?
Modelling
 What information do you need to solve this problem?
 How will you get this information?
 What assumptions about the Isle of Wight, and the people in the world
will you make in order to tackle this problem?
 What is the calculation you will need to do to solve the problem?
MEI is a registered charity, number 1058911 Hints and Possible approaches
1. Use the print out of a map to estimate area of Isle of Wight using simple shapes.
 How could you do this more accurately?
2. Use some metre sticks to make a square metre and carry out an experiment to estimate
the number of people who could fit in a square metre.
 What is the average number of people per square metre?
 How accurate do you think your value is?
3. Use the internet to research the population of the world and verify the area of the Isle of
Wight.
 World Population in 2011
=
6.93 billion people
 World Population in 1980
=
4.44 billion people
 Area of Isle of Wight
=
384 km2
4. Calculation required :
Or
Number of people
=
Area x Population density
Area required
=
Population/ Pop. Density
5. At the end remember to discuss the validity of the result, by discussing again the
assumptions you made at the start.
For example
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Not all people are the same size, most would be bigger than children
It is not possible to stand on all parts of the island
6. Can you find other islands that could be home to the whole world?
Mathematics skills that may be used
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Estimation and approximation
Area of rectangles and triangles
Compound measures like density
Standard form
Simple averages and using averages
Converting units – m2 to km2
Interpreting graphs
MEI is a registered charity, number 1058911 Cereal Packet Free Gift
Cereal manufacturers often use the marketing ploy of including a set of small toys in their
cereal boxes to get children to pressure their parents to buy the cereals to get the toy.
Suppose there are 6 toys in the set to collect.

How many boxes do you think you will have to buy before you have the full
set?

What does your answer depend on, what assumptions are you making?
Modelling
Let’s assume that your chance of getting each toy are the same = 1/6.
We can roll a die to simulate the randomness of which toy is inside the cereal packet.
Simulation

Roll a fair die and record in the tally chart the number shown?

Stop when all 6 numbers have been scored?

Count up you total number of rolls.
This is the number of packets you would have had to buy.

Repeat the simulation 3 or 4 times to see if your results are similar.
Score
First Trial
Tally
Freq.
Second Trial
Tally
Freq.
Third Trial
Tally
Freq.
Fourth Trial
Tally
Freq.
Total
Total
Total
Total
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How could you set up the simulation if,
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there were 10 toys to collect?
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the probability of getting one of the toys was much lower than the rest?
Investigate the relationship between the number of toys to collect and the number
of packets that have to be bought.
MEI is a registered charity, number 1058911