This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.
Mr Gilderdale is playing a game with his class. What rule might he have chosen? How would you test your idea?
Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?
If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?
This investigates one particular property of number by looking closely at an example of adding two odd numbers together.
Use the interactivity to sort these numbers into sets. Can you give
each set a name?
An odd version of tic tac toe
Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
You have 4 red and 5 blue counters. How many ways can they be
placed on a 3 by 3 grid so that all the rows columns and diagonals
have an even number of red counters?
This problem looks at how one example of your choice can show something about the general structure of multiplication.
Is it possible to place 2 counters on the 3 by 3 grid so that there
is an even number of counters in every row and every column? How
about if you have 3 counters or 4 counters or....?
Are these statements always true, sometimes true or never true?
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
Yasmin and Zach have some bears to share. Which numbers of bears
can they share so that there are none left over?
In this problem we are looking at sets of parallel sticks that
cross each other. What is the least number of crossings you can
make? And the greatest?
Investigate which numbers make these lights come on. What is the smallest number you can find that lights up all the lights?
Becky created a number plumber which multiplies by 5 and subtracts
4. What do you notice about the numbers that it produces? Can you
explain your findings?
If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.
Use the interactivities to complete these Venn diagrams.
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
These red, yellow and blue spinners were each spun 45 times in
total. Can you work out which numbers are on each spinner?
Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?
Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
You can trace over all of the diagonals of a pentagon without
lifting your pencil and without going over any more than once. Can
the same thing be done with a hexagon or with a heptagon?
This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?
Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?
In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.
How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?
A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6.
This article for teachers describes how number arrays can be a
useful reprentation for many number concepts.
Can you find the chosen number from the grid using the clues?
Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?
Four of these clues are needed to find the chosen number on this
grid and four are true but do nothing to help in finding the
number. Can you sort out the clues and find the number?
Follow the clues to find the mystery number.
How would you create the largest possible two-digit even number from the digit I've given you and one of your choice?
In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?
How many legs do each of these creatures have? How many pairs is
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
Try grouping the dominoes in the ways described. Are there any left
over each time? Can you explain why?
I am less than 25. My ones digit is twice my tens digit. My digits add up to an even number.
Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre.
Help share out the biscuits the children have made.
Can you place the numbers from 1 to 10 in the grid?
Read all about Pythagoras' mathematical discoveries in this article written for students.