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Happy Birthday NRICH!

To celebrate NRICH's 20th birthday, we are bringing together some rich mathematical activities that we think are 'hidden gems'.  You may not have come across these tasks before, perhaps because some of them are quite new, whereas others are rather old, but all of them are worth exploring.  Get your mathematical thinking hat on!
Up and Down Staircases
problem
Favourite

Up and Down Staircases

Age
7 to 11
Challenge level
filled star empty star empty star
One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?
Method in multiplying madness?
problem
Favourite

Method in multiplying madness?

Age
7 to 14
Challenge level
filled star filled star empty star
Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?
Ice Cream
problem

Ice Cream

Age
7 to 11
Challenge level
filled star filled star filled star
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Always, Sometimes or Never? Number

Are these statements always true, sometimes true or never true?

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

Always, Sometimes or Never? Number statement cards

Are the following statements always true, sometimes true or never true?

How do you know?

Can you find examples or counter-examples for each one?

For the 'sometimes' cards can you explain when they are true? Or rewrite them so that they are always true or never true?

The sum of three numbers is odd

If you add 1 to an odd number you get an even number

Multiples of 5 end in a 5

If you add two odd numbers you get an odd number

If you add a multiple of 10 to a multiple of 5 the answer is a multiple of 5

 

 



What about these more complex statements?

When you multiply two numbers you will always get a bigger number

If you add a number to 5 your answer will be bigger than 5

A square number has an even number of factors

The sum of three consecutive numbers is divisible by 3

Dividing a whole number by a half makes it twice as big

 

 

 

You could print off and cut out the statement cards from the top of this page and arrange them in this grid.

Alternatively, you may like to try out your ideas using the interactivities below:

Sets of Numbers

How many different sets of numbers with at least four members can you find in the numbers in this box?
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



How many different sets of numbers with at least four members can you find in the numbers in this box?

Image
Sets of Numbers

 

For example, one set could be multiples of $4$ {$8, 36 ...$}, another could be odd numbers {$3, 13 ...$}.





Baravelle

What can you see? What do you notice? What questions can you ask?

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

Baravelle poster

Look at this image for a short while before turning away.

Can you:

  • Recreate the image?
  • Describe the image?
  • Say some mathematical things about what you notice?
  • Think of some mathematical questions you would like to ask about it?

You can download a PDF of the image to look at.

Image
Baravelle

 

 

 

Tiles on a Size Ten Patio

How many tiles could you use to cover this 10 by 10 patio?
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



Imagine that round the back of where you live is a piece of ground that your family want to change. Instead of grass or plants growing there you want to have a patio. To prepare to change it the ground is dug up and levelled, it is then measured and found to be a square $10$ by $10$ [this could be feet, metres or yards it does not really matter for this challenge]. You then go to a large store that sells tiles to go on the ground. To your surprise you find that they sell tiles in ten different sizes; $1$ by $1$, $2$ by $2$, $3$ by $3$, etc. up to $10$ by $10$. You decide to go back home for a while and work a few ideas out.

You start by drawing a $10$ by $10$ square; and then you think about what sizes of tiles to use to cover it.

If you use $1$ by $1$ tiles then you find [or know already!] that you will need $100$ of them. So your shopping list would look like :-

$100$ tiles @ $1$ by $1$ > Total number of tiles $100$

Image
Tiles on a Size Ten Patio


If you use $10$ by $10$ tiles then you find [or know already!] that you will need 1 of them. So your shopping list would look like :-

$1$ tiles @ $10$ by $10$ > Total number of tiles $1$

If you use $2$ by $2$ tiles then you find [or know already!] that you will need $25$ of them. So your shopping list would look like :-

$25$ tiles @ $2$ by $2$ > Total number of tiles $25$

You could of course use a mixture of different sized tiles.

If you use $12$ tiles $2$ by $2$ and $4$ tiles $3$ by $3$ and $1$ tile $4$ by $4$ your shopping list would look like :-

$12$ tiles @ $2$ by $2$

$4$ tiles @ $3$ by $3$

$1$ tile @ $4$ by $4$ > Total number of tiles $17$

Image
Tiles on a Size Ten Patio


So you could, with these examples use $1, 17, 25$ or indeed $100$ tiles.

Your challenge starts by asking you to try to find lots of different numbers of tiles that could be used to cover this $10$ by $10$ patio.

You could think about the artistic appearance of it as well if you like. However all the patio must be covered and no cutting any of them to make different sizes or shapes ... they're actually very, very hard to cut anyway.



You could now think about these kinds of questions and perhaps you would think of other questions of your own to pursue.

  1. Are there some numbers of tiles used that actually are made up of different mixtures of tiles? [For example in our one that used $17$, could you get to use just $17$ tiles but they require a different shopping list from the one above?]
  2. Are there some numbers between $1$ and $100$ that cannot be used as the total number of tiles required?
  3. Is there some system for getting new examples that use different numbers of tiles?


polygonals

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



Polygonal numbers are those that are arranged in shapes as they enlarge (starting with $1$). Here are examples of the $5$th polygonal numbers for the first $7$ shapes Triangle ($3$) through to Nonegon ($9$).

Image
polygonals
If we were to just look at the hexagons we'd have a form of hexagonal numbers that grow  $1,6, 15, 28, 45$.
 
Image
polygonals
There is another group that are called "Centred Polygonals"  that look like these; 
Image
polygonals


and for example, the centred hexagon numbers go $1, 7, 19, 37, 61$.

 
Image
polygonals


 Now it's over to you!

 

This investigation invites you to explore these sets of numbers and explore relationships within ordinary polygonal numbers and/or centred polygonal numbers.

You could also explore relationships between ordinary polygonal numbers and the centred polygonal numbers.

 

For example, you could explore which different polygonals (both centred and ordinary) have the same number occuring in the series?



KEY QUESTIONS:

What is the relationship between ordinary triangular polygonal numbers and others?

Can you re-arrange the dots from one polygonal to make another, and then generalise?

Throughout your exploration the question "Why?" probably needs to be asked!