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Patterned numbers. Mathematical Thinking. Dynamical systems. Rich Tasks. Problem solving. Group worthy. Sequences. Pedagogy. Learning mathematics. Mathematical reasoning & proof.

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Broad Topics > Sequences, Functions and Graphs > Sequences

### Generating Number Patterns: an Email Conversation

##### Stage: 2, 3 and 4

This article for teachers describes the exchanges on an email talk list about ideas for an investigation which has the sum of the squares as its solution.

### Seven Squares - Group-worthy Task

##### Stage: 3 Challenge Level:

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

### On the Importance of Pedantry

##### Stage: 3, 4 and 5

A introduction to how patterns can be deceiving, and what is and is not a proof.

### Triangular Triples

##### Stage: 3 Challenge Level:

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

### Paving Paths

##### Stage: 3 Challenge Level:

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

### Lost Books

##### Stage: 2 Challenge Level:

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

### Regular Hexagon Loops

##### Stage: 3 Challenge Level:

Make some loops out of regular hexagons. What rules can you discover?

### Two Much

##### Stage: 3 Challenge Level:

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

### LOGO Challenge - Sequences and Pentagrams

##### Stage: 3, 4 and 5 Challenge Level:

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

### Logoland - Sequences

##### Stage: 3 Challenge Level:

Make some intricate patterns in LOGO

### Big Powers

##### Stage: 3 and 4 Challenge Level:

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

### Towers

##### Stage: 3 Challenge Level:

A tower of squares is built inside a right angled isosceles triangle. The largest square stands on the hypotenuse. What fraction of the area of the triangle is covered by the series of squares?

##### Stage: 3 Challenge Level:

What is the total area of the first two triangles as a fraction of the original A4 rectangle? What is the total area of the first three triangles as a fraction of the original A4 rectangle? If. . . .

### Sums of Powers - A Festive Story

##### Stage: 3 and 4

A story for students about adding powers of integers - with a festive twist.

### Odds, Evens and More Evens

##### Stage: 3 Challenge Level:

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

### Seven Squares

##### Stage: 3 and 4 Challenge Level:

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

### Domino Sets

##### Stage: 2 Challenge Level:

How do you know if your set of dominoes is complete?

### Polygonals

##### Stage: 2 Challenge Level:

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

### More Pebbles

##### Stage: 2 and 3 Challenge Level:

Have a go at this 3D extension to the Pebbles problem.

### Taking a Die for a Walk

##### Stage: 1 and 2 Challenge Level:

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

##### Stage: 3 and 4 Challenge Level:

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

### The Numbers Give the Design

##### Stage: 2 Challenge Level:

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

### Light Blue - Dark Blue

##### Stage: 2 Challenge Level:

Investigate the successive areas of light blue in these diagrams.

### Farey Sequences

##### Stage: 3 Challenge Level:

There are lots of ideas to explore in these sequences of ordered fractions.

### Converging Means

##### Stage: 3 Challenge Level:

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

### Intersecting Circles

##### Stage: 3 Challenge Level:

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

### Differs

##### Stage: 3 Challenge Level:

Choose any 4 whole numbers and take the difference between consecutive numbers, ending with the difference between the first and the last numbers. What happens when you repeat this process over and. . . .

### Clock Squares

##### Stage: 3 Challenge Level:

Square numbers can be represented on the seven-clock (representing these numbers modulo 7). This works like the days of the week.

### Lower Bound

##### Stage: 3 Challenge Level:

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

### Magazines

##### Stage: 2 Challenge Level:

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

### 1 Step 2 Step

##### Stage: 3 Challenge Level:

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

### Happy Numbers

##### Stage: 3 Challenge Level:

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

### The Mathemagician's Seven Spells

##### Stage: 2 Challenge Level:

"Tell me the next two numbers in each of these seven minor spells", chanted the Mathemagician, "And the great spell will crumble away!" Can you help Anna and David break the spell?

### Maxagon

##### Stage: 3 Challenge Level:

What's the greatest number of sides a polygon on a dotty grid could have?

### Remainder

##### Stage: 3 Challenge Level:

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

### Designing Table Mats

##### Stage: 3 and 4 Challenge Level:

Formulate and investigate a simple mathematical model for the design of a table mat.

### Pebbles

##### Stage: 2 and 3 Challenge Level:

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### Lawn Border

##### Stage: 1 and 2 Challenge Level:

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

### Play a Merry Tune

##### Stage: 2 Challenge Level:

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

### Mobile Numbers

##### Stage: 1 and 2 Challenge Level:

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

### Calendar Patterns

##### Stage: 2 Challenge Level:

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

### Triangular Hexagons

##### Stage: 2 Challenge Level:

Investigate these hexagons drawn from different sized equilateral triangles.

### Holes

##### Stage: 1 and 2 Challenge Level:

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

### Times Tables Shifts

##### Stage: 2 Challenge Level:

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

### Charlie's Delightful Machine

##### Stage: 3 and 4 Challenge Level:

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

### Extending Great Squares

##### Stage: 2 and 3 Challenge Level:

Explore one of these five pictures.

### Cuisenaire Environment

##### Stage: 1 and 2 Challenge Level:

An environment which simulates working with Cuisenaire rods.

### Function Machines

##### Stage: 2 Challenge Level:

If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?

### Number Tracks

##### Stage: 2 Challenge Level:

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

### A Shapely Network

##### Stage: 2 Challenge Level:

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.