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Broad Topics > Sequences, Functions and Graphs > Sequences

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Seven Squares

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Shifting Times Tables

Stage: 3 Challenge Level: Challenge Level:1

Can you find a way to identify times tables after they have been shifted up?

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Power Mad!

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Cuisenaire Environment

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

An environment which simulates working with Cuisenaire rods.

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Chairs and Tables

Stage: 1 Challenge Level: Challenge Level:1

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

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Seven Squares - Group-worthy Task

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Triangles and Petals

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

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Farey Sequences

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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What an Odd Fact(or)

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

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Tiny Nines

Stage: 4 Challenge Level: Challenge Level:1

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

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Sissa's Reward

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

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The Tomato and the Bean

Stage: 1 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

At the beginning of May Tom put his tomato plant outside. On the same day he sowed a bean in another pot. When will the two be the same height?

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1 Step 2 Step

Stage: 3 Challenge Level: Challenge Level:1

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?

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Big Powers

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

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.

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Odd Differences

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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Chain of Changes

Stage: 1 Challenge Level: Challenge Level:2 Challenge Level:2

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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Sticky Triangles

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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Domino Patterns

Stage: 1 Challenge Level: Challenge Level:1

What patterns can you make with a set of dominoes?

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Domino Sets

Stage: 2 Challenge Level: Challenge Level:1

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

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Designing Table Mats

Stage: 3 and 4 Challenge Level: Challenge Level:1

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

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Double Trouble

Stage: 4 Challenge Level: Challenge Level:1

Simple additions can lead to intriguing results...

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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.

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Odds, Evens and More Evens

Stage: 3 Challenge Level: Challenge Level:1

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

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Polygonals

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Weekly Challenge 34: Googol

Stage: 5 Short Challenge Level: Challenge Level:2 Challenge Level:2

Find the smallest value for which a particular sequence is greater than a googol.

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Weekly Challenge 16: Archimedes Numerical Roots

Stage: 5 Challenge Level: Challenge Level:1

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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Extending Great Squares

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

Explore one of these five pictures.

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More Pebbles

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

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

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Cube Bricks and Daisy Chains

Stage: 1 Challenge Level: Challenge Level:1

Daisy and Akram were making number patterns. Daisy was using beads that looked like flowers and Akram was using cube bricks. First they were counting in twos.

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Charlie's Delightful Machine

Stage: 3 and 4 Challenge Level: Challenge Level:1

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

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A Little Light Thinking

Stage: 4 Challenge Level: Challenge Level:1

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

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The Numbers Give the Design

Stage: 2 Challenge Level: Challenge Level:1

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

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Times Tables Shifts

Stage: 2 Challenge Level: Challenge Level:1

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

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Farey Neighbours

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

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Farey Approximation

Stage: 5 Short Challenge Level: Challenge Level:1

Use Farey sequences to obtain rational approximations to irrational numbers.

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Ford Circles

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Small circles nestle under touching parent circles when they sit on the axis at neighbouring points in a Farey sequence.

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Summing Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

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Holes

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

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?

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A Number Sequences Resource

Stage: 4 Challenge Level: Challenge Level:1

This resource contains interactive problems to support work on number sequences at Key Stage 4.

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Mobile Numbers

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

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?

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Street Sequences

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

Investigate what happens when you add house numbers along a street in different ways.

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Play a Merry Tune

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Sounds Great!

Stage: 1 Challenge Level: Challenge Level:2 Challenge Level:2

Investigate the different sounds you can make by putting the owls and donkeys on the wheel.

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Sums of Powers - A Festive Story

Stage: 3 and 4

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

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Triangular Hexagons

Stage: 2 Challenge Level: Challenge Level:1

Investigate these hexagons drawn from different sized equilateral triangles.

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First Forward Into Logo 12: Puzzling Sums

Stage: 3, 4 and 5 Challenge Level: Challenge Level:1

Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.

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Stretching Fractions

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?

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Vecten

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Join in this ongoing research. Build squares on the sides of a triangle, join the outer vertices forming hexagons, build further rings of squares and quadrilaterals, investigate.

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Triangles Within Pentagons

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Show that all pentagonal numbers are one third of a triangular number.

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Triangles Within Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Can you find a rule which relates triangular numbers to square numbers?