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Investigations. Pedagogy. Learning mathematics. Problem solving. Rich Tasks. Visualising. Addition & subtraction. Combinations. Mathematical Thinking. Working systematically.

Thinking Through, and By, Visualising

Stage: 2, 3 and 4

This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .

Knight's Swap

Stage: 2 Challenge Level:

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Waiting for Blast Off

Stage: 2 Challenge Level:

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Counting Cards

Stage: 2 Challenge Level:

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

Single Track

Stage: 2 Challenge Level:

What is the best way to shunt these carriages so that each train can continue its journey?

Celtic Knot

Stage: 2 Challenge Level:

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

Jigsaw Pieces

Stage: 2 Challenge Level:

How will you go about finding all the jigsaw pieces that have one peg and one hole?

Tetrafit

Stage: 2 Challenge Level:

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

Counters

Stage: 2 Challenge Level:

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

Red Even

Stage: 2 Challenge Level:

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?

Putting Two and Two Together

Stage: 2 Challenge Level:

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Painting Possibilities

Stage: 2 Challenge Level:

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

Paw Prints

Stage: 2 Challenge Level:

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Dodecamagic

Stage: 2 Challenge Level:

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

The Development of Spatial and Geometric Thinking: 5 to 18

Stage: 1, 2, 3 and 4

This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .

Nine-pin Triangles

Stage: 2 Challenge Level:

How many different triangles can you make on a circular pegboard that has nine pegs?

Map Folding

Stage: 2 Challenge Level:

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Stage: 2 Challenge Level:

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Cover the Camel

Stage: 1 Challenge Level:

Can you cover the camel with these pieces?

Coded Hundred Square

Stage: 2 Challenge Level:

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Routes 1 and 5

Stage: 1 Challenge Level:

Find your way through the grid starting at 2 and following these operations. What number do you end on?

Tessellate the Triominoes

Stage: 1 Challenge Level:

What happens when you try and fit the triomino pieces into these two grids?

Hexpentas

Stage: 1 and 2 Challenge Level:

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

Shunting Puzzle

Stage: 2 Challenge Level:

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Open Boxes

Stage: 2 Challenge Level:

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

A City of Towers

Stage: 1 Challenge Level:

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

Four Triangles Puzzle

Stage: 1 and 2 Challenge Level:

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Square Corners

Stage: 2 Challenge Level:

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Teddy Bear Line-up

Stage: 1 Challenge Level:

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

Cuboid-in-a-box

Stage: 2 Challenge Level:

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

Stage: 2 Challenge Level:

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

Display Boards

Stage: 2 Challenge Level:

Design an arrangement of display boards in the school hall which fits the requirements of different people.

Tangram Tangle

Stage: 1 Challenge Level:

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Tricky Triangles

Stage: 1 Challenge Level:

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

Neighbours

Stage: 2 Challenge Level:

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

Move a Match

Stage: 2 Challenge Level:

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

Little Boxes

Stage: 2 Challenge Level:

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

World of Tan 22 - an Appealing Stroll

Stage: 2 Challenge Level:

Can you fit the tangram pieces into the outline of the child walking home from school?

World of Tan 25 - Pentominoes

Stage: 2 Challenge Level:

Can you fit the tangram pieces into the outlines of these people?

World of Tan 21 - Almost There Now

Stage: 2 Challenge Level:

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

World of Tan 24 - Clocks

Stage: 2 Challenge Level:

Can you fit the tangram pieces into the outlines of these clocks?

Cubes Within Cubes

Stage: 2 and 3 Challenge Level:

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

World of Tan 27 - Sharing

Stage: 2 Challenge Level:

Can you fit the tangram pieces into the outline of Little Fung at the table?

World of Tan 8 - Sports Car

Stage: 2 Challenge Level:

Can you fit the tangram pieces into the outline of this sports car?

World of Tan 7 - Gat Marn

Stage: 2 Challenge Level:

Can you fit the tangram pieces into the outline of this plaque design?

World of Tan 5 - Rocket

Stage: 2 Challenge Level:

Can you fit the tangram pieces into the outline of the rocket?

World of Tan 9 - Animals

Stage: 2 Challenge Level:

Can you fit the tangram pieces into the outline of this goat and giraffe?

World of Tan 29 - the Telephone

Stage: 2 Challenge Level:

Can you fit the tangram pieces into the outline of this telephone?

Dragons and Swans

Stage: 1 Challenge Level:

A game for two players. You'll need some counters.

World of Tan 28 - Concentrating on Coordinates

Stage: 2 Challenge Level:

Can you fit the tangram pieces into the outline of Little Ming playing the board game?