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Broad Topics > Angles, Polygons, and Geometrical Proof > Squares

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Seeing Squares

Age 5 to 11 Challenge Level:

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

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Paper Patchwork 1

Age 5 to 7 Challenge Level:

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

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Data Shapes

Age 5 to 7 Challenge Level:

Sara and Will were sorting some pictures of shapes on cards. "I'll collect the circles," said Sara. "I'll take the red ones," answered Will. Can you see any cards they would both want?

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Vector Journeys

Age 14 to 18 Challenge Level:

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

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Sorting Logic Blocks

Age 5 to 11 Challenge Level:

This interactivity allows you to sort logic blocks by dragging their images.

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Jig Shapes

Age 5 to 11 Challenge Level:

Can you each work out what shape you have part of on your card? What will the rest of it look like?

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Baravelle

Age 7 to 16 Challenge Level:

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

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Eight Hidden Squares

Age 7 to 14 Challenge Level:

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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Fraction Fascination

Age 7 to 11 Challenge Level:

This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.

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Torn Shapes

Age 7 to 11 Challenge Level:

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

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Squares in Rectangles

Age 11 to 14 Challenge Level:

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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Complete the Square

Age 5 to 7 Challenge Level:

Complete the squares - but be warned some are trickier than they look!

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Square Coordinates

Age 11 to 14 Challenge Level:

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

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Ten Hidden Squares

Age 7 to 11 Challenge Level:

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

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Square It

Age 11 to 16 Challenge Level:

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

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On the Edge

Age 11 to 14 Challenge Level:

If you move the tiles around, can you make squares with different coloured edges?

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Semi-detached

Age 14 to 16 Challenge Level:

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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Fitted

Age 7 to 11 Challenge Level:

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

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Transformations on a Pegboard

Age 7 to 11 Challenge Level:

How would you move the bands on the pegboard to alter these shapes?

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Halving

Age 5 to 7 Challenge Level:

These pictures show squares split into halves. Can you find other ways?

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Square Corners

Age 7 to 11 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?

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Zig Zag

Age 14 to 16 Challenge Level:

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

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Chain of Changes

Age 5 to 7 Challenge Level:

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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A Swiss Sum

Age 16 to 18 Challenge Level:

Can you use the given image to say something about the sum of an infinite series?

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Square Within a Square Within...

Age 11 to 14 Short Challenge Level:

What fraction of this square is shaded?

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Square Surprise

Age 5 to 11 Challenge Level:

Why do you think that the red player chose that particular dot in this game of Seeing Squares?

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Seeing Squares for Two

Age 5 to 11 Challenge Level:

Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?

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Peg and Pin Boards

Age 5 to 11

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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Hidden Squares

Age 11 to 14 Challenge Level:

Can you find the squares hidden on these coordinate grids?

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Opposite Vertices

Age 11 to 14 Challenge Level:

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

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Folding Fractions

Age 14 to 16 Challenge Level:

What fractions can you divide the diagonal of a square into by simple folding?

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Logic Block Collections

Age 5 to 7 Challenge Level:

What do you think is the same about these two Logic Blocks? What others do you think go with them in the set?

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Overlapping Squares

Age 7 to 11 Challenge Level:

Have a good look at these images. Can you describe what is happening? There are plenty more images like this on NRICH's Exploring Squares CD.

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LOGO Challenge 4 - Squares to Procedures

Age 11 to 16 Challenge Level:

This LOGO Challenge emphasises the idea of breaking down a problem into smaller manageable parts. Working on squares and angles.

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Logo Challenge 3 - Star Square

Age 7 to 16 Challenge Level:

Creating designs with squares - using the REPEAT command in LOGO. This requires some careful thought on angles

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Trig Rules OK

Age 16 to 18 Challenge Level:

Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...

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Something in Common

Age 14 to 16 Challenge Level:

A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.

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Circle Box

Age 14 to 16 Challenge Level:

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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A Square in a Circle

Age 7 to 11 Challenge Level:

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

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Inside Seven Squares

Age 7 to 11 Challenge Level:

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

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Squares, Squares and More Squares

Age 11 to 14 Challenge Level:

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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Two Triangles in a Square

Age 14 to 16 Challenge Level:

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

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LOGO Challenge - the Logic of LOGO

Age 11 to 16 Challenge Level:

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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LOGO Challenge - the Humble Square

Age 7 to 16 Challenge Level:

Look at how the pattern is built up - in that way you will know how to break the final pattern down into more manageable pieces.

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Diagrams

Age 7 to 11 Challenge Level:

A group activity using visualisation of squares and triangles.

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LOGO Challenge 7 - More Stars and Squares

Age 11 to 16 Challenge Level:

Can you use LOGO to create a systematic reproduction of a basic design? An introduction to variables in a familiar setting.

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LOGO Challenge 5 - Patch

Age 11 to 16 Challenge Level:

Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

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First Forward Into Logo 1: Square Five

Age 7 to 16 Challenge Level:

A Short introduction to using Logo. This is the first in a twelve part series.

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Zooming in on the Squares

Age 7 to 14

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

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Coloured Edges

Age 11 to 14 Challenge Level:

The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?