What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Can you find the area of a parallelogram defined by two vectors?
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
Can you describe this route to infinity? Where will the arrows take you next?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Can you find sets of sloping lines that enclose a square?
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
A game for 2 players
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
It would be nice to have a strategy for disentangling any tangled ropes...
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
Can you use the diagram to prove the AM-GM inequality?
A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
What is the total number of squares that can be made on a 5 by 5 geoboard?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
To avoid losing think of another very well known game where the patterns of play are similar.
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
A collection of games on the NIM theme
Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
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?