Other tags that relate to Polycircles
Interactivities. 2D shapes and their properties. Making and proving conjectures. Visualising. Dynamic geometry. Generalising. Regular polygons and circles. Simultaneous equations. GeoGebra. Creating and manipulating expressions and formulae.There are 131 results
Broad Topics > Algebraic expressions, equations and formulae > Creating and manipulating expressions and formulae
The Medieval Octagon
Age 14 to 16
Challenge Level
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
Pareq Calc
Age 14 to 16
Challenge Level
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. . . .
Pythagoras Proofs
Age 14 to 16
Challenge Level
Can you make sense of these three proofs of Pythagoras' Theorem?
Semi-square
Age 14 to 16
Challenge Level
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?
Triangles Within Triangles
Age 14 to 16
Challenge Level
Can you find a rule which connects consecutive triangular numbers?
Triangles Within Pentagons
Age 14 to 16
Challenge Level
Show that all pentagonal numbers are one third of a triangular number.
Gutter
Age 14 to 16
Challenge Level
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Christmas Chocolates
Age 11 to 14
Challenge Level
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Nicely Similar
Age 14 to 16
Challenge Level
If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?
Partitioning Revisited
Age 11 to 14
Challenge Level
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
Regular Hexagon Loops
Age 11 to 14
Challenge Level
Make some loops out of regular hexagons. What rules can you discover?
A Tilted Square
Age 14 to 16
Challenge Level
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Magic W
Age 14 to 16
Challenge Level
Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.
Interactive Number Patterns
Age 14 to 16
Challenge Level
How good are you at finding the formula for a number pattern ?
Cubes Within Cubes Revisited
Age 11 to 14
Challenge Level
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?
Mediant Madness
Age 14 to 16
Challenge Level
Kyle and his teacher disagree about his test score - who is right?
Triangles Within Squares
Age 14 to 16
Challenge Level
Can you find a rule which relates triangular numbers to square numbers?
Attractive Tablecloths
Age 14 to 16
Challenge Level
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?
Multiplication Square
Age 14 to 16
Challenge Level
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Salinon
Age 14 to 16
Challenge Level
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
The Pillar of Chios
Age 14 to 16
Challenge Level
Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
Sitting Pretty
Age 14 to 16
Challenge Level
A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?
Three Four Five
Age 14 to 16
Challenge Level
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
Pair Products
Age 14 to 16
Challenge Level
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Number Pyramids
Age 11 to 14
Challenge Level
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Partly Painted Cube
Age 14 to 16
Challenge Level
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?
More Number Pyramids
Age 11 to 14
Challenge Level
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Simplifying Doughnut
Age 14 to 18
Challenge Level
An algebra task which depends on members of the group noticing the needs of others and responding.
Matchless
Age 14 to 16
Challenge Level
There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?
Janine's Conjecture
Age 14 to 16
Challenge Level
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
Marbles in a Box
Age 11 to 16
Challenge Level
How many winning lines can you make in a three-dimensional version of noughts and crosses?
What's Possible?
Age 14 to 16
Challenge Level
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Generating Triples
Age 14 to 16
Challenge Level
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?
Always a Multiple?
Age 11 to 14
Challenge Level
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
Magic Squares for Special Occasions
Age 11 to 16
This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.
Mindreader
Age 11 to 14
Challenge Level
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. . . .
Always the Same
Age 11 to 14
Challenge Level
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
Steel Cables
Age 14 to 16
Challenge Level
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Seven Squares
Age 11 to 14
Challenge Level
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Hallway Borders
Age 11 to 14
Challenge Level
What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?
Odd Differences
Age 14 to 16
Challenge Level
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.
Terminology
Age 14 to 16
Challenge Level
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Pick's Theorem
Age 14 to 16
Challenge Level
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.
Series Sums
Age 14 to 16
Challenge Level
Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.
Chocolate Maths
Age 11 to 14
Challenge Level
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
DOTS Division
Age 14 to 16
Challenge Level
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
One and Three
Age 14 to 16
Challenge Level
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .
NRICH is part of the family of activities in the Millennium Mathematics Project.