Other tags that relate to Pareq Exists
Creating and manipulating expressions and formulae. Visualising. Factors and multiples. Pythagoras' theorem. Constructions. Generalising. Circle properties and circle theorems. Similarity and congruence. Mathematical reasoning & proof. Triangles.There are 179 results
Broad Topics > Thinking Mathematically > Visualising
Trice
Age 11 to 14 Challenge Level:
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?
Triangular Tantaliser
Age 11 to 14 Challenge Level:
Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.
Constructing Triangles
Age 11 to 14 Challenge Level:
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
Tourism
Age 11 to 14 Challenge Level:
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.
Dotty Triangles
Age 11 to 14 Challenge Level:
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?
All Tied Up
Age 14 to 16 Challenge Level:
A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?
Corridors
Age 14 to 16 Challenge Level:
A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.
Cutting a Cube
Age 11 to 14 Challenge Level:
A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?
Tilting Triangles
Age 14 to 16 Challenge Level:
A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?
Rati-o
Age 11 to 14 Challenge Level:
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
Linkage
Age 11 to 14 Challenge Level:
Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?
Convex Polygons
Age 11 to 14 Challenge Level:
Show that among the interior angles of a convex polygon there cannot be more than three acute angles.
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?
Jam
Age 14 to 16 Challenge Level:
To avoid losing think of another very well known game where the patterns of play are similar.
Concrete Wheel
Age 11 to 14 Challenge Level:
A huge wheel is rolling past your window. What do you see?
Natural Sum
Age 14 to 16 Challenge Level:
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
Hypotenuse Lattice Points
Age 14 to 16 Challenge Level:
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
Weighty Problem
Age 11 to 14 Challenge Level:
The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
Seven Squares - Group-worthy Task
Age 11 to 14 Challenge Level:
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?
The Farmers' Field Boundary
Age 11 to 14 Challenge Level:
The farmers want to redraw their field boundary but keep the area the same. Can you advise them?
Tower of Hanoi
Age 11 to 14 Challenge Level:
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
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?
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?
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?
Sliding Puzzle
Age 11 to 16 Challenge Level:
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.
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. . . .
On the Edge
Age 11 to 14 Challenge Level:
If you move the tiles around, can you make squares with different coloured edges?
The Spider and the Fly
Age 14 to 16 Challenge Level:
A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?
Frogs
Age 11 to 14 Challenge Level:
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Picturing Triangular Numbers
Age 11 to 14 Challenge Level:
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Picture Story
Age 14 to 16 Challenge Level:
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Around and Back
Age 14 to 16 Challenge Level:
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
Shear Magic
Age 11 to 14 Challenge Level:
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
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?
Proximity
Age 14 to 16 Challenge Level:
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
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?
Cogs
Age 11 to 14 Challenge Level:
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
Counting Triangles
Age 11 to 14 Challenge Level:
Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?
Star Gazing
Age 14 to 16 Challenge Level:
Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
Cubist Cuts
Age 11 to 14 Challenge Level:
A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer?
Dice, Routes and Pathways
Age 5 to 14
This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .
Buses
Age 11 to 14 Challenge Level:
A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?
Cuboids
Age 11 to 14 Challenge Level:
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
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.
NRICH is part of the family of activities in the Millennium Mathematics Project.