PATTERNS: Using Crosses and coloured pens to create patterns

 Patterns are the basis of mathematics, if we help students find and see patterns, they are often able to use this knowledge to apply to other situations.

The multiplication array is full of patterns and we often do not ask students to find the many patterns. When you take another step there are hidden patterns in the Digital Roots of the numbers(The Digital Root is found by adding the numbers successively until a single digit is obtained. (127   -   1 + 2 + 7 = 10   -   1  + 0 = 1 1 is the digital root of 127)  The nine, three and six times tables are worth exploring with their digital roots.

For younger students we need to get them to make and draw patterns and then see if they can find underlying patterns.(Using Numerals/Numbers all the time is often too abstract for some students)

Crosses

Teachers: instead of making crosses students could colour in squares on grid paper)

    With a red and green pen you can make 4 sets of 2 crosses:  (perhaps red and green tiles/counters or blocks for some students would be better)

        XX    XX   XX   XX

    How many different sets of 3 crosses can you make?

                                                                        XXX    XXX    XXX ……….

    How many different sets of 4 crosses can you make?

    Can you see a pattern?  making a Table/list may help

    Could you predict how many sets using 5 crosses?

 

 

Now using a Blue Pen with the Red and Green Pens

        How many sets of 2 crosses can you make now?
                                                (A starter:  XX    XX    XX    XX ………….)

        How many sets of three crosses with 3 colours?
                                                ( XXX    XXX    XXX …..)

        How many sets of four crosses with three colours?
                                                (XXXX        XXXX        XXXX …….)

    Can you see a Pattern, perhaps a table will help?  

    Are you able to write your patterns in sentences?

 

 Adapted from "Bounce To It" Gillian Hatch 1984

 

Stepping Off Points #4 COUNTING TRIANGLES

 If the vertices if a square are joined in every possible way with straight lines, then 8 triangles and formed.

Four small triangles and four larger ones.

Can you find the 8 triangles?

How many triangles are formed when you join the vertices of a Regular Pentagon in every way?  

 

Make sure you get all of them some small some large and some in between

What about a Regular Hexagon? 

 

 Can you see a pattern between the number of sides and the number of triangles?

Making a table/chart sometimes helps to see patterns

Name of Shape	Number of Sides	Number of Triangles
Square	4	8
Pentagon	5
Hexagon	6
Octagon	8

 Explain your pattern in words.

Can you predict the number of triangles for a Decagon(10-sided) Icosagon(20-sided)?

Now explore the number of triangles for irregular polygons

This activity adapted from "Points of Departure" Association of Teachers of Mathematics 1989.

 

 

 

 

Stepping Off Points: #3 PROBLEMS ON A CHESSBOARD

 Problems on a Chess Board

A regular problem is to place 8 Queens on a chessboard so that no queen threatens another queen.  How can this be done (Remember that a Queen can move in all directions)

 

 

Now:  What is the greatest number of Bishops that can be placed on a chessboard so that no Bishop threatens another Bishop? (Bishops move along Diagonals)

What is the minimum number of Knights that can be placed on a chessboard so as to occupy or threaten every square?(Knights move along horizontal and vertical lines)

What is the minimum number of squares that eight Queens can command(that is to occupy or threaten?)

Change the size of the board to say a 4 x 4, 6, x 6, 7 x 7, 10 x 10, and see if there is a pattern for each of the pieces for each sized board.

 

 

 Dominoes and the Chessboard

It is easy to put dominoes (where each domino covers two squares) on a chessboard and cover all the squares.

But what happens if you have a chess board where two corner squares have been cut off.  Can you now place Dominoes to cover all squares?

 

 

 

Adapted and extended from Points of Departure by Association of Teachers Mathematics  1989