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Secondary Magazine - Issue 72: An idea for the classroom


This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 20 October 2010 by ncetm_administrator
Updated on 09 November 2010 by ncetm_administrator

 

Secondary Magazine Issue 72   sweets
 

An idea for the classroom - introducing mathematical iteration

Iteration is the repeated application of a function or process in which the output of each step is used as the input for the next step.

Here are two completely different activities that students enjoy, that are interesting in themselves and that can introduce students to the idea of iteration. 
 
Activity 1: Passing on loot

  • Several students sit in a circle.
children sitting in a circle
 
 
  • Each student has a different even number of wrapped sweets – or small objects such as multilink cubes (their loot).
  • When they are given a signal, each student passes half of his or her ‘loot’ to the student on his or her left (clockwise), and then the teacher gives any student left with an odd number of sweets an extra one to make their number of sweets even again.
  • This ‘operation’ (or step) of passing-half-of-their-sweets-clockwise-and-then-adding-one-sweet-to-any-odd-numbers-of-sweets is repeated.
  • It is repeated over and over again.

Example: 4 students start with 2, 4, 6, 8 sweets each respectively

start

sweets start

step 1

step 1

 

each student passes half their loot
to the person on their left (clockwise)
1 sweet is added to make any odd
number of sweets even




step 2

step 2


each student passes half their loot
to the person on their left (clockwise)
1 sweet is added to make any odd
number of sweets even



step 3 … … …


This process is simple iteration because the input for each step is the sweet distribution situation that was the output of (resulted from) the previous step.

Students can think about, discuss, and make and test conjectures about, the answers to questions such as:

  • what will happen to the distribution of sweets among the students when the ‘operation’ is performed over and over again?
  • will one person end up with all the sweets?
  • will everyone’s ‘booty’ grow larger and larger as the teacher gives out more and more extra sweets?
  • will the number of sweets ‘stabilise’, eventually evening out among the students?
  • perhaps an oscillating pattern will occur, with ‘clumps’ of sweets moving around the circle?
  • does what happens depend on the number of students in the circle?
  • does what happens depend on the initial distribution of sweets?

Activity 2: Nested polygons

At WolframMathWorld you can see that ‘beautiful patterns can be created by drawing sets of nested polygons such that the incircle of the nth polygon is the circumcircle of the (n + 1)st polygon’.

Explain that in a set of nested polygons each ‘new’ polygon is mathematically similar to the previous polygon, and its vertices are at the mid-points of the sides of the previous polygon. Then challenge students to create, possibly using dynamic geometry software, sets of nested regular polygons such as these:

nested polygons

Each ‘step’ is the creation of a ‘new’ polygon that nests in the previous polygon. The output of each step is the input for the next step. So this is another simple example of an iterative process.

Creating accurately each set of nested polygons is a good mathematical challenge in itself. Some students could also explore relationships between the lengths of corresponding sides, and between the areas of corresponding regions in each set of nested polygons. They might try to answer their own questions, such as:

  • what proportion of each diagram is black?
  • in each set of nested polygons what are the angles between each side of a polygon and the two sides of the previous polygon that it meets?
 
 
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