This investigation is an opportunity for children to make a table to record
and organise their results.  It would be helpful to print a series of the
triangles onto a sheet of paper and copy them for the children to draw on the
pathways. Some of the inquiries you could encourage are:-

<ul>
<li>Is there a pattern in the numbers of letters on each line? How many paths
    can be taken from each letter in the first triangle? Can a pattern be
    found to describe the number of ways the word ABACUS can be made? Is
    there a relationship between the number of letters, the number of
    pathways and the number of ways the word can be made? 
<p>
<li>
    In the second triangle, is there a pattern in the numbers of letters on
    each line? Predict if there will be more or less paths from each letter
    in this triangle. What evidence is the prediction based on?  Estimate and
    then discover how many ways the word ABACUS can be made? Can a pattern be
    found to describe the number of ways the word ABACUS can be made?
    Compare the results of the first and second triangle, how are they alike,
    how are they different?
    <p>
    
    <li>
	
The children should be able to construct a right triangle using the word
	ABACUS. Ask them to predict if the result of their explorations will
	be like or different from the other two triangles. The results are
	the same as the first triangle. Why?  Can the three types of
	triangles be named?  What is known about the properties of each
	triangle? Does knowing about different shaped triangles help explain
	the results?  
</ul>
	<p>
At this point, you might want to introduce Pascal's Triangle to the
children. Information and links for Pascal's Triangle can be found in
this month's
<a href = "magazine.htm#mag8">Mouldy
Maths section</a>.  The children could try to find
connections between patterns  that occur in Pascal's Triangle and in these
triangular arrangemenets of letters.
<p>
<ul>
  <li> The children could extend their investigation to rectangles.  How many
      different ways do they think they could write ABACUS in a rectangle
      following the conditions given?  Will the results from the triangle
      investigations help them predict the results for the rectangles?  When
      they write out the possibilities they might be surprised.  Why do they
      get the results they do?

</ul>
<p>