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:-

• 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?

• 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?

• 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?

At this point, you might want to introduce Pascal's Triangle to the children.

 The children could try to find connections between patterns that occur in Pascal's Triangle and in these triangular arrangements of letters.

• 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?