Counting on Letters
Problem
The basis of this investigation comes from discussion with some teachers in Hungary, England and Denmark, who met through our Euromaths Project, and was adapted by Merilyn Buchanan.
The letters of the word ABACUS have been arranged in the shape of a triangle.
A
B B
A A A
C C C C
U U U U U
S S S S S S
What do you notice about the pattern?
How many different ways can you find to read the word ABACUS from this triangular pattern?
A
B B B
A A A A A
C C C C C C C
U U U U U U U U U
S S S S S S S S S S S
Are there more or fewer ways to read the word ABACUS in this triangle?
Try arranging the letters ABACUS into a different shape triangle.
Can you see a pattern in the letters of the new triangle that is similar to either of the triangles above?
Record the number of ways to read the word ABACUS in your triangle. Are there more or fewer ways to read the word ABACUS in your triangle than in the other triangles?
You can find information about another triangle, Pascal's Triangle in this article.Can you find any similarities between these triangles and the patterns in Pascal's triangle?
The letters could also be arranged in a rectangle like this:
| A | B | A | C |
| B | A | C | U |
| A | C | U | S |
If you begin in the top left hand corner and move only to the right or immediately below the letter, how many ways are there to read the word ABACUS?
Arrange the letters of ABACUS into other size rectangles.
How many ways can you find to read the word ABACUS? Is there any
pattern between the number of ways you can find the word and the
size or shape of the rectangle?
Try the same problems with the word PUMPKIN.
Student Solutions
Have a go yourself, and if you discover anything interesting, e-mail us to tell us what you've done!
Teachers' Resources
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?