Classroom topics in focus: algebra as generalised arithmetic

Algebraic reasoning is foundational to all mathematical thinking. It is through algebra that we are able to explore and express mathematical structure, pattern and relationships. One of the big ideas in algebra is that of generalised arithmetic. The focus of arithmetic in the curriculum is not just about skill acquisition. Rather, it is about understanding the structure of our number system and exploring how computational methods can be expressed and generalised algebraically.

Algebra as generalised arithmetic was one of the focus topics used in the development of reSolve’s classroom resources. The two tasks below are taken from this topic and illustrate the development of algebraic reasoning.

Year 4—Number maze

The properties of odd and even numbers are explored in Year 4.

• The maze starts at 5 and finishes at 14.
• You can move through the maze using horizontal, vertical and diagonal movements.
• Add together the numbers in each box that you pass through.
• The aim is to finish with a total that is an odd number.

There are multiple pathways through this maze. Through the course of this task, students are encouraged to look at how many odd and even numbers are in each pathway. They see that an odd number of odds is always required to give an odd total.

To explain this, students look at a visualisation of why two odd numbers always add to an even and similar results. They move from a specific case to the general case where they can see that the two unpaired parts of each number can be paired to make an even number.

This task is based on a mathematical ‘magic trick’. All these so called magic tricks have a basis in logic—they can be explained! The challenge of this task is to go beyond the entertaining. Students are asked to understand why it works and to see if they can go beyond to develop another one.

• Choose two numbers.
• Write down these numbers vertically aligned, and add them. Write the answer under the second number. Then add the last two numbers in the list, and continue until there are 10 numbers in the list. For example,
• Continue adding your chain of numbers until you have 10 numbers.
• Now add up the total of the 10 numbers in your list. (In the example the total will be 825).

Time for the teacher to perform some magic!

One student is asked to come to the board and record the numbers in their list. As the student writes the 10th number the teacher quickly tells them the total.

This ‘trick’ makes the teacher look as if they have added all 10 numbers very quickly, but in fact they have just multiplied the 7th number by 11 while the student has been writing the remaining numbers on the board. Generalising the addition chain reveals a beautiful Fibonacci pattern and allows the students to see what is happening.

If a and b are the starting numbers the seventh number is 5a + 8b and the total of the ten numbers is 55a + 88b, which can be factorised to 11(5a + 8b). The door is opened here to many more mathematical investigations. How can you quickly multiply by 11? Are there other numbers in the list that can be used to find the cumulative total? Will it work with any set of numbers? What if one of the starting numbers was 0 or a negative? Can you make up your own trick based on logic and represented algebraically?

Kristen Tripet
Resource Developer, reSolve: Maths by Inquiry
​mbi@science.org.au