The purpose of this unit is to develop the students’ deeper understanding of the way numbers behave, to enable them to use everyday language to make a general statement about these behaviours, and to understand the symbolic representation of these ‘properties’ of numbers and operations.
As students solve problems using the operations of addition, subtraction, multiplication and division, they come to recognise, informally, some of the behaviours that numbers always exhibit when number operations are applied to them. They sense that this is ‘just the way numbers work’. Even though students may not be challenged to articulate generalisations which that state number properties, these behaviours become familiar. As problem-solvers, the students ‘draw’ on these properties, using relational thinking strategies, albeit in an unconscious way.
For example, a student uses a place value strategy to solve a problem such as:
There are three aliens. Each has 27 teeth. How many teeth are there altogether?
The student explains, “I said three times two is six, so three times twenty is sixty and I said three times seven is twenty one. Then I added sixty and twenty-one together, so it’s eighty-one teeth.” They are applying the distributive property of multiplication over addition. The student knows that 3 x 27 or 3 x (20 + 7) = 3 x 20 + 3 x 7 or a x (b + c) = (a x b) + (a x c).
Typically, classroom teaching and discussion is more focused on ‘finding answers’ to computations, in this case, 81, and sharing the strategies for working these out (explaining what they did), rather than being focused on identifying the number properties involved.
Number properties govern how operations behave and relate to one another, and they are essential for computation. Because algebra is the area of mathematics that uses letters and symbols to represent numbers and the relationship between them, students can be enabled, through general statements such as ‘a x (b + c) = (a x b) + (a x c)’, to see clearly the structure and nature of these number properties. It is useful therefore to have students pause to recognise and reflect more formally on ‘the way numbers work’ and in so doing to be gently introduced to variables (as yet unknown amounts), before forming and solving simple linear equations.
Generalisations can be expressed in a number of ways. Students should firstly use their own words to describe what they see happening, for example: “I just know that when you multiply, you can break one of the numbers apart, multiply the numbers separately, and then put them back together again.” The student can then be guided to connect these ideas to the elegantly simple algebraic notation that express these same ideas using letters as variables, and to recognise that is property is true for any (real) numbers. The ability to apply, recognise and understand these number properties, is foundational to ongoing success in algebra as well as arithmetic, and in mathematics in general.
The purpose of these lessons is to make the students fully aware of how the numbers are behaving. Naming each number property and writing and exploring a property using letters, is a way of helping students recognise and understand these. However, having students remember the names for each number property is not a focus of these lessons.
Links to the Number Framework
Advanced Additive/Early Multiplicative
The context for this unit of work is Enviroschools projects. A fictitious school, Kiwi School, and its projects, are used here. If possible, explore these lessons in a similar way, using the practical context of your own school projects.
Set the scene for this unit of work.
Begin the session by asking, ‘What happens when you drop a ball.’
(Answer, ‘It falls to the ground.’)
Ask several more questions about natural phenomena (such as, ‘What happens when we boil water? Answer. ‘It evaporates.’ etc.)
Explain that scientists have investigated and continue to investigate the way in which things in nature behave, and they give these things names (like ‘gravity’).
Explain that mathematicians have investigated and continue to investigate the way numbers and number operations in mathematics behave. These behaviours are called number properties and these have names too.
Write number properties on the class chart.
Tell the students that in the next three lessons they will be working in a slightly different way with maths ideas that they ‘already know’ because they have been using these, perhaps without noticing, ever since they started solving problems.
Conclude this session and unit by reflecting on student learning about number properties. Revisit the key ideas recorded on the class chart throughout the sessions.
Have students understand that we continue to apply these number properties when we solve more complex equations.