The purpose of this unit is to develop understanding of the way numbers behave, to enable the use of everyday language when making general statements about these behaviours, and to develop understanding of 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 explicitly articulate these number properties, they do develop familiarity with these behaviours. As problem-solvers, 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 are more focused on ‘finding answers’ to computations, and sharing the strategies for working these out, rather than being focused on identifying the number properties involved. However, 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 first be encouraged to 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.” Students 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, arithmetic, and 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 to 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
Advanced Multiplicative
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
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 a meaningful, culturally-responsive, and practical context that reflects your own school projects.
Te reo Māori kupu such as tāpiri (add, addition), whakarea (multiply, multiplication), āhuatanga herekore (associative property), āhuatanga tohatoha (distributive property), āhuatanga kōaro (commutative property) could be introduced in this unit and used throughout other mathematical learning
Session 1
Activity 1
Activity 2
Important number properties that we often use
Session 2
Activity 1
Activity 2
Session 3
Activity 1
Activity 2
Activity 3
Activity 4
Dear families and whānau,
In algebra we have been learning more about number properties, or the way numbers ‘behave’. Talk about and solve the equations below, and have your child explain what is ‘going on’ with the numbers as you work together.
799 + 207 - ☐ = 799
☐ + 235 = 235 + 17
25 x ☐ = 7 x 25
(125 + 16) + ☐ = 125 + 16 + 17
(15 x ☐) x 10 = 15 x (2 x 10)
8 x (19 + 3) = (☐ x 19) + (8 x 3)
Thank you.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/study-number-properties at 8:40pm on the 26th February 2024