The purpose of this unit of three lessons is to develop an understanding of how the operations of addition and subtraction behave and how they relate to one another.
Algebra is the area of mathematics that uses letters and symbols to represent numbers, points and other objects, as well as the relationships between them. We use these symbols,=, ≠, <, >, and later ≤ and ≥, to express the relationships between amounts themselves, such as 14 = 14, or 14 > 11, and between expressions of amounts that include a number operation, such as 11 + 3 = 14 or 11 + 3 = 16 – 2.
These relationships between the quantities are evident and clearly stated within an equation or expression. This highlights the purpose of an equation or expression, which is to express relationships.
The relationships between the number operations, and the way in which they behave, can be less obvious. Often these operations are part of arithmetic only, as computation is carried out and facts are memorised. However, recognising and understanding the behaviours of, and relationships between, the operations is foundational to success in algebra and arithmetic.
Early on, the relationship between addition and subtraction is explored by connecting the members of a ‘family of facts.’ These ‘fact families’ can be used to facilitate learning of basic facts and to develop a deeper understanding of the relationship between addition and subtraction.
Students often encounter conceptual difficulties in understanding the nature of the operational relationships that exist between three numbers. It is not unusual for students, when asked to write related facts, to write, for example, 3 + 4 = 7, 4 + 3 = 7, 10 – 3 = 7, 10 – 7 = 3. In these statements, equations are correct, but the relationship between 3, 4 and 7 is not understood.
As students encounter more difficult problems, and are required to develop a range of approaches for their solution, having an understanding of the inverse relationship that exists between operations is critical. To ‘just know and understand’ that subtraction ‘undoes’ addition, and that addition ‘undoes’ subtraction becomes very important. For example, it is important to know how and why problems such as 61 – 19 = ☐ can be solved by addition, saying 19 + ☐ = 61, or that the value of n can be found in 12n + 4 = 28, by subtracting 4 from 12n + 4 and from 28. By having a sound understanding of the inverse relationship between addition and subtraction (for example) students are better placed to solve equations by formal means, rather than by simply guessing or following a memorised procedure.
It is important to consider this larger purpose, as we explore ‘families of addition and subtraction facts’. Within this, an equal emphasis must be placed on both operations.
The activities suggested in this series of three lessons can form the basis of independent practice tasks.
Links to the Number Framework
Advanced counting (Stage 4)
Early Additive (Stage 5)
This unit can be differentiated by varying the scaffolding provided or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:
Situating the families of facts in familiar additive contexts will appeal to students’ interests and experiences and encourage engagement. Examples may include:
SLOs:
Activity 1
Activity 2
Activity 3
Conclude this session by summarising on the class chart, the features of a family of related facts: three number members of the family, and four equations, two of addition and two of subtraction.
SLOs:
Activity 1
Activity 2
Activity 3
Conclude by sharing and discussing student work.
SLOs:
Activity 1
Activity 2
Activity 3
Dear parents and whānau,
In maths this week, the students have been learning about the relationship between addition and subtraction. They have been exploring fact “families” with three numbers, such as 8 + 6 = 14, 6 + 8 = 14, 14 - 6 = 8 and 14 – 8 = 6. They have found out that subtraction “undoes” addition. (Subtraction is known as the inverse operation of addition, and vice versa).
They have played a game called Family Shuffle. Your child can show you how they have played this. You might like to have a turn, then make up your own fact family together and try the game again.
We hope you enjoy the game, and your discussions, as you make up and talk about new number fact families.
Thank you.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/number-families-and-relationships at 8:40pm on the 26th February 2024