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, by the very nature of an equation or expression, the purpose of which is to express a relationship.
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, recognizing and understanding the behaviours of and relationships between the operations, is foundational to success in algebra as well as arithmetic.
One way in which the relationship between the operations of addition and subtraction is often explored early on, and is made more obvious to students, is by connecting the members of a ‘family of facts.’ Whilst these ‘fact families’ have often been used to facilitate learning of basic of facts, they have not always been used well to develop a deeper understanding of the relationship between these operations.
Because of this, students often encounter conceptual difficulties in understanding the nature of the operational relationships that exist between the 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 which the 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, if responses to problems are to come from a position of understanding, rather than simply from having been taught a procedure. 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 from12n + 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’. Furthermore, in studying addition and subtraction together, an equal emphasis is placed on both operations. Unfortunately, it is not always the case that these operations are given equal emphasis in classrooms. Subtraction has been the ‘poor relation’ that receives less attention. In fact sometimes students are not helped to see that there is any relationship at all between addition and subtraction.
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)
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.
Conclude by sharing and discussing student work.