Mathematics Developmental Continuum P-10

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Equivalence in Number Sentences - More About

The ability to identify and use arithmetic relationships within number sentences to solve problems such as

99 – □ = 90 – 59

is very important for students because focusing on the structure of arithmetic relations establishes a bridge to algebra.

When completing number sentences that have missing numbers, many students can work only computationally and do not recognise or use the relationships between the number expressions on the two sides of the number sentence.

For example, to find the number in the box in 99 – □ = 90 – 59, some students can look at the whole sentence and see that because 99 is 9 bigger than 90, the number in the box needs to be 9 bigger than 59 so that the differences are equal. This is an example of relational thinking: thinking about the whole statement and how the parts must be connected.

It is essential for dealing with algebraic equations.

As another example, consider 26 + 39 = □ + 23

If students first calculate the left hand side to give 65, and then reason that they need to find the number that is added to 23 to give 65, they are working computationally. While many students will successfully find the missing number in this way, a computational approach is not giving them the necessary experience in seeing relationships between number expressions. They are not focussing on the properties of the operation of addition that is critical for algebra.

There are also students who do not understand the meaning of the equal sign, so to them, the equals sign is a cue to provide an 'answer'. These students may ignore the 23 on the right hand side and simply write 26 + 39 = 65 so the number in the box is 65.

Similarly, with number sentences involving differences, it is important that students recognise that they must keep the difference the same.

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Working with uncalculated expressions

The ability to be prepared to work with uncalculated expressions and look at the relationships within them, is very important for later mathematics.

For example:

  • you can calculate 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 directly, or you can look for patterns and use them for smart calculation (eg matching up the 1 and 19 to get 20, the 2 and 18 to get 20, the 3 and 17 to get 20 etc.
  • algebraic expressions are 'uncalculated' (e.g. you need to know x + 3 is 5 less than x + 8, even though you don't know what number it is),
  • rules for algebra are formal expressions of the informal ideas that are used here in arithmetic. For example, the algebraic identity a - (b - c) = a + c - b is a generalisation of 100 - 19 = 101 - 20 (a = 100, b = 20, c = 1).