Teaching mathematics
Teaching mathematics

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Teaching mathematics

3 The equals sign

Figure 7 Friends = love

One of the strong findings of early algebra research is that learners in the middle years need to change their perception of the equals sign. This sign is recognised from an early age (Figure 7), and it is usually understood as standing for an instruction such as ‘makes’ or ‘work it out now’. This is called the operational understanding of the equals sign and it works well for number problems. Text books, worksheets and even national examinations use the equals sign in this way:

equation left hand side 15 plus five equals right hand side
Area equals length multiplication breadth

To make progress in algebraic thinking, learners need to develop a new relational understanding of the equals sign. In algebra, the equals sign only makes sense when it is used as a way of comparing two expressions and saying that they have the same value as each other.

All of the following equations can be understood as statements that express a relationship between two quantities:

10 multiplication four equals 40
equation left hand side 10 multiplication four equals right hand side 20 multiplication two
five postfix times equation left hand side normal m postfix times equals right hand side 500 postfix times cm
equation left hand side two solidus three equals right hand side four solidus six
equation left hand side two times open normal y plus six close equals right hand side two times normal y plus 12

Two other relational symbols are the inequality signs < and >. Tasks that require learners to decide which of these symbols to use can develop a relational understanding of the equals sign.

Activity 13 Relationships between expressions

Timing: Allow 5 minutes

Which is bigger: 3ab or 2a + b? Choose values of a and b and decide. Record your results using the signs <, > and =.

Discussion

You could write:

If we choose a equals one comma b equals two then three multiplication one minus two equals one and two multiplication one plus two equals four, so one less than four.

If a equals one comma b equals four then equation left hand side three multiplication one minus four equals right hand side negative one and two multiplication one plus four equals four so three times a minus b less than a plus b.

Or a equals two comma b equals one then three multiplication two minus one equals five and two multiplication two plus one equals five

Activity 14 Reflect

Timing: Allow 5 minutes

You might have noticed the different ways in which the three responses to the previous question were written.

How would you ask your learners to record their results?

Does this ensure that they have practice in selecting and writing the correct signs, in comparing two expressions?

ab3ab2a + bWrite your inequality or equation here
12143a b < 2a + b
103
63

Discussion

Using the equals or inequality signs between numbers or symbolic expressions is correct mathematics. Learners should not write =, < or > on its own.

Using a table is often helpful for keeping track of several calculations.

ab3ab2a + bWrite your inequality or equation here
12143a b < 2a + b
10327233a b > 2a + b
6315153a b = 2a + b
TM_1

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