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Teaching mathematics
Teaching mathematics

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3.2 Always, sometimes, never true?

As previously discussed, algebraic expressions are like mathematical phrases or sentences. Equations are like mathematical statements. A mathematician will always ask whether or not an equation is true.

When equations contain no unknown quantities, use calculation to check if they are true:

three plus four minus two equals six is an equation but it is not true.

three plus 22 equals 24 plus one is an equation that is true.

When equations include unknown quantities, they may be true for only one value of the unknown:

two times n minus five equals 17 is only true when n is 11.

If n = 10 then 2n – 5 = 17 is false.

They may be true for several values:

n squared plus one equals 10 is true when n = 3 and when n = –3.

They may be true for all values of the unknown:

two times left parenthesis n plus three right parenthesis equals two times n plus six

In formal mathematics you can use the sign ≡ to show that these expressions are ‘identically equal’:

two times left parenthesis n plus three right parenthesis identical to two times n plus six

Sometimes an equation is given as a definition or a formula for working something out. We can assume that someone else has done the work of showing it is always true. For example:

a cubed equals a multiplication a multiplication a (definition).

Area of a triangle equals half base multiplication height (formula a learner could prove).

normal cap e equals normal m times normal c squared (formula that a learner could not prove!).

Activity 15 Always, sometimes or never true

Timing: Allow 10 minutes

Decide whether these equations are true for all values of the variable, or only some values or none.

Figure 9 Always, sometimes, never true?

Discussion

Trying values will help you see that there are no values of b or m that make their equations true: they are never true. Using algebraic reasoning will help us see that e = 1, t = 1.5, c = 0, q = 10 are specific values that make their respective equations true. Also, some equations are true for a range of values: k < 15, p either greater than 2 or less than -2, and x greater than 1 or less than 0. A final one, that is more subtle because it involves two variables, is that p + 12 = f + 12 is true only when p = f. All these equations are sometimes true.

Finally the equations 2n+3 = 3+2n and 16s2 = (4s)2 are true for all values of n and s. The first is because addition is commutative, and the second because squaring applies to the whole bracket. They are always true.