Properties of operations written with letters — Mathematics, 11–13
The properties of addition and multiplication work for any numbers, so an expression with letters says them all at once and lets you complete equalities.
Rules that hold for any numbers
The properties of operations are rules that work for any numbers, fractions included. With letters we write each one once: a + b = b + a and a × b = b × a (commutative), (a + b) + c = a + (b + c) (associative) and a × (b + c) = a × b + a × c (distributive: multiplying a sum means multiplying each part). Each letter holds the place of any number, so 1/2 + 4 = 4 + 1/2 and 2/3 × 9 = 9 × 2/3 are cases of the same expression.
Complete, calculate and be sure
Knowing a property lets you complete an equality without calculating both sides: if 3/4 × 8 shows up on the right, the number 8 must be inside the brackets on the left. It also speeds up mental calculation. And a property written with letters is true for every number, not only for the ones you tried, so it can be trusted where a few examples cannot.
Completing equalities with fractions
Complete 3/4 × (8 + __) = 3/4 × 8 + 3/4 × 12. In the distributive property, each number inside the brackets reappears on the right multiplied by 3/4, so __ = 12. Check: 3/4 × (8 + 12) = 3/4 × 20 = 15, and on the right 3/4 × 8 = 6, 3/4 × 12 = 9, and 6 + 9 = 15. Another: 5/6 + 4 = 4 + __. By the commutative property __ = 5/6, since the order of the parts does not change the sum. In letters: a × (b + c) = a × b + a × c, here with a = 3/4, b = 8 and c = 12.
Multiplying only the first part
Is 1/2 × (10 + 6) = 1/2 × 10 + 6? Many people multiply only the first number inside the brackets and copy the second one. It is a reasonable slip, because once the brackets are gone it looks like an ordinary sum. But the left side is 1/2 × 16 = 8 and the right side is 5 + 6 = 11. The multiplication has to reach every part: 1/2 × 10 + 1/2 × 6 = 5 + 3 = 8. In letters it is a × (b + c) = a × b + a × c, never a × b + c.
Mental calculation and areas
In mental calculation you use the distributive property without noticing: 25 × 12 = 25 × (10 + 2) = 250 + 50 = 300. It also appears in areas: a rectangle 3 cm high and 5 + 2 cm wide can be measured whole, 3 × (5 + 2) = 21 cm², or cut into two pieces, 3 × 5 + 3 × 2 = 15 + 6 = 21 cm². With letters, one line, a × (b + c) = a × b + a × c, covers every rectangle and every product of that kind.
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