L4 · Dividing Integers and Rational Numbers

Learning goal

“I can divide any two rational numbers, check every quotient by multiplying back, and explain why division by zero has no answer.”

Quotient — the result of a division; it answers "how many of the divisor fit in the dividend?" or "what factor times the divisor gives the dividend?"

Reciprocal — the number that multiplies a number to give 1: the reciprocal of 4 is 1/4, and of -3/4 is -4/3. Division will be rebuilt on reciprocals in L5.

Inverse operation — division undoes multiplication: a / b is the missing factor in b x ___ = a. Every quotient can be CHECKED by multiplying back.

Worked example

-56 / 7 and -56 / (-7). Step 1: Rewrite each as a missing factor: -56 / 7 asks "7 x ___ = -56?" The factor is negative (a positive 7 times a positive cannot give -56): ___ = -8. Check: 7 x (-8) = -56. So -56 / 7 = -8. Step 2: -56 / (-7) asks "-7 x ___ = -56?" Now the factor must be POSITIVE (two negatives multiply to a positive): ___ = 8. Check: (-7) x 8 = -56. So -56 / (-7) = 8. Conclusion: division inherits multiplication's sign rules, because division IS multiplication read backwards - different signs -> negative quotient, same signs -> positive quotient.

Check yourself — explain it

Why does -56 / (-7) have a positive answer?

Paired problem — example still visible

72 / (-9) -> "-9 x ___ = 72?" -> -8 (check: -9 x -8 = 72)

Faded completion — last step is yours

"-96 / (-12): the missing factor in -12 x ___ = -96 has sign ___ and magnitude 96 / 12 = 8, so the quotient is ___." -> positive; 8

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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