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Integers

Numbers · Integers

Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 1.0 Numbers | Sub-Strand 1.1 Integers (6 Lessons)

Lesson objectives

By the end of this sub-strand, you should be able to:

  • Add and subtract integers in real-life situations.
  • Multiply and divide integers in real-life situations.
  • Apply the correct order of operations to combined operations on integers.
  • Use integers to represent and solve everyday problems involving gain and loss, rise and fall, above and below.

Integers

A thermometer in Nyandarua can read below zero. A bank balance can be overdrawn. Both need numbers smaller than nothing, and those are what integers give us.

a) What integers are

The integers are the whole numbers together with their negatives, and zero.

The integers The integers −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 zero

They stretch forever in both directions. Numbers to the right of zero are positive. Numbers to the left are negative.

The further right a number sits, the bigger it is. That makes −2 larger than −7, which surprises many learners at first.

b) Adding and subtracting

Picture the number line. Adding moves you to the right.

Adding moves right Adding moves right −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 add 3 start

Subtracting moves you to the left.

Subtracting moves left Subtracting moves left −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 subtract 5 start

Starting at 2 and subtracting 5 takes you past zero and lands on −3.

c) When two signs meet

Sometimes a sign for the operation sits next to a sign for the number.

Two signs together Two signs together 5 + (−3) = 5 − 3 plus a negative means subtract 5 + (−3) = 2 the answer 5 − (−3) = 5 + 3 minus a negative means add 5 − (−3) = 8 the answer

Two rules cover every case. A plus next to a minus behaves like a minus. Two minuses together behave like a plus.

d) Multiplying and dividing

Here the rule is about matching signs, and it is the same for both operations.

Multiplying and dividing Multiplying and dividing (+) × (+) = + same signs give plus (−) × (−) = + same signs again (+) × (−) = − different signs give minus −24 ÷ 6 = −4 an example

Same signs give a positive answer. Different signs give a negative one.

Be careful not to confuse this with the addition rule. Adding two negatives gives a negative, but multiplying two negatives gives a positive.

e) Combined operations

When a calculation has several operations, work in the usual order. Brackets first, then multiply and divide, then add and subtract.

Combined operations Combined operations −8 + 4 × (−3) = −8 + (−12) multiply first −8 + (−12) = −8 − 12 plus a negative −8 − 12 = −20 the answer

Do not work left to right blindly. In −8 + 4 × (−3), the multiplication comes first.

f) Where this is used

Bank statements use negatives for money owed. Weather reports in the highlands use them for temperatures below freezing. A business records a loss as a negative profit.

Words to know

  • Integer -- a whole number that may be positive, negative, or zero; fractions and decimals are excluded.
  • Opposite (additive inverse) -- the integer the same distance from zero but on the other side; the opposite of -7 is 7.
  • Absolute value -- the distance of an integer from zero, ignoring direction; both -6 and 6 have absolute value 6.
  • BODMAS -- the agreed order of operations: Brackets, Of/Orders, Division and Multiplication, Addition and Subtraction.
  • Number line -- a line on which integers are marked at equal intervals, increasing to the right.

:::checkpoint Check yourself

  1. Which is larger, −9 or −3?
  2. Work out 7 + (−12).
  3. Work out −6 − (−4).
  4. Work out −5 × 3 + 10. :::

Bridge to practice

The exercises below move from direct sign rules to multi-step combined operations and real-life reasoning. Where a question involves more than one operation, write out the steps as the worked examples do rather than attempting it in one line, and check at the end whether your answer points in a direction that makes sense for the situation.

Check yourselfPractise Integers — 10 questions →Next in MathematicsMatrices →