Everything covered in this lesson, in one place - useful for revision or printing.
\[ \text{Number Line} \rightarrow \text{Operations} \rightarrow \text{Applications} \]
Integers form one connected system. We move from representing directed situations to calculating with them, explaining properties, and solving multi-step problems.
\[ -25\text{m (below)} \quad +300\text{ (deposit)} \quad 0\text{ (no change)} \]
Integers describe direction from a reference point. Below sea level is negative, above is positive. Zero means no change. The sign is part of the integer.
\[ -6 < -2 < 0 < 3 \]
Moving right means increasing. Moving left means decreasing. -5 > -12 because -5 is further right, even though 12 is numerically larger than 5.
\[ (-8)+(-11)=-19 \quad (-18)+7=-11 \]
Same signs: add values, keep sign. Different signs: find difference, keep sign of larger value. Subtraction becomes addition of the inverse: a-b = a+(-b).
\[ 250-(-40)+(-75)=215 \]
Rewrite: 250+40-75. Left to right: 290-75=215. Always show the rewriting step to avoid sign errors.
\[ \text{Same signs} \rightarrow + \quad \text{Different} \rightarrow - \]
Sign rules apply to both multiplication and division. Estimate before calculating large numbers. Check quotients by multiplying back.
\[ -2998 \times 51 \approx -150000 \]
Estimate: -3000 × 50 = -150,000. Signs differ so answer is negative. Actual answer should be close to estimate.
\[ a(b+c)=ab+ac \]
Commutative: order changes (add/mult only). Associative: grouping changes (add/mult only). Distributive: mult distributes over add/sub. Use these to rearrange and check work.
\[ 3-\{2 \times (-4)+5\} = 6 \]
Brackets first: 2×(-4)=-8, -8+5=-3. Then: 3-(-3)=6. Multiplication before addition inside brackets. Subtraction last.
\[ (-5)^2=25 \quad -5^2=-25 \]
Even exponent on negative base = positive. Odd = negative. Brackets include the negative in the power. No brackets = exponent applies to number only.
\[ -24 + (6 \times 5) - 13 + 4 = -3 \]
Start at -24m. Rise 6m/min for 5min = +30. Descend 13m = -13. Rise 4m = +4. Order: -24+30-13+4 = 6-13+4 = -3m.
\[ \text{Net rise: } 34-13=21 \quad -24+21=-3 \]
Check reasonableness: Total rise 34m, descent 13m, net rise 21m. Starting at -24 gives -3. Answer makes sense.
\[ -480 + 6(75) - 190 = -220 \]
Start -R480. Six deposits of R75 = +450. Withdrawal R190 = -190. -480+450-190 = -30-190 = -R220.
\[ \text{Integers connect representation to application} \]
You can represent contexts, calculate with all operations using correct rules, apply properties, follow BODMAS, and solve integrated problems. You have mastered Grade 8 integers.