Mathematical language: symbols and notation
Maths has its own short language. A symbol is a sign that stands for an idea: + (add), = (is equal to), < (is less than), √ (square root), π (pi). A variable is a letter that stands for a number that can change, like d for distance.
Writing "the cost is 30 plus 12 for every kilometre" as C = 30 + 12d is shorter and exact. Some rules of notation:
- 12d means 12 × d. We usually leave out the × sign between a number and a letter.
- Write the number before the letter: 12d, not d12.
- = means both sides have the same value. Do not use = to mean "then" or "next".
- Say what each letter means and its unit: C in rupees, d in km.
- Use SI units and their symbols: m, km, kg, s, °C.
Choosing a representation: words, formula, table or graph
A representation is a way of showing a mathematical idea. The taxi fare can be shown four ways:
- Words: easy for anyone to read; can be long or unclear.
- Formula: short and exact; good for working out any value.
- Table: shows exact values side by side; good for looking up.
- Graph: shows the pattern and trend at a glance; good for comparing and spotting change.
Pick the one that fits your reader and your purpose. To show a trend over time, use a line graph. To compare categories, use a bar chart. To show parts of a whole, a pie chart can work. Often the best answer uses two: a formula plus a graph.
Try it: write your daily screen time for 7 days. Show it as a table, then as a bar chart. Which one shows your busiest day faster?
Clear written working
Good working lets someone else follow and check you. Rules:
- Start by writing what you know and what you want.
- One step per line. Line up the = signs.
- Say why if a step is not obvious ("subtract 30 from both sides").
- Keep units and give the answer with a unit.
- Check: put your answer back into the first line.
Example: how far for 150?
150 = 30 + 12d
150 − 30 = 12d
120 = 12d
d = 120 ÷ 12 = 10 km
Check: 30 + 12 × 10 = 150 ✓
Judging representations critically and presenting results
Spotting misleading graphs
- Axis not starting at zero on a bar chart makes small differences look huge.
- Uneven scales (gaps of 10, then 50) bend the shape.
- Missing labels or units: you cannot tell what is measured.
- 3D or picture charts can make one item look much bigger by area.
- Cherry-picked time ranges hide the full trend.
Presenting the results of a maths inquiry
A maths report or presentation usually has: 1. Question (what you wanted to find out), 2. Method (data you collected, models or formulas you used), 3. Results (tables and graphs with titles, labels and units), 4. Conclusion (the answer in plain words), 5. Reflection (limits of your method and what you would improve). Use a poster, slides or a short paper, and practise explaining it to a classmate.
Key formulas and definitions
- Cost = start fee + rate × distance → C = a + bd
- 12d means 12 × d
- Representations: words ↔ formula ↔ table ↔ graph
- Change (%) = (new − old) ÷ old × 100
- Report = question → method → results → conclusion → reflection
Worked examples
1. Write as a formula: "a plumber charges 200 for a visit plus 150 for each hour".
Let P be the price and h the number of hours. P = 200 + 150h.
2. Use C = 30 + 12d to fill a table for d = 1, 3, 5.
d = 1: C = 30 + 12 = 42. d = 3: C = 30 + 36 = 66. d = 5: C = 30 + 60 = 90.
3. Solve with clear working: a taxi ride cost 102. How many km was it? (C = 30 + 12d)
102 = 30 + 12d → 102 − 30 = 12d → 72 = 12d → d = 72 ÷ 12 = 6 km. Check: 30 + 72 = 102 ✓.
4. Sales were 98 last year and 102 this year. A bar chart with axis from 96 makes this year's bar 3 times as tall. What is the real change in %?
Change = (102 − 98) ÷ 98 × 100 = 4 ÷ 98 × 100 ≈ 4.1%. The graph exaggerates a small rise.
5. Which representation would you choose to show the temperature of a city every hour for a day? Why?
A line graph, because it shows how the temperature changes over time and where it peaks.
6. Find the mistake: "5 + 3 = 8 × 2 = 16".
= is used as "then". 5 + 3 is not equal to 8 × 2. Write: 5 + 3 = 8, then 8 × 2 = 16.
Common mistakes
- Using = as "next": writing 5 + 3 = 8 × 2 = 16. Each = must join two things of equal value.
- Forgetting units in the final answer: write 10 km, not just 10.
- Drawing a bar chart whose axis does not start at zero, which exaggerates differences.
- Jumping steps so the reader cannot follow; one step per line and a short reason where needed.