See the problem
Many problems become much easier once you draw them. A diagram turns words into a picture that shows how the parts relate. A mathematical model goes further: it describes a real situation using maths (an equation, a table, a graph or a diagram).
Types of diagrams
| Diagram | Best for | Example |
|---|---|---|
| Bar model | parts and wholes, comparisons, fractions, ratios | "Ana has 3 times as much as Ben" |
| Number line | changes, distances, time, integers | temperature rising then falling |
| Sketch / picture | shapes, distances, positions | a ladder against a wall |
| Table | patterns, organized data | growing patterns |
| Tree diagram | combinations, probability | outfits, coin flips |
| Venn diagram | overlapping groups | students in two clubs |
| Graph | relationships between variables | cost vs distance |
Bar models
Problem: Ana and Ben share $48. Ana gets 3 times as much as Ben. How much does each get?
Draw Ben as 1 box and Ana as 3 boxes of the same size. That’s 4 boxes for $48.
| Ben | ■ | |||
|---|---|---|---|---|
| Ana | ■ | ■ | ■ |
One box = 48 ÷ 4 = $12. Ben $12, Ana $36.
Number lines
Problem: The temperature was −4 °C, rose 11 degrees, then fell 6. Final temperature?
Start at −4, jump right 11 to 7, then left 6 to 1 °C.
Venn diagrams
Problem: 30 students: 18 play soccer, 15 play basketball, 7 play both. How many play neither?
- Only soccer: 18 − 7 = 11
- Only basketball: 15 − 7 = 8
- Both: 7
- Total in circles: 26
- Neither: 30 − 26 = 4
Building a model
To model a real situation:
- Identify the quantities and how they relate.
- Make assumptions to simplify (for example, a car drives at a constant speed).
- Choose a representation: diagram, table, equation or graph.
- Use the model to solve the problem.
- Check whether the answer makes sense in the real situation.
Example: A taxi costs $3.50 plus $1.80 per km. Model: C = 1.80d + 3.50. A 12 km trip costs 1.80 × 12 + 3.50 = $25.10.
Worked examples
Example 1: A rope is cut into two pieces. One is 5 m longer than the other. Total 21 m. Lengths?
Bar model: two equal bars plus 5 m = 21 → equal parts 8 m. 8 m and 13 m.
Example 2: A snail climbs 3 m up a 10 m well each day and slides back 2 m each night. How many days to escape?
Diagram: after day 7 it’s at 7 m (net 1 per day). On day 8 it climbs 3 m to 10 m and escapes → 8 days.
Example 3: In a class of 25, 12 like cats, 15 like dogs, 4 like neither. How many like both?
21 like at least one. 12 + 15 − 21 = 6.
Common mistakes
- Drawing bars of different sizes for equal parts.
- Forgetting the overlap in Venn diagrams.
- Making unrealistic assumptions in a model.
Try it yourself
- Two numbers: one is 4 times the other, and they add to 35. Find them.
- A diver at −12 m rises 7 m then dives 3 m. Depth now?
- 40 people: 22 like tea, 25 like coffee, 10 like both. How many like neither?
Answers: 1) 7 and 28 2) −8 m 3) 3
Quick recap
- A diagram shows relationships at a glance.
- Use bar models, number lines, Venn diagrams and more.
- A mathematical model represents a real situation; state your assumptions.