Relational thinking: linking methods in a rainwater tank problem
What this lesson covers
- In relational thinking the result of one process becomes the input to the next.
- A connected chain is only as good as its weakest step, so every link must be correct.
What relational thinking means
- Separate calculations are Achieved.
- For Merit, one answer becomes the next input.
- Every link in the chain must be correct.
| Step | Find | Which then |
|---|---|---|
| 1 | roof area | feeds the rain volume |
| 2 | rain volume in litres | compared with the tank |
| 3 | tank volume | feeds the rain needed |
| 4 | rain to fill it | and days it lasts |
The problem: a rainwater tank
- A rural home runs on roof-fed rainwater.
- How much does a storm collect?
- How long does a full tank last?
| Details | |
|---|---|
| Roof | 12 m by 8 m, seen from above |
| Tank | cylinder, radius 1.5 m, height 2 m |
| Storm | 25 mm of rain |
| Use | 180 litres a day |
Link 1: roof area to water
- Roof area from above: 96 m².
- Change 25 mm into metres first.
- Water = area × depth = 2.4 m³.
| Working | Result | |
|---|---|---|
| Roof area | 12 × 8 | 96 m² |
| Rain depth | 25 mm ÷ 1000 | 0.025 m |
| Water | 96 × 0.025 | 2.4 m³ |
The links are unit conversions
- Millimetres to metres: divide by 1000.
- Cubic metres to litres: multiply by 1000.
- Get these wrong and the chain breaks.
| Equals | So | |
|---|---|---|
| 1 m | 1000 mm | divide mm by 1000 |
| 1 m³ | 1000 litres | multiply m³ by 1000 |
| 2.4 m³ | 2400 litres |
Link 2: the tank's volume
- The tank is a cylinder: V = πr²h.
- π × 1.5² × 2 = 14.137 m³.
- That is 14 137 litres.
| Working | ||
|---|---|---|
| Formula | V = πr²h | |
| Substitute | π × 1.5² × 2 | |
| Volume | 14.137 m³ | 14 137 litres |
Link 3: storm against tank
- Two answers, now in the same units.
- 2400 out of 14 137 litres.
- One storm fills about 17% of the tank.
| Value | ||
|---|---|---|
| Storm gives | 2400 litres | |
| Tank holds | 14 137 litres | |
| Fraction | 2400 ÷ 14 137 | about 17% |
Link 4: an equation for the rain needed
- Let d be the rain depth in metres.
- Water collected: 96d = tank volume.
- d = 0.147 m, so 147 mm of rain.
| Working | ||
|---|---|---|
| Water from d m | 96d m³ | |
| Equation | 96d = 14.137 | |
| Solve | d = 0.147 m | 147 mm |
Link 5: how long it lasts
- A rate: 180 litres used each day.
- 14 137 ÷ 180 = 78.5.
- Round down: 78 full days.
| Value | ||
|---|---|---|
| Full tank | 14 137 litres | |
| Daily use | 180 litres | |
| Days | 14 137 ÷ 180 = 78.5 | 78 full days |
The whole chain, in one sentence
- Area fed volume, volume fed the comparison.
- The tank's volume fed the equation and the days.
- Five links, each one explained.
Carry accuracy through the chain
- Round early and every later step inherits the error.
- Keep full values on the calculator.
- Round once, at the end, to suit the context.
Say the connection out loud
- "Using the tank volume of 14 137 litres..."
- Each step names the result it uses.
- That sentence is the evidence of the link.
What makes this Merit
- The processes are connected, not separate.
- Each part of every connection is correct.
- Excellence goes further, into assumptions.
What to take away
- Plan the chain, link it with units, round at the end.
| Relational | one answer feeds the next |
| Plan the chain | before calculating |
| Links | unit conversions join the steps |
| mm to m | divide by 1000 |
| m³ to litres | multiply by 1000 |
| Keep accuracy | round only at the end |
| Explain | each connection in words |
Your turn: four linked questions
- Each answer feeds the next question.
- Pause here and work through the chain.
- Watch the units at every link.
| Question | |
|---|---|
| Home | Roof 10 m by 9 m. Tank: radius 1.2 m, height 2.5 m. Uses 150 L a day. |
| 1 | How many litres does a 30 mm storm collect? |
| 2 | How many litres does the tank hold? |
| 3 | How many full days does a full tank last? |
| 4 | How much rain, in mm, fills the empty tank? |
Show answer
1. 10 × 9 = 90 m²; 30 mm = 0.03 m; 90 × 0.03 = 2.7 m³ = 2700 litres. 2. π × 1.2² × 2.5 = 11.310 m³ = 11 310 litres. 3. 11 310 ÷ 150 = 75.4, so 75 full days. 4. 90d = 11.310, so d = 0.126 m, which is 126 mm.
The answers
- 90 × 0.03 = 2.7 m³: 2700 litres.
- π × 1.2² × 2.5 = 11.310 m³: 11 310 litres.
- 11 310 ÷ 150 = 75.4: 75 full days.
- 90d = 11.310, d = 0.126 m: 126 mm.
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