Length
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective
Understand standard metric units of length and confidently convert between millimetres (\(\text{mm}\)), centimetres (\(\text{cm}\)), metres (\(\text{m}\)), and kilometres (\(\text{km}\)).
(a) Concrete Scenario: The Tailor at Gikomba Market
When Akinyi buys vibrant kitenge fabric at Gikomba Market in Nairobi, the vendor uses a wooden 1-metre stick. If she needs to hem a collar, she measures in millimetres (\(\text{mm}\)) or centimetres (\(\text{cm}\)). If she travels across town to deliver finished dresses, she measures the journey in kilometres (\(\text{km}\)). All these measurements describe the same fundamental physical quantity: length.
(b) Geometric Insight: The Power of 10
The metric system is a base-10 positional system. Each unit is a multiple of 10 of another:
- \(1\text{ cm} = 10\text{ mm}\) (the width of your fingernail is about \(1\text{ cm}\))
- \(1\text{ m} = 100\text{ cm} = 1000\text{ mm}\) (the height of a classroom desk is close to \(1\text{ m}\))
- \(1\text{ km} = 1000\text{ m}\) (a brisk 12-minute walk)
The Golden Rule of Conversion:
• Going from a smaller unit to a larger unit (e.g., \(\text{cm} \to \text{m}\)) \(\to\) DIVIDE (fewer large units are needed).
• Going from a larger unit to a smaller unit (e.g., \(\text{km} \to \text{m}\)) \(\to\) MULTIPLY (more small units are needed).
Key Formulas
Core Metric Relationships
Base Conversion Equalities:
\[1\text{ cm} = 10\text{ mm}\]\[1\text{ m} = 100\text{ cm} = 1000\text{ mm}\]\[1\text{ km} = 1000\text{ m} = 100{,}000\text{ cm}\]Step-by-Step Conversion Rules:
1. To a Smaller Unit (Multiply):
\[\text{km} \xrightarrow{\times 1000} \text{m} \xrightarrow{\times 100} \text{cm} \xrightarrow{\times 10} \text{mm}\]2. To a Larger Unit (Divide):
\[\text{mm} \xrightarrow{\div 10} \text{cm} \xrightarrow{\div 100} \text{m} \xrightarrow{\div 1000} \text{km}\]Perimeter Formulas with Mixed Units:
Always convert all measurements to the same unit before adding!\[\text{Perimeter of Rectangle} = 2(l + w)\]\[\text{Perimeter of Square} = 4s\]
Worked Examples
Example 1: Single-Step Direct Conversion (Easy)
Problem: A wooden ruler used in a school in Kakamega is \(45\text{ cm}\) long. What is this length in millimetres (\(\text{mm}\))?
- Identify the conversion direction: Centimetres to millimetres (larger unit to smaller unit \(\to\) multiply).
- Identify the factor: \(1\text{ cm} = 10\text{ mm}\).
- Calculate: \[45\text{ cm} \times 10 = 450\text{ mm}\]
- Answer: \(450\text{ mm}\).
Example 2: Combining Mixed Units (Medium)
Problem: Kipchoge runs morning training in Eldoret. He runs \(4.8\text{ km}\) on a trail and an additional \(750\text{ m}\) on the athletics track. What is the total distance he ran in metres?
- Convert all quantities to metres:\[4.8\text{ km} = 4.8 \times 1000\text{ m} = 4800\text{ m}\]
- Add the two distances together:\[4800\text{ m} + 750\text{ m} = 5550\text{ m}\]
- Answer: \(5550\text{ m}\).
Example 3: Multi-Step Word Problem with Cost and Unit Conversion (Hard)
Problem: A farmer in Nakuru wants to fence a rectangular goat pen measuring \(850\text{ cm}\) long and \(6.5\text{ m}\) wide with barbed wire. Fencing wire costs \(\text{KES } 120\) per metre. Calculate the total cost to fence the pen with a single strand of wire.
- Convert dimensions to consistent units (metres):\[\text{Length } l = 850\text{ cm} = \frac{850}{100}\text{ m} = 8.5\text{ m}\]\[\text{Width } w = 6.5\text{ m}\]
- Calculate the perimeter:\[\text{Perimeter} = 2(l + w) = 2(8.5 + 6.5) = 2 \times 15 = 30\text{ m}\]
- Calculate the total cost:\[\text{Total Cost} = 30\text{ m} \times 120\text{ KES/m} = \text{KES } 3600\]
- Answer: \(\text{KES } 3600\) (or \(30\text{ m}\) of fencing).
Common Mistakes
Mistake 1: Multiplying instead of dividing when converting to larger units
Incorrect: Converting \(600\text{ cm}\) to metres as \(600 \times 100 = 60{,}000\text{ m}\).
Correction: \(600\text{ cm} \div 100 = 6\text{ m}\).
Why it feels right: Learners memorize that \(1\text{ m} = 100\text{ cm}\) and see the number 100, assuming they should multiply. Remember: a metre is much longer than a centimetre, so you need fewer metres to cover the same distance.
Mistake 2: Confusing the prefix "kilo-" with 100 instead of 1000
Incorrect: Thinking \(3\text{ km} = 300\text{ m}\).
Correction: \(3\text{ km} = 3000\text{ m}\).
Why it feels right: Since "centi-" is linked to 100, students often confuse the factors. In Greek/metric prefixes, kilo- always means one thousand (\(1000\)), while centi- represents one hundredth (\(\frac{1}{100}\)).
Mistake 3: Adding numbers before converting to a common unit
Incorrect: Adding \(2\text{ m} + 50\text{ cm} = 52\text{ m}\).
Correction: \(2\text{ m} + 0.5\text{ m} = 2.5\text{ m}\) (or \(200\text{ cm} + 50\text{ cm} = 250\text{ cm}\)).
Why it feels right: Directly adding given numerals is tempting, but units represent physical scale and must match before performing arithmetic operations.
Real World
🚌 Matatu Route Planning
A commuter journey from Thika to Nairobi is \(42\text{ km}\). Transportation engineers express local stage distances in metres (e.g., \(800\text{ m}\) between bus stops) but map highway transit in kilometres.
👗 Textile & Tailoring Design
Tailors in Kisumu measure body dimensions like cuffs and waistbands in centimetres (\(\text{cm}\)) and seam allowances in millimetres (\(\text{mm}\)), while buying fabric bolts in full metres (\(\text{m}\)).
🌱 Agriculture & Irrigation
A coffee farmer in Murang'a calculates drip irrigation pipe lengths across shambas measured in metres (\(\text{m}\)), while buying pipe wall thickness specified in precise millimetres (\(\text{mm}\)).
🏃 Athletics Training
Kenyan distance runners pace their training in Iten by breaking down a \(10\text{ km}\) road race into \(400\text{ m}\) track intervals.
Practice