Base and derived quantities: why five hands give five lengths

The problem

Activity 1.1 (p. 7): one student measures the writing board with hand spans, then four or five others do the same. The answers do not match. Why, and how do we stop the confusion?

Hands, arms, feet and steps differ in size from person to person. So the same board gets different numbers. We need a standard that gives everyone the same result. That standard is a unit.

A measurement compares an unknown quantity with a widely accepted standard. It always has two parts, a number and a unit. Without the unit, the number means nothing.

From all measurable features, scientists chose a few to play the key role. These are base quantities. Every quantity described in terms of one or more of them is a derived quantity.

Base quantities are a chosen set; every derived quantity is built from them. A measurement compares with a standard and is always a number plus a unit.

Key terms

Base quantity

One of the few quantities scientists selected to play the key role.

Derived quantity

A quantity described in terms of one or more base quantities.

Measurement

Comparing an unknown quantity with a widely accepted standard.

Unit

The agreed standard, so that anyone's measurement gives the same result.

Short questions with model answers

1. Q1. Is speed a base or a derived quantity? Show why.

   • speed = distance ÷ time
   • Distance is a length, and length and time are both base quantities.

   Derived: built from length and time.

2. Q2. A student writes: 'The table is 2 long.' What is wrong?

   • The number 2 has no unit.
   • Write it with its unit, for example 2 m.

   The unit is missing; a measurement is a number and a unit.

3. Q3. Short question 1.3 (p. 25): Why do we need a standard unit for measurements?

   • Body measures such as hands and steps differ from person to person.
   • A standard unit gives the same result whoever measures, and lets countries trade and share science.

   So everyone's measurement of the same thing agrees, at home and across countries.

Common mistakes

• ✗ Writing a measurement without its unit.

  ✓ Always write number + unit: 2 m, not 2.

• ✗ Calling density or speed a base quantity because it is common.

  ✓ Check whether it is built from other quantities: density needs mass and volume, so it is derived.

• ✗ Thinking hand spans are a fine unit if you measure carefully.

  ✓ Care does not fix it; hands differ in size, so there is no common standard.

MCQs

1. 1. A measurement consists of:

   1. (a) A number only
   2. (b) A unit only
   3. (c) A number and an instrument
   4. (d) A number and a unit
   Show answer

   (d) The book names these two parts (p. 8).

2. 2. Which is a base quantity?

   1. (a) Speed
   2. (b) Time
   3. (c) Density
   4. (d) Area
   Show answer

   (b) Time is not built from another quantity; the other three are.

3. 3. Hand-span measurements of a board differ between students because:

   1. (a) Hands differ in size
   2. (b) The board changes length
   3. (c) Students cannot count
   4. (d) The board is a derived quantity
   Show answer

   (a) Different hand sizes give different numbers, which is why a unit is needed (p. 8).

4. 4. Density is described in terms of:

   1. (a) Distance and time
   2. (b) Length and temperature
   3. (c) Mass and volume
   4. (d) Speed and mass
   Show answer

   (c) Your book's own example (p. 7).

5. 5. A measurement is a comparison with:

   1. (a) Your own hand
   2. (b) A widely accepted standard
   3. (c) A friend's guess
   4. (d) The previous reading
   Show answer

   (b) That is the book's definition of measurement (p. 7).

Quick revision

• Base quantities are a chosen set; derived quantities are built from them.
• A measurement compares an unknown quantity with a standard: number + unit.
• A common unit removes the confusion of hands and steps, and lets countries share results.