A fraction as a measure, and the irreducible fraction — 11–13
A fraction measures a piece against a unit, and among all its equivalent names the irreducible one is the shortest.
What the two numbers say
A fraction is a measure. Take a unit that can be cut anywhere, such as a strip, a length or an area. The denominator tells into how many equal parts the unit is divided; the numerator tells how many of those parts you take. So 3/4 of a strip is 3 of the 4 equal parts. Change the unit and the measure changes: it is always the measure of something, compared with a unit.
The unit decides the fraction
Think of the pieces of a tangram. A large triangle covers the same area as 4 small triangles. With the small triangle as the unit, the large one measures 4; with the large triangle as the unit, the small one measures 1/4. Same pieces, different unit, different numbers. And one measure has many names (2/4, 3/6 and 4/8 are all 1/2), so it helps to agree on one: the irreducible fraction, the shortest name.
Shortening 12/18 all the way
A strip of length 1 is cut into 18 equal parts and 12 are painted: the painted part measures 12/18. Step 1: the divisors of 12 are 1, 2, 3, 4, 6, 12 and those of 18 are 1, 2, 3, 6, 9, 18. Step 2: the greatest common one is 6. Step 3: 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 12/18 = 2/3. Step 4: 2 and 3 share only the divisor 1, so 2/3 cannot be shortened: it is irreducible. The same way, 8/20 = 2/5, dividing by 4.
Stopping one step too early
Many people divide once, see smaller numbers and stop: 24/36 divided by 2 gives 12/18, and it looks finished. But 12 and 18 can still both be divided by 2, by 3 and by 6, so it is not irreducible. Do not ask “did I divide?”. Ask “do the two numbers still share a divisor other than 1?”. Dividing by the greatest common divisor, 12, gets there in one go: 24 ÷ 12 = 2 and 36 ÷ 12 = 3, so 24/36 = 2/3.
On rulers and ribbons
A ruler is a fraction machine: the unit is 1 cm, cut into 10 equal parts (millimetres). A gap of 7 mm is 7/10 of a centimetre, that is 0.7 cm, and 5 mm is 5/10, which is 1/2 cm. And 3/4 of a ribbon that is 2 m long is 1.5 m. Whenever you say a fraction, ask yourself: of what unit?
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