Lessons · Teaching · comparing fractions without a calculator
Three ways to see which fraction is bigger
With the same denominator, compare the numerators. With the same numerator, the smaller denominator is the bigger fraction. Otherwise, cross multiply or convert both to a common denominator, or compare each to a landmark like one half.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
In beta. This lesson was written for Hone and has not yet been checked by a state certification officer. Practice material, not professional advice. What that means.
What it is for
This is where a teacher is asked 'why is a fifth bigger than an eighth when eight is bigger than five?' and either has an answer or does not. The answer is a picture: the same cake cut into eight is a thinner slice than the same cake cut into five.
How to think about it
Try the cheap checks first. Same bottom? Compare tops. Same top? Smaller bottom wins. One above a half and one below? Done. Only when none of those settles it, cross multiply: for a/b against c/d, compare a times d against c times b.
Worked example
3/7 against 5/7: same denominator, so 5/7 is largerThe cheapest check, and it is always worth trying first.
3/5 against 3/8: same numerator, and fifths are bigger pieces, so 3/5 is largerThree big pieces beat three small ones.
5/9 against 4/9 and one half: 5/9 is above a half, 4/9 belowThe landmark settles many pairs with no arithmetic at all.
3/8 against 2/5: cross multiply, 3x5 = 15 against 2x8 = 16, so 2/5 is largerThe 15 belongs to the fraction it came from, which is the half people get wrong.
Your turn
Name the landmark fraction most useful for a quick comparison.
Compare each fraction to one
Solve one, graded on the server
The trap
Cross multiplying and then attaching the products to the wrong fractions. Write each product directly under the fraction whose numerator you used, or the comparison comes out backwards.