Fractions: Parts, Equivalence and Comparing

1. Fractions

Maths · Parts of a whole, made visible

Diagram: Three bars of the same length split into halves, thirds and quarters, with one part of each shaded, beside a circle split into four equal parts with three shaded.

A fraction describes equal parts of a whole. In this lesson you build fractions, compare them and use them to find parts of amounts.

2. By the end of this lesson you can

• Explain what the numerator and denominator mean
• Show a fraction as part of a shape and on a number line
• Find equivalent fractions and simplify
• Compare and order fractions
• Find a fraction of an amount

Keep these five goals in mind. The quiz at the end checks each one.

3. A fraction is equal parts of a whole

• Start with one whole
• Split it into equal parts
• Count the parts you are talking about
• Three of four equal parts is three quarters

Diagram: A bar labelled one whole is split into four equal parts, and three of the parts are shaded to show three quarters.

The parts must be equal. A bar cut into four pieces of different sizes is not cut into quarters, even though there are four pieces.

4. Numerator and denominator

• The denominator (bottom) is how many equal parts the whole has
• The numerator (top) is how many of those parts we mean
• 3/4 means 3 of 4 equal parts

Diagram: An interactive fraction builder. Buttons change the numerator and the denominator, and a bar and a circle split into that many equal parts, with the numerator's worth shaded. It starts at three quarters.

Change the numbers and watch the shapes. The denominator changes how many parts the whole is cut into; the numerator changes how many are shaded.

5. Halves, thirds and quarters

• Same whole, more parts: each part gets smaller
• One half is bigger than one third
• One third is bigger than one quarter

Diagram: Three bars of the same length. The first is split into two equal parts, the second into three and the third into four, with one part of each shaded, so one half, one third and one quarter can be compared directly.

With the same whole, a bigger denominator means smaller parts. That is why one quarter is less than one third, even though 4 is bigger than 3.

6. Fractions on a number line

• Split the line from 0 to 1 into equal steps
• Count steps from 0 to place the fraction
• Two quarters and one half are the same point

Diagram: A number line from 0 to 1 split into quarters, labelled 0, one quarter, two quarters, three quarters and 1. Three equal jumps from 0 land on three quarters, and a note shows that two quarters is the same point as one half.

A fraction is also a number with its own place on the number line, between whole numbers. Equal fractions sit at exactly the same point.

7. Equivalent fractions

• 1/2, 2/4 and 4/8 shade the same amount
• Multiply top and bottom by the same number to find one
• Divide top and bottom by the same number to simplify

Diagram: Three bars of the same length showing one half, two quarters and four eighths. The shaded part ends at the same place on every bar, showing the three fractions are equal.

Splitting every part in two doubles both the number of parts and the number shaded, so the amount stays the same. That is why you multiply the top and the bottom by the same number.

8. Write 12/18 in its simplest form

• Find a number that divides both 12 and 18: the biggest is 6
• 12 ÷ 6 = 2 and 18 ÷ 6 = 3
• So 12/18 = 2/3
• Check: 2 and 3 have no common factor except 1

A fraction is in its simplest form when the top and bottom have no common factor except 1. Dividing by the highest common factor gets there in one step.

9. Comparing fractions

• Hard to see: is 2/3 bigger than 3/5?
• Write both with the same denominator: 15
• 2/3 = 10/15 and 3/5 = 9/15, so 2/3 is bigger

Diagram: Two bars show two thirds and three fifths, which look almost the same length. Both are then split into fifteenths: two thirds becomes ten fifteenths and three fifths becomes nine fifteenths, so two thirds is bigger.

When the denominators are the same, compare the numerators. A common denominator turns a hard comparison into an easy one.

10. Two shortcuts

• Compare the numerators
• 3/8 is less than 5/8
• The parts are the same size

Same numerator:

• The bigger denominator is smaller
• 2/5 is more than 2/7
• Sevenths are smaller than fifths

If the bottoms match, the bigger top wins. If the tops match, the smaller bottom wins, because its parts are bigger.

11. Order 3/4, 2/3 and 5/6

• The lowest common denominator of 4, 3 and 6 is 12
• 3/4 = 9/12, 2/3 = 8/12, 5/6 = 10/12
• Order the numerators: 8, 9, 10
• Smallest first: 2/3, 3/4, 5/6

Rewrite every fraction with the same denominator, order the numerators, then write the answer using the original fractions.

12. A fraction of an amount

• 3/4 of 20: divide 20 into 4 equal groups
• Each group is 5
• Three groups make 15
• Divide by the denominator, multiply by the numerator

Diagram: Twenty counters are arranged into four equal groups of five. Three of the groups are shaded, showing that three quarters of 20 is 15.

To find a fraction of an amount, divide by the denominator to find one part, then multiply by the numerator.

13. Common mix-ups

• A bigger denominator means smaller parts
• Parts must be equal
• Equivalent fractions are the same amount
• Add fractions only once the denominators match

Diagram: Four cards, each with a common mistake on the front and the correct idea on the back: a bigger denominator gives smaller parts; the parts must be equal; equivalent fractions show the same amount; one half plus one third is five sixths, not two fifths.

Turn each card over to check the idea. Each of these mistakes is common in tests, so it pays to know why it is wrong.

14. Quick check

Which of these fractions is the largest?

• 3/5
• 5/8
• 7/10
• 2/3

15. Key takeaways

• A fraction is a number of equal parts of a whole
• The denominator counts the parts; the numerator counts the ones we mean
• Equivalent fractions show the same amount
• Compare fractions by giving them the same denominator
• Fraction of an amount: divide by the bottom, multiply by the top

Use these five points to check yourself against the goals from the start of the lesson.