Understanding fractions (age 9+)
Understand a fraction a/b with a > 1 as a sum of fractions 1/b (e.g. 3/5 = 1/5 + 1/5 + 1/5)
What a learner can do afterwards
- Express 5/8 as a sum of five copies of 1/8
- Show on a number line how 4/3 is built by iterating 1/3 four times
- Explain why 7/4 = 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4
1 · Read
You already know that 1/5 means one piece when a whole is cut into 5 equal parts. Today you will see that a fraction like 3/5 is just several of those pieces added up. 3/5 means 1/5 + 1/5 + 1/5.
Look at a number line marked in thirds. Start at 0 and take four hops, each hop the size of 1/3. After four hops you land past one whole, at 4/3.
Any fraction a/b is just a copies of the piece 1/b, added together. The top number tells you how many pieces to add; the bottom number tells you the size of each piece.
A fraction a/b means you add the piece 1/b to itself a times.
More to read
Decomposing fractions review
Some fractions are built from tiny equal pieces. The fraction 5/8 means five pieces of size 1/8 joined together. Each 1/8 is called a unit fraction because it has 1 on top. Count the unit pieces and you know the fraction. You can break any fraction apart this way. The fraction 4/3 is four copies of 1/3, and 7/4 is seven copies of 1/4. Seeing fractions as piles of unit pieces makes adding them much easier.
Equivalent fractions review
Two fractions can look different but name the same amount. The fractions 1/2 and 4/8 match because four eighths cover exactly one half. Pictures like bars and circles show this clearly when you shade the same length in different pieces. To test two fractions, draw them the same size and compare the shaded part. If the shaded parts match exactly, the fractions are equivalent. This trick helps you compare fractions and put them in order.
Mixed numbers and improper fractions review
Sometimes the top of a fraction is bigger than the bottom, like 4/3 or 7/4. That just means you have more than one whole. Four thirds is one whole plus one more third, so we can also write it as 1 and 1/3. A number like 1 and 1/3 is called a mixed number. You can change an improper fraction into a mixed number by grouping the pieces into wholes. Seven quarters make one whole (4/4) with 3/4 left over, so 7/4 equals 1 and 3/4. Going back the other way works too.
Visualizing equivalent fractions review
Drawings make fractions click. A bar split into 8 equal parts with 5 shaded shows 5/8 at a glance. A circle cut into 4 slices with all 4 shaded plus 3 more shaded in a second circle shows 7/4. Number lines work the same way. Mark jumps of 1/3 from zero: after 3 jumps you reach 1, and after 4 jumps you reach 4/3.
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2 · Watch
به انگلیسیTake it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.