Fractions · Lesson 2 of 9
Building and Locating Fractions
“Measure with repeated fractional units and show the resulting quantities on a number line.”
• Build halves, quarters, eighths, thirds, and sixths using paper strips. • Read a fraction as a number of copies of a fractional unit. • Explain the jobs of the numerator and denominator. • Place fractions between zero and one on a number line. • Explain why there are infinitely many fractions between zero and one.
Make your own measuring unit
A whole ruler measures lengths in centimetres, but a paper strip can become a ruler of your own. Call the complete strip one unit long. Fold it so its ends meet, press a crease, and open it. The crease divides the strip into two equal lengths, so each new length is 1/2 unit. Measuring with these smaller units lets us describe lengths that are not whole numbers.
Fold the already folded strip in half once more. When opened, it has four equal lengths, each 1/4 unit. Fold again and you obtain eight equal lengths, each 1/8 unit. Every extra halving doubles the number of pieces and halves their individual length. Keep the full strip as your reference whole throughout the activity.
You can make thirds by bringing the strip into three equal lengths and checking that they match. Halve each third to make six equal lengths. This gives a practical connection between thirds and sixths: each third contains two sixths. With circles, squares, or triangles, the same requirement remains: the intended fractional parts must contain equal amounts of the chosen whole.
Count copies of a fractional unit
Once you have chosen a fractional unit, a longer quantity can be measured by counting copies of it. Two quarter-lengths cover 2/4 of the strip, three cover 3/4, and four cover 4/4, which is the full strip. The fractional unit stays one quarter; only the number of copies changes. This distinction is central to understanding a fraction.
The top number of a fraction. It counts how many copies of the fractional unit are being taken.
The bottom number of a fraction. It tells how many equal copies of the fractional unit make one whole.
In 5/6, the denominator 6 makes the fractional unit 1/6, and the numerator 5 counts five such units. We may read the fraction as “five sixths”, “five upon six”, or “five times one sixth”. The last reading makes both jobs especially clear. A fraction can count more than enough pieces to make a whole; we do not have to stop counting when one whole is reached.
| Eighths counted | Repeated-unit description | Fraction of the whole |
|---|---|---|
| 2 | Two times 1/8 | 2/8 |
| 4 | Four times 1/8 | 4/8 |
| 6 | Six times 1/8 | 6/8 |
| 8 | Eight times 1/8 | 8/8 = 1 |
Problem
A strip is divided into four equal lengths. What length is covered by three of them?
- 1.Each length is 1/4 unit.
- 2.The three lengths give 1/4 + 1/4 + 1/4.
- 3.This is three times 1/4, so the length is 3/4 unit.
Problem
Describe quantities made from one through seven half-rotis.
- 1.One half gives 1/2; two halves give 2/2, which is one whole.
- 2.Continue counting: three halves give 3/2, four give 4/2, and five give 5/2.
- 3.Six and seven halves give 6/2 and 7/2. Each new half increases the count by one, while the denominator stays 2.
| Number of half-rotis | Addition of halves | Quantity |
|---|---|---|
| 1 | 1/2 | 1/2 |
| 2 | 1/2 + 1/2 | 2/2 = 1 |
| 3 | 1/2 + 1/2 + 1/2 | 3/2 |
| 4 | Four copies of 1/2 | 4/2 = 2 |
| 5 | Five copies of 1/2 | 5/2 |
| 6 | Six copies of 1/2 | 6/2 = 3 |
| 7 | Seven copies of 1/2 | 7/2 |
Problem
Draw or describe nine pieces, each one quarter of a roti.
- 1.Draw two whole-sized circles, each split into four equal pieces, then a third circle split into four.
- 2.Shade all eight quarter-pieces in the first two circles and one piece in the third.
- 3.The total is nine times 1/4, or 9/4 of a roti. The picture contains two complete rotis and one extra quarter.
Three sixths means three copies of 1/6, not three copies of 1/3. The numerator counts pieces; the denominator identifies their size. Keep the same whole and the same fractional unit when using repeated addition.
A number line measures distance from zero
On a number line, the distance from 0 to 1 represents one unit. To locate thirds, divide that distance into three equal intervals. The first division point is 1/3 and the second is 2/3. A fraction names both a point and the length from zero to that point. Counting intervals rather than division ticks prevents a common mistake.
To locate a fraction with denominator 5, split the unit interval into five equal intervals and count from zero. The points 2/5 and 4/5 lie after two and four intervals respectively. For eighths, the same construction gives 1/8, 2/8, 3/8, and so on up to 8/8 = 1. The distance between neighbouring points is always one of the chosen fractional units.
Problem
Locate 1/10, 3/10, and 4/5 between 0 and 1.
- 1.Divide the unit interval into ten equal lengths.
- 2.Count one length from 0 for 1/10 and three lengths for 3/10.
- 3.Each fifth covers two tenth-lengths. Therefore four fifths covers eight tenth-lengths, so 4/5 lies at the eighth division point.
There is always room for smaller fractions
A drawing has limited space, but the number line does not have only the fractions we have drawn. We could divide the unit interval into 10, 100, 1000, or still more equal parts. This creates more fraction positions. In particular, 1/2, 1/3, 1/4, and the continuing list of unit fractions are all distinct and all lie between 0 and 1.
That list never ends: whichever denominator you choose, a larger one is possible. So there are infinitely many fractions between 0 and 1. “Infinitely many” means there is no last member or finite total. A diagram with ten divisions is a convenient measuring picture, not a complete list of all fraction positions.
Quiz
In 5/6, what is the fractional unit?
Which description matches 3/8?
After three successive halvings, one full strip contains…
To place 2/5, how many equal intervals should 0 to 1 contain?
A line divided into tenths shows 4/5 at which division point after zero?
How many fractions lie strictly between 0 and 1?
Practice Problems
- Fold a paper strip into halves, quarters, and eighths. Record 2/8, 4/8, 6/8, and 8/8 as repeated additions of 1/8.
- Make thirds with a strip and then make sixths. Explain how the second set of folds changes each third.
- Continue the half-roti table for six and seven halves. Create a quarter-roti table for one through nine quarters.
- Draw pictures and addition statements for five quarter-rotis and nine quarter-rotis.
- Draw equal-part pictures for 1/3, 1/5, 1/6, and 1/8. Write how many copies of each make a whole.
- Place 1/10, 3/10, and 4/5 on one number line. Choose five other fractions between 0 and 1 and locate them.
- Explain why a number line with eight divisions between 0 and 1 does not show every possible fraction.
Key Takeaways
• A fraction a/b counts a copies of the fractional unit 1/b. • The numerator counts pieces; the denominator specifies how they divide one whole. • Folding a unit strip makes fractional lengths visible. • A fraction has a point on a number line and a length from zero to that point. • There are infinitely many fractions between zero and one.