Teach fractions in three sessions, in this order: the whole divided into equal parts, the number line with equivalent fractions, and finally comparing and simple operations. Each session lasts 30 to 40 minutes and uses paper strips of the same length. This guide is an editorial proposal: adjust the pace to what each learner’s answers show, not to the calendar.
| If the learner… | Start with | Signal to move on |
|---|---|---|
| Does not split into equal parts | Session 1: the whole and equal parts | Folds a strip into two and into four parts that match when stacked |
| Reads 3/4 but cannot place it | Session 2: number line and equivalent fractions | Marks 3/4 and 6/8 at the same point |
| Says 1/3 is greater than 1/2 | Session 3: comparing with the same whole | Explains why a third is a smaller piece |
The table works as a quick diagnostic: one question is enough to decide which session to begin with. The later sessions assume the earlier one already works.
Session 1: the whole and equal parts
Observable goal: the learner splits a strip into equal parts and says what each number in 3/4 counts. Materials: four paper strips of the same length, a ruler and colored pencils.
- Hand out two identical strips. Divide one into four parts and shade three; ask the learner to do the same with the other and stack them.
- Ask: “What stays the same and what changes?” The expected answer is that the whole is the same and each part covers the same length.
- Name the fraction: the 4 says how many equal parts the strip was divided into, and the 3 says how many are counted.
- Repeat with a strip folded into three parts, so the learner sees that equal parts can also be made without measuring.
Mistake to watch for: parts of different sizes. A drawing with four uneven pieces does not show a quarter, even though there are four. Evidence of learning: the learner points to the whole, checks that the parts match and counts the shaded parts. The English Wikipedia article on fractions says a fraction represents a part of a whole or, more generally, any number of equal parts, written as a numerator over a denominator (accessed October 17, 2026).
Session 2: the number line and equivalent fractions
Observable goal: the learner places proper and improper fractions on a number line and explains why two ways of writing a fraction name the same point. Materials: a long strip marked from 0 to 2 with a marker, and the strips from the previous session.
Mark 0 and 1 on the line, divide that stretch into four equal intervals and place 3/4. Then extend the line to 2 and ask where 5/4 falls. This step shows that a fraction is a number with a place on the line, not only a shaded region, and that it can go past 1.
For equivalent fractions, cut each quarter of the strip into two parts without touching the shaded area: there are now six eighths and the stretch is still the same. Ask what changed, which was the number and size of the pieces, and what stayed the same, which was the length covered. You can check it afterward with the equivalent fractions tool and extend the idea with the fraction number line tool.
Mistake to watch for: placing 5/4 between 0 and 1. If that happens, build 4/4 first, which completes the whole, and add one quarter. Evidence: the learner places 3/4 and 5/4 and justifies 3/4 = 6/8 with the strip.
Session 3: comparing with the same whole
Observable goal: the learner compares two fractions of the same whole and tells a fraction apart from the amount it stands for. Materials: two strips of different lengths, 12 cm and 6 cm, and a ruler.
This session’s contrast task separates two ideas that are easily confused. A fraction compares a part with its whole, and the length also depends on the size of the whole.
Worked example: a third of 12 cm versus half of 6 cm
One strip is 12 cm long and another is 6 cm long. Which is greater, a third of the first or half of the second?
- Find a third of 12 cm. Divide 12 by 3 to get 4 cm.
- Find half of 6 cm. Divide 6 by 2 to get 3 cm.
- Compare the lengths. 4 cm is greater than 3 cm.
- Compare the fractions. Of the same whole, 1/2 is greater than 1/3, because the half splits the whole into fewer pieces.
Result: a third of 12 cm is 4 cm and half of 6 cm is 3 cm. Here the greater fraction gives the shorter length.
Check: 3 × 4 cm = 12 cm and 2 × 3 cm = 6 cm, so each part returns to its whole.
Have the learners check it by folding the strips. Then ask what would have to change for the two lengths to match: the wholes just have to be equal. That gives the two comparison rules used later: same whole to compare fractions, and same denominator to compare numerators. If the learner is doing well, introduce a simple sum such as 1/4 + 1/2, which is solved by turning the half into 2/4.
The compare fractions calculator lets the learner verify their own answer after giving it. So that comparing does not depend on one particular drawing, the guide on the same whole when comparing shows why the wholes must be equal.
Questions that reveal understanding
These questions work in any session and are more useful than a finished worksheet. Note which answer fails, because it shows which model is missing.
| Question | Expected answer | If it fails |
|---|---|---|
| What is the whole in this drawing? | The complete strip, the same for both figures | Line up the strips before comparing |
| Can a fraction be worth more than 1? | Yes, 5/4 comes after 1 on the line | Build 4/4 and add a quarter |
| Why do 2/4 and 1/2 sit at the same point? | Each half contains two quarters | Fold the strip and stack the pieces |
A wrong answer shows which model is missing and does not prove the learner does not understand. The questions in the table cover the whole, the number line and equivalence, the three ideas of the sessions.
Practice before session 3
- Draw an 8 cm strip and shade 3/8. How many centimeters long is the shaded part?
- Which is greater, half of 10 cm or a fifth of 20 cm?
- Place 7/4 on a number line from 0 to 2.
See the answers
- 3 cm, because each eighth is 1 cm.
- Half of 10 cm is 5 cm and a fifth of 20 cm is 4 cm, so the half is greater.
- It falls three quarters past 1, that is, one quarter before 2.
Next steps
Once the three sessions work, the path continues with the meanings of a fraction, which separates the fraction as part of a whole, as a quotient and as an operator. To widen the models, the guide on drawing fractions compares areas, sets and number lines. If you want ready-made tasks, the fraction classroom activities gather ideas with materials and timing, and the teacher resources hub collects the rest. For practice with immediate feedback, use fraction practice.
Sources
- English Wikipedia, article Fraction, introduction (accessed October 17, 2026).
- Common Core State Standards Initiative, Grade 3, Number and Operations: Fractions: 3.NF.A.1 to 3.NF.A.3 on equal parts of a whole, fractions on a number line and equivalent fractions (accessed October 17, 2026).