3B · Chapter 9 · Lesson 2
Fractions on a Number Line
Use a chosen unit interval, equal spacing, and endpoints to place fractions.
Today's mathematical question
Between 0 and 1, a line is split into 4 equal intervals. What fraction is at the 3rd tick after 0?
Learning target
Use a chosen unit interval, equal spacing, and endpoints to place fractions.
Before starting: Prior lesson: Fractions of a Whole. Try the warm-up independently.
Prerequisite · 3 questions
Check earlier skills first
These are earlier ideas you need for today; revisit them if they are difficult.
- P1
A same-sized whole is split into 6 equal parts and 4 parts are shaded. What fraction is shaded?
Reveal explanation
Answer: 2/3
The denominator counts all 6 equal parts; the numerator counts the 4 shaded parts. In simplest form this is 2/3.
Each full bar represents the same-sized whole; shaded parts show the fraction. - P2
A same-sized whole is split into 12 equal parts and 10 parts are shaded. What fraction is shaded?
Reveal explanation
Answer: 5/6
The denominator counts all 12 equal parts; the numerator counts the 10 shaded parts. In simplest form this is 5/6.
Each full bar represents the same-sized whole; shaded parts show the fraction. - P3
A same-sized whole is split into 3 equal parts and 2 parts are shaded. What fraction is shaded?
Reveal explanation
Answer: 2/3
The denominator counts all 3 equal parts; the numerator counts the 2 shaded parts. In simplest form this is 2/3.
Each full bar represents the same-sized whole; shaded parts show the fraction.
Learn and understand
Fractions are numbers, so they have places on a number line. First identify the length of one whole from 0 to 1. If the whole interval is divided into eight equal lengths, each step equals one eighth. Three steps from zero reaches three eighths, even if the printed line has been stretched. Draw a line from 0 to 1 and split it into four equal intervals. The third step is 3/4. Count spaces between ticks, not the tick marks themselves.
Represent it three ways
- Concrete: Build equal pieces, measures, counters or shapes with real objects when useful.
- Pictorial: Mark 0 and 1 before drawing any subdivision. Divide the interval evenly and count intervals, not just interior marks.
- Abstract: Write the matching number sentence, calculation, comparison or mathematical description. Explain each number's unit.
Important vocabulary
- number line
- ordered positions representing numbers
- unit interval
- distance from 0 to 1
- unit fraction
- one equal part of a whole
Fully worked examples and checks
Try each new understanding check before revealing the explanation.
Worked example 1
Explore, calculate, verify
Between 0 and 1, a line is split into 4 equal intervals. What fraction is at the 3rd tick after 0?
- Identify the whole, the units, and what the question asks.
- Each interval represents 1/4. Moving 3 intervals from zero reaches 3/4, or 3/4 in simplest form.
- Check that the answer uses the right units and makes sense.
Answer: 3/4
Check your understanding 1
Between 0 and 1, a line is split into 10 equal intervals. What fraction is at the 7th tick after 0?
Reveal answer and explanation
7/10 — Each interval represents 1/10. Moving 7 intervals from zero reaches 7/10, or 7/10 in simplest form.
Worked example 2
Explore, calculate, verify
Between 0 and 1, a line is split into 8 equal intervals. What fraction is at the 4th tick after 0?
- Identify the whole, the units, and what the question asks.
- Each interval represents 1/8. Moving 4 intervals from zero reaches 4/8, or 1/2 in simplest form.
- Check that the answer uses the right units and makes sense.
Answer: 1/2
Check your understanding 2
Between 0 and 1, a line is split into 5 equal intervals. What fraction is at the 4th tick after 0?
Reveal answer and explanation
4/5 — Each interval represents 1/5. Moving 4 intervals from zero reaches 4/5, or 4/5 in simplest form.
Worked example 3
Explore, calculate, verify
Between 0 and 1, a line is split into 4 equal intervals. What fraction is at the 2nd tick after 0?
- Identify the whole, the units, and what the question asks.
- Each interval represents 1/4. Moving 2 intervals from zero reaches 2/4, or 1/2 in simplest form.
- Check that the answer uses the right units and makes sense.
Answer: 1/2
Check your understanding 3
Between 0 and 1, a line is split into 12 equal intervals. What fraction is at the 2nd tick after 0?
Reveal answer and explanation
1/6 — Each interval represents 1/12. Moving 2 intervals from zero reaches 2/12, or 1/6 in simplest form.
Guided · 5 questions
Guided practice
Keep the model and earlier examples visible while practising.
- G1
Between 0 and 1, a line is split into 5 equal intervals. What fraction is at the 2nd tick after 0?
Reveal explanation
Answer: 2/5
Each interval represents 1/5. Moving 2 intervals from zero reaches 2/5, or 2/5 in simplest form.
An interval from zero to one divided into equal segments. - G2
Between 0 and 1, a line is split into 3 equal intervals. What fraction is at the 1st tick after 0?
Reveal explanation
Answer: 1/3
Each interval represents 1/3. Moving 1 intervals from zero reaches 1/3, or 1/3 in simplest form.
An interval from zero to one divided into equal segments. - G3
Between 0 and 1, a line is split into 4 equal intervals. What fraction is at the 1st tick after 0?
Reveal explanation
Answer: 1/4
Each interval represents 1/4. Moving 1 intervals from zero reaches 1/4, or 1/4 in simplest form.
An interval from zero to one divided into equal segments. - G4
Between 0 and 1, a line is split into 6 equal intervals. What fraction is at the 4th tick after 0?
Reveal explanation
Answer: 2/3
Each interval represents 1/6. Moving 4 intervals from zero reaches 4/6, or 2/3 in simplest form.
An interval from zero to one divided into equal segments. - G5
Between 0 and 1, a line is split into 8 equal intervals. What fraction is at the 7th tick after 0?
Reveal explanation
Answer: 7/8
Each interval represents 1/8. Moving 7 intervals from zero reaches 7/8, or 7/8 in simplest form.
An interval from zero to one divided into equal segments.
Independent · 8 questions
Independent practice
Work these without hints and explain the method used. Write answers on paper.
- I1
Between 0 and 1, a line is split into 3 equal intervals. What fraction is at the 2nd tick after 0?
Reveal explanation
Answer: 2/3
Each interval represents 1/3. Moving 2 intervals from zero reaches 2/3, or 2/3 in simplest form.
An interval from zero to one divided into equal segments. - I2
Between 0 and 1, a line is split into 8 equal intervals. What fraction is at the 2nd tick after 0?
Reveal explanation
Answer: 1/4
Each interval represents 1/8. Moving 2 intervals from zero reaches 2/8, or 1/4 in simplest form.
An interval from zero to one divided into equal segments. - I3
Between 0 and 1, a line is split into 5 equal intervals. What fraction is at the 3rd tick after 0?
Reveal explanation
Answer: 3/5
Each interval represents 1/5. Moving 3 intervals from zero reaches 3/5, or 3/5 in simplest form.
An interval from zero to one divided into equal segments. - I4
Between 0 and 1, a line is split into 6 equal intervals. What fraction is at the 3rd tick after 0?
Reveal explanation
Answer: 1/2
Each interval represents 1/6. Moving 3 intervals from zero reaches 3/6, or 1/2 in simplest form.
An interval from zero to one divided into equal segments. - I5
Between 0 and 1, a line is split into 10 equal intervals. What fraction is at the 6th tick after 0?
Reveal explanation
Answer: 3/5
Each interval represents 1/10. Moving 6 intervals from zero reaches 6/10, or 3/5 in simplest form.
An interval from zero to one divided into equal segments. - I6
Between 0 and 1, a line is split into 10 equal intervals. What fraction is at the 8th tick after 0?
Reveal explanation
Answer: 4/5
Each interval represents 1/10. Moving 8 intervals from zero reaches 8/10, or 4/5 in simplest form.
An interval from zero to one divided into equal segments. - I7
Between 0 and 1, a line is split into 12 equal intervals. What fraction is at the 9th tick after 0?
Reveal explanation
Answer: 3/4
Each interval represents 1/12. Moving 9 intervals from zero reaches 9/12, or 3/4 in simplest form.
An interval from zero to one divided into equal segments. - I8
Between 0 and 1, a line is split into 12 equal intervals. What fraction is at the 5th tick after 0?
Reveal explanation
Answer: 5/12
Each interval represents 1/12. Moving 5 intervals from zero reaches 5/12, or 5/12 in simplest form.
An interval from zero to one divided into equal segments.
Challenge · 2 questions
Challenge and error analysis
Go beyond arithmetic: justify a claim, correct reasoning, or connect two representations.
- C1
A number line from 0 to 1 is cut into 14 equal intervals. A mark is 7 intervals from zero. Explain why it represents 1/2.
Reveal explanation
Answer: 1/2
7/14 is equivalent to 1/2 because half the equal intervals have been crossed.
- C2
A student marks 1/7 at the first tick on a 0-to-1 line but drew only 6 equal intervals. Explain and correct the model.
Reveal explanation
Answer: 7 equal intervals
To show 1/7, the full unit length must be divided into 7 equal intervals, not 6.
Exit · 3 questions
Exit check
Solve independently, then return on another day to check retention.
- E1
Between 0 and 1, a line is split into 5 equal intervals. What fraction is at the 1st tick after 0?
Reveal explanation
Answer: 1/5
Each interval represents 1/5. Moving 1 intervals from zero reaches 1/5, or 1/5 in simplest form.
An interval from zero to one divided into equal segments. - E2
Between 0 and 1, a line is split into 8 equal intervals. What fraction is at the 6th tick after 0?
Reveal explanation
Answer: 3/4
Each interval represents 1/8. Moving 6 intervals from zero reaches 6/8, or 3/4 in simplest form.
An interval from zero to one divided into equal segments. - E3
Between 0 and 1, a line is split into 8 equal intervals. What fraction is at the 5th tick after 0?
Reveal explanation
Answer: 5/8
Each interval represents 1/8. Moving 5 intervals from zero reaches 5/8, or 5/8 in simplest form.
An interval from zero to one divided into equal segments.