Year 4 Mathematics · AC9M4N04

Count by fractions and mixed numerals

Fractions are numbers that can be counted and located on a continuous number line. An improper fraction and a mixed numeral can represent the same point, such as 7/4 = 1…

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Learning goalsSay it simply

Fractions are numbers that can be counted and located on a continuous number line. An improper fraction and a mixed numeral can represent the same point, such as 7/4 = 1 3/4.

Learning routine: Choose step size → Partition each whole equally → Count beyond one → Rename → Locate and compare

Success looks like

  • Count by fractions
  • Cross whole numbers
  • Use mixed numerals
  • Rename improper fractions
  • Locate and compare on number lines
Clean visual examplesOne-page board
AC9M4N04 — Worked examples
Worked example

Count by quarters beyond one whole

Four equal intervals in each whole0123ABC

Every interval represents one quarter. Crossing a whole does not change the step size or denominator.

  1. Mark 0, 1, 2 and 3 at equal distances; split each whole into four equal intervals.
  2. Count quarter-steps from 0: 1/4, 2/4, 3/4, 4/4 = 1, 5/4 = 1 1/4. Point A is 5/4.
  3. Continue with the same spacing: B is 7/4 = 1 3/4 and C is 10/4 = 2 1/2. The number of quarters per whole never changes.
Worked example

Rename between improper fractions and mixed numerals

Improper fractionWhole groupsMixed numeral7/44/4 + 3/41 3/411/510/5 + 1/52 1/59/28/2 + 1/24 1/2

Count how many complete denominator-sized groups fit in the numerator, then keep the remainder over the same denominator.

Worked explanation: For 7/4, separate one group of four quarters and three quarters more: 7/4 = 4/4 + 3/4 = 1 3/4. For 2 2/3, two wholes contain six thirds, and two more thirds give 8/3.

These are different names for the same number-line positions, not multiplication of the whole and fractional parts.

Worked example

Count backwards across a whole

Start at 1 1/4 and move left by quarters: 1 1/4, 1, 3/4, 1/2, 1/4, 0. Think of 1 as 4/4 when you cross it. The step is still one quarter.

Quick check: One third before 2 is 1 2/3 because 2 can be renamed as 1 whole and 3 thirds.

Worked example

Read a line that does not begin at zero

Equal intervals on a number line23P

The whole unit from 2 to 3 has three equal intervals. Point P is two thirds after 2, so it is 2 2/3 = 8/3. The left endpoint is 2; it must not be relabelled as zero.

Look for: Students count spaces, not tick marks, and use the given whole-number labels.

Worked example

1. Human fraction line

Mark equal quarter intervals on a floor number line and stand at 3/4, 5/4, 1 1/2 and 9/4.

Equal intervals on a number line0123ABCD

Check the positions: A = 3/4; B = 5/4 = 1 1/4; C = 1 1/2; D = 9/4 = 2 1/4.

Worked example

2. Count and rename

Count by thirds from 0 to 3, recording improper fractions and mixed numerals at every whole and beyond.

0→1/3→2/3→3/3 = 1→4/3 = 1 1/3→5/3 = 1 2/3→6/3 = 2→7/3 = 2 1/3→8/3 = 2 2/3→9/3 = 3
Worked example

3. Missing-point challenge

Use interval size and neighbouring labels to identify missing fractions on partially labelled number lines.

Equal intervals on a number line1231 1/2AB

Answers: A = 1 1/4 and B = 2 3/4. Use the quarter-intervals and the labelled whole numbers.

Curriculum examplesCopied content

Content description: count by fractions including mixed numerals; locate and represent these fractions as numbers on number lines.

The following teaching examples cover the curriculum elaborations.

E1: Count quarters beyond one whole

Cut or draw two identical paper strips in quarters. Lay down the pieces one at a time: one quarter, two quarters, three quarters, four quarters or one whole, five quarters or one and one quarter. Continue to eight quarters or two wholes.

Check: Six quarters are 1 2/4 = 1 1/2. Students should show one complete whole and the remaining parts.

E2: Align halves, quarters and thirds

One unit is the same length on all three linesHalves012Quarters012Thirds012

Draw three aligned number lines with the same 0, 1 and 2. Partition each whole into halves, quarters and thirds. The common point at 1 is 2/2 = 4/4 = 3/3, and the point at 2 is 4/2 = 8/4 = 6/3.

On the halves and quarters lines, 1/2 = 2/4 and 1 1/2 = 3/2 = 6/4 align too. Halfway points fall between the third-ticks; they are not whole numbers of thirds.

E3: Rename and locate

Mark 5/4 on a quarter number line: four steps reach 1, and the fifth reaches 1 1/4. Label the same point with both names. Reverse the process for 2 2/3: two wholes contain 6 thirds, then another 2 thirds give 8/3.

Check: The value and position stay unchanged when the number is renamed.

E4: Count tenths across one whole

Partition a line from 0 to 2 into ten equal intervals per whole. Count 8/10, 9/10, 10/10 = 1, 11/10 = 1 1/10, 12/10 = 1 2/10. Count backwards over the same points.

Check: One tenth before 1 is 9/10, and one tenth after 1 is 1 1/10. The denominator remains 10.

Questions and answersWith answers

Core idea: Fractions are numbers that can be counted and located on a continuous number line. An improper fraction and a mixed numeral can represent the same point, such as 7/4 = 1 3/4.

Remember

  • Count by fractions
  • Cross whole numbers
  • Use mixed numerals
  • Rename improper fractions
  • Locate and compare on number lines

Important questions

  • Count by fifths from 4/5 to 9/5. 4/5, 5/5 = 1, 6/5 = 1 1/5, 7/5 = 1 2/5, 8/5 = 1 3/5, 9/5 = 1 4/5.
  • Rename 7/4 as a mixed numeral. 1 3/4. Four quarters make a whole, leaving three quarters.
  • Rename 2 2/3 as an improper fraction. 8/3. The two wholes contain six thirds, with two thirds more.
  • Locate 5/2 on a number line. Mark 2 1/2, halfway between 2 and 3. Five half-steps from zero reach this point.
  • Explain why 8/4 = 2. Each whole contains four quarters, so eight quarters make two whole units.
Practice and reviewReady for practice
  • Denominator changes after one whole — The denominator keeps naming the same-sized parts.
  • Mixed numeral treated as multiplication — 1 3/4 means one whole and three quarters, not 1 × 3/4.
  • Unequal number-line intervals — Equal fraction steps require equal distances.
  • Improper fractions cannot be numbers — They are valid numbers greater than or equal to one.

Before this lesson: Students recognise unit fractions and equal partitions. Support: Use physical half and quarter strips alongside a line from 0 to 2. Core: Count forwards and backwards across wholes; locate and draw fractions, mixed numerals and matching improper fractions. Extend within this code: Read a line beginning at 2 or count jumps of 2/3 or 3/4 while keeping the unit intervals fixed.

Boundary: Renaming supports counting and number-line representation. Formal algorithms for adding unlike fractions are not the target.

  1. Count backwards by thirds from 2 1/3 to 1 1/3. Review hint: Rename the whole when crossing it. Answer: 2 1/3, 2, 1 2/3, 1 1/3.
  2. A line from 1 to 3 has eight equal spaces. What is one space? Review hint: How many spaces fit in one whole? Answer: 1/4; there are four equal spaces per whole.
  3. Draw and label 7/3 on a line from 2 to 3. Review hint: Six thirds make 2. Evidence: A point one third after 2, labelled 7/3 = 2 1/3.

Use the Topic Guide, then project the Classroom View. Complete the written worksheet tasks before Practice and Test. Drawing tasks need an adult to inspect equal intervals, the whole-number scale and actual labelled points.

Curriculum alignmentStart here
Exit ticketCheck understanding
  1. Count by quarters from 3/4 to 7/4. 3/4, 4/4, 5/4, 6/4, 7/4.
  2. Rename 7/4 as a mixed numeral. 1 3/4, because 4/4 makes one whole and 3/4 remains.
  3. Where is 5/2 on a number line? At 2 1/2, halfway between 2 and 3.
  4. Draw a line from 1 to 2 in thirds and mark 1 2/3. Show three equal spaces and place the point at the second tick after 1. Check the actual drawing, not just the label.

Mastery evidence: Students keep equal spacing on the number line and can rename between improper fractions and mixed numerals without changing the value.

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