Math // Mathematical Thinking

Understanding fractions

Move fluently between real-world situations, diagrams, and symbolic equations involving three-digit numbers and fractions, explaining what each part represents

Ages
7 to 8
Type
Thinking skill
Builds on
3 topics
Leads to
1 topic

Quick check

When the learner works through a maths problem involving fractions or three-digit numbers, can they switch between drawing a diagram, using a number line, and writing an equation — and explain what each one shows?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Write an equation with three-digit numbers to match a measurement or money word problem
  • Draw a bar model to represent a fraction problem and use it to solve
  • Explain how a number line diagram relates to the quantities in a word problem

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

3 topics
  • Connecting RepresentationsRequired

    Mathematical Thinking // Ages 6 to 7

    Age 7-8 quantitative reasoning builds on age 6-7

  • Numbers on a number lineHelpful

    Addition & Subtraction // Ages 7 to 8

    Number line representations exercise moving between diagrams and equations

  • Fractions on a number lineHelpful

    Fractions // Ages 7 to 8

    Placing fractions on number lines exercises connecting representations

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Understanding fractions

Ages 7 to 8 // Thinking skill

Leads to

1 topic
  • Fractions on a number lineRequired

    Mathematical Thinking // Ages 8 to 9

    Age 7-8 quantitative reasoning is prerequisite to age 8-9 level

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

For 7–8 year olds, connecting representations widens to include the NUMBER LINE, bigger (three-digit) numbers, money/measurement, and fractions. The core objective: move fluently among a real situation, a number line, a bar model, and a symbolic equation, explaining what each part represents and keeping every number tied to its referent (including units). The pivotal new representation is the number line, which shows a fraction as a NUMBER with a magnitude — a length/position — not two whole numbers stacked. The load-bearing misconceptions: placing 3/4 between 3 and 4 (reading numerator and denominator as separate whole numbers), and counting tick marks/parts without seeing that numerator and denominator together name one quantity. The operational equals-sign habit (left-to-right compute; drop units) also persists into multi-step money/measurement sentences. Teaching keeps every symbol tied to its referent and uses partitioning on number lines plus bar models to focus on quantity rather than count of parts.

Common misconceptions

  • Places 3/4 between 3 and 4, reading numerator and denominator as two separate whole numbers
  • Counts tick marks or parts without seeing that numerator and denominator together name one quantity
  • Partitions with the wrong number of marks (four hash marks for fourths) or treats the whole line as the unit
  • Keeps the operational equals-sign habit in multi-step money/measurement sentences and drops units off the numbers

Useful representations

  • Number line (fractions as lengths/positions; whole numbers labelled as fractions; equivalent fractions stacked above/below)
  • Bar/strip diagram for fractions and multi-step money/measurement problems
  • Part-whole and comparison bar models with an unknown
  • Symbolic equation with three-digit numbers and units tied to the story

Teaching practices

  • Use the number line as the central tool for fractions-as-numbers: partition 0–1 into equal parts, mark whole numbers as fractions too (e.g. 5/5), and contrast where 3 and 4 sit versus where 3/4 sits (IES Rec 2)
  • Use unmarked-between-endpoints number lines and 'what quantity is this?' questioning to prevent counting parts without understanding that numerator and denominator name a single number (IES caution)
  • Keep every number in an equation tied to its referent in the money/measurement story — units included — so translation stays meaningful (MP2)
  • Draw bar models to expose fraction and multi-step structure before computing (NCETM: reveals structure, precursor to algebra), then map the diagram to the equation
  • Translate in all directions among situation, bar/number line, and equation, explaining what each representation shows

Activities to try

  1. 1

    Place a fraction on a number line as a length

    Why it works: A fraction is a single number with a magnitude — a length/position from 0 — not a numerator and a denominator parked side by side as two whole numbers.

    1. Given a number line that runs 0 to 4, decide where 3/4 belongs before marking it.
    2. If you first place it between 3 and 4, judge how far that point is from 0 — it stretches past 3, far too long for 'three quarters' (an interactive tool can light up this length, but you can equally read the distance on a fixed, accurately drawn line).
    3. Drag or redraw 3/4 back between 0 and 1, partition that first unit into four equal parts, and mark three of those steps from 0.
    4. State the single amount that point names in one sentence (e.g. 'three of four equal parts of one whole, which is less than 1').
    5. Use a whole-number/fraction view to contrast where 3 and 4 sit versus where 3/4 sits.
  2. 2

    Partition an unmarked line and name the quantity

    Why it works: A point on the line names one quantity; the numerator and denominator together specify a single number, and partitioning into n equal parts needs n−1 interior marks.

    1. Start from a number line with only its endpoints (0 and 1) marked — no interior tick marks.
    2. Cut the space from 0 to 1 into the number of equal parts needed for the fraction you want (e.g. four parts for fourths).
    3. Say how many marks you had to add between 0 and 1 and why that many (fourths need three interior marks).
    4. Point to the exact spot for the fraction and state what single amount that point represents — do not just count marks.
  3. 3

    Build a bar model and equation for a money/measurement story, units attached

    Why it works: Equations should be structurally faithful to the situation — each number carries a referent and unit, quantities must share a unit before combining, and '=' means 'the same amount as', not 'compute left to right'.

    1. Read a measurement or money story, e.g. 'A ribbon is 245 cm long; 3/4 of a metre is cut off.'
    2. Draw a bar model that shows the whole, the parts, and where the unknown sits.
    3. Before combining any numbers, convert so every quantity is in the SAME unit — e.g. 3/4 of a metre = 75 cm — so you are not subtracting 3/4 from 245.
    4. Write a matching equation in which each number keeps its referent and a unit, using the common unit you converted to (for this ribbon, every quantity in cm, e.g. 245 cm − 75 cm = ? cm; use £ or pence consistently when the story is about money).
    5. Say aloud what each number in the equation points back to in the situation, then solve using the bar (245 cm − 75 cm = 170 cm).

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