Math // Fractions

Fractions of a whole (age 9+)

Solve word problems involving multiplication of a fraction by a whole number using visual models and equations

Ages
9 to 10
Type
Method
Builds on
1 topic
Leads to
1 topic
Standards
1 code

Quick check

If each section of a garden uses 3/4 of a bag of compost and there are 4 sections, can the learner work out how many bags are needed in total?

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.
  • Each person eats 3/8 of a pizza and there are 5 people — how many pizzas are needed?
  • A ribbon is cut into pieces of 2/3 metre; how long are 4 pieces altogether?
  • Between what two whole numbers does 6 × 3/4 lie?

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

1 topic
  • Multiplying fractionsRequired

    Fractions // Ages 9 to 10

    Must multiply fraction by whole number before solving word problems

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Fractions of a whole (age 9+)

Ages 9 to 10 // Method

Leads to

1 topic

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

Learners solve word problems that multiply a fraction by a whole number (n × a/b = na/b) by treating the product as n copies of the unit-fraction multiple, accumulating them on a bar or number line, converting the improper result to a mixed number, and interpreting that result in context — distinguishing exactly how much is used (e.g. 1 7/8 pizzas eaten) from how many whole units must be bought (2 pizzas). Core examples: 5 people each eating 3/8 of a pizza; four ribbon pieces of 2/3 metre; estimating that 6 × 3/4 lies between 4 and 5.

Common misconceptions

  • Multiplies the denominator too (4 × 2/3 → 8/12), scaling the unit as well as the count
  • Adds the whole number to the fraction instead of multiplying (3/8 + 5)
  • Leaves the answer as an improper fraction with no contextual meaning, or fails to round 'pizzas needed' up
  • Chooses the wrong operation because the repeated-groups structure is not read

Useful representations

  • Number line with repeated fraction jumps of a/b
  • Repeated-groups bar (n copies of the fraction) accumulating past wholes
  • Improper → mixed conversion of the product
  • Context interpretation card (eaten vs needed)

Teaching practices

  • Model as repeated equal groups — n copies of a/b — accumulating on a bar or number line so the product is built, not just computed
  • Multiply only the numerator by the whole number (n × a/b = na/b); keep the denominator as the unchanged unit
  • Estimate between which whole numbers the product lies before computing (6 × 3/4 is between 4 and 5)
  • Convert the improper product to a mixed number and INTERPRET it in context (round pizzas needed up to a whole)
  • Distinguish 'how much is eaten' (exact mixed number) from 'how many must be bought' (rounded up)

Activities to try

  1. 1

    Build the product by stacking repeated fraction groups

    Why it works: Multiplying a fraction by a whole number is repeatedly adding that fraction: n copies of a/b accumulate to na/b, so only the numerator (count of unit fractions) grows while the denominator stays the fixed unit.

    1. Take a catering problem: 5 friends, each eats 3/8 of a pizza — work out the total eaten by building copies, not by jumping to a rule.
    2. Stack five copies of a 3/8 bar (or draw five jumps of 3/8 on a number line) and track the running total: 3/8, 6/8, 9/8, 12/8, 15/8, noting when the total crosses a whole.
    3. Record the result as the improper fraction 15/8 and convert it to the mixed number 1 7/8.
    4. State in words: the top number (count of eighths) grows while the denominator stays eighths — the unit does not change.
    5. Repeat the build for 4 × 2/3 (thirds accumulating to 8/3 = 2 2/3) to generalise.
  2. 2

    Interpret the result: eaten versus bought

    Why it works: An improper-fraction product must be read back into the situation: the exact mixed number answers 'how much', but 'how many whole units are needed' may require rounding up — and some quantities (like length) sensibly stay as mixed numbers.

    1. Using the computed total 1 7/8 pizzas eaten, decide how many whole pizzas must be ordered, given you cannot buy part of a pizza.
    2. Explain why 1 7/8 rounds UP to 2 here, and why the question 'how much is eaten' keeps the exact 1 7/8.
    3. Answer a contrast pair: 'How much pizza is eaten?' (1 7/8, exact) and 'How many whole pizzas must be bought?' (2, rounded up).
    4. For the ribbon problem, state 4 × 2/3 = 8/3 = 2 2/3 metres and decide whether rounding is needed (it is not — a length can be a mixed number).
    5. Before computing another problem, estimate between which two whole numbers the product lies, e.g. 6 × 3/4 is between 4 and 5.
  3. 3

    Check and consolidate with targeted questions

    Why it works: Fluent learners compute na/b (numerator only), convert to a mixed number, interpret in context, estimate between whole numbers, and transfer the method to a new word problem.

    1. Answer: 4 × 2/3 = ? Choose and justify from 8/12, 8/3, 2 2/3, explaining why 8/12 is wrong (denominator scaled) and showing that 8/3 and 2 2/3 are the same value.
    2. Fill in the blank: n × a/b = __/b, stating the rule in your own words.
    3. Estimate on a scale 'between which two whole numbers is 6 × 3/4?' and place it.
    4. Solve a fresh word problem (e.g. a new fraction × whole context) end to end: build, compute na/b, convert, and interpret whether to round.
    5. Explain in a sentence what a party host should order for a group, separating 'eaten' from 'bought'.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
4.NF.4.c

Math // Fractions

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