Cleerelearn

Tag: gcse

  • Why Do You Multiply Probability by the Total?

    You’re in the middle of a probability question and it says something like:

    The probability of rolling a 6 is 0.3.
    The dice is rolled 200 times.
    Estimate how many times it lands on 6.

    And everyone (including your teacher) immediately does this:

    0.3 × 200

    …and somehow gets the answer.

    But if you’re thinking:

    “Why are we multiplying? Where did that come from?”

    Good. That question is the difference between copying a method and actually understanding probability.

    Let’s make it make sense.

    The idea: probability is “out of 1”, frequency is “out of lots”

    A probability is basically saying:

    “Out of 1 attempt, what fraction of the time would this happen?”

    So if the probability is 0.3, that means:

    “This should happen about 30% of the time.”

    Now if you don’t do it once…
    You do it 200 times…

    You’re asking a different question:

    “If it happens 30% of the time, how many times is that out of 200?”

    That’s why we multiply.

    A simpler example first: coins

    If you flip a fair coin:

    • Probability of tails = 0.5
    • If you flip it 200 times, how many tails should you expect?

    Half of 200 is 100.

    So you do:

    0.5 × 200 = 100

    That multiplication isn’t random.

    The rule (this is the whole topic)

    If something has probability pp
    and you try it nn times,

    then the estimated number of times it happens is:

    expected frequency (or relative frequency) = probability × total trials

    Back to the dice question

    Probability of a 6 = 0.3
    Number of rolls = 200

    So the estimate is:

    0.3 × 200 = 60

    Meaning:

    If you rolled this biased dice 200 times, you’d expect about 60 sixes.

    Not guaranteed. Just the best estimate.

    Why this works (the “feel” for it)

    Think of it like this:

    • Probability tells you the “rate”
    • Total tells you “how many chances you have”
    • Multiply them to get “how many hits you expect”

    It’s the same logic as:

    “If 30% of the class is left-handed, and there are 200 students… how many left-handed students would you expect?”

    Same exact maths.

    Quick check you can do in your head

    Before you even calculate, do this:

    • 0.3 is about a third
    • a third of 200 is about 66
    • so the answer should be somewhere around 60–70

    So 60 makes sense.

    This is how you stop silly mistakes.

    Final thought

    You multiply probability by the total because:

    Probability is a proportion.
    The total tells you how many chances you have.
    Multiplying applies the proportion to the total.

    That’s all relative frequency really is:
    turning probability into an estimate.

  • Why Does My Child Struggle With Maths Exams But Not In School or at Home?

    Your child seems fine in lessons.
    Homework gets done.
    They can explain things out loud.

    Then the test results come back and they don’t reflect any of that.

    This is one of the most common concerns parents raise:

    “My child can do it at home and in school, so why don’t the results show it?”

    It’s tempting to assume something must be wrong with ability, confidence, or effort.

    In most cases, it isn’t any of those.

    The key difference isn’t intelligence it’s how maths is tested

    In school, children are usually taught methods.

    For example:

    • how to add fractions
    • how to multiply
    • how to use the column method

    And when they’re asked directly to do those things, many children are fine.

    If you put in front of them:

    Add 27+16\frac{2}{7} + \frac{1}{6}

    they may be able to do it — especially if they’ve practised that method.

    But most maths tests (especially SATs, 11+ and GCSE’s) don’t stop there.

    Where tests are harder than Lessons

    Instead of asking what to do, tests often ask children to work out what needs to be done.

    For example:

    Jonathan eats two-sevenths of a cake.
    Lena eats one-sixth of the same cake.
    How much of the cake is left?

    Mathematically, this still involves adding fractions.

    But cognitively, it’s much harder.

    Now the child has to:

    • read carefully
    • decide what the maths actually is
    • remember the method
    • carry it out
    • realise they then need to subtract from one

    That’s a lot going on at once, especially for a Year 5 or Year 6 child.

    “But they know how to add fractions…”

    This is the most frustrating part, they do know how!
    They just haven’t been trained to recognise when to use it.

    This is why parents often say:

    “If you tell them what to do, they can do it.”

    That’s true, and also the problem.

    Why this affects maths more than English

    English is more fluid.

    Children read, write, talk, and interpret language every day. Even when they’re unsure, they can often have a go.

    Maths doesn’t feel like that.

    When a child sees a maths question they don’t immediately recognise, they often think:

    “I don’t know this.”

    Not:

    “I need to work out what this is asking.”

    Once that thought kicks in, confidence drops quickly.

    And maths is very unforgiving emotionally, answers are right or wrong, and children know it instantly.

    Why getting stuck destroys confidence

    In class, children rarely feel “stuck” when practising methods.

    They might make a mistake, but they still know what step comes next.

    Problem-solving questions are different.

    They create a pause.
    A moment of uncertainty.
    A feeling of not knowing where to start.

    Many children interpret that moment as:

    “I can’t do maths.”

    Even though what’s really happening is:

    “I haven’t seen this type of question before.”

    Why this issue shows up in SATs, 11+ and GCSE papers

    In primary school tests:

    • Paper 1 is arithmetic (straight calculations)
    • Papers 2 and 3 are problem solving and reasoning

    It’s very common for children to do well in arithmetic but struggle badly in the other two papers.

    That doesn’t mean they’re weak at maths.

    It means they haven’t had enough exposure to:

    • unfamiliar questions
    • questions where the maths is hidden
    • questions that require thinking before calculating

    The fix is simpler than most parents expect

    This usually isn’t about learning more maths.

    It’s about:

    • seeing lots of different question styles
    • becoming comfortable with uncertainty
    • learning that being stuck doesn’t mean being incapable

    Past papers help enormously, not because they’re magic, but because they:

    • normalise problem-solving
    • reduce panic
    • build confidence through familiarity

    The most important thing for parents to know

    If your child struggles with maths exams, it does not mean:

    • they aren’t logical
    • they aren’t capable
    • they “just aren’t a maths person”

    It usually means they’re being tested on application, not recall and that shift hasn’t been made clear to them yet.

    That’s a teaching gap, not an ability gap.

    Final thought

    Maths becomes difficult not when the numbers get harder but when the thinking changes and no one explains that change.

    Once children understand that maths tests are asking them to decide what to do, not just do it, a lot of the fear starts to disappear.

    And confidence comes back surprisingly fast.

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