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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 9: Probability (Core)
Recommended prior knowledge
Students should understand:
1. The definition of the probability of an event occurring as the number of times the event can occur divided by the total number of events. That a probability
can only be greater than or equal to 0 and less than or equal to 1 (100%)
2. Can manipulate fractions, decimals and percentage and convert between them
Context
This is the first of two statistics units. This must be taught after Unit 1 and could be used to reinforce fraction decimal and percentage skills. It could be taught early in
the course but should then be revisited. This could be taught as a complete unit or as two blocks (9.1, 9.2 and 9.3) and 9.5. The second block could therefore be
taught later in the course. Students who are following the extended syllabus will move through this faster but need to have all these skills in place before working on
the extended units, or applying them to problems.
Outline
The content allows discussion of the difference between the probability of an event occurring and what actual happens, the difference between experimental and
theoretic probability and some tools to work out probability. Within the suggested teaching activities ideas are listed to identify and remediate misconceptions and to
pull learning through to the required standard. The learning resources give both teaching ideas, summaries of the skills and investigative problems to develop the
problem solving skills and a depth of understanding of the mathematics, through exploration and discussion.
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Cambridge IGCSE Mathematics (US) 0444
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Syllabus ref
9.1
CCSS:
S-CP1
Learning objectives
Probability P(A) as a fraction,
decimal, or percentage
Significance of its value,
including using probabilities to
make fair decisions
Suggested teaching activities
Learning resources
This gives a comprehensive guide to probability and the steps of
development and is full of activities and ideas for teaching the topic.
www.counton.org/resources/ks3framewor
k/pdfs/probability.pdf
Notes and exemplars
Includes an understanding that the probability of an event occurring is
1 – the probability of the event not occurring.
Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections,
or complements of other events (“or,” “and,” “not”).
The knowledge and use of set notation is not expected.
www.counton.org/resources/ks3framewor
k/pdfs/probability.pdf
http://nrich.maths.org/4304
http://nrich.maths.org/4334
Teaching activities
Shuffle a pack of 0-9 cards and reveal the top card. Ask the class to vote
whether the next card will be higher or lower. And have a recorder note the
outcome versus the class decision. Continue through the whole pack.
Discuss briefly number of times class is correct – you want this to be
incorrect so rig if necessary. Then give out a recording sheet so they
students can record what has already gone and play again. e.g. The
numbers 0 -9 repeated in 8 rows. So they can cross of the numbers that
have already been used and ring the card currently being held up. Rig it from
time to time so that either higher or lower is impossible and so that the
strongest possibility isn’t the next card that appears and discuss. Most
students will record the fractions for higher and lower and compare
instinctively but the activity gives an opportunity to discuss certainty and
impossibility and whether the event with the highest probability has to win.
Have a bag containing a total of 10 cubes of two or three different colours.
Pull one out, reveal it and return it and repeat 20 times (students should
record the results in a frequency table). Ask class to estimate the number of
each colour in the bag. Then reveal the contents or make another twenty
recordings to see if the result refines better. Discuss the number of repeats
needed to give accurate results.
9.2
CCSS:
S-IC2
v1 2Y01
Relative frequency as an
estimate of probability
Scattered throughout the learning resource listed are examples where fair
and unfair can be discussed.
Notes and exemplars
Decide if a specified model is consistent with results from a given datagenerating process.
e.g. using simulation, e.g. a model says a spinning coin falls heads up with
Cambridge IGCSE Mathematics (US) 0444
www.bbc.co.uk/schools/ks3bitesize/maths
/handling_data/relative_frequency/revise1.
shtml
2
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
probability 0.5.
Would a result of 5 tails in a row cause you to question the model?
www.counton.org/resources/ks3framewor
k/pdfs/probability.pdf page 283
Teaching activities
This first task gives a good visual image of some of the conundrums of
probability.
Probability Art – Use cm squared paper. A square will be coloured red for a
H and Green for a Tail. Start in the top left hand corner of the paper with the
page turned landscape. Toss the coin and colour the first square. Toss again
and colour the next square. Continue until at least one row is complete.
Discuss with the class whether to snake to the next row or to go to the left
hand side. Discuss the total red and greens (hopefully approximately 50%)
but with no pattern in the reds and greens. Discuss the fact the coin has no
memory and the probability does not tell you which event will occur next as
each is independent.
9.3
CCSS:
S-IC2
Expected number of
occurrences
Relative Probability is best demonstrated with things that cannot calculated
by theoretic probability. Tossing a drawing pin (in a sealed jar) see the p283
of the KS3 framework document, or using a page of text and working out the
relative probability for a chosen vowel or consonant. Changing to a different
type or age range text can then be compared.
General guidance
A discussion is needed to distinguish between the probability of an event
occurring given the probability and it actually occurring – e.g. the science of
weather forecasting. However, it is also necessary to teach that if the
probability of an event is 0.3 and the experiment is repeated 500 times then
you would expect the event to happen 150 times, in spite of the fact that for
each instance 0.3 is less than 0.7 and so the event is less likely.
Teaching activities
Set up variety of probability experiments that have a theoretic probability.
Get each group of five pairs to work on one experiment. Work out the
theoretic probability and the experimental probability for 100 goes. Compare
the theoretic and the experimental probability and pool the results of the
group to get 500 results. Discuss the outcomes for the different experiments
and when the theoretic and experimental converge.
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
9.5
Possibility diagrams
Notes and exemplars
Simple cases only.
www.bbc.co.uk/schools/ks3bitesize/maths
/handling_data/probability/revise7.shtml
CCSS:
S-CP1
Tree diagrams including
successive selection with or
without replacement
Teaching activities
Work out the range of outcomes in two way tables with two objects involved.
E.g. possible outcomes when dice are added, or menu options when there
are three main courses and three desserts etc.
www.bbc.co.uk/schools/gcsebitesize/math
s/data/probabilityhirev1.shtml
http://nrich.maths.org/6033
http://nrich.maths.org/7541
Use tree diagrams to show how events combine, noting that you add the
ends of branches but multiply along branches to get probabilities for
combined events. Note where branches add to 1 (100%). Encourage
students not to simplify fractions until the end but to leave denominators the
same to simplify working and provide checks that the correct parts of the
diagram add to 1(100%).
Students find it difficult to decide what to put on the tree diagram so a variety
of problems that requires them to choose is essential, rather than questions
which give pre-labelled branches.
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