Teaching and learning plan on introduction to probability

Teaching and learning plan on introduction to probability
Teaching & Learning Plans
Plan 1: Introduction to Probability
Junior Certificate Syllabus
Leaving Certificate Syllabus
The Teaching & Learning Plans
are structured as follows:
Aims outline what the lesson, or series of lessons, hopes to achieve.
Prior Knowledge points to relevant knowledge students may already have and also
to knowledge which may be necessary in order to support them in accessing this new
topic.
Learning Outcomes outline what a student will be able to do, know and understand
having completed the topic.
Relationship to Syllabus refers to the relevant section of either the Junior and/or
Leaving Certificate Syllabus.
Resources Required lists the resources which will be needed in the teaching and
learning of a particular topic.
Introducing the topic (in some plans only) outlines an approach to introducing the
topic.
Lesson Interaction is set out under four sub-headings:
i.
Student Learning Tasks – Teacher Input: This section focuses on teacher input
and gives details of the key student tasks and teacher questions which move the
lesson forward.
ii.
Student Activities – Possible and Expected Responses: Gives details of
possible student reactions and responses and possible misconceptions students
may have.
iii. Teacher’s Support and Actions: Gives details of teacher actions designed to
support and scaffold student learning.
iv. Checking Understanding: Suggests questions a teacher might ask to evaluate
whether the goals/learning outcomes are being/have been achieved. This
evaluation will inform and direct the teaching and learning activities of the next
class(es).
Student Activities linked to the lesson(s) are provided at the end of each plan.
Teaching & Learning Plan 1:
Introduction to Probability
Aims
• To familiarise students with the ways in which we talk about uncertainty
and look at everyday situations in which probability arises
• To engage students in activities that will give them contact with the main
ideas of probability
• To rehearse the language and patterns associated with probability
Prior Knowledge
Prior knowledge and experience of handling fractions and percentages is required.
Students have prior knowledge of some of the ideas and language patterns of the topic
of probability from the primary school curriculum, third class upwards, but the topic
may need to be revisited to ensure that all students know the basics. Students may have
certain ‘misconceptions’ based on intuition and personal experience. Experimentation is
required, where students count and analyse outcomes, thereby constructing their own
meanings by connecting the new information to what they already believe. Students
accept new ideas only when they see that their old ideas do not work: for example,
finding out experimentally that 6 is not the hardest number to get when throwing a fair
die, and that all outcomes of such a throw are equally likely.
When working together cooperatively in small groups, to test a hypothesis for
example, students will be using the language of the topic thus improving their ability
to communicate effectively using correct terminology. They can then move on from
the experimental approach, where they calculate the relative frequency of an event,
which tends towards the probability for an infinite sequence of trials, to the theoretical
approach, which is based on logical reasoning.
Learning Outcomes
As a result of studying this topic, students will be able to
• distinguish certain from uncertain events
• describe events as being more or less likely from experience
• order events from least likely to most likely and justify their choice
• use a scale from 0 to 1 to informally place everyday chance-related events
• represent and interpret probabilities as fractions, decimals and percentages
• represent the probability of an event as a fraction or decimal between 0
and 1 or as a percentage
• list all possible outcomes for practical experiments such as rolling one die
• determine the probability of an event using the results of an experiment
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Teaching & Learning Plan 1: Introduction to Probability
Relationship to Junior Certificate Syllabus
Sub-topics
Ordinary Level
1.5 Counting
Listing outcomes of experiments in a
systematic way
1.6 Concepts of probability
The probability of an event occurring:
students progress from informal to
formal descriptions of probability.
Predicting and determining probabilities
Decide whether an everyday event is
likely or unlikely to occur.
Relationship to Leaving Certificate Syllabus
Sub-topics
Foundation Level
Ordinary Level
1.2 Concepts of
probability
Decide whether an
everyday event is likely or
unlikely to occur.
Estimate probabilities from
experimental data.
Recognise that probability
is a measure on a scale of
0-1 of how likely an event
is to occur.
Associate the probability
of an event with its longrun, relative frequency.
1.3 Outcomes of
random processes
© Project Maths Development Team 2009
Apply the principle that, in
the case of equally likely
outcomes, the probability
is given by the number
of outcomes of interest
divided by the total
number of outcomes.
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Teaching & Learning Plan 1: Introduction to Probability
Introducing the Topic
Students need to get into a frame of mind for learning probability by looking at the
language of uncertainty and then trying to order phrases used to describe uncertainty,
leading to being able to quantifying it.
The following examples could be used to explore misconceptions:
• What is the most difficult number to get when throwing a fair die?
• Random events should have outcomes which appear random; for example,
in the lotto theory tells us that any of the six numbers is equally likely to
turn up, yet more people choose randomly spaced numbers than numbers
which form a pattern like 1,2,3,4,5,6 etc.
• The likelihood of 2 consecutive numbers appearing in any Lotto draw
(which is > 50%) could easily be investigated by reference to a number of
recent draws.
Real Life Context
The following examples could be used to explore real life contexts.
Looking at statistics from the Census, questions like:
• How long will I live?
• Will I get married?
• How many children will I have?
These questions can be answered with some degree of certainty based on population
statistics.
Life assurance companies work out how much to charge for their premiums based on
tables of life expectancy. Why are some premiums cheaper than others?
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»» When, in our everyday
lives, would knowing the
chance or likelihood of an
event occurring affect our
actions? Do you have any
suggestions?
»» In this lesson we will be
investigating ideas about
uncertainty or chance or
probability.
Student Learning Tasks:
Teacher Input
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• Health insurance premiums
are less if you don’t smoke
and have no diagnosed
serious illnesses since the
probability of you being
sick is less.
• Car insurance companies
charge premiums based on
the probability that you
will have an accident.
• Flying with a particular
airline – what is its safety
record to date?
• Taking a new drug – what
are the chances of it curing
the disease?
• Betting on a horse –
knowing the odds.
Student Activities: Possible
and Expected Responses
Teacher’s Support and
Actions
Lesson Interaction
Teaching & Learning Plan 1: Introduction to Probability
KEY: » next step
• student answer/response
»» Did students come up with
several varied suggestions?
Checking Understanding
4
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»» Depending on the event,
place the uncertain ones
nearer the “certain to
happen” or “certain not
to happen” side of the
diagram.
»» (in the area marked
uncertainty) 3 events
which might or might not
happen.
»» 3 events which are certain
not to happen or to have
happened.
»» 3 events which are certain
to happen.
»» Student Activity 1A, fill in:
»» We will begin by looking
at the language used in
uncertainty and chance,
and move on to how
numbers are assigned to
uncertainty.
Student Learning Tasks:
Teacher Input
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Student Activities: Possible
and Expected Responses
• student answer/response
»» Did students come up with
several varied suggestions?
Checking Understanding
KEY: » next step
»» Take selections from each
group and put these on the
board.
»» Walk around to see what
students are writing down;
if they are struggling, ask
questions which will give
them a hint of an example.
»» Divide class into small
groups. Each member of
the group should have
a clear task. Make sure
there is a time limit on the
activity.
»» Distribute Student Activity
1.
Teacher’s Support and
Actions
Teaching & Learning Plan 1: Introduction to Probability
5
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Student Learning Tasks:
Teacher Input
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• Might/might not happen
1. Ireland will win a gold
medal in the next Olympics.
2. It will rain tomorrow.
3. Roy Keane will manage the
Irish soccer squad (students
basing their convictions on
experience or intuition).
KEY: » next step
»» Take selections from each
group and put these on the
board.
• student answer/response
»» Did students come up with
several varied suggestions?
»» Walk around to see what
students are writing down;
if they are struggling, ask
questions which will give
them a hint of an example.
• Certain to happen/have
happened:
1. I have been born.
2. The sun rose this morning.
3. There is a Chinese
Olympian who is 6’ 7”.
• Certain not to happen/have
happened/impossible:
1. Ireland will not host the
next Olympics.
2. Finding a triangle with 4
sides.
3. Someone in our class was
born during the First World
War.
4. Michael Jackson is still
alive.
Checking Understanding
Teacher’s Support and
Actions
Student Activities: Possible
and Expected Responses
Teaching & Learning Plan 1: Introduction to Probability
6
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»» Take down the selection
of class phrases from the
board into the table on
Student Activity 2A.
»» Write down at least 5
phrases on the bottom of
Student Activity 1B.
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»» Students copy from
the board the selection
of phrases (about 12)
suggested by class.
• student answer/response
»» Did students have lots of
different suggestions once
they understood the task?
Checking Understanding
KEY: » next step
»» Distribute Student Activity
2.
»» Check if everyone
»» Students will fill out
understands the task.
phrases on Student Activity
1B:
»» Walk around to see what
students are writing down.
• a dead cert
• a very good chance
If anyone is struggling, ask
• likely
questions which will give
• almost certain
them a hint of an example.
• no chance
• more than likely
»» Take selections from each
• extremely likely
group and record them on
• 50/50
the board/flipchart/laptop.
• a small chance
• extremely unlikely
• never
• not in a month of Sundays
• fairly likely
»» Working in pairs,
brainstorm phrases used
to describe the probability
or chance of an event
occurring on a scale from
absolute certainty to no
chance at all.
Teacher’s Support and
Actions
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 1: Introduction to Probability
7
© Project Maths Development Team 2009
»» Now think in terms of
how you might interpret
this lack of precision in
different situations, for
example the plane you’re
on probably won’t crash
vs. it probably won’t rain
tomorrow. Would you be
happy with the phrase
“probably won’t” in both
situations?
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»» Students will be looking for
more precision in the case
of a plane not crashing –
in the form of a numeric
representation of the
phrase “probably won’t”.
»» Check if everyone
understands the task.
»» From this list of phrases
»» Students then fill in one
identify an event which can
event for each phrase on
best be described by each
Student Activity 2B.
of these terms.
• student answer/response
»» Do students recognise
the need for a numeric
representation of the
phrase “probably won’t”?
»» Were there many and
varied suggestions and
were they appropriate to
the phrases?
Checking Understanding
KEY: » next step
»» Take selections from each
group and put on the
board. Invite students to
agree or disagree, but
explain that they must
have a valid reason for
doing so.
»» Walk around to see what
students are writing down;
if they are struggling, ask
questions which will give
them a hint of an example.
Teacher’s Support and
Actions
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 1: Introduction to Probability
8
• >50%
»» What is the range of the
answers?
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• Maybe 70% , 80% but
greater than 50%
»» Say that something
probably will happen, that
it is not a “dead cert” but
that it has a very good
chance; can you assign a
percentage to this event?
© Project Maths Development Team 2009
• 0%
• 100%
Student Activities: Possible
and Expected Responses
»» In terms of percentages,
how would you describe
something which had no
chance of happening?
Write this in on ‘The
Probability Scale’ (Student
Activity 3A).
»» Write this in at the correct
position on ‘The Probability
Scale (Student Activity 3A).
»» In terms of percentages,
how would you describe a
“dead cert”, or something
which was definitely going
to happen?
»» Let’s try to get more
precision by using
percentages.
Student Learning Tasks:
Teacher Input
• student answer/response
»» Have students understood
that phrases like “has a
good chance” are imprecise
but yet have a bias towards
the upper end of the scale?
»» Are students getting the
idea of limits of 0% and
100% for the range of
probabilities of an event.
Checking Understanding
KEY: » next step
»» Emphasise the idea of a
possibility of a range of
answers but note that all
answers are greater than
50%.
»» Emphasise no chance.
»» Distribute Student Activity
3.
»» Ask the class and give them
a moment to think before
asking one student.
Teacher’s Support and
Actions
Teaching & Learning Plan 1: Introduction to Probability
9
• <50%
• 0% to 100%
• 100%
• 50%
• ½
»» What is the range of the
answers?
»» What range of percentages
can we use to represent
the chance or likelihood of
something happening, to
cover all possibilities?
»» Consider a whole bar
of chocolate – what
percentage of the bar are
we looking at?
»» If we toss a coin, what is
the chance of getting a
‘tail’?
»» What is another way of
expressing this chance?
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• Maybe 20%, 10%
»» Assign an estimated
percentage to the
phrase “probably won’t
happen?”(snow on St.
Patrick’s day ).
© Project Maths Development Team 2009
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
• student answer/response
»» Do students understand
that chance can be
represented by a range
from 0% to 100%?
»» Again, have students
understood the idea of a
possibility of a range of
answers but all less than
50%?
Checking Understanding
KEY: » next step
»» If students are comfortable
with the idea of fractions,
it may be possible to
explore the rolling of a fair
die.
»» Lead the class to the idea
of a fraction as expressing
the chance of something
happening.
»» Ask the class; wait a short
while and then ask an
individual student. If
students can’t answer, take
them back through the
previous questions.
Teacher’s Support and
Actions
Teaching & Learning Plan 1: Introduction to Probability
10
• 0
»» Instead of giving 0% as
the chance of something
happening what number
could we use?
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»» Show me where 5/8 would
be placed. Has anyone
anything different?
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• student answer/response
»» Can students apply what
they have learned about
the probability scale and
can students give a range
of numbers including both
decimals and fractions?
»» Do students understand
that probability can be
represented on a scale of 0
to 1 as well as from 0% to
100%?
»» Are students associating
a 0% chance with the
number 0?
Checking Understanding
KEY: » next step
Note: (Ensure that a good
selection of proper fractions and
decimals are included). Note that,
by contrast with “likely”, we
can all agree with the placing of
these values.
»» Write up correct values on
the board.
•
•
•
•
•
•
•
•
»» Give examples of numbers
which can represent the
possibility of something
happening?
»» Position them on ‘The
Probability Scale1 on
Student Activity 3A.
»» Ask most students in the
class, each time asking the
class to verify if they are
correct, and why.
• 0% to 100%
or
• 0 to 1
»» So now you have two ways
of representing a scale of
probabilities. What are
they?
0.9, 1, 0, .5, .3, ¾, 0.75
1 = dead cert
0 = never
0.5 = evens
0.8 = quite likely
0.1 =very unlikely
0.2 = quite unlikely
0.4 = not a good chance
»» Ask the class; then select
an individual student to
answer.
• 0 to 1
»» Ask the class; then select
an individual student to
answer.
»» Ask the class; then select
an individual student to
answer.
Teacher’s Support and
Actions
»» Between what ranges of
numbers can I represent
the chance of something
happening to cover all
possibilities?
»» Now write this in on ‘The
Probability Scale (Student
Activity 3A).
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 1: Introduction to Probability
11
»» Students draw the line and
fill in the numbers.
»» Using a ruler, draw in your
copy ‘The Probability Scale’
line segment from Student
Activity 3A.
»» Label this “The Probability
Scale”. Mark in the
numbers listed on the
board.
»» Circulate, supporting
students who have
difficulty with the task.
»» Suggest fractions and
negative numbers.
»» Ask most students in the
class. Then ask the class to
verify if the answers are
correct, and why.
»» Ask the class; then select
an individual student to
answer.
Teacher’s Support and
Actions
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• student answer/response
»» Are there many
students having
difficulty understanding
percentages etc?
»» Have all students been
successful in drawing and
marking the scale?
»» Can students give a
range of numbers
including negative
numbers?
»» Can students apply what
they have learned about
the probability scale?
Checking Understanding
KEY: » next step
»» Pin a large scale onto the
»» Students write in each
»» The line on Student
board and have cards with
phrase or number onto the
Activity 3B represents a
the various options written
appropriate spot on the scale.
scale from 0 to 1. Working
on them (or draw in on the
in pairs write in each item
board).
»» Students might start with
from Box A at the most
percentages, which they
appropriate position on the
»» Check if everyone
are most familiar with, and
line. (Use arrows.)
understands the task.
proceed to phrases and then
fractions.
•
•
•
•
•
»» Can you give examples
of numbers which cannot
represent the chance of
something happening?
-1
5, 7,
2000,
23.6,
9/8
• No, because the chance of
something happening must
be a number between 0 and
1.
»» If I say that the chance of it
raining tomorrow is 2.5 – is
this possible?
»» If I say that the probability
of one of you flying to
Mars tomorrow is 3 – is this
possible?
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 1: Introduction to Probability
12
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»» Why do you think it should
be changed?
»» Has anyone anything
different?
»» Does anyone think they
should be changed?
»» Does everyone agree with
these placements?
»» Thank student who does
board work.
»» When everyone is finished
a volunteer student fills in
the phrases on the board.
»» Attach the phrases to the
large scale on the board.
(or write in on the board).
• student answer/response
»» Is the student body in
agreement with the
placement of the phrases?
»» Has everyone completed
the task?
Checking Understanding
KEY: » next step
»» If a student disagrees they
must give a reason. Brief
class discussion to achieve
consensus.
»» Walk around to see what
students are writing down.
Some students may have
difficulty here changing
percentages or decimals
to fractions or vice versa,
and as you walk around
identify and guide those
students.
»» Students may leave the
phrases until last as they
are imprecise.
»» To keep the diagram
clear you could put all
percentages in a line,
fractions underneath on
another line and then
phrases on another line.
Teacher’s Support and
Actions
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 1: Introduction to Probability
13
© Project Maths Development Team 2009
»» Now write down
three ideas you
have learned about
probability and at
least one question.
»» Give 3 numerical
representations of a
50/50 chance?
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• 0.5, 1/2, 50%
• %, fractions and decimals
between 0 and 1
KEY: » next step
• student answer/response
4. Not everything in mathematics is
certain!
3. Probability can be represented by
a fraction or a decimal between
0 and 1 or by a percentage e.g.½,
0.5, 50%.
2. The Probability scale is between 0
and 1.
1. Probability is about uncertainty
and how to assign numbers to
uncertainty (phrases being too
imprecise) given some information
about the particular situation.
»» Ask a student who was
previously unsure to call
out and explain how he/
she did the conversions
from one form to another.
»» Place students who can
do this competently
with students who have
difficulty. The better
student can have a
supporting role when the
other student is asked to
explain his/her answer.
»» Now think how you
have represented
probability
numerically.
¾= 75%=0.75
0.375=3/8=37.5%
87.5%=7/8=0.875
0.125=1/8=12.5%
0.25=1/4=25%
»» Are students aware of these ideas?
•
•
•
•
•
Checking Understanding
Reflection
»» For each of
the numerical
representations of
probability in Box
A write it in the
two other possible
forms, for example
as a fraction/decimal/
percentage.
Student Learning Tasks: Student Activities: Possible Teacher’s Support and
Teacher Input
and Expected Responses
Actions
Teaching & Learning Plan 1: Introduction to Probability
14
Teaching & Learning Plan 1: Introduction to Probability
Student Activity 1
Student Activity 1A
Area of
Uncertainty
Certain not to
happen
Certain to happen
1.
1.
2.
2.
3.
3.
Student Activity 1B
Phrases used to describe uncertainty
1.
2.
3.
4.
5.
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Teaching & Learning Plan 1: Introduction to Probability
Student Activity 2
Student Activity 2A
Student Activity 2B
Phrases used to describe uncertainty
(examples from the class)
An event associated with each phrase
(examples from the class)
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
10.
10.
11.
11.
12.
12.
Student Activity 2C
Order the above phrases on the scaled line segment below – from least likely to most likely.
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16
Teaching & Learning Plan 1: Introduction to Probability
Student Activity 3
Student Activity 3A
The Probability Scale
Student Activity 3B
The Probability Scale
0
1
Box A
Extremely
unlikely
1/4
Probability of
getting an odd
number when
rolling a die
50:50
1
3/8
87.5%
0.25
0.125
1 in 4
chance
Extremely
likely
75%
0
¾
Certain
Equally
likely
1/2
Impossible
Place the above phrases, numbers and percentages at the correct position
on the probability scale.
Find and write down instances from TV, radio, or in the newspaper
which illustrate how probability affects people’s lives.
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