Teaching and learning plan on tossing coins

Teaching and learning plan on tossing coins
Teaching & Learning Plans
Plan 4: Outcomes of Coin Tosses
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 4:
Outcomes of Coin Tosses
Aims
• To learn that by tossing two coins and observing the results (two heads,
two tails, one head and one tail) not all results are equally likely
Prior Knowledge
Students should have prior knowledge (from T&L Plan 1 and/or from primary school)
of some terms associated with chance and uncertainty. They should be familiar with
probability expressed as fraction or decimal in the range 0 to 1, or as a percentage in
the range 0% to 100%.
Learning Outcomes
As a result of studying this topic, students will be able to
• list the two possible outcomes when tossing a coin and calculate the
probability of getting a head or a tail
• list all the possible outcomes when tossing 2 coins
• relate the number of outcomes to the fundamental principle of counting
• determine the probability of an event using the results of an experiment
and use this to predict the result of a repetition of the experiment, for
equally likely outcomes
• use the term ‘independent events’
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1
Teaching & Learning Plan 4: Outcomes of Coin Tosses
Relationship to Junior Certificate Syllabus
Sub-topics
Ordinary Level
Higher Level
1.5 Counting
List all possible outcomes of an
experiment.
Apply the fundamental
principle of counting.
1.6 Concepts of Estimate probabilities from
probability experimental data.
Recognise that, if an
experiment is repeated, there
will be different outcomes and
that increasing the number
of times an experiment is
repeated generally leads to
better estimates of probability.
1.7 Outcomes
of simple
random
processes
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.
Use binary/counting methods
to solve problems involving
successive events where only
two possible outcomes apply
to each event.
Relationship to Leaving Certificate Syllabus
Sub-topics
Foundation
Level
1.1 Counting
List outcomes of an
experiment.
Apply the
fundamental
principle of
counting.
1.2 Concepts of
probability
Ordinary Level
Estimate probability from experimental
data; appreciate that, if an experiment is
repeated, there will be different outcomes
and that increasing the number of times
an experiment is repeated generally leads
to better estimates of probability.
Associate the probability of an event with
its long run frequency.
Calculate expected value.
Resources Required
Coins for students.
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2
• ½
© Project Maths Development Team 2009
»» Fill in on Student Activity
1B.
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• 15 heads, 15 tails
»» If you were to toss a coin
30 times predict and write
down how many heads you
would expect to get and
how many tails.
»» Fill in on Student Activity
1A.
• ½
»» Given that there are 2
possible outcomes, heads
and tails, what is the
probability of getting a
head when you toss a coin?
Probability of a tail?
»» Are the outcomes equally
likely and why?
• Yes because of the
symmetry of the coin.
• 2 – a head and a tail
»» Toss a coin. How many
outcomes are there and
what are they?
»» Fill in the Table on Student
Activity 1.
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
• student answer/response
»» Can students work out the
expected value?
»» Can all students fill out
their tables correctly?
Checking Understanding
KEY: » next step
»» Circulate checking written
predictions and then ask a
student for an answer.
»» Circulate to see if students
are filling in the table
correctly and ask a student
for the answer.
»» Emphasise that students
must think about this
before calculating
probabilities.
»» Distribute Student Activity
1.
»» Distribute some coins to a
few students first.
Teacher’s Support and
Actions
Lesson Interaction
Teaching & Learning Plan 4: Outcomes of Coin Tosses
3
»» Results should be close to
predicted values.
Student Activities: Possible
and Expected Responses
»» Are students keeping
tallies correctly?
Checking
Understanding
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KEY: » next step
• student answer/response
»» Do students
understand that
the events are
independent of each
other?
• No because the outcomes
are independent – the
outcome of one trial does
not influence the outcome
of any other trial.
»» If I get four heads in a row does
that increase the chances of
getting a tail next time?
»» Can you think of any other real
life situations which have only 2
possible outcomes?
»» Do students see the
connection between
relative frequency
and probability?
• Because of the larger
number of trials the
relative frequency
is approx. equal to
probability.
»» Enter results from the whole
class on the board.
»» Were most students
»» Circulate and look at tables
able to make correct
of results, asking different
predictions?
groups how many heads and
tails they got.
»» Circulate and remind
students about the time
limit, checking progress.
»» Distribute coins to the rest
of the class.
Teacher’s Support and
Actions
»» Why are the results now closer
to the predicted values?
»» Check against your prediction.
Complete in Student Activity 1F.
»» Fill results into your own master »» Results should show closer
agreement with predicted
table Student Activity 1E and
values.
complete the table.
»» Compare your result with your
prediction and fill in Student
Activity 1 D.
»» Now, working in pairs, toss a
coin 30 times and fill in the
table on Student Activity
1C. You have a time limit of
________.
Student Learning Tasks: Teacher
Input
Teaching & Learning Plan 4: Outcomes of Coin Tosses
4
• HH,TT,HT,TH
• Yes, as the coins are
symmetrical
»» Given 2 coins make a list of
all the possible outcomes
you could get using the
table in Student Activity
2A.
»» Are all outcomes equally
likely?
© Project Maths Development Team 2009
• A head and a tail (HT or
TH) – 14 times
• TT- 7 times
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• Predict and write down the
outcomes you might get
if you were to toss 2 coins
simultaneously 28 times.
(Why didn’t I pick 30?)
• HH – 7 times
• Probability of head and tail
= 2/4 =1/2
»» the probability of
getting a head and a
tail?
»» Fill in on Student
Activity 2B.
• Probability of 2 tails = ¼
»» the probability of
getting 2 tails?
»» Using the information from
the table:
»» what is the probability
• Probability of 2 heads =¼
of getting 2 heads?
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
»» Circulate and check
predictions asking for
justifications.
KEY: » next step
• student answer/response
»» Can students calculate the
theoretical probability
given equally likely
outcomes?
»» Have they got 4 possible
outcomes?
»» Circulate and check the list
of outcomes, questioning
where necessary.
• Emphasise equally likely
outcomes
»» Do students see HT as a
different outcome to TH?
Checking Understanding
»» Distribute Student Activity
2.
Teacher’s Support and
Actions
Teaching & Learning Plan 4: Outcomes of Coin Tosses
5
»» Students should get approx
twice as many H and T
combinations as HH or TT.
»» Working in pairs, toss 2
coins simultaneously 28
times and record your
results in table (Student
Activity 2C).
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• Coin 1 has 2 possible
outcomes and coin 2 has
2 possible outcomes so
total number of possible
outcomes is 2x2=4.
»» How does this fit in with
what you learned in the
fundamental principle of
counting?
© Project Maths Development Team 2009
• Since one head and one tail
can occur as HT or TH, the
number of outcomes for
HH and TT should be half
that for a head and a tail.
»» The table only gives one
option for heads and tails
combination, while the
original table showed HT
and TH. Will this make a
difference to the results?
Why?
• student answer/response
»» Can students relate back to
the fundamental principle
of counting?
»» Can students see that
as the number of trials
increases the relative
frequency approaches the
probability?
»» Have students been able to
make correct predictions
and justify them?
»» Can students see that there
are 2 possible ways of
getting a head and a tail?
Checking Understanding
KEY: » next step
»» Enter results from the
whole class on the board.
»» Results should now be
closer to predictions.
»» Fill in results on master
table Student Activity 2D
from the board.
»» Are these results in
agreement with your
predictions?
»» Ask a number of groups to
call out their results while
walking around check
others.
»» Results should be close to
predictions.
Teacher’s Support and
Actions
»» Do your results agree with
your predictions?
»» Your time limit is________.
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 4: Outcomes of Coin Tosses
6
• HH—250
»» If you tossed 2 coins
simultaneously 1000 times
how many of each of the
possible outcomes would
you expect?
© Project Maths Development Team 2009
3. Calculate expected value
for a large number of trials.
»» Write down any questions
you may have.
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4. Note that the outcome
of any one trial does not
affect the outcome of any
other trial – independent
events.
2. How to use the list of all
possible outcomes to judge
fairness.
1. How to list all possible
outcomes for tossing 1 coin
and tossing 2 coins
»» Write down anything you
found difficult.
Reflection
»» Write down 3 things you
learned about probability
today.
»» Is a game ‘fair ‘or ‘unfair’ if • Unfair! B has twice the
A wins if he gets two heads
probability of winning.
and B wins if the outcome
is one head and one tail.
• Make it fair by allowing A
to win for HH or TT.
»» How would you make the
game ‘fair’?
• A head and a tail – 500
• TT—250
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
• student answer/response
»» Are they using the
terminology with
understanding and
communicating with each
other using these terms?
»» Have all students learned
and understood these
items?
»» Do students recognise
the need for a numeric
representation of the
phrase “probably won’t”?
»» Are students able to
calculate the expected
value given the
probability?
Checking Understanding
KEY: » next step
»» Circulate and take
note particularly of any
questions students have
and help them to answer
them.
Teacher’s Support and
Actions
Teaching & Learning Plan 4: Outcomes of Coin Tosses
7
Teaching & Learning Plan 4: Outcomes of Coin Tosses
Student Activity 1
Student Activity 1A
Tossing a coin
Possible outcome
Number of times it can appear
Probability of getting a head________________
Probability of getting a tail________________
Student Activity 1B
Tossing a coin 30 times
I predict that I will get _________________ heads and _________________ tails.
Student Activity 1C
Results of the experiment
Outcome
Tally
Frequency
Relative frequency
Student Activity 1D
Do the results agree with my predictions?
Student Activity 1E
Master table of class results
Outcome
Heads
Tails
Counts from each group
Frequency
Relative frequency
Student Activity 1F
Do the results agree with my predictions?
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8
Teaching & Learning Plan 4: Outcomes of Coin Tosses
Student Activity 2
Student Activity 2C
Tossing 2 coins simultaneously: List all the
possible outcomes in the table.
Coin 1
Coin 2
To help you visualise what’s happening list the
possible outcomes on the diagrams below.
H
Coin 1
Coin 2
Student Activity 2C
Using the information from the table,
What is the probability of getting 2 heads?________________________________
What is the probability of getting 2 tails?________________________________
What is the probability of getting one head and one tail?________________________________?
Student Activity 2C
Table for experiment
Outcome
HH
HT or TH
TT
Tally
Frequency
Student Activity 2C
Master table for experiment
Outcome
HH
HT or TH
TT
Frequency (from each group)
© Project Maths Development Team 2009
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Total
Relative frequency
9
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