Teaching and learning plan on rolling two dice

Teaching and learning plan on rolling two dice
Teaching & Learning Plans
Plan 3: Fair Trials with Two Dice
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 3:
Fair Trials with Two Dice
Aims
• To learn that not all results are equally probable
• To consider a given set of rules and the outcomes that produce a ‘win’ for
one of two players and to be able to determine whether the game is ‘fair’,
i.e. whether each of the two players has the same chance of winning
• To consider the theoretical probability of each result and devise a set of
rules to make the game ‘fair’
Prior Knowledge
Students should have prior knowledge (from T and 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
• investigate further what the concept of fairness means in a game with 2
dice
• list all the possible outcomes for throwing 2 dice using a two way table
• relate the number of outcomes to the fundamental principle of counting
• come up with rules for a game which make it fair/unfair
• construct a probability table
• understand the relationship between an event and its complement
• 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
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Teaching & Learning Plan 3: Fair Trials with Two Dice
Relationship to Leaving Certificate Syllabus
Sub-topics
Foundation
Level
Ordinary Level
1.1 Counting
List outcomes of an
experiment.
Count the
arrangements of
distinct objects.
Higher Level
Apply the
fundamental
principle of
counting.
1.2 Concepts of Recognise that
probability probability is a
measure on a scale
of 01- of how likely
an event is to occur.
Discrete
probability
(as relative
frequency)
Probability of
desired outcomes in
problems involving
experiments, such
as, dice throwing.
1.3 Outcomes
of 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.
Recognise the role
of expected value
in decision making
with a focus on fair
games.
Extend the
students’
understanding of
the basic rules of
probability.
Resources Required
Counters, 2 dice and a Game Sheet for each pair of students
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2
Student Activities: Possible
and Expected Responses
© Project Maths Development Team 2009
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»» Students may think that
»» Think about this for a few
since 11 numbers have
minutes and write down
been listed, and since A
which player you think will
can win on 6 of these and
win the game most often if
B only on 5, that A is more
it is played many times, and
likely to win than B.
why you think this.
»» B wins if sum i.e. is 5, 6,
7, 8, 9. Play the game on
Game Sheet 1.
»» A wins if sum (i.e. outcome)
is 2, 3, 4, 10, 11 or 12.
»» Students A and B
alternately roll the die,
each time adding the
scores on each die to get
the outcome. They place a
counter on each outcome.
»» In each group of two, one
person is nominated as A
and one as B. Players A
and B take turns to roll
the die, and the winner is
determined by the sum of
the numbers on the faces
as follows:
Student Learning Tasks:
Teacher Input
Checking Understanding
KEY: » next step
»» Ask for a show of hands on
who thinks A will win and
who thinks B will win.
»» Distribute counters and
Game Sheet 1 and dice to
each pair, checking that
students have a written
prediction and a reason for
it.
• student answer/response
»» Distribute Student Activity »» Can students record their
1 so that students can write
initial results in their
their prediction (Student
copies?
Activity 1A) and keep a
count of the number of
times each player wins
(Student Activity 1B).
Teacher’s Support and
Actions
Lesson Interaction
Teaching & Learning Plan 3: Fair Trials with Two Dice
3
»» The students should find
that A wins only about 1/3
of the time.
»» Students check this against
their prediction, analysing
any difference and fill in
Student Activity 1C.
»» The game is played until
one person has reached
the bottom of the grid on
Game Sheet 1 and students
count the number of times
A won and the number of
times B won and fill out
the Master Sheet on the
board also.
»» When you are finished fill
out Student Activity 1B
and fill in your results on
Master Sheet 1 (results
from the whole class) on
the board.
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• student answer/response
»» Did they react to this as
being unfair?
»» How many students in the
class were able to make
accurate predictions with
valid reasons?
»» Were students able
to conduct the game
successfully and did they
notice B winning much
more often?
Checking Understanding
KEY: » next step
• No. Fair – if A and B have
»» Give students a minute to
an equal chance of winning
think and then ask one
otherwise unfair.
student.
»» Does this game appear to
be fair?
© Project Maths Development Team 2009
• “1 “can never be an
outcome as the smallest
sum possible is 2.
»» As you circulate check
predictions against
outcomes of the
experiment, and ask
students about their
justifications.
»» Place Master Sheet 1
(results from the whole
class) on the board.
»» Check that each pair
understands what they are
doing and occasionally,
as you circulate, ask
“hands up how many A’s
are winning?” and then
“hands up how many B’s
are winning?” so students
are aware of an overall
trend.
Teacher’s Support and
Actions
»» Which number on each
die cannot be a possible
outcome?
»» The relative frequency is
now closer to 1/3.
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 3: Fair Trials with Two Dice
4
© Project Maths Development Team 2009
»» Students fill up the table
as per example in Student
Activity 2A.
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»» Individually, fill in the
2-way table for the sample
space for the sum achieved
on throwing 2 dice
(Student Activity 2A).
»» Try this for a few minutes.
• student answer/response
»» Are all students able to fill
in the table successfully
and with understanding?
»» How many of the students
were able to come up with
a suitable table?
»» Do students understand
the concept of fairness?
Checking Understanding
KEY: » next step
»» Circulate to check that
students understand how
to fill it in.
»» Distribute Student Activity
2.
»» Circulate looking at
student suggestions and
giving helpful hints. On
finding a suitable table
ask students to show and
explain it to the class.
Students may need help
designing the table.
»» Could you design a table to »» Some students will
probably come up with the
show you all the possible
2 way table and show it to
outcomes.
the class.
»» The set of all the possible
outcomes is called the
‘Sample Space’.
»» Give students a minute to
think and then ask one
student.
»» Give students a minute to
think and then ask one
student.
• No, because the first time
4 is from die 1 and the
second time it’s from die 2.
»» In how many ways could an • 4+5
outcome of 9 be achieved? • 5+4
• 3+6
• 6+3
»» Is the outcome 4+5 the
same as the outcome 5+4?
• All outcomes are not
equally likely.
• Students might say that 2
»» Give students a minute to
can only be got 1 way (1+1)
think and then ask one
whereas 9 for example can
student.
be got from (3+6), (4+5).
»» Why is it not fair?
Teacher’s Support and
Actions
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 3: Fair Trials with Two Dice
5
• If there are 6 different
outcomes from die 1,
and 6 different outcomes
from die 2, then the
total number of possible
outcomes is 6x6 = 36.
• 12 of the 36 outcomes give
a win for A whereas 24 of
the outcomes give a win
for B.
»» Students articulate their
misconceptions.
»» Can you relate this back to
the fundamental principle
of counting in a previous
lesson?
»» Going back to the rules of
the game, for how many
outcomes will player A
win and for how many
outcomes will player B
win? Student Activity 2C.
»» Go back to your prediction
again, Student Activity 1C,
and if it was not consistent
with the outcome write
down why it was different.
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• 36.
»» How many possible
outcomes are there?
Student Activity 2B.
© Project Maths Development Team 2009
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
• student answer/response
»» Are students able to
understand in cases where
their predictions were in
error why this was so?
»» Have students been able
to recall and use the
fundamental principle of
counting?
Checking Understanding
KEY: » next step
»» Circulate and note
misconceptions, and ask a
selection of students who
thought A would win to
explain why they thought
that and why they now
think differently.
»» Ask the question and wait
for someone to volunteer
an answer. If it’s the same
person answering all the
time, ask another student,
leading them to the
answer if they don’t know
it, by referring back to an
example they did when
learning the fundamental
principle of counting.
»» Circulate and look at the
answers being filled in,
asking questions where
necessary.
Teacher’s Support and
Actions
Teaching & Learning Plan 3: Fair Trials with Two Dice
6
© Project Maths Development Team 2009
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»» In the theoretical approach, when
throwing a die, there are six possible
outcomes which are all equally likely
to appear due to the symmetry of
the die, and given that it is a fair die
(not loaded) so we have no reason to
assume that any number will appear
more often than another. A2 is one
of the 6 possible outcomes, so the
likelihood of it turning up is 1 out of
6.
»» The approach in Teaching & Learning
Plan 2, and above in Student Activity
1B, is known as the ‘experimental’ or
‘empirical’ approach to calculating
probabilities. In this case the
probability of an event is the value that
the relative frequency tends to in an
infinite number of trials.
• For example. Player A
wins if the outcome is
2, 3, 4, 5, 8 or 10 and B
wins if the outcome is
6, 7, 9, 11 or 12.
»» Different groups
present their new rules
for class approval.
»» Write down new rules for the game,
Student Activity 2D, which will make
it fair and an explanation of why it is
now fair.
»» Is more than one set of rules possible?
Student Activities:
Possible and Expected
Responses
Student Learning Tasks: Teacher Input
• student answer/response
»» Are students able to
make up more than one
set of rules satisfying
the criterion for a fair
game?
Checking Understanding
KEY: » next step
»» Circulate and when
groups are finished ask
a couple of groups to
explain their new rules.
Teacher’s Support and
Actions
Teaching & Learning Plan 3: Fair Trials with Two Dice
7
• As the number of trials
increases the relative
frequency becomes almost
equal to the theoretical
probability.
• How many times it
occurs divided by the
total number of possible
outcomes.
• All outcomes are equally
likely.
»» What is the relationship
between the experimental
approach to calculating
probability and the
theoretical approach?
»» How do we calculate the
theoretical probability
of each outcome in this
sample space?
»» What assumption are we
making?
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»» What is the sum of the
• 1
probabilities for the sample
space?
»» Assuming that both of the • Using the 2 way table
(Student Activity 2A)
dice are fair, and all 36
showing the sample space
outcomes are equally likely,
constructed for the game,
what is the probability of
the sum being 5?
there are 4 outcomes for
5, so the probability of
getting a 5 is 4/36.
»» Construct a probability
table (Student Activity 3)
for the sum of 2 dice using
Student Activity 2A.
»» Are all 36 outcomes equally • Yes.
likely here?
Student Activities: Possible
and Expected Responses
Student Learning Tasks:
Teacher Input
• student answer/response
»» Did all students get 1?
»» Did students fill out the
table correctly?
»» Can students recall how to
calculate probability for
equally likely outcomes?
Checking Understanding
KEY: » next step
»» Ask students if they
remember this happening
in the last lesson with
outcomes from 1 die?
»» Walk around observing
students as they fill out the
probability table.
»» Asks students to recall and
then after a short pause
asks an individual.
Teacher’s Support and
Actions
Teaching & Learning Plan 3: Fair Trials with Two Dice
8
© Project Maths Development Team 2009
»» Fill in answer on Student
Activity 3.
• student answer/response
»» Do students see that
the experimental results
approach the theoretical
values for probability as
the number of games
increases?
»» Do students understand
the relationship between
an event and its
complement?
»» How many students got
the correct answer without
having to add up all the
other probabilities?
Checking Understanding
KEY: » next step
»» Circulate checking that
Student Activity 3 is
being completed, asking
questions of students who
are having difficulty.
• They are very close and the
relative frequency becomes
closer to the theoretical
probability as the number
of games increases.
»» Compare this with
experimental result for
relative frequency.
»» Fill in answer on Student
Activity 3.
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»» If a student cannot get the
answer without adding up
all the other probabilities,
tell them to do that first
and then see if they can
then find the other way by
inspecting their answer.
• 1-1/3=2/3
»» Use the probability that
A wins to calculate the
probability that B wins,
without adding.
»» Fill in answer on Student
Activity 3.
»» For students who have
difficulty: ask them to
count how many of the
outcomes give A a win, and
how many outcomes are
there in total.
• 12/36=1/3
»» Going back to the original
rules for the game, what
is the probability that A
wins?
Teacher’s Support and
Actions
»» If a student cannot get the
answer without adding up
all the other probabilities,
tell them to do that first
and then see if they can
then find the other way by
inspecting their answer.
Student Activities: Possible
and Expected Responses
»» If the probability of getting • 1-2/36=34/36
a 3 is 2/36, what is the
probability of not getting a
3 without adding up all the
other probabilities? Write
down the answer.
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 3: Fair Trials with Two Dice
9
Student Activities: Possible
and Expected Responses
© Project Maths Development Team 2009
»» Write down any questions
you may have.
»» Write down anything you
found difficult.
»» Write down 3 things you
learned about probability
today.
Reflection
»» Emphasise that you are
dealing with equally likely
outcomes.
Teacher’s Support and
Actions
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• 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?
»» Can students calculate an
expected value for a large
number of trials?
Checking Understanding
KEY: » next step
• How to:
»» Circulate and take
1. list all possible outcomes
note particularly of any
(sample space) for
questions students have
throwing 2 dice
and help them to answer
2. use the list of all possible
them.
outcomes to judge fairness
3. construct a probability
table
4. calculate the complement
of an event
5. calculate expected value
for a large number of trials.
»» If the probability of a sum
• 100x1/6 i.e. approx 17
of 7 occurring is 6/36=1/6,
(1/6 of the time you expect
how many 7’s would you
a 7.)
expect to get if the dice are
tossed 100 times?
Student Learning Tasks:
Teacher Input
Teaching & Learning Plan 3: Fair Trials with Two Dice
10
Teaching & Learning Plan 3: Fair Trials with Two Dice
Game Sheet 1
1
A wins A wins A wins B wins B wins B wins B wins B wins A wins A wins A wins
2
3
4
5
6
7
8
9
10
11
12
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Teaching & Learning Plan 3: Fair Trials with Two Dice
Student Activity 1
Student Activity 1A
Prediction
Player _____ will win most often because:
Student Activity 1B
Play the game and record the results below:
Use Tally Marks to Help Total (Frequency)
You Keep a Score
Relative Frequency
(total no. of wins/total
no. of games)
Player A Wins
Player B Wins
Totals
Student Activity 1C
Did your predicted results agree with your actual results?
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Teaching & Learning Plan 3: Fair Trials with Two Dice
Master Sheet 1
(Results from the Whole Class)
Play the game and record the results below:
Use Tally Marks to Help Total (Frequency)
You Keep a Score
Player A Wins
Relative Frequency
(total no. of wins/total
no. of games)
2+3+...
Player B Wins
Totals
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13
Teaching & Learning Plan 3: Fair Trials with Two Dice
Student Activity 2
Student Activity 2A
Complete the table to show all of the possible outcomes.
Two way table showing the sample space.
Number
thrown
1
1
2
3
4
5
6
1+1=2
2
3
4
5
5+2=7
6
Student Activity 2B
How many possible outcomes are there?__________________
Student Activity 2C
Original Rules:
Player A wins when the sum is 2, 3, 4, 10, 11 or 12.
Player B wins when the sum is 5, 6, 7, 8 or 9.
For how many outcomes will player A win?__________________
For how many outcomes will player B win?__________________
Student Activity 2D
New Rules:
Player A wins when the sum is __________________________
Player B wins when the sum is __________________________
Why I chose these new rules:
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Teaching & Learning Plan 3: Fair Trials with Two Dice
Student Activity 3
Sum
Frequency
2
1
Probability
no. of outcomes in the event
=
no. of outcomes in the sample event
1
36
3
4
5
6
7
8
9
10
11
12
© Project Maths Development Team 2009
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15
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