Unit Plan Assignment
Margaret Cooper, Daljit Dhaliwal, & Ross Dolan
PED 4187
Professor Lubna Al-Shaama
February 26, 2010
Course: Mathematics of Data Management (MDM4U)
Grade: 12
Unit: Counting and Probability
Prerequisite: Functions, Grade 11, University Preparation, or Functions and Applications,
Grade 11, University/College Preparation. (Also see Figure 1)
Time: Unit plan for 20 hours (16 classes at 75 minutes each)
Course Description:
This course broadens students’ understanding of mathematics as it relates to managing
data. Students will apply methods for organizing and analysing large amounts of information;
solve problems involving probability and statistics; and carry out a culminating investigation
that integrates statistical concepts and skills. Students will also refine their use of the
mathematical processes necessary for success in senior mathematics. Students planning to
enter university programs in business, the social sciences, and the humanities will find this
course of particular interest.
Figure 1: Mathematics pathways with minimum recommended entrance grades. Created by
Ross Dolan and taken with permission from the Merivale High School mathematics website.
Mathematical Process Expectations:
The mathematical process expectations of Problem Solving, Reasoning and Proving,
Reflecting, Selecting Tools and Computational Strategies, Connecting, Representing, &
Communicating will be integrated into student learning throughout this unit.
Big Ideas of the Unit: (Nelson 2003)
 The sample space of an experiment is the collection of all possible outcomes of the
experiment.
 The event space is the collection of all outcomes of an experiment that corresponds to a
particular event.
 Venn diagrams can be used to organize and solve counting problems.
 The probability of an event occurring can be experimental or theoretical.
 The experimental probability of an event occurring is calculated using the ratio number
of times the desired event occurred: total number of trials.
 The expected value of a discrete random variable is estimated by averaging its values
over a large number of trials.
 The Additive Principle is used to determine the probability of either of two events
happening (one event OR the other event)
 The Multiplicative Principle is used to calculate the probability of independent events
and for calculating the conditional probability of an event B occurring, given that event A
has occurred.
 Factorial notation is useful when using counting techniques.
 Formulas for permutations (e.g.
matters.
) pertain to situations where order
 Formulas for combinations (e.g.
) pertain to situations where order
does not matter.
 Simulations can be designed and carried out to estimate the theoretical probabilities of
situations for which the theoretical probability is difficult or impossible.
Unit Description: (OAME)
In this unit, students will:
 solve problems involving the probability of distinct events;
 solve problems using counting techniques for distinct items;
 apply counting principles to calculating probabilities;
 explore variability in experiments;
 demonstrate understanding of counting and probability problems and solutions by
adapting/creating a children’s story/nursery rhyme in a Counting Stories project;
 explore a significant problem of interest in preparation for the Culminating
Investigation.
Profile/Needs of the Learner:
Compared to other Grade 12, University Preparation courses, the pathway to this course creates
a more diverse group of students. Basic concepts of probability are introduced in Grade 8 Mathematics
so starting with Counting and Probability allows students to build on their prior knowledge of
probability. This offers students, who may not have excelled in more algebraic approaches, an
opportunity for success early in the course. Building on this prior knowledge using a more formal
mathematical approach, will begin to train students to use more complex mathematical processes. The
concepts behind Universal Design for Learning (Ontario Ministry of Education 2005) will be implemented
in the design of unit activities. This approach to teaching incorporates activities to target the three
different learning styles (visual, auditory, and kinaesthetic). The diverse student population in the
course makes it particularly important to utilize a large variety of approaches and activities. These will
include hands-on activities, the use of a variety of technologies, and student centered inquiry activities.
Social interaction and communication of ideas is integral to the development of mathematical
understanding. As such, opportunities for collaboration and communication will be provided.
Overall Expectations:
A1. Solve problems involving the probability of an event or a combination of events for discrete
sample spaces;
A2. Solve problems involving the application of permutations and combinations to determine
the probability of an event.
E1. Design and carry out a culminating investigation that requires the integration and
application of the knowledge and skills related to the expectations of this course.
E2. Communicate the findings of a culminating investigation and provide constructive critiques
of the investigations of others.
Specific Expectations:
A1.1 recognize and describe how probabilities are used to represent the likelihood of a result of
an experiment (e.g., spinning spinners; drawing blocks from a bag that contains different
coloured blocks; playing a game with number cubes; playing Aboriginal stick-and-stone games)
and the likelihood of a real-world event (e.g., that it will rain tomorrow, that an accident will
occur, that a product will be defective)
A1.2 describe a sample space as a set that contains all possible outcomes of an experiment, and
distinguish between a discrete sample space as one whose outcomes can be counted (e.g.,
all possible outcomes of drawing a card or tossing a coin) and a continuous sample space
as one whose outcomes can be measured (e.g., all possible outcomes of the time it takes to
complete a task or the maximum distance a ball can be thrown)
A1.3 determine the theoretical probability, P (i.e., a value from 0 to 1), of each outcome of a
discrete sample space (e.g., in situations in which all outcomes are equally likely),
recognize that the sum of the probabilities of the outcomes is 1 (i.e., for n outcomes,
P + P + P +… + P = 1), recognize that the probabilities P form the probability distribution
associated with the sample space, and solve related problems
A1.4 determine, through investigation using class generated data and technology-based
simulation models (e.g., using a random-number generator on a spreadsheet or on a graphing
calculator; using dynamic statistical software to simulate repeated trials in an experiment),
the tendency of experimental probability to approach theoretical probability as the number
of trials in an experiment increases (e.g., “If I simulate tossing two coins 1000 times
using technology, the experimental probability that I calculate for getting two tails on
the two tosses is likely to be closer to the theoretical probability of than if I simulate
tossing the coins only 10 times”)
A1.5 recognize and describe an event as a set of outcomes and as a subset of a sample space,
determine the complement of an event, determine whether two or more events are mutually
exclusive or non-mutually exclusive (e.g., the events of getting an even number or getting an
odd number of heads from tossing a coin 5 times are mutually exclusive), and solve related
probability problems [e.g., calculate P(~A), P(A and B), P(A or B)] using a variety of strategies
(e.g., Venn diagrams, lists, formulas)
A1.6 determine whether two events are independent or dependent and whether one event is
conditional on another event, and solve related probability problems [e.g., calculate P(A and B),
P(A or B), P(A given B)] using a variety of strategies (e.g., tree diagrams, lists, formulas)
A2.1 recognize the use of permutations and combinations as counting techniques with
advantages over other counting techniques (e.g., making a list; using a tree diagram; making a
chart; drawing a Venn diagram), distinguish between situations that involve the use of
permutations and those that involve the use of combinations (e.g., by considering whether or
not order matters), and make connections between, and calculate, permutations and
combinations.
A2.2 solve simple problems using techniques for counting permutations and combinations,
where all objects are distinct, and express the solutions using standard combinatorial notation
[e.g., n!, P(n, r),
( )]
A2.3 solve introductory counting problems involving the additive counting principle (e.g.,
determining the number of ways of selecting 2 boys or 2 girls from a group of 4 boys and
5 girls) and the multiplicative counting principle (e.g., determining the number of ways of
selecting 2 boys and 2 girls from a group of 4 boys and 5 girls)
A2.4 make connections, through investigation, between combinations (i.e., n choose r) and
Pascal’s triangle [e.g., between ( ) and diagonal 3 of Pascal’s triangle]
A2.5 solve probability problems using counting principles for situations involving equally
likely outcomes2. Solving Problems Using
E1.1 pose a significant problem of interest that requires the organization and analysis of a
suitable set of primary or secondary quantitative data (e.g., primary data collected from a
student-designed game of chance, secondary data from a reliable source such as E-STAT),
and conduct appropriate background research related to the topic being studied
E2.2 Present a summary of the culminating investigation to an audience of their peers within a
specified length of time, with technology (e.g. presentation software) or without technology
E2.3 Answer questions about the culminating investigation and respond to critiques (e.g., by
elaborating on the procedures; by justifying mathematical reasoning)
E2.4 Critique the mathematical work of others in a constructive manner.
Schedule of Lessons and Assessments:
Day
Lesson Title
Summary of Activities
1
Introduction
to Probability
2
Probability
and “Throw
Sticks”
3
Using
Simulations
4
“And”, “Or”
events
5
Pick the Die
6
Let’s Make a
Deal and
Conditional
 Investigate probabilities of distinct events when all
outcomes have equal probability (e.g. coin toss, dice
roll)
 Introduce vocabulary and concepts (e.g. outcomes,
events, trials, experimental probability, theoretical
probability).
 Reflect on the differences between experimental and
theoretical probability and assess the variability in
experimental probability
 Recognise that the sum of the probabilities of all
possible outcomes in the sample space is 1.
 Students will play an Apache stick and stone game in
order to generate data.
 This data will be used to investigate probabilities of
distinct events (outcomes, events, trials,
experimental probability, theoretical probability
 Develop some strategies for determining theoretical
probability (e.g., tree diagrams, lists)
 Use reasoning to develop a strategy to determine
theoretical probability
 Use mathematical simulations to determine if games
are fair
 Reflect on how simulations can be used to solve real
problems involving fairness
 Determine whether two events are dependent,
independent, mutually exclusive or non-mutually
exclusive
 Verify that the sum of the probabilities of all possible
outcomes in the sample space is 1.
 Use non-transitive dice to compare experimental and
theoretical probability and note the tendency of
experimental probability to approach theoretical
probability as the number of trials in an experiment
increases
 Draw tree diagrams for events where the branches in
the tree diagram do not have the same probability
 Use the Monty Hall problem to introduce conditional
probability
 Use Venn diagrams to organize data to help
Curriculum
Expectations
A1.1, A1.2,
A1.3, A1.5
A1.1, A1.2,
A1.3,
A1.5
A1.1, A1.2,
A1.4
A1.3, A1.5,
A1.6
A1.4, A1.6
A1.6
Probability

7
Counting

Arrangements
and
Selections


8
Counting
Permutations


9
Counting
Combinations


determine conditional probability
Use a formula to determine conditional probability
Solve problems that progress from small sets to more A2.1
unwieldy sets and using lists, tree diagrams, role
playing to motivate the need for a more formal
treatment.
See examples where some of the distinct objects are
used and where all the distinct objects are used.
Discuss how counting when order is important is
different from when order is not important to
distinguish between situations that involve, the use
of permutations and those that involve the use of
combinations.
Develop, based on previous investigations, a method A2.1, A2.2
to calculate the number of permutations of all the
objects in a set of distinct objects and some of the
objects in a set of distinct objects.
Use mathematical notation (e.g., n!, P(n, r)) to count.
A2.1, A2.2
Develop, based on previous investigations, a method
to calculate the number of combinations of some of
the objects in a set of distinct objects.
Make connection between the number of
combinations and the number of permutations.
 Use mathematical notation (e.g., (
10
Introducing
the Counting
Stories
Project
11
Pascal’s
Triangle
)) to count
 Ascribe meaning to. ( ). ( ) .( )
 Solve simple problems using techniques for counting
permutations and combinations, where all objects
are distinct.
 Introduce and understand one culminating project,
Counting Stories Project (e.g. student select children’s
story/nursery rhyme to rewrite using counting and
probability problems and solutions as per Strand A).
 Create a class critique to be used during the
culminating presentation.
• Pascal’s triangle
- Investigating Pascal’s triangle numerically
n
- Investigating Pascal’s triangle using  
r
- Relationship between patterns and combinations
- Investigating pathway problems
• Examples of patterns and combinations (From:OAME-
A1.1, A1.3,
A1.5, A1.6,
A2.1, A2.2,
A2.3, A2.4,
A2.5, E2.3,
E2.4
A2.4
MDM4U document)
- Number of ways of choosing elements:
 n
n
 n
n  n 
   1;    1;    1;    
; etc
 n
0
1
n n  r 
• The total number of selections of r elements from n
elements is made up of selections that either include
a particular element or not.
E.g.
 n   n  1  n  1
   
  

r
r

1
r
  
 

12
Mixed
Counting
Problems
13
Counting
Stories
Project
14
Probability
15
Counting
Stories
Project
16
Culminating
Investigation
• Review permutations and combinations
- Distinguish between the use of permutations and
combinations
- Make connections between situations involving their
use
• Solve counting problems using counting principles –
additive, multiplicative
 Ask students to work on counting stories project.
(Example: Goldilocks)
 Counting techniques and probability strategies
- When order does not matter
- When order matters
 Solve probability problems using counting principles
involving equally likely outcomes
 Students will complete counting stories project.
• Students identify a significant problem of interest for
Culminating Investigation
- Identify problem (area of interest) and formulate
research question
- Pose problem
A2.3
A1.1, A1.3,
A1.5, A1.6,
A2.1, A2.2,
A2.3, A2.4,
A2.5, E2.3,
E2.4
A2.5
A1.1, A1.3,
A1.5, A1.6,
A2.1, A2.2,
A2.3, A2.4,
A2.5,
E2.3,
E2.4
E1.1
- Find sources of data
- Determine importance and relevancy of data
- Design plan to study the problem
• Brainstorm ideas, e.g., mind mapping, for organization
and analysis of data related to a significant problem.
Rationale of Design:
Various strategies will be used to reinforce learning:
 Hands on activities and real world applications
 Visualization
 Shared math
 Independent math
 Conferencing (in-class discussions)
 Word wall
 Exploring differentiated strategies
 Think-pair and share
Assessment Strategies:
 Curriculum expectations will be informally assessed through observation of student
participation in group activities.
 All formal assessment of curriculum expectations will be done through the evaluation of
projects and investigations. This is a more authentic form of assessment than a
traditional unit test. Students will carry out a counting stories project to demonstrate
their learning of the material from the unit. This will also give them practice in carrying
out and communicating a project. The feedback from this project can be used when
carrying out their culminating investigation. This culminating investigation project will
be introduced at the end of this unit so that they have time to formulate a solid question
and collect or find data to answer it. Three full class periods will be provided for
students to work on the Counting Stories Project and another period will be provided to
start their Culminating Investigation.
Student input and choice will be accommodated wherever possible:
 This unit includes the creation/adaptation of children’s story or nursery rhyme in a
counting stories project. Students will be able to choose whatever story they want. If
students are feeling creative, they can even write their own story.
 As part of culminating investigation for this topic, students will formulate and explore a
problem in their area of interest.
Strategies for Students with Exceptionalities:
 The adaptations and modifications required for students with IEP’s will be met.
 Students identified as autistic will be taught depending upon their best learning style.
 Students that are visually impaired will be provided with the handouts and other
material with larger print. Lessons will be tape recorded and provided to the student. For
the counting stories and culminating activities, they can record their thoughts and ideas
(e.g. using Kurzweil, tape recorder etc).
 Use of visual aids would be helpful for students identified with a hearing impairment.
 Students with ESL needs will be provided with additional help, visual and audio cues to
increase their learning.
 Students identified as gifted will be provided with additional information and challenging
problems
Lesson Outlines:
Lesson 1: Introducing Probability
Curriculum Expectations: A1.1, A1.2, A1.3, A1.5
Approach:
Use coin tosses and dice rolls to discuss experimental and theoretical probability. Discuss the
relevance of sample size and variability in experimental probability. Introduce vocabulary such
as trial, experiment, sample space, event space, and outcome. Introduce idea of uniform
distribution. The fact that the sum of the probabilities of all possible outcomes in the sample
space is 1 will be introduced.
Group Exercise:
In pairs, students will play the game HOPPER (TIPS4M exercise 1.2.1 The HOPPER Game) and
tally results to generate experimental probabilities. Results will be pooled for the class.
Consolidation:
HOPPER will be debriefed using a tree diagram in order to discuss the theoretical probabilities
(when probability of each even is the same). The probability of complementary events will be
discussed and the idea of fair games introduced.
Lesson 2: Introducing Probability and “Throw Sticks”
Curriculum Expectations: A1.1, A1.2, A1.3, A1.5
Approach:
Further explore probability by having students play an Apache stick and stone game (found at
http://illuminations.nctm.org/LessonDetail.aspx?id=L585) to generate data. Tree diagrams will again be
used to discuss theoretical probability.
Group Exercise:
Students will generate and summarize data from their games. They will then develop strategies
to determine the theoretical probabilities of all the possible outcomes.
Consolidation:
Debrief the game using tree diagrams and lists to identify all possible outcomes in the sample
space. The theoretical probability of each outcome will be determined and compared to the
point value associated with the outcome in the game. Students will then have to re-design the
game so that the two coincide. Students will then be introduced to the concept of expected
value.
Lesson 3: Using Simulations
Curriculum Expectations: A1.1, A1.2, A1.4
Approach:
Mathematical simulations will be used to determine if games are fair. Students will then design
a simulation to solve a real problem involving fairness.
Group Exercise:
Use the TIPS4M exercise where students, in pairs work through simulations of “Rock, Paper,
Scissors” and World Series data to determine whether each are fair games.
Consolidation:
Debrief World Series activity and ask students whether or not the data suggest the games are
rigged. Take the use of simulations to uncover fraud further by introducing data on Lottery
fraud (i.e. how the government determined that lottery ticket distributors were cheating).
Lesson 4: “And”, “Or” Events
Curriculum Expectations: A1.3, A1.5, A1.6
Approach:
Students will determine whether two events are dependent, independent, mutually exclusive
or non-mutually exclusive. The fact that the sum of the probabilities of all possible outcomes in
the sample space is 1, will be reinforced.
Group Exercise:
In small groups, students will work through a series of examples to identify independent,
dependent, mutually exclusive and non-mutually exclusive events. They will then develop
strategies to determine the probabilities of each type of event.
Consolidation:
As a class, the examples will be debriefed and formulae will be developed to calculate the
probabilities of the various events discussed.
Lesson 5: Pick the Die
Curriculum Expectations: A1.4, A1.6
Approach:
Students will use non-transitive dice to compare experimental and theoretical probability and
note the tendency of experimental probability to approach theoretical probability as the
number of trials in an experiment increases. Students will draw tree diagrams for events where
the branches in the tree diagram do not have the same probability.
Group Exercise:
Students will play the TIPS4RM game Pick the Die. They will be working to determine the best
strategy for winning the game (either picking the die first or waiting to pick the die second).
Data regarding wins will be recorded.
Consolidation:
Bring together all student data and determine the best strategy for winning. Have students
develop a means of determining theoretical probability associated with the game (e.g. tree
diagram where probability along all branches is not the same). Have students write a journal
reflection on the game and the best strategy to win.
Lesson 6: Let’s Make a Deal and Conditional Probability
Curriculum Expectation: A1.6
Approach:
Use the OAME TIPS4RM exercise to introduce conditional probability and then proceed to
introduce the formulae that support this subject.
 Use the Monty Hall problem to introduce conditional probability
 Use Venn diagrams to organize data to help determine conditional probability
 Use a formula to determine conditional probability
Group Exercise:
Students will investigate the probability of two events occurring together at the same time.
Students will construct Venn diagrams to pictorially understand the relationship amongst
dependent conditions.
Use the TIPS4RM exercise 1.6.1 Let’s Make a Deal!, with the class to demonstrate conditional
probability. A chart will be used to record the class data. The teacher will lead a discussion
about the probabilities found by the students.
Consolidation:
Use a tree diagram to record the always stick strategy and compare it to the tree diagram for
the always switch strategy to convince students about the correct strategy. Use the game as
reference for a discussion on conditional probability.
After the game introduce various problems from the textbook and walk through the formulae
for conditional probability. Textbook chapter linkage – Mathematics of Data Management
section 4.4, Nelson (2003)
The conditional probability of event B, given that event A has occurred, is given by
The Multiplication law for conditional probability will also be presented. This measures the
probability of events A and B occurring, given that event A has occurred. The formula for this is:
Assign homework for practice with the formulae.
Lesson 7: Counting Arrangements and Selections
Curriculum Expectations: A2.1
Approach:
Use the OAME TIPS4RM exercises to introduce Counting Arrangements and Selections and then
proceed to introduce the theory to consolidate the lesson.
 Solve problems that progress from small sets to more unwieldy sets and using lists, tree
diagrams, role playing to motivate the need for a more formal treatment.
 See examples where some of the distinct objects are used and where all the distinct objects
are used.
 Discuss how counting when order is important is different from when order is not important
to distinguish between situations that involve, the use of permutations and those that
involve the use of combinations.
Group Exercises:
Using the OAME TIPS4RM exercises:
 Students will arrange themselves in groups of 4; students choose a president, vicepresident, secretary and treasurer for their group. How many different ways can this be
done? Students draw tree diagrams on large paper to represent this situation.
 Choose three students to come to the front of the room. Try to choose people who are
wearing different types of outfits. As a class, construct a tree diagram of all the possible
combinations of outfits that can be made from the clothes the students are wearing. For
example: (red shirt (person 1), blue jeans (person 2), and running shoes (person 3).
Students discuss what changes when you add more choices. (4 people, include socks).
Continue with investigating putting all students in the class in a line. Students attempt
to make a tree diagram and discuss the problems with the construction. Start over

again using only 5 people from the class to be put in a line. “How many choices do we
have for the first, second, third, fourth, and fifth?” Students discuss and compare the
total number of choices for each experiment.
Examine a Postal Code. In Canada, we use the code LNL NLN. How many different
possibilities for postal codes are there? How is this different from the previous example
(numbers and letters can be repeated). Pairs complete this exercise.
Key ideas to communicate:
The class discussion should focus on when to use a tree diagram and when to not. This should
lead to the concept of Multiplicative Principle for Counting Ordered Pairs, Triplets, and so on.
Consolidation:
Textbook chapter linkage – Mathematics of Data Management, section 4.5, Nelson (2003).
A tree diagram is used to represent the outcome of an experiment that is the result of a
sequence of simpler experiments. If we assume that the outcome for each experiment has no
influence on the outcome of any other experiment. The total number of outcomes is the
product of the possible outcomes at each step of the sequence. If a is selected from A and b is
selected from B, then:
This leads us to the Multiplicative Principle for Probabilities of Independent Events. If a and B
are independent events, then:
However, when order matters, we use factorial notation:
Assign homework for practice with the concepts.
Lesson 8: Counting Permutations
Curriculum Expectation: A2.1, A2.2
Approach:
Use the OAME TIPS4RM exercises to introduce Counting Permutations and then proceed to
introduce the theory to consolidate the lesson.
 Develop, based on previous investigations, a method to calculate the number of
permutations of all the objects in a set of distinct objects and some of the objects in a set of
distinct objects.
 Use mathematical notation (e.g., n!, P(n, r)) to count.
Group Exercises:
Using the OAME TIPS4RM exercises:
 Students complete BLM 1.8.3 working in pairs and using the labelled cards. Students
should understand the meaning of permutations, factorial notation and how to calculate
total number of possible arrangements using P (n, r).

Use BLM 1.8.4 to help students recall prior learning on counting techniques and assist
them in investigating the concept of factorial notation. After students have completed
the page, discuss solutions with students.
Consolidation:
The class discussion should focus on having n objects to choose from, but we only want to
select some rather than all. Textbook chapter linkage – Mathematics of Data Management,
section 4.6, Nelson (2003).
A permutation is an ordered arrangement of objects selected from a set where P(n,r). This
represents the number of permutations possible in which r objects from a set of n different
objects are arranged. :
The number of permutations can be calculated as:
Where we have a set of n objects in which a are alike, b are alike, c are alike, and so on.
Assign homework for practice with the concepts.
Lesson 9: Counting Combinations
Curriculum Expectations: A2.1, A2.2
Approach:
Use the OAME TIPS4RM exercises to introduce Counting Combinations and then proceed to
introduce the theory to consolidate the lesson.
 Develop, based on previous investigations, a method to calculate the number of
combinations of some of the objects in a set of distinct objects.
 Make connection between the number of combinations and the number of permutations.
 Use mathematical notation (e.g., (
 Ascribe meaning to. (
). (
) .(
)) to count
)
 Solve simple problems using techniques for counting permutations and combinations,
where all objects are distinct.
Group Exercises:
Using the OAME TIPS4RM exercises:
 Students work through the problem on BLM 1.9.2 and discuss the similarities and
differences between this problem and the previous day’s work on permutations.

Students in small groups work on the investigation on BLM 1.9.3 – A Novel Idea.
Solutions are recorded on chart paper and shared with the whole class.

Each group should be given a piece of chart paper and a marker. Assign to each group
n n n
 ,  ,   . Have students discuss and reason what they think each of these
n 1 0
combinations represent. Have students create a problem that could be modeled by
each combination.
Consolidation:
Textbook chapter linkage – Mathematics of Data Management, section 4.7, Nelson (2003).
A combination is a collection of chosen objects for which the order does not matter. C(n,r), also
written as ( ), represent the number of combinations possible in which r objects are selected
from a set of n distinct objects:
Assign homework for practice with the concepts.
Lesson 10: Introduction to the counting stories project
Curriculum Expectations: A1.1, A1.3, A1.5, A1.6, A2.1, A2.2, A2.3, A2.4, A2.5, E2.3, E2.4
Approach:
Use the OAME TIPS4RM exercises for the culminating project. Students may select their own
exercises for the culminating project as desired.
 Introduce and understand one culminating project, Counting Stories Project (e.g. student
select children’s story/nursery rhyme to rewrite using counting and probability problems
and solutions as per Strand A).
 Create a class critique to be used during the culminating presentation.
Group Exercises:
Using the TIPS4RM exercises:
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Using BLM 1.10.2, introduce the count stories project to students, and discuss the
description of the task and the assessment rubric (BLM 1.10.3).
Using the SMART™ Notebook file, PowerPoint files, or BLM 1.10.4, and BLM 1.10.5
develop the counting story exemplar with student input. At the end of the
presentation, model writing a component of the story with student input.
In small groups, students complete an additional component of the story, e.g.,
independent events, dependent events, mutually exclusive events, non-mutually
exclusive events or complementary events. Ensure that each group completes a
different missing component, including mathematical justification
Lesson 11: Pascal’s Triangle
Curriculum Expectation: A2.4
Approach:
Use the OAME TIPS4RM exercise to introduce Pascal’s triangle and then proceed to introduce
the patterns is that produces Pascal’s triangle.
 Use the blank Pascal triangle worksheets and discover the numerical patters
 Think-Pair-Share Pascal’s Pizza party
 Use a formula to consolidate Pascal’s triangle
Group Exercise:
Working in pairs, students will investigate the combinatory patterns using BLM 1.11.2 and BLM
1.11.3 worksheets.
Use the TIPS4RM exercise BLM 1.11.4 Case of the stolen jewels!, with the class to extend their
knowledge of Pascal’s triangle. BLM 1.11.5 worksheet would be used for student practice.
Consolidation:
First six rows of Pascal’s triangle are:
1
1
1
1
1
1
2
3
4
5
1
1
3
6
10
1
4
10
1
5
1
The pattern that produces Pascal’s triangle is given by:
t (n, r) = t(n-1, r-1) + t(n-1, r)
n!
A number in a triangle can be found by n C r =
r!(n  r )!
There are different patterns found in Pascal’s triangle (pathway problems):
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Sum of the numbers in any row is equal to 2 to the nth power (2 n), where n is the
number of the row.
If the first element in the row is prime number, all the numbers in that row are divisible
by it.
Hockey stick pattern
Magic 11 patterns: If a row is made into a single number, the number is equal to 11 to
the nth power.
Fibonacci sequence can also be traced in Pascal’s triangle
Sierpinski’s triangle (Extended activity only)
Lesson 14: Probability
Curriculum Expectation: A2.5
Approach:
Use the OAME TIPS4RM exercise and manipulatives (Linking cubes, counters, dice, chart paper
etc) to solve problems using counting techniques involving equally likely outcomes.
 Use BLM 1.14.1 – 1.14.5 worksheets
 Manipulatives to solve problems
Group Exercise:
Working in pairs, students will investigate the marble mystery using BLM 1.14.3 and share their
strategy and solutions with the whole class.
Use the TIPS4RM exercise BLM 1.14.4 to determine experimental (Can use linking cubes) and
theoretical probability.
Consolidation:
The strategies and solutions of the whole class would be posted and everyone will be given an
opportunity to walk around and reflect on alternate approaches for the final answer. This
would be followed by a discussion on the connections made to counting techniques, probability
and real-world events.
As a follow up to this consolidation, students will be asked to look at BLM 1.14.5 in pairs.
Resources:
Mathematics of Data Management (University Preparation), Nelson (2003)
OAME/OMCA material:
www.oame.on.ca/main/staging9.phpcode=grspecres&ph=12&sp=MDM4U
OCDSB Material:
www.ocdsb.edu.on.ca/Secondary_Websites/teacher_res/secondary/tecint/mdm4u.htm
McGraw Hill Student Resources
Project Resources/Ideas:
EStat
Statistics Canada
Sample Datasets