Unit Plan AssignmentMargaret Cooper, Daljit Dhaliwal, & Ross Dolan

PED 4187Professor Lubna Al-Shaama

February 26, 2010

Course: Mathematics of Data Management (MDM4U)Grade: 12Unit: Counting and ProbabilityPrerequisite: 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. ) pertain to situations where order matters.

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 ofan 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, anddistinguish 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 spaceas one whose outcomes can be measured (e.g., all possible outcomes of the time it takes tocomplete 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 adiscrete 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 distributionassociated with the sample space, and solve related problemsA1.4 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 graphingcalculator; 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 timesusing technology, the experimental probability that I calculate for getting two tails onthe two tosses is likely to be closer to the theoretical probability of than if I simulatetossing 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 mutuallyexclusive 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 and5 girls) and the multiplicative counting principle (e.g., determining the number of ways ofselecting 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) andPascal’s triangle [e.g., between ( ) and diagonal 3 of Pascal’s triangle]A2.5 solve probability problems using counting principles for situations involving equallylikely outcomes2. Solving Problems Using E1.1 pose a significant problem of interest that requires the organization and analysis of asuitable set of primary or secondary quantitative data (e.g., primary data collected from astudent-designed game of chance, secondary data from a reliable source such as E-STAT),and conduct appropriate background research related to the topic being studiedE2.2 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 technologyE2.3 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 Curriculum Expectations

1 Introduction to Probability

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.

A1.1, A1.2, A1.3, A1.5

2 Probability and “Throw Sticks”

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

A1.1, A1.2, A1.3, A1.5

3 Using Simulations

Use mathematical simulations to determine if games are fair

Reflect on how simulations can be used to solve real problems involving fairness

A1.1, A1.2, A1.4

4 “And”, “Or” events

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.

A1.3, A1.5, A1.6

5 Pick the Die 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

A1.4, A1.6

6 Let’s Make a Deal and Conditional

Use the Monty Hall problem to introduce conditional probability

Use Venn diagrams to organize data to help

A1.6

Probability determine conditional probability Use a formula to determine conditional probability

7 Counting Arrangements and Selections

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.

A2.1

8 Counting Permutations

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.

A2.1, A2.2

9 Counting Combinations

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., ( )) to count

Ascribe meaning to. ( ). ( ) .( ) Solve simple problems using techniques for counting

permutations and combinations, where all objects are distinct.

A2.1, A2.2

10 Introducing the Counting Stories Project

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.

A1.1, A1.3, A1.5, A1.6, A2.1, A2.2,A2.3, A2.4, A2.5, E2.3, E2.4

11 Pascal’s Triangle

• Pascal’s triangle- Investigating Pascal’s triangle numerically

- Investigating Pascal’s triangle using (nr )

- Relationship between patterns and combinations- Investigating pathway problems

• Examples of patterns and combinations (From:OAME-

A2.4

MDM4U document)- Number of ways of choosing elements:

(nn )=1 ;(n0 )=1 ;(n1)=1; (nn)=( nn−r );etc

• 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.

(nr )=(n−1r−1 )+(n−1r )12 Mixed

Counting Problems

• 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

A2.3

13 Counting Stories Project

Ask students to work on counting stories project. (Example: Goldilocks)

A1.1, A1.3, A1.5, A1.6, A2.1, A2.2,A2.3, A2.4, A2.5, E2.3, E2.4

14 Probability Counting techniques and probability strategies- When order does not matter- When order matters Solve probability problems using counting principles

involving equally likely outcomes

A2.5

15 Counting Stories Project

Students will complete counting stories project. A1.1, A1.3, A1.5, A1.6, A2.1, A2.2,A2.3, A2.4,

A2.5, E2.3, E2.4

16 Culminating Investigation

• Students identify a significant problem of interest for Culminating Investigation - Identify problem (area of interest) and formulate

research question- Pose problem

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, vice-president, 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., ( )) to count

Ascribe meaning to. ( ). ( ) .( )

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

(nn ) ,(n1 ),(n0) . Have students discuss and reason what they think each of these 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:

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 BLM1.11.3 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 11 2 1

1 3 3 11 4 6 4 1

1 5 10 10 5 1

The pattern that produces Pascal’s triangle is given by:t (n, r) = t(n-1, r-1) + t(n-1, r)

A number in a triangle can be found by n C r=

n !r! (n−r )!

There are different patterns found in Pascal’s triangle (pathway problems):

Sum of the numbers in any row is equal to 2 to the nth power (2n), 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 theirstrategy 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:EStatStatistics CanadaSample Datasets