### Probability Progressions.ppt - Mathematics resources for learning

```Google “Starter of the Day”
Pot of Colours from Transum
• http://www.transum.org/Software/SW/Starter_of_the_day/st
arter_March18.ASP
A fun way to begin the day.
Mathematics and Statistics
Progressions in Probability
Ngatea Teacher Day, June 15th
Hauraki Plains College
Jim Hogan, Sandra Cathcart, Brenda McNaughton
Faculty of Education
Team Solutions, Auckland University
Objectives de jour
• To enhance teacher knowledge and understanding of
probability progressions to senior levels
• To experience a selection of probability activities for
use in the classroom to enhance pedagogy of NZC
• To develop teacher capability for scheme design
War plan
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20 minutes on progression activity
20 minute report back
30 minutes on activities
10 minutes scheme design ideas
10 minutes feedback
• Total 90 nec minits
Progressions
• Use the A3 photocopies sheet as a resource.
• List main idea(s) at each level
• Describe one key activity/learning event for
that level
• 20 minits
Level 1 Probability
• The key idea of probability at level 1 is beginning to explore chance
situations. At level 1 students are exploring games of chance that they
play and outcomes from statistical investigations. They are discussing the
different outcomes that are possible. Students are developing early
probability language and describing outcomes. For example an outcome is
certain, possible, impossible or an outcome always happens, never
happens, sometimes happens.
• Learning Activity
– Lonely Pig (NZMATHS)
– Describing events with own probability language (will, won’t, might). Using a
dice.
• “Dad will not be bringing me to school tomorrow. He never brings me to school.”
Level 2 Probability
The key idea of probability at level 2 is beginning to recognise that some
events are more likely than others in chance situations. At level 2
students are recognising the different likelihood of outcomes for games
of chance that they play. They are describing outcomes from statistical
investigations, for example, children in our class were more likely to have
bananas in their lunch today than any other fruit (15/24 children had
bananas in their lunch). Students are developing more sophisticated
nuances with probability language to describe outcomes, for example,
more likely, less likely.
• Learning Activity
– Addition facts using two 0-9 dice
– Memory card game.
Level 3 Probability
•
At level 3 students are beginning to explore one-stage chance situations, for
example tossing a drawing pin, throwing a die from a board game, tossing or
spinning a coin, rolling a pencil, or tossing a “pass the pigs” pig. They are listing
possible outcomes for situations. Students are recording their results and plotting
frequencies of outcomes. Students are comparing their experimental results with
others in the class, acknowledging that results may vary. Students are recognising
that in some chance situations outcomes are not equally likely, for example tossing
a pig or drawing pin.
•
Key Skill
– Systematic recording of outcomes.
– Recognising fair and unfair games
•
Learning Activity
-
Dicey Differences Year 6 Nat STD
Random walking in 2 dimensions.
Battleships, Yahtzee (1 – 6), Numahtzee (0 – 9)
• Key idea
Level 4 Probability
– Using simple fractions and ratios to describe
probabilities. Independence of events. Writing and
• Key Skill
– Connecting 9 out of 10 as 90% = 9/10 =0.9
– ½ is guessable. 1/10 is much harder than 1/6.
• Learning Activities
– Card collecting. How many do you need to buy?
– Dicey Differences Year 8 Nat STD
• Key idea
Level 5 Probability
– Long run probability. Experimental v theoretical
probability expectations. Everyday event chance.
• Key Skill
– Using random numbers, tree diagram, logic.
• Learning Activity
– Inventing games of chance
– Random walks
– Chance in Monopoly
Level 6 Probability
• Key idea
– Random selections. Sample size matters.
Independence of events.
• Key Skill
– Sample size of about 30 is a good choice.
– Conditional prob using tree diagrams
• Learning Activity
– Graphing the long probability of outcome like tossing
a coin.
NCEA L1 • 91037
AS90137 – Demonstrate understanding of chance and data (4crE)
Key Ideas
PPDAC. Interpreting statistical tables and graphs
Analysing investigations with multivariate and time series data.
Probability concepts in discrete situtations.
Situations include two dice, cards, experimental and theoretical comparison.
Language of probability such as sample space, event, experiment, outcomes.
Variation and simple expectation. Simulation using a computer. See
specifications doe externals on NZQA website!
Probability appears to have been high-jacked by statistics!
NCEA L1 • 91038
AS90138 – Investigate a situation involving elements of chance (3cr I)
Key Ideas
PPDAC.
Comparing experimental and theoretical situations using displays and
patterns, features.
Situations include two dice, cards, experimental and theoretical comparison.
Language of probability such as sample space, event, experiment, outcomes.
Variation and simple expectation. Simulation using a computer.
This is the practical version of probability.
Level 7 Probability
• Key ideas
– Continuous probability, Normal distributions, simulating
events, random number generation.
• Key Skills
– Using normal tables
– Properties of random numbers
• Learning Activity
– Simulate a situation like twins in a population.
– Simulate to find how many cereal boxes do I need to
collect the card set?
NCEA L2 • 91267
AS91267 – Apply probability methods in solving problems(4crE)
Key Ideas
Interpreting risk and relative risk. Calculating?
Using normal distribution. Exploring experimental distributions and relative
frequencies. Two way tables and prob trees. Informal conditional prob.
Situations include two dice, cards, experimental and theoretical comparison.
Language of probability such as sample space, event, experiment, outcomes.
Variation and simple expectation. Simulation using a computer. See
specifications doe externals on NZQA website!
NCEA L2 • 91268
AS91267 – Investigate a situation involving elements of chance using a
simulation (2cr Int)
Key Ideas
Random numbers and simulation.
The language of probability such as trial. Designing and doing a
simulation to investigate a situation
PPDAC in the sense of getting data, displays, communicating findings
of the situation.
See sen sec guides.
Level 8 Probability
• Key ideas
– Conditional probability, expected values.
– Poisson, binomial and normal applications
– Random variables, expectation
• Key Skill
– Using tables
– Recognising model application
• Learning Activity
– Explore simulation of practical situation.
– Expected return from LOTTO.
NCEA L3 • 3.13
AS????? – Apply probability concepts in solving problems(4crE)
Key Ideas
Randomness, independence, combined, MU, conditional prob,.
Prob distrib tables and graphs
Two way tables, trees and Venn diagramms
True? v model v experimental.
Language.
See specifications for externals on NZQA website! Sample assessments
also available.
NCEA L3 • 3.14
AS????? – Apply probability distributions in solving problems(4crE)
Key Ideas
Discrete and continuous probability distributions
Mean and sd of random variables
Expected values
Theoretical v model v experimental.
Situations include Poisson, Binomial and Normal.
Language.
See specifications for externals on NZQA website! Sample assessments
also available.
Record Ideas
• Record all the brilliant ideas for learning about
probability.
• Record key progressions.
• There are thousands of probability resources
and do not forget the human interaction of a card game!
Websites
• NZMATHS NZC http://www.nzmaths.co.nz/nzc-and-standards
• Good progression, resources and examples
• NZMATHS SNP http://www.nzmaths.co.nz/snpactivities?parent_node=
• 20 lessons on random numbers and prob expts, language.
• NZMATHS http://nzmaths.co.nz/key-mathematicalideas?parent_node
• Key Mathematical ideas – good.
• Senior Secondary Guides
• http://seniorsecondary.tki.org.nz/Mathematics-and-statistics
• “PROBABILITY GAMES”
• http://www.free-training-tutorial.com/probability-games.html
• STARTER OF THE DAY
• http://www.transum.org/Software/SW/Starter_of_the_day/Similar.asp?ID
_Topic=30
• POT OF COLOURS
• http://www.transum.org/Software/SW/Starter_of_the_day/starter_Marc
h18.ASP
• BOOK 9 NDP – MONTE HALL AND SIMULATION
Logo Programming
• Random walking
• http://www.alancsmith.co.uk/logo/
• Helen Chick’s work (Melbourne MAV)
• http://staff.edfac.unimelb.edu.au/~chick/hom
epage_files/
Nice Problems to Solve
• Birthday Problem
How many people do you need to have a probability of 50% or
so that two have a birthday on the same day?
• Breaking the Stick
If I break a stick randomly into 3 pieces what is the probability
the pieces will form a triangle?
Who uses probability?
• Scientists, social scientists, researchers,
• Quantum Mechanics involves “probability
density functions” which deals with the
likelihood of where one particle might be.
• Insurance, actuaries, teachers.
SCHEMES and THINGS
On to more serious things.
Rule 1 of scheme writing
- There are no rules.
- It is all about what works for you.
Scheme
Is not a lesson plan.
It is an outline of how the learning for a
particular topic is going to be approached
An example
A n NZC scheme.
Mine actually!
The lesson plans are in the plan book, if
required, and are probably best written after the
lesson.