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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 • • • • • 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 talking about this. • 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 GOOGLE • “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 - for your students - with your resources - in your classroom/school. 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. • Please fill in the evaluation form • Thank you and enjoy probability.
