Flooding likely?

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Purpose

The purpose of this unit, which provides an integrated social sciences/statistics and probability context, is develop students' understandings of skills and knowledge of interpreting statistical and chance situations. Within this, students investigate flooding as an example of a natural catastrophe.

Achievement Objectives
NA5-5: Know commonly used fraction, decimal, and percentage conversions.
S5-3: Compare and describe the variation between theoretical and experimental distributions in situations that involve elements of chance.
S5-4: Calculate probabilities, using fractions, percentages, and ratios.
Specific Learning Outcomes
  • Combine probabilities to find the probability of an outcome from a complex chance situation.
  • Run a simulation to find an experimental probability
  • Understand and describe the differences and advantages between theoretical and experimental probabilities.
  • Find a theoretical probability and express it as a percentage.
  • Develop an understanding of dependence and independence in chance situations. 
Description of Mathematics

In this unit, students solve theoretical and model practical probability problems in the context of natural disasters. This encompasses the following mathematical ideas:

  • considering a probability as a proportion of specified outcomes to the total number of outcomes in a probability distribution
  • expressing a probability as a percentage, fraction or a decimal
  • designing and carrying out a probability simulation to find the experimental probability of a probability situation. This is more complex than theoretical probability calculations at this level.
Opportunities for Adaptation and Differentiation

This cross-curricular, context-based unit aims to deliver mathematics learning, whilst encouraging differentiated, student-centred learning. 

The learning opportunities in this unit can be further differentiated by providing or removing support to students, and by varying the task requirements. Ways to differentiate include:

  • roaming and supporting students in a variety of groupings to ensure they understand the task at hand, the skills needed to succeed, and can apply these skills in a suitable process
  • varying the amount of structured scaffolding and guided teaching you provide to students when investigating new tasks
  • providing opportunities for students to create their own problems related to a relevant context
  • providing extended opportunities for students to revise and apply learning from throughout the unit
  • modelling the application of ideas at every stage of the unit
  • strategically organising students into pairs and small groups in order to encourage peer learning, scaffolding, and extension
  • allowing access to calculators to decrease the cognitive load required in each stage of the unit
  • working alongside individual students (or groups of students) who require further support with specific areas of knowledge or activities.

With student interest engaged, mathematical challenges often seem more approachable than when presented in isolation. Therefore, you might find it appropriate to adapt the contexts presented in this unit. For example, instead of investigating floods, you might investigate earthquakes or forest fires. Consider how the type of natural disaster that you chose to investigate might make links to current events, other curriculum areas (e.g. shared texts, science foci), and your students' lived experiences. Your students might be further engaged by investigating the probability of these events occurring, in relation to specific locations.

Structure

The first session is an introductory activity that is aimed to spark the imagination of students, to introduce the need for a particular idea or technique in mathematics that would enable them to explore deeper into that context. It is expected that rich discussion may be had around the context and around the nature of the mathematics involved.

Following the introductory session, each subsequent session in the unit is composed of four sections: Introducing Ideas, Building Ideas, Reinforcing Ideas, and Extending Ideas. 

Introducing Ideas: It is recommended that you allow approximately 10 minutes for students to work on these problems, either as a whole class, in groups, pairs, or as individuals. Following this, gather the students together to review the problem and to discuss ideas, issues and mathematical techniques that they noticed during the process. It may be helpful to summarise key outcomes of the discussion at this point.

Building Ideas, Reinforcing Ideas, and Extending Ideas: Exploration of these stages can be differentiated on the basis of individual learning needs, as demonstrated in the previous stage of each session. Some students may have managed the focus activity easily and be ready to attempt the reinforcing ideas or even the extending ideas activity straight away. These could be attempted individually or in groups or pairs, depending on students’ readiness for the activity concerned. The students remaining with the teacher could begin to work through the building ideas activity together, peeling off to complete this activity and/or to attempt the reinforcing ideas activity when they feel they have ‘got it’. 

It is expected that once all the students have peeled off into independent or group work of the appropriate selection of building, reinforcing and extending activities, the teacher is freed up to check back with the ‘early peelers’ and to circulate as needed.

Importantly, students should have multiple opportunities to, throughout and at the conclusion of each session, compare, check, and discuss their ideas with peers and the teacher, and to reflect upon their ideas and developed understandings. These reflections can be demonstrated using a variety of means (e.g. written, digital note, survey, sticky notes, diagrams, marked work, videoed demonstration) and can be used to inform your planning for subsequent sessions.

The relevance of this learning can also be enhanced with the inclusion of key vocabulary from your students' home languages. For example, te reo Māori kupu such as tūponotanga
(probability, chance), tūponotanga whakamātau (experimental probability), tūponotanga tātai (theoretical probability), ōrau (percent), pāpono whakawhirinaki (dependent event), and pāpono wehe kē (independent event) could be introduced in this unit and used throughout other mathematical learning.

Activity

Introductory Session

The aim of this activity, which presents an opportunity to develop the knowledge of statistical and probability processes needed to discuss the likelihood of various chance situations occurring, is to motivate students towards the integrated geographical/mathematics context, and to inform teachers of students' understandings. 

  1. Introduce the following context to students: The severity of a natural catastrophe (eg. a one in 10 year or one in 100 year event, etc.) is defined by the number of years that would be expected to elapse between two such events. A 1 in 10 year flood is expected to occur once within any given ten year period, meaning that the probability, or chance, of a 1 in 10 year flood occurring in any given year is 10%. It is unlikely, but still possible, for two such floods to occur in a given year. 

  • Pose the following question to students: What is the probability of two 1 in 10 year floods occurring in the first two years of a ten year period?  

As students work, observe their capacity for solving a problem involving combined probabilities. Use these observations to gauge your students' positions on the Interpreting Statistical and Chance Situations learning progression.

  1. Discuss, drawing attention to the following points:

  • What other chance situations are you aware of with similar likelihoods of various outcomes?
  • How might a probability be used to predict and determine what will happen?
  • What are the different formats of probability (fraction, decimal, percentage, ratio)?
  • What is an independent event? How can probabilities be combined for independent events? 
  • What is meant by theoretical and experimental probabilities; differences, practicalities, etc.?

Session Two

This session focuses on combining probabilities to find the probability of an outcome from a complex chance situation.

Introducing Ideas

  1. What is the probability, or chance, of two 1 in 10 year floods occurring in two consecutive years? 

  2. Discuss, drawing attention to the following points:

  • How could we calculate this event occurring? 

  • How would our answer change if the question was two such floods in three years? 

Building Ideas

  1. Explain: We will work through how to find the probability of two 1 in 10 year floods occurring in three consecutive years? 
  2. Work through the following tasks in a manner appropriate to the students in your class:
  • What is the probability of a 1 in 10 year flood occurring in any year?
  • What is the probability of a 1 in 10 year flood not occurring in any year?
  • List the sample space for this probability distribution. This is a list of all the ways the floods could occur over a three year period (e.g. flood, flood, no flood).
  • Apply the probabilities of the event of flooding or not flooding for each arrangement of two floods over the three years. For example flood, flood, no flood would be 1/10 of 1/10 of 9/10, which is 0.1 x 0.1 x 0.9.
  • Find the probability of two 1 in 10 year floods occurring in three consecutive years. This is the sum of the probabilities found for the last question.

Reinforcing Ideas

  1. Provide time for students to work through the following tasks:

  • What is the probability of a 1 in 5 year flood occurring in any given year?
  • Find the probability of one 1 in 5 year floods occurring in five consecutive years.
  • Find the probability of two 1 in 5 year floods occurring in five consecutive years. 

Extending Ideas

Provide time for students to work through the following tasks:

  • What is the probability of a 1 in 20 year flood occurring in any given year?
  • Find the probability of one 1 in 20 year floods occurring in five consecutive years.
  • Find the probability of two 1 in 20 year floods occurring in three consecutive years.
  • What is the probability of a 1 in n-year flood occurring in any given year?
  • Find the probability of one 1 in n-year floods occurring in three consecutive years. 

Session Three

This session focuses on running a simulation to find an experimental probability.

Introducing Ideas

  1. Explain to students:
  • The probability of a 1 in 10 year flood occurring in any given year is 0.1.
  • The probability 1 in 10 year floods occurring in the three out of ten consecutive years is 0.0574. 
     
  1. Pose the following question to students: How could this probability have been found? 
     
  2. Discuss, introducing the idea of probability simulations by using random numbers and theoretical probability for a flood to occur in any given year. 
  • How can the experimental probability be as accurate as possible? We can increase the number of trials – 50 is a reasonable number.

Building Ideas

  1. Support students to use a die to run a probability simulation of 1 in 6 year floods occurring in the three out of ten consecutive years. Students should run at least 30 trials to find the experimental probability of the simulation.

Reinforcing Ideas

  1. Support students to use random numbers to run a probability simulation of 1 in 10 year floods occurring in the three out of ten consecutive years. Students should run at least 30 trials to find the experimental probability of the simulation.

Extending ideas

  1. Support students to use random numbers to run a probability simulation of 1 in 20 year floods occurring in the three out of ten consecutive years. Students should run at least 30 trials to find the experimental probability of the simulation.

Session Four

This session focuses on understanding and describing the differences and advantages between theoretical and experimental probabilities.

Introducing Ideas

  1. Introduce the following context to students: 

  • A meteorologist ran a storm model to give an experimental probability that an approaching storm will result in a major flood of 0.324.
  • A statistician calculated the theoretical probability of the same storm resulting in a major flood as being 0.2978.
     
  1. Support students to comment on the similarities and/or differences between these two probabilities. Can they both be correct?
  2. Discuss, drawing attention to the following points:

  • These numbers are similar – with what precision (number of d.p.) can we say the probability of the approaching storm causing a major flood would be?
  • What are the advantages and disadvantages of each method of gaining a probability?
  • What would the effect of increasing the number of trials in the meteorologist’s simulation have on the agreement between the two probabilities.

Building Ideas

  1. Have students consider this situation and then complete the following tasks: The probability of a southerly storm event next weekend is 50%. One sixth of all storms of this type result in flooding. 
  • Find the experimental probability that there will be a flood next weekend. Run at least 30 trials in your simulation.
  • Calculate the theoretical probability that there will be a flood next weekend.
  • Compare your experimental probability with the theoretical probability, that you calculated for the same event.

Reinforcing Ideas

  1. Have students consider this situation and then complete the following tasks: The probability of a storm event next Wednesday is 25%. One tenth of all storms in recent history have resulted in flooding. One third of all floods are likely to be severe. 
  • Find the experimental probability that there will be a severe flood next Wednesday. Run at least 30 trials in your simulation.
  • Calculate the theoretical probability that there will be a severe flood next Wednesday.
  • Compare the two probabilities. 

Extending ideas

  1. Support students to comment on the similarities and/or differences between the theoretical and experimental probabilities for whether there will be a severe flood next Wednesday. 
  • What do you expect the results of the simulation to be if you ran 50, or 100 trials?

Session Five

This session focuses on finding a theoretical probability and expressing it as a percentage.

Introducing Ideas

  1. Introduce the following problem to students: If the probability of a 1 in 10 year flood occurring in any given year is 10%, what is the probability of a 1 in 100 year flood occurring in any given year? Give the solution in % (as the probability is given in %).

  2. Discuss: 

  • What proportion is a year of the total time period specified in a 1 in 10 year event? In a 1 in 100 year event? 

  • How does this proportion relate to a probability expressed as a percentage?  

Building Ideas
  1. Support students to set up and complete the table below. They should work out the percentage probability of an event occurring in any year and give the frequency of its occurrence.
Years between eventsProbability of event
in a given year
Percentage probability
of event in a given year
200  
1001/1001 %
501/50 
20  
101/1010 %
5  
2  

Reinforcing Ideas

  1. Provide time for students to work through the following questions:

  • What is the percentage probability of a 1 in 200 year flood occurring in any given year?
  • What is the percentage probability of a 1 in 1000 year flood occurring in any given year?
  • What is the percentage probability of a 1 in 50 year flood occurring in any given year?

Extending Ideas

  1. Support students to give a general rule for the percentage probability of a 1 in n year flood occurring in any given year.  

Session Six

This session focuses on showing understanding of dependence and independence in chance situations. 

Introducing Ideas

  1. Introduce the following problem to students: If a farm experiences two 1 in 100 year floods in a single decade, should the owners rush out and buy a lottery ticket?

  2. Discuss, drawing attention to the following points:

  • What is your reasoning in terms of chance situations?

  • Are the events of the two floods independent? This is a debatable topic given the factors involved in storm systems and the trends of global warming. 

  • What about the following events: a flood and a lottery? Will the probability of a win on the lottery be affected by the weather? Are they independent events?

Building Ideas

  1. Introduce the following context to students: A local raffle has 100 tickets. The probability of sunshine on the day the raffle is drawn is ½.

  2. Provide time for students to work through the following tasks:

  • What is the probability of buying one ticket and winning the raffle?
  • What is the probability of buying the winning ticket and it being a sunny day?
  • What is the probability of buying the winning ticket and/or it being a sunny day?

Reinforcing Ideas

  1. Introduce the following context to students: The local lottery has 10,000 tickets. The probability of rain on the day the lottery is drawn is 20%.
  2. Provide time for students to work through the following tasks:

  • If all the lottery tickets are sold, what is the probability of winning the lottery on one ticket?
  • What is the probability of winning the lottery on one ticket on a rainy day?
  • What is the probability of having a ticket, but not winning the lottery, and it not raining on the day of the draw?
  • What is the probability of either a win on a lottery ticket or rain on the day of the draw?

Extending Ideas

  1. Support students to look at the sum of their answers to the last three questions. 
  • What do you notice? 
  • Could you use that knowledge to find another way of answering the last question? 
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Level Five