How far until it stops? Investigating stopping distances using the PPDAC statistical enquiry cycle
Unit Outline
Introduction
This unit of work uses the PPDAC (Problem, Plan, Data, Analysis and Conclusion) statistical enquiry cycle to investigate stopping distances in cars. The unit of work is aimed at Level 5 of the curriculum (most likely a Year 10 class). The context is one that should appeal to students due to its relevance to them as they and their peers prepare to learn to drive. The unit allows them to investigate factors which affect stopping distances, such as speed and road conditions. This will give more meaning to applying appropriate following distances and the ‘2 second rule’.
The investigation analyses data collected during a studyon total car stopping distance with different road conditions and different initial speeds.The total stopping distance of a car can be broken down into two major components: the reaction distance and the braking distance. The reaction distance takes account of the time it takes for a driver to perceive that they need to stop and the amount of time it takes for them to react to the situation. The braking distance depends on the vehicle reaction time and the vehicle braking capability. Although the study is fictitious, the findings in terms of averages for the different stopping components under different conditions reflect reality.
The stopping distance investigation ‘How far before it stops?’ includes a template with scaffolding to guide students through the components of the PPDAC cycle in the context described above. It can be printed for students to write on directly, projected in a classroom so that students can write out the answers in their books or can be distributed electronically for students to type into directly. The in-class investigation assumes some prior statistical knowledge, which is described below with resources and teaching suggestions. Detailed lesson plans with suggested duration are also included and can be adapted to meet the needs of the learners. It is expected that the lessons will consist of discussion, analysis and some drafting of responses but the students will write up their answers in more detail for homework. Alternatively more time could be allowed in class for the project to be completed. An assessment rubric for the in-class investigation has been developed and includes a self-assessment component to allow students to reflect on their own learning. The in-class project could be used independently of the lesson plans as a homework-based assessment tool at the end of a unit of work. However, the depth of the conclusion sections is greatly enhanced by in-class discussions so that students candraw on their peers’ and their teacher’s knowledge.
Prior knowledge (with websites for resources and teaching suggestions):
- Calculating averages, quartiles and inter-quartile range
- NZ Maths statistical investigations: nzmaths.co.nz/statistical-investigations-units-work?parent_node=
- NZ Census at School classroom activities:
- Collect data about students in the class, e.g. heights, distance from home to school, number of hours studying the Road Code so far, number of hours supervised driving time they think they should have before sitting their first driving test (compare this to the NZTA recommended time of a minimum of 120 hours). Calculate statistics using this data.
- Graphing data
- NZ Maths statistical investigations: nzmaths.co.nz/statistical-investigations-units-work?parent_node=
- NZ Census at School classroom activities:
- NZ Census at School data viewer:
- NZ Assessment Resource Banks (Mathematics): arb.nzcer.org.nz/searchmaths.php
- Box and whiskers graph, dot plot, histogram, stem and leaf graphs of class data.
- Random sampling
- NZ Maths statistical investigations: nzmaths.co.nz/statistical-investigations-units-work?parent_node=
- NZ Census at School data viewer:
- Useclass data or select a random selection of students from the class (asking them to stand up) to demonstrate simple random sampling and/or systematic sampling
- Population and variables
- NZ Census at School classroom activities:
- PPDAC cycle
- NZ Census at School classroom activities:
- Posing investigative questions
- NZ Maths statistical investigations: nzmaths.co.nz/statistical-investigations-units-work?parent_node=
- NZ Census at School classroom activities:
- Use class data to pose summative, comparative and relationship questions.
Curriculum links:
Learning area
Mathematics and Statistics – Statistics Level 5
Values
Excellence
Innovation, inquiry and curiosity
Community and participation
Respect
Achievement objectives
Level 5 Statistical Investigation:
Plan and conduct surveys and experiments using the statistical enquiry cycle:
- Determining appropriate variables and measures;
- Considering sources of variation;
- Gathering and cleaning data;
- Using multiple displays, and re-categorising data to find patterns, variations, relationships, and trends in multivariate data sets;
- Comparing sample distributions visually, using measures of centre, spread, and proportion;
- Presenting a report of findings.
Key competencies based on achievement objectives related to each competency specific to Mathematics and Statistics (source: Team Solutions New Zealand, AucklandUniversity).
Thinking:
- Think logically
- Justify
- Co-construct knowledge
- Investigate
- Discern if answers are reasonable
- Interpret
- Deal with uncertainty and variation
- Make connections
- Hypothesise
- Seek patterns and generalisation
- Explore and use patterns and relationships in data
- Demonstrate and develop relational understanding
- Evaluate
- Analyse
Using language, symbols and text:
- Understand mathematics as a language
- Interpret statistical information
- Process and communicate mathematical ideas
- Know, use and interpret specialised vocabulary
- Communicate findings
- Use ICT as appropriate
- Interpret visual representations such as graphs, diagrams
- Use appropriate units
- Demonstrate statistical literacy
Managing self:
- Work independently
- Self-assessment – What can/can’t I do
- Manage time effectively
Relating to others:
- Listen actively
- Share ideas
- Accept being wrong as part of learning
- Work cooperatively
- Communicate thinking
- Think-pair-share
- Remain open to learning from others
Participating and contributing:
- Share strategies and thinking
- Work in groups with everyone contributing
- Contribute to thinking groups
- Build on prior knowledge
- Contribute to a culture of inquiry and learning
Learning Intentions
- Identify suitable variables for the statistical investigation.
- Identify the population.
- Describe the problem of interest.
- Calculate the summary statistics for relevant data.
- Graph relevant data.
- Analyse the data.
- Compare/contrast the data.
- Explain the likely result of repeating the sampling process.
- Explain why different groups and organisations would be interested in the findings of your investigation.
- Peer-critique each others’ work.
- Generate an investigative question.
- Hypothesise the answer to the investigative question.
- From the analysis carried out, reflect on and justify your findings.
- Reflect on how reasonable the results of the investigation are.
- Generate a question that could be further investigated based on your investigation.
- Present your findings in a way that will inform others.
ICT resources
New Zealand Maths statistics investigations
nzmaths.co.nz/statistical-investigations-units-work?parent_node=
NZ Census at School classroom activities
NZ Census at School statistical investigation and the PPDAC cycle
NZ Census at School informal inference
NZ Census at School data viewer
NZ Assessment Resources Banks (Mathematics)
arb.nzcer.org.nz/searchmaths.php
Websites on stopping (and following) distances
subheading of ‘Safety’)
Youtube videos on stopping distances
Website with information on Fathom software
softwareforlearning.tki.org.nz/Products/Fathom
Excel spreadsheet of stopping distances
Available on the NZTA website
‘How far until it stops?’ lesson plans
Lesson 1: Introduction to the ‘How far until it stops?’ investigation
Content:- Introduction to the ‘How far until it stops?’ investigation
- Stopping distances
- PPDAC cycle: Stopping distances problem
Activities:
- Introduce the investigation.
b)Use think-pair-share discussion to find out students’ prior knowledge about stopping distances in cars using the sentence starter: ‘The stopping distance of a car depends on...’
c)Have students read through the introduction to the investigation ‘How far until it stops?’ in the student workbook.
d)Discuss the problem being investigated. Use think-pair-share to discuss why this study might have been carried out.
e)Discuss the population for this investigation.
- Problem
b)Students to pose a comparative question for this investigation and identify the variables and population.
c)Students’ questions are then checked and if required students are given feedback on their question and given the opportunity to rewrite it (this may take more than one lesson).
Notes for teachers:
- There are a number of factors which affect the overall stopping distance. Some of these are:
- Perception time
- Reaction time
- Vehicle reaction time
- Vehicle braking capability
- How good the brakes are
- Grip of the tyres on the road
- Conditions of the road surface
- Weather conditions
- Weight of the car and its contents
- Example of a description of the problem being investigated: ‘The problem being investigated is stopping distances (in metres) for a variety of cars, road conditions and drivers in New Zealand’.
- Students should make two attempts to pose their own comparative question. An example of the format that students can use is: ‘Do total stopping distances in wet conditions collected in the NZ road and vehicle safety study tend to be further than the total stopping distances in dry conditions collected in the NZ road and vehicle safety study?’.
- Ideally questions should be written to reflect the hypothesis although this is not necessary. The question above implies that the hypothesis is that stopping distances in wet conditions are further than those in dry conditions (for the NZ road and vehicle safety study).
- A suitable comparison investigative question is one that reflects the population, has a clear variable to investigate, compares the values of a continuous variable across different categories, and can be answered with the data.
- Questions need to be checked by the teacher and feedback given as required.
- If students are unable to write a question, even with guidance, they should then be given a question to investigate. It is useful to make a note of this as this should be considered when awarding the final grade for the investigation.
- Students who were able to write a question will then use one of their approved questions for the statistical investigation.
- You may wish to have all students use the same question in order to simplify the monitoring of the work and the marking. However, more valuable discussions can be had at the end if students are working on different questions and this will be more interesting to students.
- Population for this investigation is: Trials measured during the NZ road and vehicle safety study.
- Students must identify the two variables for their investigative question, the categorical variable (road conditions or initial speed) and the continuous variable (reaction distance, braking distance or total stopping distance).
- Students could complete part or all of this investigation in pairs or threes. Such use of cooperative work will enhance students understanding and communication of context through shared knowledge as well as help them through the PPDAC statistical investigation cycle.
Resources:
- Copy of the investigation for each of the students or an electronic copy to project in the classroom.
- NZ Census at School classroom activities
- Websites on stopping (and following) distances
(see subheading of ‘Safety’)
- You tube videos on stopping distances
Lesson 2: ‘How far until it stops?’ investigation: Plan, Data and Analysis
Content:- PPDAC cycle: Stopping distances: Plan, Data and Analysis
Activities:
- Plan
b)Discuss with students what data they will need to use from the sample of 120 trials given to them.
c)Students to write up their hypothesis and describe the data that they will use.
- Data
b)Discuss the larger stopping distances, why might these have occurred?
- Analysis
b)Students calculate summary statistics and construct graphs, using appropriate technology if available (e.g. Fathom) or by highlighting the relevant trials on a printout of the dataset provided and carrying out this process by hand.
Notes for teachers:
- Depending on the amount of time available for this investigation, an extra lesson could be included for students to spend class time on the internet researching stopping distances. Alternatively this could be done as homework prior to this lesson in order to aid students in making their hypothesis and increasing their knowledge of the context (which will also increase the depth of their discussion in the conclusion).
- Students need to appreciate that the dataset they have been provided with is a random sample of all the trials of stopping distances carried out in the NZ study. They need to use all of the data appropriate to their question and not sample further.
- As the sample is random the students can assume that it is representative of all the trials carried out in the study.
- When considering whether the data needs cleaning students need to consider whether the values seem reasonable. Avoid the use of the word ‘outlier’ to describe extreme values in this dataset as none of the values presented can be considered outliers. All values are reasonable and the data does not need cleaning.
- Students are likely to point out some of the larger values and question whether these are valid. Discuss possible reasons for these values. For example a large reaction distance may be due to the driver being distracted (e.g. by using their mobile phone). A large braking distance may be due to worn tyres, which reduces the grip that the tyres has on the road.
- Graphs must allow for direct comparison, i.e. share the same axes.
- The choice of graphs will depend on whether technology is being used to produce the graphs. If suitable ICT resources are available then it is useful for students to become familiar with using programs such as Fathom to produce the statistics and graphs, especially if this is how the analysis is carried out in the assessment of AS 1.10 in your school.
- A side-by-side box and whisker graph allows many features to be compared such as the middle 50% of the data, medians and overlap of the data. A dot plot can be overlaid or stacked with the box and whisker graph in order to provide insight into the distribution of the data. Overlaying these two types aids students in developing a deeper understanding of what the box and whisker graph represents.
- Students should aim to have two or more graphs. However, one graph is sufficient.
- Avoid calculating and discussing the range or focusing on the maximum and minimum values as these do not provide evidence when answering the investigative question.
Resources:
- Copy of the investigation for each of the students or an electronic copy to project in the classroom.
- Excel spreadsheet of stopping distances (available from the NZTA website)
- NZ Census at School classroom activities
- NZ Assessment Resource Banks (Mathematics)
- Websites on stopping (and following) distances
(see subheading of ‘Safety’)
- You tube videos on stopping distances
- Website with information about Fathom software
Lesson 3: ‘How far until it stops?’ investigation: Analysis
Content:- PPDAC cycle: Stopping distances analysis
Activities:
- Analysis
b)Have students draft 3-5 statements about what they notice about their summary statistics and graphs.
c)Peer-critique of statements, either as a class or in small groups. In each case discussing what makes the statement good or how it could be improved.
d)Students to write final versions of 3-5 analysis statements after considering the feedback given by their peers (may be completed for homework).
Notes for teachers:
- There are a number of features of the graphs that the students can comment on. The investigation template guides them through these features.
- When discussing what students notice about their statistics and graphs, repeat back to students their ideas, alteringtheir words to model correct statistical language, ensuring it is also in context and includes values.
- Students do not need to write an analysis statement about each of the listed features, instead they should focus on what is relevant for their analysis and what will help them answer their question.
- Analysis statements must be made in the context of stopping distances.
- Analysis statements should include values and units.
- Analysis statements should make it clear that it is the sample being referred to, not the population.
- An example of an analysis statement is: ‘In the sample analysed, the median total stopping distance for dry conditions is 21.1 m, which is 8m less than the median stopping distance for wet conditions (29.1 m). Each median value lies outside of the middle 50% of the data for the other road condition.’ Further examples can be seen in the assessment schedule.
- Avoid analysis statements about the range and extreme values as these do not add to the discussion.
- Peer-critiquing and class discussion encourages students to consider what makes a quality analysis statement.
- If students are doing different investigations, rather than a single question for the whole class, it may be worthwhile to start by showing students a single set of summary statistics and graphs and discuss analysis statements based on these. Then students can write and peer-critique statements based on their own results.
- Some of the literature has students writing analysis statements starting with the words ‘I notice…’ This can be incorporated into the instructions for writing analysis statements if desired.