The Numbers Don’t Lie

Purpose

The purpose of this unit is for students to learn and apply ideas about sample to population statistics, including a rule of thumb for making the call, to data collected as part of their physical education (PE) lessons.

Achievement Objectives
S5-1: 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...
Specific Learning Outcomes
  • Be able to use technology (CODAP) to process data and produce graphs.
  • Analyse dot plots and box and whisker plots.
  • Understand and use the “median outside the overlap” rule of thumb.
  • Write a report that answers a comparison investigative question.
Description of Mathematics

This unit uses many of the same ideas as the Level 4 unit Crunch the Coach.

The overall aim is for students to make a judgment about whether year 9s or year 10s show better improvement during a term of physical education classes. It could also work with any two groups within a class such as “boys or girls”.

The statistics learning is based on the Statistical Investigations series (Part 1Part 2Part 3) by Matt Regan, Maxine Pfannkuch and Pip Arnold that was developed in conjunction with the TLRI research project: Building students’ inferential reasoning: statistics curriculum levels 5 and 6.

  • Students collect and process data about their catching skills.
  • Students use initial data to learn about dot plots, distributions, box and whisker plots.
  • Students use the Karekare College population from CensusAtSchool to learn about sampling and sample to population inference.
  • Students use pre and post PE results for their class to learn about arrow graphs.
  • Students use both classes’ improvement data to report on the comparative progress of the two classes and use this to make an inference about all junior students at the school.

Associated Achievement Objectives:

Health and Physical Education:

Personal Health and Physical Development

  • Experience a range of personally enjoyable physical activities and describe how varying levels of involvement affect well-being and lifestyle balance.

Movement Concepts and Motor Skills

  • Acquire and apply complex motor skills by using basic principles of motor learning.
Required Resource Materials
Activity

This unit is presented as a series of sessions, with most activities roughly corresponding to approximately 45-50 minutes. Each mathematics lesson is alternated with a PE lesson. At the beginning and end of the unit these are closely correlated but in the middle there is room for each subject area to focus on its own specific skills. At the end of the unit there is a greater focus on the statistical report.

The PE lessons are focussed on catching and throwing skills; these can be in the context of any catching and throwing game you choose.

It is assumed all students know and are able to follow the PPDAC cycle. It is also assumed that students have learned about dot plots but have not yet learned how to draw a box and whisker plot.

The statistical software used is CODAP. It is browser based and tutorials in a NZ context can be found at http://karekareeducation.co.nz/category/codap/.

This unit could be run by two classes at the same school, or at different schools and results compared as part of the analysis (see sessions 6 and 11).

Session 1

Two lessons

PE focus

SLOs:

  • Students explore the idea of control variables in a statistics and PE context

Activity 1: Introduction to the PE focus – throwing and catching

We are going to explore (1) what factors/attributes might affect our ability to throw and catch, (2) if instruction and practice improve our ability to throw and catch and (3) quality data collection.

Introduce the PE focus for the unit of work – throwing and catching. Discussion around the three things we will explore as listed above and how all of these connect to support improving our ability to throw and catch.

An initial PE activity where students are using the skills of throwing and catching. Drawing students’ attention to the techniques of throwing and catching and thinking about what attributes the better throwers and catchers might have. This could be a game of dodgeball.

Following the initial PE activity, explain how the baseline data for throwing and catching will be collected.

Activity 2

Discussion about the three tests: standing catch, moving catch, and throwing. Students brainstorm and discuss ideas about variables and factors that have to be controlled for in these tests.

Students then take part in the testing and data collection. Data collection sheets are used to record student results for each test. 

Collecting baseline data for throwing and catching

There are three different pieces of information that are collected.

  1. Stationary catch
  2. Moving catch
  3. Throwing

Each student will need their own data sheet (Copymaster 1) to collect their baseline data. The data sheet could be used to collect final data (and mid-point if wanted) as well. Each student also needs a data card (Copymaster 2) to record the combined score for each of the three pieces of information.

Instructions for set up to collect throwing and catching data

Stationary catch

Students will catch the ball at three different distances from the person throwing. There will be three throws at three different distances (10m, 20m, 30m) to catch, a total of nine catches per student.

The students line up at each of the three distances and the teacher throws the ball three times to each student. The student records the number of catches for each distance on their own data card. 

Moving catch

There will be three balls to catch at each of three positions. The starting positions are approximately 20m from the person batting the ball. The person batting the ball will aim to get the ball to land abut 5m from the student forcing a need to move to catch. Students line up behind each of the three positions and receive three balls to catch. Students record the number of moving catches in each position.

Throwing

The students get five throws from the throwing zone. Distance and accuracy are important. Depending on where the throw lands depends on the score. The gym is split up into zones based on a blue grid similar to the one below. The 12 larger zones should all be the same size, with the 3 higher scoring zones at the end around half the size. Students make their five throws and the score for each throw is recorded.

At the end of the session have each student complete their data card with their name, year, gender, and a total score for each of standing catch, moving catch and throwing. Collect in the data cards. The data will be input along with other data collected in the next session.

Session 2

SLOs:

  • Students learn how to organise data into a spreadsheet.
  • Students use the data to make personal goals for the unit.

Preparation: Prior to the lesson the teacher needs to make a copy of the data cards, one set for each group. The teacher should also have prepared a blank spreadsheet with the student names and space to enter each student’s data. Each row of the spreadsheet corresponds to one person, the first column is a unique identifier such as a name and each other column contains a single variable. 

Activity 1

The teacher introduces the job of Data Entry Administrator using a website such as www.careers.govt.nz. In this session, students will be data entry administrators. Their task is to enter the data from the previous lesson into a spreadsheet and save it as a csv file. This is a good time to learn that the tab key moves across columns and the enter key moves down. They can work in pairs, one person reading the data and one entering it, or work individually using a ruler to keep track of the rows.

Activity 2

Students are then taught/reminded about smart goals (Specific, Measurable, Achievable, Realistic, and Timely) and encouraged to write one or two goals for improving their small ball skills during the unit.

Activity 3

Students practice small ball skills in a PE context.

Session 3

SLOs:

  • Students learn to make box and whisker plots
  • Students see what the whole class data looks like

Preparation: Some pre-prepared axes drawn on the board or on a PowerPoint that will allow students to quickly and easily place their data cards in position on the whiteboard.

Activity 1

The teacher and class discuss how difficult it is to quickly get a good idea of where the class is at just by looking at the numbers on the data sheets. This activity involves creating a picture/graph to help us. The teacher hands out the data cards, each with a small amount of bluetack.

For the first variable of interest, such as standing catch, students come up and place their cards on the axis on the whiteboard. A class discussion can be had about the shape of the distribution. The teacher can then demonstrate how the median and quartiles are found and what they mean. The end result is the distribution and box and whisker plot for the whole class are on the board. Students could sketch this into their books.

This activity is repeated for the other variables of interest. Each time the students come up and move their own cards. If the white board is wide enough, it is possible to fit two or three distributions up and this allows for discussion about how different tests have led to different (or similar) distributions.

Activity 2

Discussion around how this data could inform PE learning.

Students practice small ball skills in a PE context.

Session 4

SLOs:

  • Students produce digital box and whisker plots.

Preparation: The teacher needs to have created a single data file so that all students are working with the same data. This can then be converted to a CSV file, imported and saved in CODAP. Share a link to this CODAP file so that students can easily access it when required in the lesson. 

Activity 1

The teacher demonstrates how to use CODAP to produce a distribution and box and whisker plot for the first variable of interest. Having entered data themselves and taken part in the data card activities it will be easier for students to understand what the table is about and that each dot on the graph represents a person and the position of the dot is based on data about that person.

Students then have a go at producing the same graph themselves. Students could also colour the graphs by gender (though at this stage we are not making comparisons).

Activity 2

Students are tasked with producing a graph for each variable of interest and copying the graphs into a document file such as a Google Doc.

Activity 3

Continuing to think about how the data is informing us, students practice small ball skills in a PE context.

Session 5

SLOs:

  • Students learn to analyse distributions for whole groups.

Preparation: The teacher needs to have printed the relevant graphs or created a booklet for students to work in. Alternatively, students could type their analyses directly into their document file made in the previous session.

By the end of this session, students should have a solid understanding of what the data looks like for the class and how it links to the physical activities and training they are doing.

Activity 1

Using class ideas and discussion, the teacher leads an analysis for the whole group using one of the variables of interest. This could result in a “model analysis”. The four categories for discussion are shape, centre, spread, unusual/interesting.

Activity 2

Students then write analyses for each of the remaining variables of interest.

Activity 3

Continuing to think about how the data is informing us, students practice small ball skills in a PE context.

Session 6

SLOs:

  • Students learn to explore data and create graphs that compare two sub-groups.

Preparation: The teacher needs to have created a file that includes data from the competing class. This file could be saved in CODAP and then a link sent to students.

Activity 1

The teacher hooks students into the lesson by introducing them to doing their first comparison and seeing how they compare with the other class (if there is one).

The teacher starts with the problem they are going to investigate: e.g. “I wonder if the Year 10 class tends to have better initial standing catch scores than the Year 9 class”.

The teacher then demonstrates how to use CODAP to show a comparison of the two classes for this variable of interest and then leads a class discussion into what this means.

Students then use CODAP to reproduce this graph.

Activity 2

Using class ideas and discussion, the teacher leads a more detailed analysis for the comparison. This could result in a “model analysis”. The four categories for discussion are shape, centre, spread, unusual/interesting. The discussion can finish with a conclusion to the posed investigative question for the group. This could be followed up by starting to get students wondering about whether this is true just for these two classes or might be true for the whole junior school.

Activity 3

Continuing to think about how the data is informing us, students practice small ball skills in a PE context.

Session 7

SLOs:

  • Students write an analysis for a comparison question.

Activity 1

The teacher leads a discussion where students think about other comparison investigative questions they could ask of the data. By the end of the discussion a number of well written investigative questions could be up on the board. 

Activity 2

Students are tasked with writing a comparison investigative question, creating a graph to compare the variable of interest, and then writing an analysis. They can then write a conclusion for the group and be prompted to start wondering about whether this might be true for the whole junior school. There is also the opportunity for students to share their findings with each other as they will not all work on the same problem.

Activity 3

Continuing to think about how the data is informing us, students practice small ball skills in a PE context.

Session 8

SLOs:

  • Students learn to take samples.
  • Students learn how to make a call about the population based on their sample.

Throughout this session the PE context has now moved to practicing and preparing for the final challenges against the competing class.

Activity 1

Prior to this lesson, the teacher needs to have made enough sets of the Karekare College data cards for one set per pair of students as well as pre-prepared grids for heights and time to school boxplots. Note that gender on the data cards is indicated by the colour of the card so boys need to be printed on a different colour to girls.

Introduce the purpose of this session, that is to try and see if can we make a call about all year 9s and 10s based only these two classes.

The teacher should emphasise that the class is now going to step outside of the real life situation into a learning space. The learning done here will then be applied back to the real-life situation. Students are going to use another high school’s data from CensusAtSchool to learn about how to do make a population inference. It is a learning situation because students will be able to take multiple samples, in real life they only get one sample.

Introduce Karekare college by handing out data cards to pairs/groups of students. Describe the population to the students. Tell the students that there are 13 variables altogether and see what variables they can work out. The variables all come from the 2009 CensusAtSchool survey. Provide students with a link to the 2009 CensusAtSchool questionnaire or provide paper copies of the survey questions that were used. When they identify a variable, they should give a reason why they think it is a particular variable.

The variables on the data cards are indicated in the table below.

Ethnicity

Age in years

Year level

Transport to school

Time to school in mins

Height in cm

How carry school bag

Weight of school bag in grams

Popliteal length in cm

Fitness level

Index finger length in mm

Ring finger length in mm

Get students to pose some investigative questions that could be answered using the available variables. Talk about possible “sub” populations. Do not expect students to know the term sub-populations rather that populations can be defined e.g. girls at Karekare College.

Students can pose some comparison investigative questions that could be asked about different “sub” populations of students at Karekare College. Introduce the question “Do boys tend to be taller than girls at Karekare College?” this is the first investigative question we are going to answer about Karekare College.

Students can now start to start to draw out cards and record the heights on the pre-prepared grid. Hopefully someone will suggest that using all the cards will take a long time. So what can be done instead? Hopefully someone will suggest that they just take some students to get a picture of the school. Suggest that each group selects a handful of students from Karekare College. This is called a sample. Ask “Why might a sample be a better idea than using the whole population?” Some reasons for sampling: practicable, takes less time. A sample can tell us about the population.

Each pair of students takes a sample of about 30 boys and 30 girls and draws a box plot on the pre-prepared grid. Once they have completed this, they can go and see other groups results and hopefully notice that each sample has produced different statistics. Ask the class who tends to be taller at Karekare College, boys or girls, based on their samples. There will be a mix of answers. By a show of hands the class can see who says boys and who says girls.

Some groups will show boys have a higher median height, others will show girls have a higher median height – this is a problem. Obviously different samples are showing different statistics and in this case the samples are leading to different conclusions. The story is inconsistent and we don’t know who tends to be taller.

The same process is repeated to answer the question “Do students who walk to school tend to take longer than students who bus at Karekare College? The students are given time to think about whether they think this is true. With samples of about 30 busers and 30 walkers, students complete a box and whisker plot on a pre-prepared grid. Students then report back to the class. All groups will see the bus median higher than the walk median and have the same conclusion – bus students tend to take longer to get to school. Different samples are still showing different stats but the message is consistent and we can probably be pretty sure bus students tend to take longer to get to school.

HOWEVER in the real world we only get to take one sample so how would we know which situation we are looking at? This is the motivation for Activity 2.

Activity 2

Using the work from the previous activity, students can now develop a way to “make a call”. Develop these ideas using lessons 10, 11 and 12 from the Statistical Investigations on Census at School. (Lessons 9-12)

Activity 3

Students go back to their work from Session 6 and apply their learning by making a call for the whole junior school based on their class data.

Session 9

SLOs:

  • Students develop their own investigative questions.

Activity 1

Problem

As a class, the students are now ready to be able to evaluate the improvement of all participating groups. They can start posing comparison questions about the improvement of different groups. E.g. “I wonder if Year 10s tend to show better improvement in throwing score than Year 9s at our high school?” The teacher tidies up and records a list of well-written investigative questions on the board.

Students choose an investigative question to work on and write a hypothesis for it.

Activity 2

Plan

Students group together based on their posed investigative question, decide on how data can be collected, and organise to do this. Alternatively, if the data collection will be organised by teachers there needs to be some discussion with the class about the factors taken into account when running the tests.

Activity 3

Data

Students collect data as part of their PE lessons.

Session 10

SLOs:

  • Students analyse improvement data.

Activity 1

Data

As a class, the students are now ready to be able to evaluate the improvement of all participating groups. The first task is to process the data. This can happen on paper initially, for just their class, or a smaller group of students. Using the sheet, students calculate the improvement scores for each student. This helps generate discussion about what a positive improvement score means, what a zero score means, and what a negative score means. This is an ideal time to celebrate the best improvers in the class.

Activity 2

Analysis

Students use CODAP to process data and produce a graph that helps them answer their comparison investigative question. They can then write an analysis of this.

Activity 3

Conclusion

Students make a call about who tends to show better improvement for all junior students based on the “median outside the overlap” rule of thumb. There is also the opportunity to discuss other factors for this and whether the results of their investigation match what they would expect (their hypothesis).

Session 11

SLOs:

  • Students reflect on the results of their investigations.

Activity 1

Students report back to the class on their findings or the teachers can summarise the results for each of the comparison investigative questions.

Students and teachers come up with ideas on what the results mean for the PE department and what they could do about them.

Activity 2

Students take part in a PE challenge using the skills they have been focusing on. This could be against the other class doing this unit, if there is one.


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