Prior experiences

Before working on this unit, students should have engaged in practical measurement exercises where they measured items of varying length using metres, centimetres and millimetres. They should also know the relationship between metres, centimetres and millimetres. This knowledge will be further developed throughout the following sessions, however, it is also an important prerequisite to collecting the required data (i.e. measurements).

Session 1 

In this session we introduce the problem and start collecting data.

1. Introduce the session within a context that is relevant to your students. 
2. Show the students a photocopy of the giant’s hand. Ask the question:
   We know the length, width and span of the giant’s hand… using this information, can we determine their height?
3. Discuss the possible relationship between their hand size and their body height. Record students’ ideas on a class chart. 
   Encourage students to suggest that it might be worth measuring and recording: body height, hand length, hand width, hand span.
4. Gather information to investigate the possible relationships.
5. Discuss the size of the sample needed to provide reliable information (the larger the sample size, the more reliable the data.
6. Discuss the range of ages (junior, middle, adolescent, adult) and genders (male, female) as variables.
7. Briefly discuss how to measure accurately with the tools provided (e.g. no gaps between measurements, start the measurement at 0, measure in a straight line). Provide the students with a table to use when recording data, or support them to construct one. Get the students to measure themselves and a partner. Ask students to record this data in their table. Collate it, as the teacher, on a spreadsheet (ensuring that students do not see each other’s data).  You could share data with other classes to get a larger sample of people.
8. The students could also gather more data from their family and whānau. This would give a greater range of ages and heights.

Session 2

In this session we create scatter plots of the relationship between height and hand measurements.

1. Discuss:
   What relationship are we trying to determine? (Relationship between height and hand size.)
   How can we determine whether there is a relationship between the various pieces of information we have gathered?
   How best could we show that relationship?
    
2. Show the students how to create a scatter plot. Get them to plot some data manually as well as with a digital tool as this will give them a better idea of how the computer generates scales for the axes.
   
    
3. Develop scatter plots for each of the following: height versus hand width, height versus hand length, and height versus hand span.
    
4. Discuss the appearance of the graphs.
   Can you see any patterns or relationships?
   Using the information on the graph can you predict a person’s height or hand size? Try using the sentence “if they are X cm tall, I think their hands will be X cm in length because the data shows that…”
   What relationships are you identifying? A person is (?) times their hand length, or their hand width is one-tenth of their height?
   Which hand measure is the best predictor of height? Why?
   Can you use this information to work out the giant’s height?
    
5. You may like to discuss the amount of variance (i.e. how far the plots are spread out - this shows how much a random variable differs from its expected value) and the range within which the relationships fall. This gives an opportunity to discuss lines of best fit and apply them to constructing the giant.
   
    
6. Explore whether the prediction will be different if you know that the giant is young or old, male or female. Excel allows you to sort the data in ascending or descending order and draw a scatter plot of only the junior children, or males, etc… if you want.

Session 3

In this session we create a silhouette of the giant.

1. Discuss possible relationships between other body parts.
2. Determine a list of body parts to be measured.
   Will the giant be 2-dimensional or 3-dimensional?
3. Can we use a simple length measurement to predict the size of other parts of the giant? For example: head length/width, foot to hip length, shoulder width, foot length, arm length, waist circumference, head circumference, or thigh/calf circumference.
4. Record the predicted measurements on the board.
5. Get the students to measure the identified measurements for themselves and a partner.  They should clearly identify which relationship they are trying to predict (e.g. relationship between height and shoulder width etc.) Ask students to record this data in a table and then use it to create a scatter plot. Discuss the importance of keeping the data confidential by not attaching any names to the data. If you shared data with other classes in the first session (when recording height), then you should also share the data from this session. 
6. Repeat the same discussion from the previous session. Ask:
   Can you see any patterns or relationships?
   Using the information on the graph, can you predict a person’s … or …? 
   What relationships are you identifying? A person is (?) times their …., or their … measurement is …?
   Which of your measures is the best predictor of the other variable? Why?
   Can you use this information to work out the giant’s height or to to determine the size and proportion of the giant?
7. Encourage students to explain the relationships they see in their data, and use these to predict a specific part of the giant (e.g. head width). Collate these predictions.
   Using the class’s predictions, make a 2-dimensional silhouette of the giant using newspaper or butcher's paper. Look up the tallest known man and woman and compare your giant to these people.

Session 4

In this session we discuss the accuracy of our findings.

1. Discuss:
   How accurate is your information?
   Using only a hand print, what other measurements can you predict? 
   How accurate were your predictions?
2. Discuss how a small change in the length of hand measurement makes a big difference to the predicted height.
3. Ask: Should we measure the hand length in centimetres or millimetres if we’re going to use it to predict the giant’s height? Why?
4. For several people use the relationship you have established for hand length and height. Measure their hand lengths in both centimetres and millimetres and use both measurements to predict their height. Does the accuracy of the hand length measure give a better prediction of height?
5. Ask: If we used thumb length as the predictor for height would it be more or less accurate than hand length? Why?
6. Get the students to investigate these questions in small groups and report back their results.

Session 5

In this session students are challenged to investigate other things we might be able to find out about the giant from their hand print. Challenge students to carry out an investigation in pairs or small groups. Support them to work through the steps of the PPDAC cycle. Support them to present their findings in an engaging and informative manner (this could provide a link to explanation writing, visual presentation skills, using digital tools to create a video or speech, and oral language skills).

Students will have many ideas about this such as:

• How many steps will the giant take to walk from your house to school?
• How long will it take them?
• What size would the giant’s house be?
• How much would the giant eat each day?
• What would their food bill be for a week?
• How big would the giant have been when they were a baby?
• How many balls of wool would you need to knit the giant a jersey?

Prior experiences

Before working on this unit, students should have engaged in practical measurement exercises where they measured items of varying length using metres, centimetres and millimetres, and measured time using minutes and seconds. They should understand how to measure accurately, and may have some experiences of converting between metric units. Through these experiences, they should have developed some understanding of the relationship between metres, centimetres and millimetres and between minutes and seconds.

Getting started

1. Introduce the topic of paper planes to the students by framing the concept in a relevant context (e.g. as part of an instructional writing unit, as part of an investigation into transportation methods). Tell the students there will be a competition at the end of the week to see which plane can fly the longest. They will all be designing their own planes to enter. Encourage them to think about the features of a paper plane that would help it to fly the longest. Record students' ideas on a chart. 
   What might be meant by the longest? Is this the longest distance, or the longest time, or some other longest? 
   What features would a plane that can fly a long way have? 
   What features would a plane that can fly for a long time have? 
   If you were to make a plane to fly a long way/long time what else would you need to consider?
2. Allow students time with paper to carry out initial experiments with planes and then brainstorm their ideas about features that affect the distance or time a plane will fly. Discuss these features and introduce the word variable.
   • Identify the different numerical variables (i.e. variables that can be measured or counted) that could be involved in the activity.  
     • how far the plane flies
     • how long the plane flies for
     • the length of the plane
     • the wingspan of the plane
     • the number of paper clips on the plane.
   • Identify the response or outcome variable that we will be measuring, either length of flight or time of flight, and then discuss how we will measure it (groups might want to do both and the class competition could include both, with their places in each combining to get an overall winner)
     • Flight time is the length of time from when the plane leaves the thrower's hand to when it lands on the ground, Which units of seconds, minutes or hours would be most appropriate for time? Why?
     • Flight length – throwers must stand behind a line. The place the plane touches the ground is identified (not where it ends up after skidding along the ground). The distance from the line to where the plane touches the ground is measured. Which units of measure, millimetres, centimetres or metres would be most appropriate for distance? Why?
   • Identify the possible explanatory variables (the feature of the plane that we will change) and then discuss how they will be measured
     • e.g. wingspan – measure from tip to tip
3. Have students work in pairs to experiment with the different units for measurement, and then facilitate a discussion about the units to be chosen for the class competition. 
   Which units allow for the greatest accuracy? Why?
4. If necessary, model accurate measuring using the tools you have provided and the units discussed. This could also be done as a quick, paired activity.
5. Set criteria for the materials to be used to make the planes in the following sessions. These criteria need to include a limit on the size of the paper that can be used and details on the numbers and amounts of other materials that can be used e.g. paper clips, tape or staples.

Exploring

Over the next few days have students work in pairs or small groups to carry out investigations using the following steps. Model each of these steps, in the correct order, before giving students the opportunity to start their investigations. Pay attention to the gaps that appear in students’ knowledge and use these to plan small group or individualised teaching sessions. Students may want to research the flight of paper planes before they chose the focus of their investigations. 

Investigation Steps

1. Make an assertion (a thoughtful statement) on a variable that will affect flight distance/time.
2. Choose a basic design for your paper plane, then modify (change the features of) the variable for this design to provide a variety of different models, based on your assertion.
3. Decide on how to test each plane, e.g. how many times will you throw the plane at each distance, how will you decide which outcome measure best represents for the variable (feature) chosen
4. Collect data by trialling each plane and recording the distance/time it flies alongside the variable (e.g. wingspan, plane length) you are testing. This can be done by recording the data in a spreadsheet or in a table in a statistical software package such as CODAP.
5. Plot data on a scatter plot to establish whether there is a relationship between the variables you are investigating. This can easily be done using CODAP or spreadsheet software, see Cars for ideas on plotting scatter plots using CODAP.
6. Check your assertion. Was your original idea correct? Modify your plane design accordingly.

Example Investigation

1. Assertion: a longer wingspan will help a paper plane fly a greater distance.
2. To test the assertion about wingspans several planes with varying wingspans are required.
3. Decided to test each plane three times and record the middle measurement.

Data collected as below:

Data plotted as below 

As the wingspan increases so does the distance flown.

As investigations are carried out the following points may need to be discussed with the students.

• For the investigations to be a fair test only one variable can be altered across each of the planes to be tested. The planes need to be the same in every respect other than the feature being tested.
• The number of trials needed for each plane should be considered carefully. In scientific contexts 3-5 trials are often used, pragmatically because of time constraints. This is usually sufficient to see if there are any outliers. At higher levels the mean is also taken, but for this level we will use the middle value when placed in order (median).
• The best way to record and plot the data.
• Is all data plotted or an average e.g. the middle value (median) for each different set of trials?

Reflecting

1. Hold a competition to see which plane flies the longest. Ensure accurate measurements are taken of distances flown or times taken.
2. After the competition reflect on the most successful planes.
   What evidence did we have that those planes would be the most successful? 
   If we were going to hold another competition which features could we combine to produce a very successful plane?
   Was there anything fair or unfair about our investigation?
3. If time permits, students could be given the opportunity to create a display that demonstrates their mathematical understanding in an appropriate format (e.g. video, oral or digital presentation, poster or infographic, written summary, by acting it out).
    

Although this unit is set out as five sessions, to cover the topic of statistical investigations in depth will likely take longer. Some of the sessions, especially sessions 4 and 5 could easily be extended as a unit in themselves. Alternatively, this unit could follow on from a unit on data presentation to give students an appreciation of practical applications of data display.

Session 1

Session 1 provides an introduction to statistical investigations. The class will work together to answer the investigative question – How many brothers and sisters do people in our class have? Be sensitive to the needs of your class - if this context is inappropriate for your students , then it may need to be altered.

1. Explain to the class that their job for maths this week will be to gather information or data on the class, summarise the information, or data, collected and then present this as a report which will be sent home to parents and displayed in the class. 
2. Ask students whether they can explain what the word statistics means.
   Explain that statistics concerns the collection, organisation, analysis and presentation of data in a way that other people can understand what it shows.
3. Explain that the class will work in small groups, each of them with the job of finding out information about the class.
4. First we will work as a whole class to answer the investigative question:
   How many brothers and sisters do people in our class have?
5. Ask the students what information we need to get from everyone in the class to answer our investigative question.  
   Students might suggest that we ask how many siblings they have, or they might suggest we ask how many brothers, how many sisters and how siblings they have altogether. 
   The idea of asking about brothers and sisters separately allows for a deeper exploration of the data and a more in depth answer to our investigative question. 
6. Agree as a class to ask about the three pieces of information. See if anyone can suggest how we could collect the data.
7. Working with student ideas, move towards a solution whereby each student records their information on a piece of paper.  
   Sticky notes could be a good way to collect this information from the students as it will allow rearrangements of the data quickly.
8. Suggest that the students divide their paper into three as shown in the diagram below to answer three survey questions. 
   • How many brothers do you have?
   • How many sisters do you have?
   • How many siblings do you have (or total number of brothers and sisters)?
     
9. Get the students to fill in their responses for their brothers and sisters. Check what a response of zero means – in one or all the sections (no brothers/no sisters, no siblings/only child)
10. Work with a partner to check that the information is correct and in the correct place. A good way to do this is for the partner to take the piece of paper and describe to another student the number of brothers and sisters the student has. 
    For example: 
    Pip records the following information about her brothers and sisters.  She gives it to her partner.  Her partner, Kaycee shares this information with another student.  Kaycee says that Pip has three brothers and one sister.  Altogether Pip has four siblings.
    
11. Collect all the pieces of paper (or sticky notes) and ask:
    How can we use the pieces of paper (or sticky notes) to show someone else how many brothers and sisters people in the class have?
    How can we show the information so that people can easily understand what it is showing?
    Hopefully, someone will suggest a more organised list, or counting the number of 0s, the number of 1s etc and writing sentences to explain how many there are of each.
12. Carry out these suggestions to show how much clearer they make the information.
13. Ask for suggestions for other ways to show the information. If nobody suggests it, introduce the idea of using a dot plot.
14. Demonstrate how to draw a dot plot of the information, ensuring that you highlight important features of dot plots; axis, scale and labels on the axis, title (use the investigative question), and accurately plotted points.
15. Students could draw their own versions as a practice exercise. It may be useful to provide a template with an appropriate scale for students to use.
16. Encourage students to draw separate graphs showing just brothers and just sisters as well.
17. Now that we have made a display of the data, in this case a dot plot, we need to describe the dot plot.
18. Ask the students what things they notice about the data. Record these ideas on the board. Write the words “I notice…” on the board or chart paper and capture ideas under this. They might notice:
    • What is the most common number of siblings/brothers/sisters?
    • The largest number of siblings, the smallest number of siblings
    • Where most of the data lies e.g. most of our class have 2-4 siblings.
19. Work with the students to tidy up their statements to ensure that they include the variable and reference the class. For example:
    • The most common number of siblings for people in our class is 2.
    • The largest number of siblings in our class is 7.
    • Most of the people in our class have between 2 and 4 siblings.
20. Explain that over the next few days students will be investigating some other ideas about the class, making their own graphs to show the information and describing what the information shows.

Session 2

This session is ultimately about choosing an appropriate topic to investigate about the class. There will be a real need for discussion about measurable data and realistic topics that can be investigated in the given time frame. It would be a good idea to provide the students with a list of topics if they get stuck, but they should be encouraged to try and come up with something original where possible.

1. Recap the previous session’s work, discussing how the information was collected, how it was presented, and how it was discussed.

PROBLEM: Generating ideas for statistical investigation and developing investigative questions

2. Explain that in this session students will work in small groups to come up with three topics to explore about the class.  The topics need to be ones that they can collect information from the class about and therefore complete the investigation.
3. Discuss criteria that the topics must meet.
   • Is this a topic that the students in our class would be happy to share information with everyone?
   • Would the topic apply to everyone in the class?
   • Is the topic interesting or purposeful?
4. Put students into small groups and give them a few minutes to come up with some ideas that they think they might use. Try to group together students with different levels of competence and encourage tuakana-teina. Encourage them to think of topics that use categories and topics that use counts (e.g. number of siblings). Ideally they should have at least one of each across their three topics.
   If groups are having trouble thinking of ideas, you could try writing a list of suggestions on the board but limiting groups to using one of your ideas only, to encourage them to think of their own. Some ideas could be:
   • Favourite flavour of ice cream/pizza/soft drink etc.
   • Favourite pet
   • Number of pets
   • Colour of eyes
   • Shoe size
   • Favourite native bird
   • Favourite beach
   • Cultural identity
   • Birthday month
   • Home language
   • Number of skips (using a skipping rope) in 30 seconds
   • Number of hops in 30 seconds
   • How they travel to school
   • Number of (whole) hours sleep the previous night
   • Number of languages students speak
   • Number of letters in their first name
   • Number of letters in their first and last names
   • Number of items in their school bag
5. Once groups have decided on their topics, work with them to pose investigative questions.  Model examples of these to help the students pose their own. 
   • How many brothers and sisters do people in our class have?
   • What are Room 30’s favourite pets?
   • What are Room 30's favourite native birds?
   • What eye colours do the people in our class have?
   • What cultures are present within Room 30?
   • How did Room 30 students get to school today?
   • How long are our class’s first names?
   • When are Room 30’s birthdays?
6. Once they have posed their investigative questions, share them as a class, and ensure that they are all appropriate, checking in on the criteria specified in 3.
7. If groups need to change any of their investigative questions, give them time to do so now.

PLAN: Planning to collect data to answer our investigative questions

8. Explain to the students that they need to think about what question or questions they will ask to collect the information they need to answer their investigative question.

9. Explain that these questions are called survey questions and they are the questions we ask to get the data. Work with groups to generate survey questions. For example: 

   • If the investigative question is: “What are Room 30’s favourite pets?”, ask the students how they could collect the data. 
   • A possible response is to ask the other students “What is your favourite native bird?”
   • Also, the students might want to ask, “What is your favourite native bird out of Tūi, Kiwi, Kerēru, or Kea?” You could challenge them as to if this would really answer the investigative question and suggest that possibly they might change the survey question to allow for other answers.

   Possible survey questions are:

   • What is the colour of your eyes?
   • How did you travel to school today?
   • What language do you speak at home?
   • What culture do you mostly identify with?
   • What month is your birthday?

   In these examples you can see that the survey question and investigative question are very similar, but there are key differences that make it an investigative question (What are Room 30’s favourite native birds? – overall about the class data) rather than a survey question (What is your favourite native bird? – asking the individual).

10. Ensure that all groups record their investigative and survey questions for the next session.

Session 3

Data collection is a vital part of the investigation process. In this session students will plan for their data collection, collect their data and record their data and summarise using a tally chart or similar for analysis in the following sessions.

PLAN continued: Planning to collect data to answer our investigative questions

1. Get the students to think about how they will record the information they get. Options may include:
   • Tally chart
   • Writing down names and choices
   • Using predetermined options
   • Using a class list to record responses
2. Let them try any of the options they suggest.  They are likely to encounter problems, but this will provide further learning opportunities as they reflect on the difficulties and how they can improve them.

DATA: Collecting and organising data

3. Students collect data from the rest of the class using their planned method.  Expect a bit of chaos. Possible issues aka teaching opportunities include:
   • Predetermined options
     • What happens for students whose choice is not in the predetermined options?
     • What if nobody likes the options given and they end up with a whole lot of people choosing the 'other' category and only have tally marks so they cannot regroup to new categories?
   • Using tally marks only
     • The discussed issue about the “other” category
     • Have less tally marks than the number of students in the class 
       • and they think they have surveyed everyone 
       • or they do not know who they have not surveyed yet
     • Have more tally marks than the number of students in the class
   • Possible solutions to the above issues could be (generated by the students)
     • Recording the name of the student and their response and then tallying from the list
     • Giving everyone a piece of paper to write their response on, then collecting all the papers in and tallying from the papers
4. Regardless of the process of data collection we are aiming for a collated summary of the results. 
   

Session 4

In this session the students will work on creating data displays of the data collected in the previous session.

ANALYSIS: Making and describing displays

Numerical data – displaying count data e.g. “How many…” investigative questions

1. Show the dot plot created in Session 1 of numbers of siblings.
2. Discuss how it was made and what needs to be included on it.
3. Get students to identify which one (or more) of their investigative questions involves count data. Choose one of these to work on first if they have more than one.
4. Give students time to work on their first graph, providing support as required. Providing pre-drawn axes may be useful, but students may still need help selecting an appropriate scale to use and placing the “dots”.
5. After all students have completed one of their graphs, bring the class together to share what students have done.
6. Discuss and compare graphs between groups.

Categorical data – displaying data that has categories e.g. “What…” investigative questions

7. Get students to identify which one (or more) of their investigative questions involves category data.  Choose one of these to work on first if they have more than one.
8. Ask students if they can remember how to graph categorical data (you may have already done some work on using pictographs or bar graphs e.g. Parties and favourites). Reference back to this previous work and discuss how it was made and what needs to be included.
9. Give students time to work on their categorical data graph, either a pictograph or a bar graph. You may want to encourage a bar graph depending on how much statistics you have already done prior to this unit.
10. After all students have completed one of their categorical data graphs bring the class together to share what the students have done.
11. Discuss and compare graphs between groups.
12. Send students to work on the last graph of their three.

Session 5

Session 5 is a finishing off session. Students should be given time to complete their graphs if they have not already, and to write statements about what the graphs show.

1. Give groups time to finish graphs as required.
2. Students should also write statements under each graph telling what the graph shows. Ideas for describing graphs were discussed in session 1.  Refer to these ideas.  In addition, starters for these statements could be given:
   • The most popular…
   • The most common…
   • The least popular…
   • The least common…
   • Most students in our class…
   • The largest number of…
   • The smallest number of…
3. Check their descriptive statements for the variable and the group. For example, favourite pets and our class; travel to school and Room 30.
4. Discuss with the students whether there is any action we should take as a result of any of the information we have found out in our investigations. 
5. Ask if there are any conclusions we can make from the investigations we have done.
6. Students could compile their displays as a booklet to take home to their families entitled "About our class" or similar. Alternatively, create a class display of the findings, or share them with another class.

Session 1

1. Explore children's prior knowledge and experience with the process and ingredients of hāngī and other culturally-relevant celebrations involving the preparing and sharing of food. You may want to refer to some articles and link the statistical investigation to literacy. 1) School Journal Level 2 August 2011- "Hāngī and Hogays" by Iona McNaughton, p.18 or 2) School Journal Part 3 Number 3 2011-"Puia Hāngi - Cooking with Steam" by Henarata Ham and the children of Room 8, Hirangi School, p.29. 
2. Pose the following problem to the class:
   The school has decided to put down a hāngī for all students to experience. Previously when the school did a hāngī a lot of food was wasted because students didn’t like certain foods. You have been given the task of finding out what food should be put in the hāngī so that there is the least amount of wastage.
3. Ask:
   What foods are cooked in a hāngī? What would best be cooked using this process?
4. Discuss how you could find out what vegetables and meat the students in your school prefer.
5. Pair students to discuss ways that they could go about collecting this data. Ensure they think about how they will collect the data (e.g. table, tally chart), what they will ask (their survey question), who they will ask (their sample), and why they will ask their sample their chosen question (what will it tell them?)
6. Share the various solutions and discuss the merits of each strategy. Encourage students to think about which methods will be most systematic - meaning organised and able to be followed according to a plan. Make a class decision by voting or a similar means to decide the method that will be undertaken.
7. Design the questionnaire. Look at bias, and pose such statements as:
   We are thinking of having mutton in the school hāngī, do you like mutton yes or no?’
   What is wrong with this question?  Explain that this question shows bias because the surveyor is already thinking about having mutton. Because they have said this, it may affect how the person answering the questionnaire might answer. This means that they have not given an equal chance to both possible answers to the question.
   How could we reword the question to eliminate the bias?
8. Sample survey the class to model the process of recording students' choices, by using a tally chart. If appropriate, direct students to use a tally chart, and provide a table, or help them to construct one. Alternatively, you could give students the option of collecting the data with a method they discussed earlier. Digital tools such as Google Forms or Padlet could be used alongside hard-copy tables.
9. Ensure that the questionnaire questions are written down and are consistent as different students will ultimately be surveying different classes. Why is it important that the way in which we collect the data is consistent?

Session 2

Prior to this session you would need to have organised for students to visit different classes to gather the data from each student.

1. Remind students of the investigation and what we are trying to find out. 
   Why are we trying to find this out?  (reduce food wastage)
2. Group the students so that they are divided evenly around the classes in the school. They are to gather the required data and return to class.
3. On their return, ask them to graph their data. If necessary, you could teach them to construct a specific type of graph (e.g. bar graph, dot plot, pie chart, strip graph, pictograph using a key) using a digital or hard-copy graphing template. Alternatively, you could allow students to graph their data using a method of their choosing. Ensure you support students to include the important features of their chosen graph type by asking questions such as: 
   What labels are needed on your graph? 
   What would be an appropriate title for your graph? 
   Can you tell me what your graph shows?
4. What statements can you make about the preferred meat and vegetables that need to go down in the school hāngī for your class? You might show one group’s example and model using specific points of the data to make summary statements (for example, I can see that 2 people said they preferred carrots, this was the least preferred vegetable).
5. Students conclude the session by making statements about what their data shows. What assertions can you make?
   Direct students to make a presentation (e.g. a video, infographic, set of Google slides) that shows the questions they asked and why, their ‘raw’ data, their graph, and 3-5 summary statements. 

Session 3 and 4

1. Provide the opportunity for each group to share their findings about the preferred vegetables and meat for each class they visited. Allow students to give and receive constructive feedback to and from their peers in response to the presentations. After all of the groups have shared, display all of the graphs and ask questions such as;
   What similarities and differences can you see between these classes?
   Is it easy comparing bar graphs and pie graphs? 
   What should we have done?
   What trends are starting to appear? 
   Which meat and vegetables seem to be most favoured?
2. Using the groups’ original tally charts collate the totals to get a whole school total for each vegetable and meat choice.
3. Students could move into new groups or remain in the previous groups and be given the task of displaying the new data in another specified way, or in a way chosen by the students. You could use this as an opportunity to teach students about a new graph type, or to consolidate knowledge of a previously-visited graph type.
4. From their findings students should write a statement to sit alongside their graph to answer the original problem for investigation. What food should be put down in the hāngī?

Session 5

1. Students should be given the opportunity to share their graphs and discuss their findings with the rest of the class.
   Ask reflective questions such as:
   Which graph best shows this data?
   Which graphs are easiest to interpret?  Why? 
   Which graph is least effective? 
   What kind of data would this graph be better for?
2. Other possible questions that could be investigated may include: 
   • How many students can be fed from one chicken?  How many chickens would need to be purchased to feed 46 students/112 students? 
   • Each potato can be cut into three pieces, how many potatoes will we need to purchase to feed 236 students?  

 

A roadside ‘coffee cart’ offers drinks and muffins for sale.

To make a healthy profit, they need to make just enough muffins to nearly sell out by the end of the day.

Use the sales figures for the past three months to suggest how many muffins they should make for a typical day.

Sales figures (XLSX, 12KB)

The following prompts illustrate how this activity can be structured around the phases of the Statistical Enquiry Cycle.

Problem

The problem section is about what data to collect and who to collect it from and why it’s important.

• What is my  investigation question? Why do I wonder about that? (The question might be framed as “I wonder how many muffins the cart operators should bake each day?)
• What variables should the cart owners consider? (Students might acknowledge that total sales and months (time) are key variables but they might also separate weekdays and weekend days into two groups.)
• Is my question a summary, comparison, or relationship question? (The question involves a relationship between time and number of muffins sold.)
• Why is my question important? (The answer will affect the profitability of the cart and whether it can operate financially.)

Plan

The planning section is about how students will gather the data.

• How will I go about answering this question?
• What data will I need? Is all the data available?
• How will I organise and display the data to look for patterns and relationships?
• Will I need to calculate some statistics? Which statistics? How will those measures be useful?
• Can I predict possible answers to my question, even before I look at data? How?
• What might my answer look like? (Students should realise that they need to provide a recommended number of muffins, supported by evidence and an argument.)

Data 

The data section is concerned with how the data is managed and organised.

• What format will I use to organise my data as I display it?
• What is the best way to look at trends over time? Why?
• What digital or written tools will help my organise my data? (Consider software for display, since the data are in a spreadsheet that can be imported into online software.)
• How will I protect the data I gather or retrieve so it is safe?
• Is some data ‘dirty’ (untrustworthy)? How will I clean my data?

Analysis

The analysis section is about exploring the data and reasoning with it.

• Have I reflected back on my question so I am clear what I am trying to find out?
• Do I get ideas about patterns, differences, relationships and trends from just ‘eyeballing’ the table of data?
• Are my important variables categoric, discrete numeric, or continuous (measurements)?
• How can I display different sortings of my data to look for patterns, differences, relationships and trends? 
• What tools will help me to display the data in different ways? What ways are appropriate to my data and help answer my question?
• How might I describe the trends over time? What language might I use to describe what I see? Are there trends or the beginning of a seasonal pattern?
• Are there differences among statistics at different times in terms of centre, shape and spread? Which measures should I use?
• What variation can I see in my data? What might be the causes of that variation?
• Can I make preliminary statements about my findings, starting with “I noticed that...”

Conclusion

The conclusion section is about answering the question in the problem section and providing reasons based on their analysis.

• Have I answered my original question? If not, why not? (There should be a number of muffins per day recommended, possibly given for weekdays and weekends.) 
• How might I convince someone else I have answered the question?
• What displays best show what I have found out? Explain why these displays are best.
• Is my summary of findings clearly written so others can understand it?
• Have I used displays and measures to support my ideas?
• Do I go back to the context to suggest why the relationships and trends occur?
• Do I say about the limitations of what I have found out? What can I say and what can’t I say?
• What strategies and tools proved the most useful in my inquiry?
• What other things have I learned and what further questions do I have?

Examples of work

Work sample 1

The student calculates relevant sample statistics and considers trends in the time series data to make a judgement about the number of muffins to bake.

Click on the image to enlarge it. Click again to close. 

Work sample 2

The student uses a combination of calculating sample statistics, creating data display and considering trends on weekdays and on weekends, to make a judgement about how many muffins to bake.

Click on the image to enlarge it. Click again to close. 

This unit is set out to cover the topic of statistical investigations in depth will likely take 1-2 weeks. Some of the sessions may take more than one classroom session to complete. There is an introduction session followed by five sessions that follow the statistical enquiry cycle (PPDAC cycle) as described in the New Zealand Curriculum. Data detective posters showing the PPDAC (problem, plan, data, analysis, conclusion) cycle are available to download from Census At School in English and te reo Māori.

While this unit plan uses the five phases of the PPDAC cycle as a step by step process, in reality when using the PPDAC cycle one often moves between the different phases. For example, students might need to revisit the investigative question (problem) as a result of the planning phase. 

Session 1: Introduction

This session provides an introduction and purpose to statistical investigations. The teacher will need to provide the students with plenty of magazines, newspapers and websites that have some good examples of how data can be presented effectively and perhaps some examples of poorly displayed data. This could be collated into a chart or slideshow. Prior to the session, ask the students to spend some time at home looking through magazines and newspapers to find examples of statistics to bring in for the session.

1. Start the session with a class discussion to get the students thinking about whether or not we have a need for statistical investigations, and who uses the information?
   What is a statistical investigation?
   Can you think of an example when we might need to carry out a statistical investigation?
2. Organise the students into groups of two or three. Give out magazines, newspapers and website links and ask the students to find some examples of statistics.
3. Ask the students to look closely at the examples they have selected. Ask them to consider the following questions;
   Who has done the research for/carried out this investigation?
   Who will benefit from the results of this investigation?
   Is it clear to you what the purpose of the investigation is?
   What do you like about the way that the information is presented?
   Does it help you in any way to understand the information better?
   Do you think the information could have been presented in a different way to help the audience understand the findings? If so, what would have made it better?
4. Use a class discussion to share ideas from each group. Have the students all come up with the same ideas? Try and steer the students towards the conclusion that the best way to present the information depends on the information itself. They might notice that category data is displayed differently to numerical data.

Session 2: PROBLEM (Generating ideas for statistical investigation and developing investigative questions)

This session is ultimately about choosing an appropriate topic to investigate. You will need to discuss what data is actually measurable within your context and realistic topics that can be investigated in the given time frame. It would be a good idea to provide the students with a list of topics (perhaps relating to a current school issue, relevant curriculum area, or your students' cultural backgrounds and interests). Encourage students to come up with something original where possible.

1. Set the scene by recapping the discussion from the previous day about the purpose of a statistical investigation. The purpose of a statistical investigation is to identify a problem or issue that can be explored using data. The process includes “designing investigations, collecting data, exploring and using patterns and relationships in data, solving problems and communicating findings” (New Zealand Curriculum, 2007, p.26).
2. Set the students up to decide on a broad, relevant topic to investigate. This could include an initial brainstorming session in small groups and then the sharing of ideas as a class. Make sure the students know to choose a topic that will have some benefit or serve a purpose. Ideas to help include:
   • An issue across the school e.g. litter, uniform, parking, traffic, drop off/pick up zones
   • About the class e.g. pets, favourites, number of…, use of devices,
   • Something specific to the community e.g. options for a gala, market stall, Matariki celebration, best time for whānau to visit and see what is happening in class
   • Finding information about a particular activity e.g. sport involvement; hobbies and interests
   • Behaviours e.g. fridge pickers, tv watching, online learning
3. Once the initial brainstorming of ideas is done, interrogate the topics with the following questions:
   • Is this an area that the students in our class would be happy to share information with everyone? Or is it an area that our target group (e.g. whānau) would be happy to share information with us. If not reject the idea [ethics].
   • Can we collect data to answer an investigative question based on this topic or issue? If not reject the idea [ability to gather data to answer the investigative question].
   • Would you be able to collect the data to answer the investigative question in the timeframe we have specified? If not reject the idea [ability to gather data to answer the investigative question].
   • What would be the purpose of asking about this topic or issue? If it is not purposeful then reject the idea [purposeful or interesting].
   • Would the investigative question we pose involve everyone in the group (e.g. the class or another defined group)? If not reject the idea [does not involve the whole group].
4. Organise students into groups and have them select a topic or issue to focus on.
5. Support students to develop an investigative question(s) based on their topic or issue. If necessary, you could develop a few investigative questions as a class, before asking students to do this in their groups.
   These are the questions we ask of the data; it will be the question(s) we explore using the PPDAC cycle. 
   • Prompts to help with posing investigative questions are:
     • What is the variable that you want to ask about?
     • Describe the group that you are asking about?
     • Do you want to describe something (summary) or compare something (comparison)?
       • Summary questions have one variable and one group e.g. How much litter is around the school after lunch? [litter after lunch, around the school]; What pets do the students in our class have [pets, our class]?
       • Comparison questions have one variable and two or more groups e.g. How does the amount of litter that is around the school compare between after recess and after lunch? Does the traffic outside the school in the afternoon tend to be more than the traffic outside the school in the morning?
6. Check the investigative questions that students have posed. Collate them (e.g. write them on the board, type into a google doc or write on sticky notes to be pinned up). As a class check each investigative question for the variable and the group to be asked, against the remaining criteria:
   • Is the question purposeful? This should have been sorted in the generating topics for investigation stage.
   • Is the question about the whole group? Check that it is not just finding an individual or smaller group of the whole group. This too should have been sorted in the generating topics for investigation stage.
   • Is the question one that we can collect data for? This again should have been sorted in the generating topics for investigation stage. 
   • Is it clear that the question is a summary or comparison question?
7. Collect in the final investigative questions. Label who posed them in preparation for the next session. Double check the investigative questions before the next session as poorly posed investigative questions can hinder the subsequent phases.

Session 3: PLAN (Planning to collect data to answer our investigative question)

Data collection is a vital part of the investigation process. The teacher will need to stress to the students, once again, the importance of being consistent in the collection of their data. There will also need to be sufficient discussion around efficient methods for data collection and recording.

1. We need to plan to collect the data. Explain to the students that all the data will be collected using one of the following methods, depending on what data they need to collect. They might use an online survey form (e.g. google forms), and/or a paper survey, or tables (online or hard copy). Consider the skills and knowledge already developed by your students, and which method will best, in reflection of this. Ultimately, the class should move towards collecting individual data in individual rows of a spreadsheet or table. 
2. To answer our investigative questions, we need to collect specific information or data using data collection/survey questions. In this phase of the cycle we are planning to collect our data. This means we need to pose data collection/survey questions. 
   Fundamentally, data collection and survey questions are the same – they are both questions we ask to get the data. 
   • Survey questions are those we pose for a questionnaire to survey people e.g. What is your favourite colour? How did you travel to school today? Do you like eggs? People answering the questionnaire record their own responses and we collate these once all the questionnaires are complete
   • Data collection questions as those we pose for other data collection situations e.g. if we are going to collect data about the make and colour of cars passing the school then we might pose the data collection questions – what is the make of the car; what is the colour of the car and record these in a table.
3. Ask students what they think would be useful to consider when they pose their data collection/survey question(s). Gather a few key ideas to help them with this. For example:
   • The question needs to be specific
   • Keep wording simple and short
   • Avoid questions that ask about more than one thing
4. Support students to pose their data collection/survey questions. They should also think about any specific instructions, e.g. if they were going to collect information the amount of litter around the school they may need to define what they consider to be litter, what are the different areas they will collect from, how they will count the litter e.g. by number of pieces of litter, by weight, by plastic bags full.

Managing surveys: depending on the target groups and how you plan to manage the survey process there are a few options here to choose from.

Option 1: an online questionnaire is developed for each group that will be surveyed. This following should be considered:

• What is the group? E.g., the class; the parents of the class; teachers in the school; students in another class (e.g. another year level)
• Does the questionnaire contains all the survey questions from across the class that pertain to that group?
• How will the questionnaire link be sent to participants and collected by students?
• Is any identifying information collected? All responses should be anonymous – teachers will need to manage this carefully.

Option 2: a paper questionnaire is developed for each group that will be surveyed. Similar considerations to the online questionnaire are needed, except that a paper copy will need to be printed for each person to fill out. These should be collected up and brought back to the class if the people who have filled them out are not in the class

Other data collection methods

Depending on the topics, students might be collecting data about litter, cars, pedestrian traffic. These are not things that we would use a questionnaire for so the students will need to think about a plan to collect the data. They may decide to use a pre-prepared table or grid to do this. The table should be set up so that the information for each of their data collection questions for a single object can be recorded in a single row. For example:

Collecting information about vehicle make and colour – students might also think to collect the vehicle type too.

• Set up a table with four columns: 

  Number plate	Vehicle type	Vehicle make	Vehicle colour
  AAA123	Car	Audi	Blue
  BBB456	Ute	Ford	Gold
  CCC789	Car	Holden	Red
  DDD111	Truck	Isuzu	White
  123AA	Motorcycle	Suzuki	Red

• Record in a single row the information about one car
• They should also consider in their planning how long they will collect the data for and where (this will form the “group” – data about the vehicles driving past the school from 1-2pm on 24 September).

Students need to check with the teacher before commencing data collection to ensure that their method of collection is the most appropriate and will result in data that is useful for analysis.

Session 4: DATA (Collecting and organising data)

1. Provide time for students to collect and record their data, according to their plan. Regardless of the method of collection our end aim is for students to have their data tabulated with the data from a single person or object in a single row. 
2. Provide modelling and support for students as they enter their data into a spreadsheet. This should be tabulated with the variables across the top and the data listed in rows below, the table in the example about vehicle make and colour shows the structure. Consider the following:
   • If data is in an online questionnaire, give the students only the data pertaining to their investigative questions
   • For paper questionnaires the data should be collected into a spreadsheet for their questions only
   • If a paper copy of a table was used this should be transferred into a spreadsheet
3. Check for any data input errors
4. Save as a .csv file

Note for teachers: 
Students will use their .csv file to make their displays in the next session. If it is not possible for them to save as a .csv then the teacher may need to do this and share with them or set up the CODAP document with their data and share a link to this. See the video or written instructions on how to do this. Note the video and the instructions include getting started with CODAP too.

Session 5: ANALYSIS part 1 (Using an online tool to make data displays)

In this session the students will be introduced to using an online tool for data analysis. One suggested free online tool is CODAP. Feel free to use other tools you are familiar with. This session is written with CODAP as the online tool and assumes students have not used CODAP before.

If you do not want to use an online tool, then continue to Making Displays, and construct paper versions of bar graphs and dot plots.

Learning how to use CODAP

1. Allow the students some time to get familiar with CODAP. Using the Getting started with CODAP example is a good starting point. This has a built-in video that shows the basic features of CODAP and gets you started using the tool. Other support videos can be found here. 
   
   The main features that students need to be familiar with are how to draw a graph and how to import their data. More on importing data into CODAP can be found here.
   
   Bar graphs for categorical data
   
   CODAP by default makes a dot plot for both categorical and numerical data. If the data is categorical the bar graph icon (configuration icon) can be selected to fuse the dots into bars, shown in the two pictures below. The graphs are showing the habitats of mammals.
      
   Students should be encouraged to try different things out with the data to get further insights as to what the data might show them. For example, for the above data about mammals students might want to see what happens to the diet for different habitats. They can drag the diet attribute onto the top axis of the graph (and to get different colours they can drag the diet attribute into the middle of the graph to make a legend) and the following display will result.
    
   This gives a deeper insight into the data. You will find that students at this age are comfortable with using CODAP once they have had a little time to play with the software.
   
   Dot plots for numerical data
   
   When using CODAP for numerical data a dot plot is the default setting. For example, sleep in hours for mammals shown below.
   
   The data can be split into groups by dragging a categorical attribute to the vertical axis. To explore the sleep by the different habitats, drag habitat to the vertical axis, or to explore sleep by the different diets, drag diet to the vertical axis. The following graphs result.
    
    

Making displays for the data they have collected to answer their investigative question

1. Now that the students are familiar with CODAP they can make displays with their own data to help them to answer their investigative questions. Have students label their graphs using their investigative question.
2. Graphs can be exported by using the camera icon or students can take a screen grab of the graph to put into another document. Alternatively, students can use the text feature in CODAP and write their descriptions in there. As we are heading towards a presentation it is most likely that they will use their graphs in another document for the presentation.

Session 6: ANALYSIS part 2 (Describing data displays)

1. To describe the display, encourage students to write “I notice…” statements about their displays. Initially accept all statements as encouraging the idea of noticing is valuable for both statistics and other aspects of the mathematics curriculum. If students are not sure what to notice the teacher can prompt further statements by asking questions such as:
   • What do you notice about the most common number of…?
   • What do you notice about the largest number… the smallest number…?
   • What do you notice about where most of the data lies…?
   • What do you notice about the most popular… least popular…?
   • What do you notice about how the data for the litter after lunch is different to the data for the litter after recess (more specific example for a comparison) …?
2. Check the “I notice…” statements for the variable and reference to the group. For example: “I notice that the more than half the vehicles that went past our school from 1-2pm on 24 September were cars.” This statement includes the variable (types of vehicles) and the group (past our school 1-2pm on 24 September). Support students to write statements that include the variable and the group.

Session 7: CONCLUSION (Answering the investigative question and reporting findings)

This last session will focus on the final presentation of the data each group has found out. Encourage the students to be constantly evaluating what they are doing. Explain that it is fine to discover that a particular way of presentation is not working, and that it is a good idea to adjust.

1. Use this time to finish presenting information in graphs, tables, or any other format.
2. Present information in a way that includes the important parts of their investigation. Provide time and opportunity for your students to present this information using tools that are relevant and engaging for different students (e.g. as a video, poster, digital animation, speech).
   • Topic chosen
   • Investigative question(s)
   • Survey/Questionnaire/Data collection method/questions
   • Group data was collected from
   • Results – tables/graphs and descriptions of the data
   • Conclusion – answer to their investigative question
   • Call to action?
3. Have groups of students share their finished presentations with the class.
4. Evaluation: (Peer and Teacher)
   • Give feedback, including constructive criticism.
   • Is the information easy to understand?
   • Could we make it any clearer?
5. Talk about who could use the information that has been presented. Can we send it to anyone outside school? For example, investigations related to a road safety issue could be forwarded to the local council.