In this unit we conduct a number of investigations using a party or favourites theme. Students count, compare, organise, analyse, display and interpret data and at the same time, apply early additive strategies for combining numbers.
- Pose investigative questions.
- Plan for and collect category data.
- Display data in tally charts, pictographs and bar graphs.
- Make statements about data displays.
- Answer the investigative question.
At Level 2 you can expect students to be posing (with teacher support) a greater range of questions, including investigative questions and survey questions. They will also be helped to understand some of the issues involved in conducting surveys and learn new methods for collecting data. While at Level 1 students collected data and chose their own ways to display their findings, at Level 2 they will be introduced to pictographs, tally charts and bar graphs. More emphasis here will also be placed on describing the data and the making of sensible statements from both the student’s own displays and the displays of others.
At Level 2 students should be generating broad ideas to investigate and the teacher works with the students to refine their ideas into an investigative question that can be answered with data. Investigative summary questions are about the class or other whole group. The variables are categorical or whole numbers. Investigative questions are the questions we ask of the data.
The investigative question development is led by the teacher, and through questioning of the students identifies the variable of interest and the group the investigative question is about. The teacher still forms the investigative question but with student input.
Survey questions are the questions we ask to collect the data to answer the investigative question. For example, if our investigative question was “What ice cream flavours do the students in our class like?” a corresponding survey question might be “What is your favourite ice cream flavour?”
As with the investigative question, survey question development is led by the teacher, and through questioning of the students, suitable survey questions are developed.
Analysis questions are questions we ask of displays of data as we start to describe it. Questions such as: what is the most common? the least common? how many of a certain category? what is the highest value (for numerical data)? lowest value (for numerical data)?
In a pictograph the pictures are drawn on uniform pieces of paper. This means that the number of objects in each category now bears a direct relationship to the size of each category on the display. An example is shown in the diagram below.
In a further development the pictures can be displayed on a chart with axes and titles. The vertical axis can be numbered to match the pictures.
In a bar graph equal-width rectangles (bars) represent each category or value for the variable. The height of these bars tells how many of that object there are. The bars can be vertical, as shown in the example, or horizontal.
The example above shows the types of shoes worn in the class on a particular day. There are three types of shoes: jandals, sneakers, and boots. The height of the corresponding bars shows that there are six lots of jandals, 15 lots of sneakers and three lots of boots. It should be noted that the numbers label the points on the vertical axis, not the spaces between them. Notice too, in a convention used for discrete data (category and whole number data), there are gaps between the bars.
A tally chart provides a quick method of recording data as events happen. If the students are counting different coloured cars as they pass the school, a tally chart would be an appropriate means of recording the data. Note that it is usual to put down vertical strokes until there are four. Then the fifth stroke is drawn across the previous four. This process is continued until all the required data has been collected. The advantage of this method of tallying is that it enables the number of objects to be counted quickly and easily at the end.
In the example above, in the time that we were recording cars, there were 11 red cars, four yellow cars, 18 white cars and five black ones and 22 cars of other colours. Microsoft Excel is a program available on most types of computers that allows data to be entered onto a spreadsheet and then analysed and graphed very easily. There are a number of freely available tools for graphing data, for example CODAP – Common Online Data Analysis Platform, is an online statistical tool that is accessible from a young age.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:
- Direct students to collect category data or whole number data – whole number is harder
- Give students summarised data to graph rather than them having to collect it and collate it
- Give students a graph of the display and ask them to “notice” from the graph rather than having them draw the graph
- Write starter statements that students can fill in the blanks to describe a statistical graph e.g. I notice that the most common XXXX is ________, More students chose _______ than chose _______.
The context for this unit can be adapted to suit the interests and experiences of your students. For example:
- Planning for a class party
- Planning for a special occasion – e.g. deciding on the type of food, the activities to include, venue
- Kai festivals e.g. Motueka Kai Fest, Hokitika Wildfoods Festival, Kāwhia Kai Festival
- The favourite example can be adapted to explore any favourites
- Packet of balloons – different shapes and colours if possible
- Sheets of A4 cut into eighths
- Prepared bar graph outlines
- Multi packs of chips (popcorn and cups)
- Party props: hats, candles, cards, sweets, blind fold
Session 1: Balloons investigation
Today we will make a pictograph of our favourite balloon shapes. We are going to answer the investigative question “What different balloon shapes do the students in our class like?”
- Take a bag of balloons and spread out. Discuss shapes. Suggest the investigative question “What colour balloons do the students in our class like?”
- tudents choose favourite shape (or colour if different shaped balloons are not available) and draw it on a piece of paper (one eighth of an A4).
- As a class, discuss ways to display the data. If matching pictures in 1:1 lines (pictograph) is not suggested, teacher will need to direct them to this.
- Students each attach their drawing to the class chart.
- Ask the students what they notice about the information shown on pictograph. Use the prompt “I notice…” to start the discussion. These “noticings” could be recorded as "speech" bubbles around the chart.
- Talk about the need to label the axes and give the chart a title so that others could make sense of the display. A good idea is to write the investigative question as the chart title.
- Ask analysis questions, to extend the noticing, about the results that require students to combine sets:
How many students liked long wiggly balloons?
How many students liked long straight balloons?
How many students liked long balloons altogether?
How can you add the numbers together?
How many students liked balloons that were not long?
How many more students liked long wiggly balloons than long straight balloons? (Model and reinforce the use of subtraction or addition rather than counting on or back to solve this type of question.)
Try to find analysis questions that will allow students to use strategies such as near doubles and adding to make 10s.
Session 2: Birthday Party investigation
This birthday party investigation is described in full as a possible model for teaching and developing ideas for each of the stages of the statistical enquiry cycle at Level 2. In New Zealand we use the PPDAC cycle (problem, plan, data, analysis, conclusion). You can find out more about the PPDAC cycle on Census At School New Zealand.
Check in: if the birthday party context is not suitable for your students you can find another context, the process described here will work for other contexts.
PROBLEM: Generating ideas for statistical investigation and developing investigative questions
- Ask the students to think about the topic of birthday parties. Explain that we will be going to collect some information to answer different investigative questions about birthday parties that we are going to pose.
Using the starter “I wonder…” ask the students what they wonder about birthday parties. On the board or on a chart record their ideas. For example:
- What are our favourite birthday cakes?
- What birthday games we like?
- How many birthday parties kids have gone to? (check in on this, will it be a tricky question to answer, could it cause distress?)
- Where kids like to have their birthday parties?
- What birthday presents we want?
- What types of food we like?
- How many people we want at our birthday party? (this too might cause distress, as may other questions, as the teacher you will have a reasonable idea of the questions that will be ok and those that will not).
Using the “I wonder” prompt helps with generating investigative questions, questions we ask of the data.
- Depending on what the students give you in the brainstorming session depends on how much work is needed to tidy up the investigative questions. New Zealand based research has identified six criteria to support the development of and/or critiquing of investigative questions. These criteria are used in the example below. The teacher asks questions of the students to identify the information needed e.g. variable, group and with this information develops the investigative question.
- The intent of the investigative question is clear – we need to pose summary investigative questions (about category or whole number data)
- The variable of interest is clear e.g. favourite birthday cake flavour, number of people at our party
- The group we are interested in is clear e.g. our class, Room 30, Kauri class
- We can collect data to answer our investigative question
- The investigative question considers the whole group e.g. How many birthday parties have the children in Room 30 been to this year? – considers the whole group; whereas What is the most number of birthday parties that a child in Room 30 has been to this year? – does not and is not an investigative question (it is an example of an analysis question and asks about an individual)
- The investigative question is interesting and/or purposeful – in this case if the ideas are generated by the students then we would expect them to be interesting to the students.
For the birthday cakes example some possible questions are:
- What will we wanting to find out about? (Favourite birthday cakes)
- Who are we going to ask? (Our class)
- Do you think we could find this out by asking our class? (Yes)
- Are you interested in knowing about favourite birthday cakes? (Yes (if no, then ditch the question).)
For each of the ideas generated in part 1, possible investigative questions are:
- What are favourite birthday cakes for children in Room 30?
- What are Room 30’s favourite birthday games?
- How many birthday parties have the children in Room 30 been to this year?
- Where do Room 30 children want to have their birthday parties?
- What presents do Room 30 children want for their birthday?
- What are favourite birthday foods for the children in Room 30?
- How many people do Room 30 children want at their birthday parties?
- Each group selects one of the investigative questions to explore.
PLAN: Planning to collect data to answer our investigative questions
- 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.
- 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 the favourite birthday cakes for children in Room 30?”, ask the students how they could collect the data.
- A possible response is to ask the other students “What is your favourite birthday cake?”
- This might lead to a discussion about whether they mean the flavour and/or the style (a possible way to extend this for some students), e.g. I might respond my favourite birthday cake is a dinosaur, when they might have been meaning the flavour e.g. chocolate, banana etc.
- This might also mean that the investigative question needs adjusting to: What are favourite birthday cake flavours for children in Room 30?
- Also, the students might want to ask, “What is your favourite birthday cake out of chocolate, banana, ice-cream or sponge?” 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 your favourite birthday cake flavour?
- What is your favourite game to play on your birthday?
- How many birthday parties have you been to this year?
- Where do you want to have your birthday party?
- What presents do you want for your birthday? (this could give multiple answers, may want to change to what is the present you most want…)
- What is your favourite birthday food?
- How many people do you want to have at your birthday party?
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 favourite birthday cakes for the children in Room 30? – overall about the class data) rather than a survey question (What is your favourite birthday cake flavour? – asking the individual).
- 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 pre-determined options
- Using a class list to record responses
- 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
- Students collect data from the rest of the class using their planned method. Expect a bit of chaos. Possible issues that lead to useful teaching opportunities include:
- Pre-determined options
- What happens for students whose choice is not in the pre-determined options?
- What if nobody likes the options given and they end up with a whole lot of people choosing other? They only have tally marks so they cannot regroup to new categories.
- Using tally marks only
- The discussed issue above about the “other” category
- Have fewer 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 please)
- 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
- Pre-determined options
- Regardless of the process of data collection we are aiming for a collated summary of the results.
ANALYSIS: Making and describing displays
- Taking their summarised information the students make a pictograph to help to answer their investigative question. As for the balloon activity we want to have uniform pieces. Provide:
- Squares of paper all of the same size for students to create their own pictures
- Chart paper
- Students give the chart a title – a good option is the investigative question.
- Students make the pictograph by gluing enough pictures to represent the data they collected.
- Teacher roams questioning for understanding and ensuring that students can correctly construct a pictograph.
- Once students have completed their pictograph they should write at least three “I notice…” statements about their pictograph. They can write the “I notice…” statements onto the chart paper as well.
- Teachers can prompt further statements by asking questions such as:
- What do you notice about how many students liked cakes that were not chocolate?
- What do you notice about the number of birthday parties attended? Did you notice the greatest number of birthday parties? The least number of birthday parties?
- Emphasise questions that require students to operate with the numbers in their displays.
- Check the “I notice…” statements for the variable and reference to the class. For example: “I notice that the most favourite birthday cake flavour for Room 30 children is carrot cake.” This statement includes the variable (favourite birthday cake flavour) and the class (Room 30 children). Support students to write statements that include the variable and the group.
- Get students to leave their charts on their desks. Hand out post it notes to the students and get them to wander around the class and to look at all the other graphs. Encourage them to add “I notice…” statements to the graphs of others by using the post it notes.
CONCLUSION: Answering the investigative question
At the end of the session get each group to share their chart. They should state their investigative question and then the answer to the investigative question. The answer should draw on the evidence from their pictograph and their “I notice…” statements.
For example: What are favourite birthday cakes for children in Room 30?
Answer: The most popular birthday cake flavour for Room 30 is chocolate cake. 15 students in our class had chocolate as their choice. The other flavours that were liked included carrot cake, banana cake and ice-cream cake. Carrot cake was the least popular cake flavour for Room 30.
Extending: If I (the teacher) was to make a cake for the class what flavour should I make?
Session 3: Chips or popcorn
The previous session involved the full PPDAC cycle. In this session today we are going to look at using tally marks to record the number of chips in a snack bag or the number of pieces of popcorn in a small cup and a bar graph to display the data. We are focusing on the data collection and analysis phases.
- Display a snack bag of chips (or a small cup of popcorn) and ask the students to guess how many chips (popcorn) they think are in the bag (cup).
- Pose the investigative question: How many chips are in XXX brand snack bags? (How many popcorns are in the small cups?
- We are going to collect data to answer our investigative question by counting how many chips are in each of the snack bags I have here (count the number of popcorns in the small cup).
- How should we do that? Elicit ideas including counting them all. Ask how we could count them and keep a track? Accept all ideas including using tally marks to keep a track.
- Teacher models using tally marks to track how many chips (popcorns) she/he eats.
- Distribute individual bags of chips to small groups.
- Students eat chips and use tally marks to record the number of chips in each bag by adding the total of the tally marks each student in the group recorded. (Record the number of popcorns in the cup, but don’t combine, will use each students cup count as a data point – this gives more data points than the chips do, unless you give each child their own bag of chips.)
- Gather the total tallies on the board or a chart.
- Using a prepared bar graph outline the teacher constructs a bar graph with the information from the individual total tallies.
- Discuss features of the graph and summarise the information shown.
What was the most common number of chips (popcorns)?
What was the least common number of chips (popcorns)?
How many more chips (popcorns) were there in the packet (cup) with the most than there were in the one with the least?
- As a class challenge, try to work out how many chips (popcorns) the class ate altogether.
How many chips (popcorns) did the boys eat?
How many chips (popcorns) did the girls eat?
Discuss strategies for adding the numbers together (for example: combine the numbers that add to 'tidy' numbers; add the tens and then the ones; use doubles or near doubles).
Session 4. Party props.
Today we will plan our own investigation from a party prop display.
- Use "party props" to generate discussion about parties.
- Brainstorm possible investigative questions using the prompt “I wonder…”. You may need to model possible investigative questions as you display party props. For example:
Here are some balloons. Gosh I’m hopeless at blowing up balloons. It probably takes me 12 or more breaths to blow it up. How many breaths does it take you?
I wonder how many breaths it takes our class to blow up a balloon.
Generate a few ideas using this process and following the ideas from session 2.
- In pairs students select an investigative question to explore and plan how they are going to collect the data. Use the ideas from session 2 to support students to plan for data collection.
- Once the data is collected the pairs need to display the data using either a pictograph or a bar graph. Ask them to write a couple of ”I notice…” statements to accompany their graph.
- Share survey results. Students can be challenged to answer the questions on each other's data displays.
Session 5. Favourites
In this session we will undertake a statistical investigation using the idea of favourites as our starting point. The big ideas for the investigation are detailed in session 2. Ideas to support the specific context are given here.
Brainstorm with the students different things that they have a favourite of. You might also use the starter “I wonder what are favourite _________ for our class?”
Using the ideas developed previously identify 10-15 favourites to be explored and develop investigative questions for pairs of students to explore.
Investigative questions might be:
- What are favourite sports that the children in our class play?
- What are our class’s favourite waiata?
- What are Room 30’s favourite kai?
As the students have had some practice with planning previously allow them some freedom to plan for their data collection. Check in on the survey questions they are planning to ask.
Students collect the data that they need to answer their investigative question, remember that they will have potentially inefficient methods they employ to do this. Be prepared for some chaos. Use any resulting errors or problems to improve their data collection methods.
Get the students to display the data to answer their investigative question. They may use a pictograph or a bar graph. Remind them to label with the investigative question and to write “I notice…” statements about what the data shows.
Allow time for pairs to present their findings by giving their investigative question and then answering it using evidence from their displays and noticings.