Session 1-2 

In these sessions we investigate the volume of corn kernels before and after popping. The amount of popcorn to be made is based on the batch size using a popcorn maker, which is usually about 1/2 a cup of kernels. We use non-standard units to measure the volume of both kernels and popped corn. We think about why the measurements vary (because the units being used vary) and what could be done to improve the consistency of the measurements (use a standard unit).

1. Present the students with a small container (about 1/2 cup size). Have the students measure the volume of the container using spoonfuls of kernels. Use a variety of spoons from teaspoons to large salad and serving spoons.
2. Record the different measurements of volume on a chart with illustrations of the spoons used.
   Ask: Why are we coming up with different numbers of spoonfuls needed to fill the container? 
   Record students’ responses, encouraging them to compare the sizes of spoons.
3. Make the popcorn – popcorn makers are the easiest to use in the classroom setting, but other methods provide the same results.
4. As the popcorn pops ask students to make predictions about how much popcorn there will be. Explore some containers including ice-cream containers, bowls and boxes and ask the students to identify a container that they think will be the same volume as the popped corn. (Note that 1/2 cup of kernels makes about 4 litres of popcorn in a popcorn maker.)
5. Once the popcorn is popped, tip it into the selected container to check the students’ prediction, then find a container that is a good fit for all of the popped corn.
6. Get the students to consider the original measure used for the kernels, i.e. 1/2 cup, and to think about how many of these would be filled by the popped corn. Have the students measure the popped corn with the measure used for the kernels. For example, they might find that a half-cup scoop of corn kernels produces 32 half-cup scoops of popped corn.

Session 3

In this investigation the students revisit the results from the previous investigation, and the idea of measuring the popcorn using a litre measure is explored.

1. Revisit the results of the previous volume investigations and talk about the volume of kernels used, and  how much popcorn was made. Focus on the fact that the volume of kernels used was different when measured with different sized spoons, and introduce the idea of a standard measure. Note that standard measures are useful because they enable accurate communication, suggesting that if you were going to send instructions for making popcorn to another class, spoonfuls of corn kernels would not be a useful way to measure, because they might be using different sized spoons than you.
2. Introduce the litre as a standard measure of volume, and ask students to identify things they know that use litres (for example, 2L milk bottles, 2L ice-cream tubs, 1L bottles of juice). Show some litre containers to the class (soft drink bottles could be cut down to a litre or half litre quite easily).
3. Have the students measure the volume of the container which the popped corn fitted into using the litre measure (water, rice, wheat or sand could be used for this task, it doesn’t have to be popcorn.)
4. Get the students to explore the other containers available and measure them in the same way using the litre container and counting how many fit into each bowl, ice cream container and box. Have the students label and order the containers and identify any that would have fitted the batch of popcorn.

Session 4

In this investigation the students think about a standard serving size for the popcorn. The students will find out how many litres of popcorn will be needed for everyone in the class to get one serving.

1. Discuss the question: How much popcorn do you like to eat when you go to the movies? 
2. Have lots of small containers available so that students can choose the size of container which represents the amount of popcorn they would like. Try to come to a consensus about the size of a share of popcorn.
3. Choose one container, which represents the size of a serving of popcorn (having some containers which are 1/2 or 1/4 litre size and directing towards those will make the measurement easier, but this isn’t necessary).
4. Use the serving size container and talk about how many servings you would need for the class. Students may like to consider making some popcorn for the principal, or a neighbouring class. Use water to measure out the right number of serves – a large plastic bucket or container will be needed.
5. Use the litre containers to find out how many litres of popcorn will be needed for everyone to get a serving. Measure by filling the litre container and counting.
6. Look back at the results of Session 3 and have the students work out how many batches of popcorn the class will need to make. Some students may be able to calculate this easily, others will have to use the container that the popped corn fitted into to help them work it out.

Session 5

On the final day of the unit the students make cones to fit one serving of popcorn into. The batches of popcorn will be made and the students will be able to measure out their serving to eat.

1. Provide materials to make cones to hold the popcorn. Ask the students to construct a cone that will hold the agreed serving size of popcorn, then check the volume of their cone by measuring. The volume of the cone can easily be adjusted by making the cone wider or narrower or by cutting from the top.
2. Make the batches of popcorn as the students work on the containers. Have the students fill their cone using the serving size container as the measure.
3. When all the containers are filled, ask the students to record some facts about making the popcorn to share. 
   What volume of corn kernels have been used?
   How many batches of popcorn have been made?
   How many servings were made?
   How many litres of popcorn have been made?
   How much is left over?
   If you were making some popcorn for your whānau how many batches would you need to make?
4. Discuss responses together.

Getting Started

1. Introduce the session by asking the students to work through several equal group (set) problems first and then ask them to pose their own problems. For example:

   • There are 8 cars. Each one can take 2 people on the school trip. How many people are there altogether?
   • There are 6 bags of shellfish (kaimoana). Each bag contains 4 pipi. How many pipi are there altogether?
   • There are 7 waka in the race. Each waka holds 3 students. How many students are in the race?
      

   When writing these problems, consider what times tables your students are confident in applying to word problems. Also consider how students might benefit from working in pairs (tuakana-teina).
   
   The students can represent these and similar ‘equal sets’ problems with:

   • towers of interlocking cubes 
   • threading beads 
   • jumps on the number line 
   • interlocking cubes on a number track 
   • drawing a picture to show the number of waka and the corresponding number of students. 
      

   Note: It is important to link the examples (where possible) to the structure of repeated addition of equivalent sets as multiplication.  For example: 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28 or 7 x 4 = 28. Discuss what the numbers 4, 7, and 28 refer to and what the operations symbols + and x refer to. The multiplication symbols can be thought of as meaning ‘of’. For example, 7 x 4 = 20 means seven sets of four.
    

2. Now ask the students to make up word problems using the problem structure above with different answers. For example, “Write a multiplication problem with an answer of 24”.
    

3. Use several sets of ice-cream containers (all with the same number of items in them) with the contents of each covered except for one.   Ask the students to write story problems for each example.

   • What strategy do the students use to solve the problems?
   • Do they try to count the contents of each ice-cream container by ones; that is, those that are visible and those that are concealed?
   • Do they use skip counting, e.g. 3, 6, 9, …, or repeated addition, e.g. 3 + 3 = 6, 6 + 3 = 9, …?
   • Do they apply multiplication facts, e.g. 5 x 3 = 15 so 6 x 3 = 18 (3 more)?
      

   Be aware that students’ choice of strategy depends on the connection between the conditions of each problem and the number resources that they have available. Expect that the strategies used by individual students will vary.

Exploring

Over the next three days the students are exposed to a variety of different types of story problems. They are encouraged to model the problems using different equipment and explain their answers to others. They think about the most efficient ways of solving the problems. It is important that students are provided with opportunities to build up multiplication facts to 10 and then to 20. Some students may solve these problems without equipment, using the number knowledge they have available.

Rate problems

1. On the first day, work through several measurement rate problems. Draw on the multiplicative strategies students used previoulsy, and model how to work out the answer when necessary. It is good if students notice that the situations are structurally similar to the ‘equal sets’ problems from the previous session. Measurement quantities, especially time, are more intangible than ‘bags of’ or ‘packets of’ in equal sets situations. Acting out problems can support students to see the common structure.
    
2. Use these measurement rate situations:
   [Name] can write their name in 10 seconds. How long does it take to write their name four times?
   You might select a student to role play writing their name and use an analogue clock (less battery) and/or stacks of cubes to track the time in seconds
   
   [Name] drinks four cups of water each day. How many cups does he/she drink in one week?
   Use plastic cups to build up the equal sets of four cups that are involved in this problem. Use another material to track the number of days.
   
   [Name] puts five pieces of harakeke into every bundle. How many pieces of harakeke does he/she need for eight bundles?
   
   [Name] puts three spoons of Milo in each cup. How many spoons of Milo does he/she need for 10 cups?
   
   These problems are more accessible than the time related tasks. Shirts can be cut out of paper and buttons represented by counters. Cups and plastic spoons can be used to model the Milo problem. Both quantities in each rate are tangible.
    
3. Copymaster 1 has some rate problems for students to solve. The problems contain a mixture of tangible and intangible units.
   The students can create similar types of problems with pictures and pose the problems to each other. Encourage them to explain their strategies to each other.
    
4. After time solving the problems, gather the class.
   Discuss what is the same about all the problems you have just worked on.
   Do students express the idea of a rate as being a “for every” relationship?
   Now ask the students to make up similar word problems and to pose their problems to each other. Encourage them to explain their answers to each other. 

Multiplicative comparison

1. On the second day of exploration use PowerPoint 1 to expose your students to comparison situations. The first preference of students may be to look for additive relationships. For example, here is a correct additive response to the question on Slide One.
   How much taller is Jill’s apartment block than Jack’s apartment block?
   S: Jill’s apartment block has 12 floors and Jack’s has four floors. Jill’s block is eight floors higher.
   Students may not offer a ‘times as many’ multiplicative answer. If that occurs pose this problem:
   Jill says that her apartment block is three times higher than Jack’s block. I wonder what she means?
    
2. Slides Two and Three show the additive and multiplicative comparisons that can be made.
   Look for students to note the inverse relationships:
   S: Jack’s apartment block is eight floors less (shorter) than Jill’s.
   S: Jack’s apartment block is one third of the height of Jill’s.
3. Work through the other slides of PowerPoint 1 looking to see if students identify the common structure of finding difference (additive) and finding the scale factor (multiplicative).
    
4. Copymaster 2 contains many multiplicative comparison problems. Some problems are in the form where the relationship is required while others require application of a given scale factor. Students are also encouraged to write equations to represent the situations. Let the students solve the problems, with the support of materials like counters if they need it. After a suitable time, share the answers as a class.
   
   Do your students:
   • Recognise the meaning of “times as many.”?
   • Represent the situations correctly with materials?
   • Identify the scale factor and the set to be scaled?
   • Record the equations correctly?

Arrays

1. On the third day of exploration work through several array problems based on situations in which there are equal groups. When modelling arrays, it may be helpful to talk about the lines across as ‘rows’ and the lines up and down as ‘columns’.  Teams with the same number of members in each are often used during the school day. 
   Pose problems such as:
   The students are lined up in 3 teams for sport. Each team has 6 members. How many students are there altogether?
   Encourage the students to draw representations of problems like this using three rows (one for each team) and six columns (one for each team member). Alternatively, use the students as the objects in the problem. If students draw the situation as three columns of six it opens discussion of the commutative property since 6 x 3 = 3 x 6.
    
2. Other problems you might use include:
   • A tray of eggs has five rows and six columns. How many eggs are in the tray altogether?
   • The kapa haka group is arranged in four rows of children. There are ten children in each row. How many children are there altogether?
   • A chocolate block has six columns, with four pieces in each column. How many pieces are in the whole block?
   • A tray of rēwana bread for the school whānau day has seven columns, with six loaves of rēwana bread in each column. How many loaves of rēwana bread can fit on the whole tray?
      
3. The students can model these and other array problems with:
   • pegboards:  pegs on a pegboard can be used to illustrate arrays in multiplication. 
     For the problem above this could be talked about as 3 rows of 6 pegs or 3 sixes, 
     or 3 rows of 6, or 3 x 6 = 18
     By turning the pegboard a quarter turn, the array still has a total of 18 pegs.
     This could be talked about as 6 columns of 3 pegs or 3 rows of 6 pegs or 6 threes or 6 columns of 3 or 6 X 3 + 18.
   • interlocking cubes
   • colouring grid paper
   • diagrams.
4. Copymaster 3 contains some problems related to arrays. Encourage your students to record an equation for each array, to show how they found the total number of objects. 

Reflecting

On the final day of the unit we play a game called 'Array Trap' in which the students use graph paper to plot arrays.

For this activity you will need:

• a sheet of graph paper for each player (at least 30 x 20 squares);
• 3 x ten-sided dice - each side has a different digit on it (you can use 3 x standard 1-6 dice, but the facts are limited)
• different coloured felt pens.

Players take turns to:

1. Roll the dice and choose two numbers, for example 8 and 2;
2. Mark out a rectangle of that size, for example 8 rows of 2, or 2 columns of 8; or a different pair of factors with the same area, for example, four columns of four.
3. Write the multiplication basic fact in the rectangle, for example 8 x 2 = 16 or 2 x 8 = 16;
4. Players take turns until one player is trapped. That is, they are unable to find space for the array they have rolled.

Discuss strategies for the game such as:

• placing large arrays near the edge to maximise the space available for more arrays
• renaming the factors so the array fits a given space.

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.

As each pede is developed, help ākonga focus on the number patterns involved by creating tables as below. Similar tables can be drawn for each type of pede.

Use of a hundreds board may help ākonga visualise the number patterns more easily and help them to predict which numbers will be part of the patterns.

The conclusion of each session is an ideal time to focus on the number patterns involved. Questions to develop number knowledge include:

Which number comes next in the number pattern for this pede? How do you know?
Which number will be before 20 in this pattern? (or another number as appropriate)
How do you know?
What is the largest number you can think of in this pattern? How do you know? 
Could a pede with 20 squares be a Spotted Pede? Why / Why not?
Could a pede with 32 squares be a 2-pede? Why / Why not?
Are there any numbers that could be Spotted Pedes and Humped Back Pedes? What are they? How did you work that out?

Session 1

Here we explore number patterns related to the mythical creatures that live on the planet Elsinore. The patterns involve skip counting by 2s.

Introduce the idea of a mythical creature with a picture book like Taniwha by Robyn Kahukiwa, or Zog by Julia Donaldson. Consider whether a local iwi member could come to talk to your class about the history of taniwha in your local area. On planet Elsinore there lives a strange collection of creatures. There is the Humped-Back Pede. The Humped Back 1-pede looks like this. Can you see their eye? And the Humped Back 2-pede looks like this. They have an eye too. (Show your ākonga the pictures below.) Ask ākonga to work individually or in pairs (tuakana/teina model) to make a Humped Back 3-pede with the green tiles, or draw a picture.
Can you work out how many squares a Humped Back 4-pede has?

Gather ākonga together to talk about the creatures that they drew. Explore the number pattern of counting in 2s that comes from the Humped-Back Pedes. Also ask questions like:
Can you tell me how many green squares a Humped Back 5-pede will have?
Can you tell me how many green squares a Humped Back 7-pede will have?
Can you tell me how many green squares a Humped Back 10-pede will have?
How many feet has a Humped-Back Pede with 12 squares?
How many feet has a Humped-Back Pede with 18 squares?
How many feet has a Humped-Back Pede with 20 squares?
Can you tell me how to get the number of squares that a Humped-Back Pede with a particular number of feet has?
Can you tell me how to get the number of feet that a Humped-Back Pede with a particular number of squares has?

Session 2

Here we investigate some more mythical creatures that live on planet Elsinore. The patterns here involve skip counting by 3s.

1. There are other creatures on the planet Elsinore. They look as if they have been made up from squares. The ones with one foot are called 1-pedes. The ones with two feet are called 2-pedes and the ones with 3 feet are called 3-pedes. (Show ākonga the picture below.)
   
2. Did you know that ‘pede’ means ‘foot’?
   How many feet would a 4-pede have? What about a 5-pede?
   Can you tell me how many squares a 1-pede has?
   How many squares does a 2-pede have?
   What about a 3-pede?
   (Put the numbers of squares beside the creatures as ākonga answer the questions, or create a table)
3. Can someone tell me what a 4-pede looks like?
   How many feet will it have?
   How many squares will it have?
   Can someone make one for me with these square tiles?
   Does everyone agree with that?
   (Write 11 under the 4-pede.)
4. Let’s have a look at the number of squares that these Pedes have. Count them out. 2, 5, 8, 11.
   I wonder what sort of Pede comes next? (5-pede.)
   How many squares does a 5-pede have? (14)
   I wonder what sort of Pede comes next? (6-pede.)
   How many squares does a 6-pede have? (17)
   Is there any pattern in the number of squares that Pedes have? (Add on 3 for each extra foot. Talk about skip counting by 3s.)

Session 3

Here we explore patterns further. Where we are particularly interested in linking the number of squares on a creature and the number of feet it has.

1. Here we are going to explore the Spotted Pedes. This is what they look like. Talk about the number of squares they have, and the number of blue and red squares.  Record these beside each of the Spotted Pedes.
   
2. Ask ākonga to draw or make the Spotted 4-pedes? As they work ask them the following questions:
   How many red squares does a Spotted 4-pede have?
   How many blue squares does a Spotted 4-pede have?
   How many squares are all together? How did you work that out?
   Why are there more blue squares than red squares? How many more?
3. Repeat with Spotted 5-pedes and Spotted 6-pedes.
   How many red squares does a Spotted 5-pede have?
   Can you tell me how many blue squares a Spotted 5-pede has? Draw it.
   How many red squares does a Spotted 6-pede have?
   How many more blue squares does a Spotted 6-pede have? Draw it.
4. For those ākonga who need challenging, ask them about the Spotted 10-pede?
   These ākonga could also be encouraged to make a table of there mathematical thinking.
   How many red squares does a Spotted 10-pede have?
   How many blue squares does a Spotted 10-pede have?
5. At the end of the session spend time sharing findings as a class (mahi tahi). Ask ākonga:
   What did you find out about the Spotted Pedes?
   What patterns did you find?
   (Try to get them to see that they have as many red squares as they have feet. This means that it is very easy to find out how many red squares they have.)

Session 4

1. We are going to find out about the Big-Headed Pedes. This is what they look like.
   
2. Ask ākonga to work with a partner (tuakana/teina model could work well here) to draw the next three Big-Headed Pedes (4, 5, 6). As they work, ask ākonga the following questions:
   How many yellow squares does a Big-Headed Pede 4-pede have?
   How many yellow squares does a Big-Headed Pede 5-pede have?
   How many yellow squares does a Big-Headed Pede 6-pede have?
   Do you need to draw the creatures to work out how many squares they have? Why or why not?
   Could you work out how many yellow squares a Big-Headed Pede 10-pede would have?
3. At the end of the session spend time sharing findings as a class (mahi tahi). Ask them:
   What did you find out about the Big-Headed Pedes?
   What patterns did you find?
   (Try to get them to see that they have three more yellow squares than they have feet. This means that it is very easy to find out how many yellow squares they have.)
   I saw a Big-Headed Pede with 16 yellow squares. How many feet did she have?

Session 5

1. Can you tell me what kinds of Pedes we have been talking about this week?
   Discuss the Pedes, the Humped-Back Pedes, the Spotted Pedes and the Big-Headed Pedes.
2. Ask ākonga to work with their partners (tuakana/teina model) to invent a Pede of their own. Ask ākonga to record the first three pedes on one piece of paper and the other three on a second sheet. Ask them to also invent a name for their creature.
3. Pairs could then swap the first three Pedes to see if they can work out the next three Pedes for each other’s creature. They can then check with each other to see if they arrived at the same Pedes for the 4, 5 and 6 creatures.
4. As time allows, ākonga could swap pedes with other pairs.
5. As the pairs work, ask them to discuss the various patterns that they have produced. Ask them questions such as:
   How many squares does one of your 5-pedes have?
   Can you tell me the number of squares an X Pede with 10 feet (or some other relatively large number) has?
   If I had 15 squares, what is the Pede with the largest number of feet that I could draw?
6. Collate the classes Pedes into a book of Pede problems that can be worked on by ākonga during any choosing time.

Prior Experience

The activities are mostly open ended so they cater for a range of achievement levels. It is expected that students have some experience with naming and classifying basic geometric shapes, with measuring weight in kilograms, and with translating, reflecting and rotating shapes. They should also have place value knowledge to at least 200.

Session One

1. The Māori New Year is celebrated at a different time each year. That is because the date depends on two events, the rising of the star cluster Matariki and the arrival of a new moon. In June, Matariki and the other six or eight stars of the cluster become visible in the Eastern Sky about 30 minutes before dawn. This is known as the rising of Matariki as, for the month prior to Matariki, it is below the horizon. After the rising of Matariki, Māori look for the next new moon to signal the New Year. The week prior to the new moon, excluding the night of no moon, is when Matariki is celebrated. Slide 1 of the PowerPoint shows the best known seven stars of Matariki. Slide 2 shows how to find Matariki should you want to organise a pre-dawn star spotting expedition. 
2. The Tūwharetoa legend of Tamarereti has connections to Matariki. Versions of the story vary but all name him as responsible for creating the stars in the night sky. Slides 3-8 tell the legend in abbreviated form. You may like to show an animated version of the story.
3. Tamarereti cast the shining stones into the heavens on his journey across Lake Taupō. The stones stuck in the dark night sky to become the stars. Ranganui, the sky father, put Tamarereti’s waka into the sky in honour of his deeds, and the waka appears today as the Milky Way. The Southern Cross and Pointers make up the anchor and rope of this great canoe (see slide eight).
4. Show the students slide 9 which shows a satellite picture of Lake Taupō. Click to discuss where Tamarereti drifted to while asleep, cooked his fish, then set off from to return to his village.
5. Select a student to act out the next part of the lesson. 
   What the legend does not tell you is that Tamarereti collected 200 bright shiny stones and put them at the bottom of the waka.
   Have a ‘waka’ with 200 counters ready for the student to act out the story. Any narrow container will make a good waka.
6. Tell the student to start rowing, then grab a small handful of stones to throw into the sky. Remember that the stones have to last the whole journey so Tamarereti cannot use them up all at once. Ask the student to cast the counters onto a sheet of art paper so the whole class can see.
7. Ask: Is there a way to group these stones to count them easily?
   Look for students to suggest ways to group the counters. Combinations that add to ten are especially useful. 
8. Tell students: Tamarereti is being careful not to use all of his stones up because the Taniwha will eat him if he is unable to see. He wants to know how many stones he has left. How could he work that out?
9. Let the students work out the remaining number in pairs. Then share the different ways the answer could be achieved. Look for part-whole strategies rather than counting back. For example if 15 stones are thrown into the sky subtracting ten then five is a good strategy. If 19 stones are thrown then subtracting 20 and adding one is effective. 
10. Ask: What might Tamarereti scratch into the side of his waka to keep track of his number of stones?
11. Invite suggestions from the students about how to record the number of stones. Write on a map of Tamarereti’s journey the first stone toss or use the animation on Slide 5 to show how the journey might be recorded. Let the students work in pairs to act out and record Tamarereti’s journey across Lake Taupō. Each pair will need a container, 200 counters and a copy of Copymaster 1. The number of counters can be lessened or increased to vary the challenge. Expect students to manage the distance to go on the map and the number of stones left. They should record the results of their calculations as each handful of stones is cast into the heavens. The grid system on Copymaster 1 could be used to create coordinates so students can indicate the position of Tamarereti’s waka each time he casts out shining stones.
12. After a suitable period, bring the class back together to discuss the strategies they used to calculate the remaining stones. Use an empty number line or two connected hundred bead strings to illustrate strategies as students suggest them. 
    • Back through ten (173 – 16 = 157)
      
    • Tidy numbers and compensation (145 – 29 = 116)
      
    • Standard place value (156 – 34 = 122)
      
13. You might use the work samples students produce as evidence of their additive thinking.

Session Two

1. Slide 11 of the PowerPoint shows the phases of the moon. Ask the students why the moon (marama) changes appearance. Some may know that the change is caused by the moon’s orbit around the Earth and the extent to which the half of the moon lightened by the sun’s rays is visible. You can demonstrate this with a ball and a lamp.
2. The phases of the moon are important to Māori as they indicate which days are best for traditional food gathering, particularly fishing. Slide 12 shows a page from Mathematics Across Cultures (1992). Ask the students to interpret the calendar.
   • Why does the month only have 30 days? That is the length of one moon orbit of the Earth (actually 29.5 days).
   • When are the best days to fish in the lunar month? (The red days which are days 18, 24, and 25)
   • When are the worst days of the month to fish? (The first 2 days of the new moon, days 6-7, 10, 16, 20-22, and the last two days of the old moon). 
   • So what are the best days to fish during Matariki? Matariki is celebrated in the last quarter of the lunar cycle but not on the day of the new moon. 
3. Use the timeanddate.com website to capture the lunar calendar for the current month. Give each student a copy of the calendar and ask them to make a puzzle for a classmate. They do that by cutting the calendar up into jigsaw pieces. Set the maximum number of pieces to eight and tell the students to use the straight lines of the calendar to cut along. They can cut vertically or horizontally so shapes like an L or a Z are encouraged.
4. Once they have cut up their calendar, students give their pieces to a partner to reassemble. Look to students to attend to the progression of days at the top of the calendar, the maximum of seven days in each row, and the sequence of whole numbers to put the puzzle back together. Have the students glue their completed calendar into their mathematics book.
5. With their calendar intact students can answer these questions:
   How do we find out the date of the full moon from this calendar? 
   So when will the last quarter start? 
   When are the good days for fishing? 
   When will the new moon appear?
   So when does the New Year start? 
6. Tell the students that, in honour of Matariki, they are going fishing. If the day is not a good fishing day, wish them luck. If it is a good day for fishing, say you are expecting a lot of success. The fishing game can be played in two ways:
   • Cut the fish out (see Copymaster 2) and attach a paper clip to each fish. Make a fishing rod using a stick, a piece of string and a magnet (magnetic strip is a relatively cheap way to do this). Students capture a fish by getting it to stick to the magnet.
   • Cut out the fish cards and turn them upside down. Players take turns to choose a fish.
7. The game can be played in pairs or threes. The object of the game is for each player to gather fish that add to 100. They do that as often as they can. At any time players can trade fish with each other to make 100.
8. Once the students have played the game on Copymaster 2, gather the class to share another legend. Māui was known as a trickster. It was Matariki, the New Year, and it was very cold outside. Māui’s brothers were getting bored (again) so he decided to play a trick on them. He made up the second set of fishing cards (see page 2 of Copymaster 2). The brothers tried for a long time to make 100 with the fish. They could not. Can you?
9. Let the students try the second fishing game to see if they can do better than Māui’s brothers. It is actually impossible to make 100 with the cards but see if your students can figure out why. They may need to take the game home to their whānau to see if anyone can explain how Māui’s clever trick works. All of the numbers on the fish are answers to the nine times table so the totals must always be in the nine times table (multiples of nine). 100 is not a multiple of nine.

Session Three

Matariki is a time of cultural pursuits and feasting to celebrate the New Year ahead. The hāngī or earth oven has particular significance at the time of the new moon after the rise of Matariki in the eastern pre-dawn sky. Matariki is the star at the bow of Te Waka o Rangi and her travels around the sky for eleven months of the year are exhausting. It is said the steam of the first hāngī in the New Year rises into the sky and replenishes the strength of Matariki. From the offerings she gathers strength to lead the giant canoe for another year. Without Matariki at the bow the canoe cannot travel and Taramainuku cannot cast his net to gather the souls of the departed. At the New Year the names of the dead are called out so the souls of the departed may be cast into the heavens as stars.

There are many resources already available about hāngī.

“Preparing for the hāngī” is a Level 3 activity from the Figure It Out series.
“Hanging out for hāngī” is a unit at Level 3 that develops a statistical investigation around deciding which foods to cook. 

The notes below are an adaptation more suitable for Level 2 students.

1. Tell your students about the types of food that are usually cooked in a hāngī. Chicken, pork and lamb are the most common meats used and the vegetables tend to be root crops like kūmara, potato and pumpkin. Stuffing is also popular. Before your class can plan the hāngī you will need to find out what people like to eat.
2. Your investigation question is “What hāngī foods do people in our class like to eat?”
3. Copymaster 3 has a photocopy sheet of ‘choice squares’. Put a container such as a shoebox or 2L plastic ice-cream container in the centre of the room. That is where the data will be placed. Show the students the first page of the Copymaster.
4. Ask: If you want to eat any of these foods at our hāngī you need to cut out that square and put it into the box. Should there be some restrictions on what you can eat?
   Students might mention that people should not eat every meat and every vegetable. Agree on some restrictions like one or two meats and up to three vegetables. Point out that stuffing is a yes or no choice.
5. Explain that the data will be used to order the food. “If someone chooses two meats while another person chooses only one meat, how will we deal with that?” Students might suggest that a person choosing two meats can put in one half of each square while a person choosing one meat might put in the whole square. 
6. Give the students time to make their choices and put the squares of the food they choose into the container. It is important that they cut out squares rather than the food within the squares as scale is important for possible data displays. Once you have brought the class together in a circle on the mat, empty the container of squares.
7. Ask: How might we organise these data so we can order food for the hāngī?
8. Students should suggest putting the squares into categories so get a few students to sort the data into piles. Ask, “How might we show the data so the number of squares for each food is easier to see?” After some discussion you should end up with a picture graph made with the squares. Managing the half squares should provoke a discussion about how large fractions such as five halves are. You might glue the squares in place on a large sheet of paper and add labels and scale to the axes. The graph might also be given a title. If relevant, you could use a digital graphing tool (e.g. Microsoft Excel, Google Sheets) to create a spreadsheet and bar graph.
   
9. Once the data display is complete, put the students into small groups to discuss “How might we use this data to order food for the hāngī?” After a suitable time, gather the class to share ideas. Expect students to consider the idea of a portion, that is how much of a food is reasonable as part of a meal. For example, one pumpkin is too much for a single portion so a fraction such as one eighth or one tenth is more sensible.
10. Share the information about meat (see PowerPoint slide 13) for a poster about this information). The poster has some questions for the children to consider. Have a set of kitchen scales available to identify objects around the room that weigh the same as a lamb chop or a size 14 chicken. You might use the scales to count in lots of 100 grams to find out how many portions are in one kilogram of meat.
11. Ask the students to work with a partner to decide how much of each food to buy. Look for them to consider the data on preferences you have collected, the information about portions of meat and their estimates of how much of each vegetable is required for each portion. You may decide to pool the data across several classes to make the task more challenging and avoid having a lot of pork left over! The students should produce a shopping list with clear working about how they decided on each amount.
12. Share the shopping lists and agree on suitable amounts of each food. The amounts of vegetables are likely to be expressed as numbers of whole vegetables, e.g. two pumpkins, which will add interest to the next part of the lesson – working out the cost per person. Copymaster 3 has a fictitious flyer from the local butcher and fruit and vegetable shop so that the students can create a budget for the hāngī food (see also Slide 13 of the PowerPoint). Allow students to use a calculator if they need to. Some may like to use a spreadsheet to keep track of their budget. Students will need to convert from numbers of vegetables into kilograms by estimating. For example, four or five good sized potatoes weigh 1 kilogram. Students may realise that they need a recipe for stuffing so they can calculate how much bread to order. Let them search for a stuffing recipe. Onions are an important ingredient in stuffing.
13. The final part of the budget is to work out a cost per person. This is a sharing context. The total cost, say $75, is divided equally among all the people in the class. Look for students to realise that the operation needed is division. You may need to link to simpler sharing problems so they connect the equal sharing to division and can write an equation for the solution, e.g. 75 ÷ 25 = 3. Talk about the meaning of the numbers in the equation, e.g. 3 represents $3 per person.

Session Four

Matariki was a time when food was already stored, and it was cold outside. So whānau (families) spent time together engaging in cultural pursuits such as storytelling, arts and games. Whai (string games) were popular with tamariki (children) and adults alike, especially when they involved co-operation. Whai has a long history and is common to many indigenous cultures around the world, including the indigenous tribes of North America. Traditionally whai was played with twine made from flax. The best man-made fibre to use for whai is nylon since it slides and flexes, and is soft on your hands. It is commonly used to form lines for brickwork so is available at most hardware stores in a variety of colours. Nylon string is usually available in craft shops.

1. Ask your students to make a tau waru (number 8) loop by wrapping the string loosely around their palms eight times, cutting the string, knotting it with minimal wastage, and trimming any loose ends. 
2. Whai relies on algorithms that are standard procedures. Algorithms are an important part of mathematics. Processes that initially take some time to master become standardised routines. The more complicated whai rely on the some basic moves being well known by the maker. That is where you should start with your students.
3. Dasha Emery has created an excellent series of videos that give clear instructions about making well known whai. A good first move for students to learn is called “Opening A” which is a standard algorithm. This opening is the start of many whai, particularly those that end in diamond shapes. Play the video at the link below which talks students through Opening A. Dasha refers to this pattern as kotahi taimana (one diamond). Let students practise the opening until they have it mastered: 
   • Kotahi taimana - One diamond (YouTube)
4. Next follow the instructions in this video to create te kapu me te hoeha (cup and saucer): 
   • Cup and saucer (YouTube)
     At 1:08 it is easier to think of going over two strings and ‘picking up the third string’ in that move. Note that the move where you use your mouth to shift the bottom of two strings over your thumbs (2:00 - ) is called ‘Navajoing your thumbs’ and is another algorithm common in whai. 
5. The next whai your students might make is ngā taimana e rua (two diamonds) which builds on Opening A. Tell your students to make the standard opening before the video starts. Alert them that somewhere in the video they will need to Navajo their thumbs. Work through the video, with students supporting each other to create the ngā taimana e rua pattern. Alternatively give the students copies of Copymaster 4 that has the instructions in graphic form. The written form will be much harder to interpret. 
   • Two of diamonds
6. Put the students into small groups of about three or four. Provide them with opportunity to become class experts in a particular whai pattern. Their job will be to teach the rest of the class how to make that pattern. Another YouTube chanel with many examples of string patterns is available at the link below:
   • Mom's Minivan YouTube channel
7. Once your students have practised their whai in small groups invite them to teach others how to make their pattern. You could do this as a whole class or create expert-novice pairs.
8. For an interesting geometry challenge consider looking for different shapes within a whai pattern. For example the gate pattern looks like this
   
9. Ask your students what shapes they can see in the figure. Here are a few possibilities:
   
10. Some interesting points might arise such as:
    • A three sided polygon is called a triangle, irrespective of the length of the sides, size of angles or orientation. The same is true of all five sided polygons being called pentagons and all six sided polygons being called hexagons.
    • Non-regular means that all the sides and angles are not equal so a regular polygon, such as a square, must have equal sides and angles.
    • The prefix ‘tapa’ means sides and the number name tells how many of those sides are in the shape, e.g. tapatoru means three sides (very helpful).

Session Five

In this session students learn to play the traditional Māori game Mū Tōrere which is like a form of draughts. The original game is sometimes referred to as the wheke (octopus) game or the whetū (star) game due to the shape of the board. It is appropriate that students learn to play the game at the time of Matariki, since the Māori New Year is a time of engaging in cultural pastimes. The board (see Copymaster 5) has been altered to include the nine or seven stars of Matariki, depending on the version of the game that is played. A digital version of this game is available online - search for “Mū Tōrere - HEIHEI Games”.

1. Introduce the Mū Tōrere Ngāwari (easy) version first. The game is played in pairs with each player needing three counters for the easy game and four counters for the original game. Their counters should be of one colour. The rules are included on the game boards. 
2. Let the students explore the easy game in pairs. Tell them that they need to record the winning position if one of them wins. Using black and grey for the counter colours can help to identify the arrangements that create wins. After they have played awhile bring the students together to share the winning positions. Create a set of diagrams. In these examples grey wins.
   
3. While they may look different the winning arrangements are actually the same, and are just rotations or reflections of one another. That can be demonstrated by putting the patterns on cards and turning them.
4. Ask: What must be true for a player to win in the easy game?
   The winner must occupy the centre circle, the opponent’s stones must be clustered together around the hexagon and the winner must have the ends of the cluster blocked off. You might try to find a winning arrangement by separating the loser’s stones into a group of two and one but there is no way for the other player to stop them moving.
5. Transfer to the original game that has the same set of moves but more winning arrangements. Ask the students to create winning arrangements on their board prior to playing the game. Create a gallery so the students can look for similarities and differences. Here are winning positions for black. Notice how all four, three and one, and two and two configurations of grey can all result in a victory to black but the winner must always occupy the centre. Discuss the similarity of winning arrangements created by students as the diagrams are reflected or rotated.
   
6. Is it possible to trap a player that has four ones, or a two and two ones (as shown below)? Try colouring in four circles grey to achieve a trap. It is not possible.
   
7. Once winning positions have been analysed let the students play the game. Competitive games go for over thirty moves so tell your students to be patient and think ahead. An interesting idea is that players can always create a draw if they know what they are doing. Is that true?