## Late level 2 plan (term 2)

Planning notes
This plan is a starting point for planning a mathematics and statistics teaching programme for a term. The resources listed cover about 50% of your teaching time. Further resources need to be added to meet the specific learning needs of your class, to support your local curriculum and to provide sufficient teaching for a full term.

## Popcorn

Level Two
Geometry and Measurement
Units of Work
This unit is made of a number of popcorn investigations, which provide both a purposeful and enjoyable measuring context. The focus of the unit is introducing the students to the need for a standard unit for measuring volume.
• Use non-standard volume units (cups, spoons, bowls) to fill a container and count the number used.
• Recognise the need for a standard unit of volume.
• Measure to the nearest litre and half litre by using litre containers to fill and count.

## Multiplication stories

Level Two
Number and Algebra
Units of Work
In this unit students explore the different situation types to which multiplication can be applied. Particularly, they engage with rate, comparison and array problems.
• Pose different types of word problems.
• Explain their mathematical thinking in solving problems.
• Use a variety of equipment to model their solutions.

## Planning a statistical investigation (Level 2)

Level Two
Statistics
Units of Work
In this unit students will identify how to plan and carry out a statistical investigation, looking at facts about their class as a context.
• Write investigative questions for statistical investigations and design a method of collection of data.
• Display collected data in an appropriate format.
• Make statements about implications or possible actions based on the results of an investigation.
• Make conclusions on the basis of...

## Pede patterns

Level Two
Number and Algebra
Units of Work
This unit is about generating number patterns for certain ‘insects’ from the mythical planet of Elsinore. Each ‘Pede’ is made up of square parts and has a number of feet. The patterns range from counting by 2s and 3s to being the number of feet plus three.
• Continue a simple pattern.
• Generalise the pattern.

## Matariki - Level 2

Level Two
Integrated
Units of Work
This unit builds the learning of mathematics around celebrations of Matariki, the Māori New Year. The sessions provide meaningful contexts that highlight Māori culture and provide powerful learning opportunities that connect different strands of mathematics.

Session One

• Use place value based strategies to subtract single and two digit numbers.

Session Two

• Interpret a calendar to make decisions about dates.

Session Three

• Gather and sort data to make decisions about quantities of food to order.
• Calcul...
Source URL: https://nzmaths.co.nz/user/75803/planning-space/late-level-2-plan-term-2

## Popcorn

Purpose

This unit is made of a number of popcorn investigations, which provide both a purposeful and enjoyable measuring context. The focus of the unit is introducing the students to the need for a standard unit for measuring volume.

Achievement Objectives
GM2-1: Create and use appropriate units and devices to measure length, area, volume and capacity, weight (mass), turn (angle), temperature, and time.
Specific Learning Outcomes
• Use non-standard volume units (cups, spoons, bowls) to fill a container and count the number used.
• Recognise the need for a standard unit of volume.
• Measure to the nearest litre and half litre by using litre containers to fill and count.
Description of Mathematics

When students can measure effectively using non-standard units, they are ready to move to the use of standard units. The motivation for moving to this stage, often follows from experiences where the students have used different non-standard units for the same volume. This allows them to appreciate that consistency in the units used allows for easier and more accurate communication.

The usual sequence used in primary school is to introduce the litre as a measurement of volume before using cubic centimetres and cubic metres.

Students’ measurement experiences must enable them to:

• develop an understanding of the size of a litre and 10 millilitres. (1 millilitre is very small and difficult to appreciate however it can be demonstrated with an eyedropper)
• estimate and measure using litres and millilitres
• develop an understanding of the size of a cubic metre and a cubic centimetre
• estimate and measure using cubic metres and cubic centimetres.

The standard units can be made meaningful by looking at the volumes of everyday objects. For example, the litre milk carton, the 2-litre ice-cream container and the 100-millilitre yoghurt pottle. Students should be able to use measuring jugs and to say what the measuring intervals on the scale represent.

This unit can be differentiated by varying the scaffolding provided or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:

• have students use non-standard units to measure, as needed
• provide additional activities for students to practice measuring volume. For example, set up a measuring station with a variety of containers, material which can be used to measure (sand, beads, cotton balls), and both non-standard and standard measures (a variety of spoons, cups, and smaller containers). Challenge students to find the container with the greatest/smallest volume, or to find a set of 3/5 containers and place them in order of increasing volume, recording their results with diagrams and descriptions
• work in small groups with students who need additional support, measuring and recording together.

Including the process of making popcorn in a broader context within the class or school will encourage engagement. For example, the popcorn could be part of a shared lunch for the class, a popcorn stall could be set up as a fund-raiser or as part of a school fair, or the popcorn could be shared at a community event.

Required Resource Materials
• Popcorn kernels (1kg of popcorn makes 10, 1/2 cup batches).
• A popcorn maker (or pot with a lid or microwave and dish)
• Cardboard containers of varying sizes and shapes
• Plastic bowls, ice cream containers and cups
• Spoons of varying sizes
• Standard cups and measuring spoons
• Various containers, which hold a litre or have a litre and half litre marked on them
Activity

#### 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, and 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. Pose the question: How much popcorn do you like to eat when you go to the movies? and discuss.
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 rice, water, sand or wheat 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

In 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?
4. Discuss responses together.

## Multiplication stories

Purpose

In this unit students explore the different situation types to which multiplication can be applied. Particularly, they engage with rate, comparison and array problems.

Achievement Objectives
NA2-1: Use simple additive strategies with whole numbers and fractions.
NA2-2: Know forward and backward counting sequences with whole numbers to at least 1000.
Specific Learning Outcomes
• Pose different types of word problems.
• Explain their mathematical thinking in solving problems.
• Use a variety of equipment to model their solutions.
Description of Mathematics

The basic concept of multiplication is an important one because of its practicality (How much do 4 ice creams cost at \$2 each) and efficiency (It is quicker to determine 4 x 2 than to calculate 2 + 2 + 2 + 2). Multiplication is used in many different situations. Here students think about multiplication as a short way to find the result of repeated addition of equal sets. They do so by solving rate problems, comparison problems and array problems.

A rate problem involves a statement of ‘so many of one quantity for so many of another quantity.’ All multiplication situations contain some form of rate but at this level, the problems are usually about equal sets or measurement. Take this example:

“Lena buys six bags of biscuits. Each bag contains four biscuits. How many biscuits does she buy altogether?”

This is an equal sets problem that contains the rate ‘four biscuits for every bag.’ A measurement rate problem is usually something like this:

“Hone’s kumara plant grows five centimetres each week after it sprouts. How long will his plant be after six weeks?”

The rate in Hone’s problem is “five centimetres for every week.” Comparison problems involve the relationship between two quantities. For example:

“Min’s apartment block has three floors. Anshul’s block has 12 floors. How much taller is Anshul’s block than Min’s?”

An additive answer is 12 – 3 = 9 floors. A multiplicative answer is 4 x 3 =12 so Anshul’s block is four times higher than Min’s. An array is a structure of rows and columns. For example, this chocolate block has two rows of five pieces (2 x 5 or 5 x 2).

An array is a structure of rows and columns. For example, this chocolate block has two rows of five pieces (2 x 5 or 5 x 2).

Array problems can help students to see the commutative property of multiplication, for example, that 5 x 2 = 2 x 5. In other words, the order of the factors does not affect the product (answer) in multiplication.

As well as thinking about multiplication in a variety of situations, students are encouraged to use a variety of materials to solve the problems. Using a variety of materials can help students see the multiplicative structure that is common to a variety of problems and assist them to transfer their understanding to situations which are new to them.

This unit can be differentiated by varying the scaffolding provided to make the learning opportunities accessible to a range of learners. For example:

• accept students’ use of counting strategies to solve multiplicative problems, as needed
• have students use materials or diagrams to support their thinking, as needed
• work in small groups with students who need additional support, solving problems together.

Focus on familiar contexts which include multiplicative situations to appeal to students’ interests and experiences and encourage engagement. Examples may include:

• lines of students in kapa haka groups
• collecting bags of pipi or other shellfish
• crews of students racing in waka ama
• loaves of bread for a school or community event
• groups of people travelling in vans, cars or buses
• preparing bundles of harakeke for weaving.
Required Resource Materials
• Interlocking cubes
• Number track
• Graph paper
• Ten-sided dice (or normal dice)
• Copymasters One, Two and Three
• PowerPoint One
Activity

#### 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 has 2 people in it. How many people are there altogether?
• There are 6 fish bowls. Each bowl contains 4 goldfish. How many goldfish are there altogether?
• There are 7 tables. Each table has 3 legs. How many legs altogether?

2. The students can represent these and similar ‘equal sets’ problems with:
• towers of interlocking cubes
• jumps on the number line
• interlocking cubes on a number track
• drawing a picture to show the number of tables and the corresponding number of legs.

3. 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.

4. 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”.

5. 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)?

6. 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 strategies of individual students will be variable.

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. 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 her name in 10 seconds. How long does he/she take to write his/her 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] sews five buttons on every shirt. How many buttons does he/she need for eight shirts?

[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 One 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 One 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 time higher than Jack’s block. I wonder what she means?

2. Slides Two and Three shows 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.
1. Work through the other slides of PowerPoint One looking to see if students identify the common structure of finding difference (additive) and finding the scale factor (multiplicative).

2. Copymaster Two 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.

• 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 carpark has four rows. Ten cars park in each row. How many cars can be parked altogether?
• A chocolate block has six columns, with four pieces in each column. How many pieces are in the whole block?
• A ‘Connect Four’ board has seven columns, with six holes in each column. How many counters can fit on the whole array?

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 Three 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, 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

## Planning a statistical investigation (Level 2)

Purpose

In this unit students will identify how to plan and carry out a statistical investigation, looking at facts about their class as a context.

Achievement Objectives
S2-1: Conduct investigations using the statistical enquiry cycle: posing and answering questions; gathering, sorting, and displaying category and whole-number data; communicating findings based on the data.
Specific Learning Outcomes
• Write investigative questions for statistical investigations and design a method of collection of data.
• Display collected data in an appropriate format.
• Make statements about implications or possible actions based on the results of an investigation.
• Make conclusions on the basis of statistical investigations.
Description of Mathematics

It is vital when planning statistical investigations that the students understand the importance of the way that they plan, collect, record and present their information. If they are not consistent in the way they carry out any of these steps, they could alter their findings, therefore making their investigation invalid.

In this unit the students will first look at choosing investigative questions to explore, making sure that the topic lends itself to being investigated statistically. They will then collect their data using structured recording methods. Once they have collected and recorded their data, they will present their findings using appropriate displays and making descriptive statements about their displays to answer the investigative question.

Dot plots

Dot plots are used to display the distribution of a numerical variable in which each dot represents a value of the variable.  If a value occurs more than once, the dots are placed one above the other so that the height of the column of dots represents the frequency for that value. Sometimes the dot plot is drawn using crosses instead of dots.

Investigative questions

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

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.

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:

• directing students to collect category data or whole number data – whole number is harder
• giving students summarised data to graph rather than them having to collect it and collate it
• giving students a graph of the display and ask them to “notice” from the graph rather than having them draw the graph
• writing 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.

Required Resource Materials
• Paper and pencils
• Presentation materials
Activity

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 dealing with data presentation and description, 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?

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 and then present as a report which will be sent home for parents.
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 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 can ask how many siblings, or they might suggest we ask how many brothers, how many sisters and how many 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 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 (sticky notes) and ask:
How can we use the pieces of paper (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; axes, scale and labels on the axes, 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, capture all the 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

1. 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..
2. 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?
3. Put students into small groups and give them a few minutes to come up with some ideas that they think they might use. 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 vegetable
• Favourite type of music
• Handedness
• Birthday month
• 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
4. 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 eye colours do the people in our class have?
• How did Room 30 students get to school today?
• How long are our class’s first names?
• When are Room 30’s birthdays?
5. 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.
6. 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

1. 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.
2. 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 pet?”
• Also, the students might want to ask, “What is your favourite pet out of cat, dog, fish, bird?” 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?
• How many letters are in your first name?
• 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 pets? – overall about the class data) rather than a survey question (What is your favourite pet? – asking the individual).
3. Ensure that all groups record their investigative and survey questions for 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 pre-determined 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

1. Students collect data from the rest of the class using their planned method.  Expect a bit of chaos. Possible issues aka 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 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 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
2. 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

1. 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.
2. 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.
3. 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.
4. After all students have completed one of their categorical data graphs bring the class together to share what the students have done.
5. Discuss and compare graphs between groups.
6. 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 pet 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.

## Pede patterns

Purpose

This unit is about generating number patterns for certain ‘insects’ from the mythical planet of Elsinore. Each ‘Pede’ is made up of square parts and has a number of feet. The patterns range from counting by 2s and 3s to being the number of feet plus three.

Achievement Objectives
NA2-6: Communicate and interpret simple additive strategies, using words, diagrams (pictures), and symbols.
NA2-8: Find rules for the next member in a sequential pattern.
Specific Learning Outcomes
• Continue a simple pattern.
• Generalise the pattern.
Description of Mathematics

Patterns are the basis of much mathematics. There is always a need to find a link between this variable and that variable. This unit provides an introduction to pattern in the context of ‘insects’. The students are given practice in finding the next insect in a sequence. This leads to the main aim, which is for the students to begin to see the link between the number of feet that certain insects have and the number of squares that make them up.

One of the things that is deliberately attempted here is for them to see the link above in both directions. So not only do they get practice in linking squares to feet but they also are asked to try to find the number of feet that an insect with a certain number of squares has.

This unit provides an opportunity to develop number knowledge in the area of Number Sequence and Order, in particular development of knowledge of the Forward Number Word Sequence and skip counting patterns.

This unit can be differentiated by varying the scaffolding provided or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:

• Provide students with additional time to explore the patterns by drawing and counting pede, before expecting them to continue the patterns using only numbers.
• Have students work with pede that involve simple number patterns. Other simple alternatives include:

The context of this unit can be adapted to address diversity, and appeal to students’ interests and experiences to encourage engagement. For example, students may like to colour their pede with their team's favourite colours, or decorate them with koru or other patterns.

Required Resource Materials
• (Magnetic) tiles
• Squares of coloured paper
Activity

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

 Humped Back Pede Number of Feet Number of squares 1 2 2 4 3 6 4 8 5 10

Use of a hundreds chart will help students 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 or Humped Back Pedes? What are they? How did you work that out?

#### Session 1

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

On the planet Elsinore there live a strange collection of insects. There is the Humped-Back Pede. The Humped Back 1-pede looks like this. Can you see his eye? And the Humped Back 2-pede looks like this. He has an eye too. (Show them the pictures below.) Ask the students to work individually or in pairs to make a Humped Back 3-pede with the green tiles.
Can you work our how many squares a Humped Back 4-pede has?

Gather the students together to talk about the insects that they drew. Explore the number pattern of counting in twos that comes from the Humped-Back Pedes. Also ask them 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 of the mythical insects that live on the planet Elsinore. The patterns here involve skip counting by 3s.

1. There are other insects 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 the students 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?

(Put the numbers of squares beside the insects as the students answer the questions.)
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? (A 5-pede.)
How many squares does a 5-pede have? (14.)
I wonder what sort of Pede comes next? (A 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 threes.)

#### Session 3

Here we explore patterns further. Here we are particularly interested in linking the number of squares on an insect 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. Now ask the students to draw 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 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 students who need challenging ask them about the Spotted 10-pede?
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. Ask them
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

Here we look at more patterns, this time related to the Big Headed Pedes. Here the number pattern involves starting with 4 and adding 1 each time to get the next number of squares.
1. Here we are going to explore the Big-Headed Pedes. This is what they look like.
2. Ask the students to work with a partner to draw the next three Big-Headed Pedes (4, 5, 6). As they work ask them 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 insects 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. Ask them
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

The students now work at making up their own Pedes.
Discuss the Pedes, the Humped-Back Pedes, the Spotted Pedes and the Big-Headed Pedes.
2. Tell them that you want them to work with their partners to invent a Pede of their own. Ask the students 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 insect.
3. Pairs could then swap the first three Pedes to see if they can work out the next three Pedes for each other’s insect. They can then check with each other to see if they arrived at the same Pedes for the 4, 5 and 6 insects.
4. As time allows get the students to 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 the students during any “free” or “choosing” time.

## Matariki - Level 2

Purpose

This unit builds the learning of mathematics around celebrations of Matariki, the Māori New Year. The sessions provide meaningful contexts that highlight Māori culture and provide powerful learning opportunities that connect different strands of mathematics.

Specific Learning Outcomes

Session One

• Use place value based strategies to subtract single and two digit numbers.

Session Two

• Interpret a calendar to make decisions about dates.

Session Three

• Gather and sort data to make decisions about quantities of food to order.
• Calculate with measures, including money.

Session Four

• Recognise shapes in a figure.
• Follow a set of instructions for movement.

Session Five

• Use symmetry to recognise when winning positions are the same.
Description of Mathematics

#### Specific Teaching Points

Session one involves subtracting single digit and two digit numbers starting at 200. As students take handfuls or counters from their waka they will need to anticipate how many counters remain. Students will use need to use place value to calculate in ways other than counting back. The session notes recommend using a linear model for representing the calculations. A bead string is ideal and can be mounted along the edge of a whiteboard so jumps can be recorded on the whiteboard.

The session notes recommend linking two strings end on end to form a line of 200 beads. An important strategy in the activity is ‘back through ten’. For example, a student has 93 counters left and removes a handful of 17 counters. How many do they have left?

On the bead string this calculation can be modelled like this:

93 – 7 can be carried out in two steps, take away three to get to 90 then take away four to get to 86. This is a ‘back through ten’ strategy so it applies to using any decade number as a benchmark. Of course a student might take away a the ten of 17 first.

Session three involves dealing with like measures, e.g. dividing or multiplying weights. Actually measuring objects with devices like kitchen scales is important to the development of students’ understanding of the measurement system. For example, students will need to find out how many kilograms of kūmara will need to be ordered for the hāngi. If possible bring a few kūmara along so students can experiment to find out how many kūmara make up one kilogram in weight. They will then need to use division or multiplication to calculate how many kūmara they need. So if 24 kūmara are needed and four kūmara weigh one kilogram then 24 ÷ 4 = 6 kilograms will need to be purchased.

Session four develops important geometry ideas out of whai (string figures). A common issue with the learning of geometry is that students form prototypical views of shapes. For example, they might consider an equilateral triangle to be the only shape that is a triangle. All of the shapes below are triangles:

The issue of prototypical ideas will also apply to other polygons such as hexagons and octagons.

These three shapes are all hexagons. Note that the bottom hexagon is concave as it has two internal angles greater than 180°. It is important to discuss the defining characteristics of a class of shapes like hexagons. The only required property is that the shape is closed by six sides.

Session five also involves an important mathematical idea, distinctness. Rotating or reflecting a shape does not change its properties, except orientation (direction it is facing). The idea is fundamental to determining if given shapes are similar or different. For example, all of the quadrilaterals below are similar even though they look different. They can all be mapped onto each other using translation (shifting), reflection (flipping), and rotation (turning).

Similarity is applied in Session Five by looking for different winning positions. If the positions are reflections or rotations of one another then they are not considered to be distinct.

Required Resource Materials
Activity

#### 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 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 one of the PowerPoint shows the best known seven stars of Matariki. Slide two shows how to find Matariki should you want to organise a pre-dawn star spotting expedition.
2. The Tūwharetoa legend of Tamarereti has connection 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. Yo umay like to show an animated version of the story.
3. So 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 nine 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 Five 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 co-ordinates 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 eleven 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 Twelve 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 whanau 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 hāngi 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 year are exhausting. It is said the steam of the first hangi 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.

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

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

1. Tell your students about the types of food that are usually cooked in a hāngi. 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 hāngi you will need to find out what people like to eat.
2. Your investigation question is “What hāngi 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 icecream 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 hāngi 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 of 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 hāngi?
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.
9. Once the data display is complete put the students into small groups to discuss “How might we use these data to order food for the hāngi?” 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. To support their thinking have a set of kitchen scales available so you can find 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 fruiterer so that the students can create a budget for the hāngi 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 weight 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. So if instructions tell you to “Navajo your thumbs” that means a common move that is attributed to a tribe of indigenous Americans. 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 commonly used to form lines for brickwork so is available at most hardware stores in a variety of colours. Nylon string is also 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:
4. Next follow the instructions in this video to create te kapu me te hoeha (cup and saucer):
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.
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:
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, dependent on the version of the game that is played.

1. Introduce the Mū Tōrere Ngāwari (easy) version first since it is simpler and is a good lead in to the more complex original game. 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 gameboards.
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 of card 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 of. 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 to students create as the diagrams are reflected or rotated.
6. Is it possible to trap a player that has four ones, or 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?