## Late level 2 plan (term 4)

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.

## Fun and fitness

Level Two
Integrated
Units of Work
The purpose of this unit is to have students design a PE game, use standard measures of length, and conduct a statistical investigation into the safety factors and the health benefits of their game.
• Understand how running a distance contributes to fitness and wellbeing.
• Create a personal benchmark for 1 metre and for 1 kilometre.
• Accurately use three measuring devices to measure a distance of more than 3 metres.
• Correctly record length measurements using abbreviations.
• Understand how...

## Letter patterns

Level Two
Number and Algebra
Units of Work
In this unit we look at the number patterns we get from letters and numbers. We keep track of the numbers involved by drawing up a table of values. It’s important here to look for the pattern and see how the number of tiles changes from letter to letter.
• Draw the next shape in a pattern sequence.
• See how the pattern continues from one shape to the next.
• Draw up a table of values.

## Voting vitality

Level Two
Statistics
Units of Work
In this unit, which explores the context of voting, students will become familiar with and apply the five keys steps of carrying out a statistical investigation.
• Pose investigative questions
• Design data collection methods
• Collect and collate data.
• Display collected data in an appropriate format and make statements about the displays of data.
• Make conclusions based on a statistical investigation.

## Fraction bits and parts

Level Two
Number and Algebra
Units of Work
This unit explores the beginnings of proportional thinking by introducing fractions and associated language. The purpose for this unit is to make, name, and recognise wholes, halves, third parts, fourth parts and fifth parts of a variety of objects.
• count in fractions forwards and backwards to a named whole number.
• recognise the whole of an object, part of an object and equal parts and their names.

## In the garden

Level Two
Geometry and Measurement
Units of Work
This unit uses the context of a garden to explore Level 2 symmetry and transformation concepts (translation, reflection and rotation).
• Investigate shapes that tessellate.
• Make geometric patterns by translating, reflecting and rotating shapes.
Source URL: https://nzmaths.co.nz/user/75803/planning-space/late-level-2-plan-term-4

## Fun and fitness

Purpose

The purpose of this unit is to have students design a PE game, use standard measures of length, and conduct a statistical investigation into the safety factors and the health benefits of their game.

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.
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
• Understand how running a distance contributes to fitness and wellbeing.
• Create a personal benchmark for 1 metre and for 1 kilometre.
• Accurately use three measuring devices to measure a distance of more than 3 metres.
• Correctly record length measurements using abbreviations.
• Understand how many metres are in one kilometre.
• Understand that rules are designed to ensure fairness and safety.
• Create and write instructions for a PE game, giving consideration to fitness, safety and enjoyment.
• Accurately measure and record the length of a given outdoor space.
• Plan and carry out a statistical investigation, answering an investigative question and presenting findings.
• Recognise the need for small units of length measure (millimetres).
Description of Mathematics

This unit of work assumes prerequisite knowledge gained at level one: the student can recognise the attribute of length, knows that measurement units are countable and that they can be partitioned and combined. When measuring length they realise that there should be no gaps or overlaps.

In these lessons the student is developing an understanding of a linear scale, and recognising that it is made up of units of equal size that are known as ‘standard units’, because they are understood by everyone.

They are learning to accurately reposition a metre ruler when required to measure a length longer than the ruler. In becoming familiar with metre and centimetre units of measure, the student is learning to express parts of metres as centimetres and to use the abbreviations m and cm when recording length measures. Students come to understand that 1000 metres are equal to 1 kilometre, and develop a personal benchmark for one metre and one kilometre measurements.

Further to the development of measurement skills and knowledge, the student participates in planning and collecting appropriate data to answer a question that has been composed with the support of the teacher (as required). The student sorts the data and presents these using a dot plot, at the same time refining key understandings of the investigative process. They can answer the investigative question and can suggest consequences of their findings.

Associated Achievement Objectives

Health and Physical Education
Positive attitudes

• Participate in and create a variety of games and activities and discuss the enjoyment that these activities can bring to themselves and others.

Safety management

• Identify risk and use safe practices in a range of contexts.

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

• Simplify measuring tasks by using whole metres and half metres rather than a measurement scale.
• Brainstorm ideas for fitness games with students and have them choose one of the ideas as a starting point for their work.
• Have students work in small groups or pairs with the additional support of teacher, as needed, to develop a game and  carry out a statistical investigation.

The focus of this unit is designing a fitness game. Encourage students to consider their friends and classmates when planning, and to create a game that will appeal to them and be fun to play. This could be achieved by incorporating favourite elements from other games, or items of current interest.

Required Resource Materials
• A set of metre rulers
• 1 centimetre cubes
• Retractable tape measures of at least 10 metres in length
• Measuring wheels
• PE equipment including small and large balls
Activity

Whilst this unit is presented as sequence of five sessions, more sessions than this will be required between sessions 3 and 4. It is also expected that any session may extend beyond one teaching period.

#### Session 1

This session is about playing a familiar game and evaluating its health benefits and safety considerations.
Students of this age may be challenged to accurately measure their pulse rate. Therefore these lessons use an alternative ‘indicator’ of the effects of exercise: that is that the intensity of worthwhile exercise should prevent you from singing, but should not prevent you from talking.

SLOs:

• Understand how running a distance contributes to fitness and wellbeing.
• Create a personal benchmark for 1 metre.
• Accurately use three measuring devices to measure a distance of more than 3 metres.
• Correctly record length measurements using abbreviations.
• Understand how many metres are in one kilometre.
• Establish a person benchmark for 1 kilometre.

Activity 1

Begin the lesson with singing a favourite song that has a more challenging vocal range.

Activity 2

1. Explain to the students that they will be undertaking their regular fitness run (a distance of up to half a kilometre). Ask what are the effects/benefits of this, and elicit specific statements that should include: “We get puffed”, “Our heart beat/pulse speeds up”, “It’s good for us”. In response, explain that the increase of heartbeats per minute (hpm) is because their body physically needs to circulate oxygen more quickly as they exercise. Science has shown this is good for us.
Explain that today, when they each return from their run they should (individually) immediately sing the first verse of the song from 1 (above), and then talk to a classmate about their run.
2. Have students complete their run and this task.
3. Ask: ‘Who was able to sing the song immediately?’
‘Who was able to talk to their classmate?’
Count the responses for each and record these on the class chart. Discuss the results, explaining that being unable to sing immediately shows that they exercised well and their bodies will benefit.

Activity 3

Ask, “How far did you run today?” and “How can we find out?”
Make available 1 centimetre cubes, meter rulers, a 10 metre + tape measure and a measuring wheel.
Have the students tell what they know about the metre ruler. Establish that it is called a metre ruler. It is 1 metre long. (If centimetres have already been introduced and used, have students line up 1 cm cubes along the ruler to confirm that 1 metre = 100 centimetres).

Activity 4

Develop a personal benchmark by asking: “Who can jump 1 metre?”
Have student pairs measure a 1 metre length on the carpet/floor, marking this with chalk.
Highlight that the measure begins at 0 and ends at 100. Discuss the ‘extra’ space at each end of the ruler.
Have student pairs check if each person can each jump 1 metre.
Agree that when we think about how big 1 metre is, we can think of it as one big personal jump.

Activity 5

1. Introduce the tape measure and measuring wheel, explaining and showing how each measures 1 metre and multiples of 1 metre.
Highlight the 1 metre personal benchmark by asking:
If the tape measure is 10 metres, about how many of your jumps is that?
If we measured 100 metres with the wheel, about how many of your jumps is that?
2. Write ‘standard measure’ on the class chart and ask what it means. Elicit responses and point out that standard units have been created to allow consistency and communication of measures. We understand each other’s measurements if we use the same measures.
Explain that the standard units used in New Zealand, and in most countries in the world, are metric units.
3. Write centimetre, metre and kilometre on the class chart. Explain that when we write them often, we want a quick way to record them. Model cm, m and km abbreviations.

Activity 6

Have a student model both the correct and an incorrect way to measure using a metre ruler. Highlight how to mark the beginning and end of the measure and how to correctly replace the metre ruler, when measuring a distance greater than 1 metre.
Have several students measure a length that is more than a metre, read the measure aloud, and record this on the class chart.

Activity 7

Explain that student pairs will participate in two measuring tasks to become familiar with the measuring tools. Emphasise that their recording should use the correct abbreviations.

1. Show and have students make a recording sheet, for example:

 Measurement from ... to Metre ruler Measuring tape

Have them measure at least three different lengths around the classroom, hall, or other designated area, using a metre ruler and a measuring tape, checking that they get the same measure using each tool.
2. Clarify the exact fitness course route, the start and end points, and set relevant boundaries. Have student pairs take turns using the measuring wheel to measure the distance around the course and to then record the result.

Activity 8

Conclude the session by sharing measurement results and reviewing the fitness course distance. Discuss how many metres in 1 kilometre. Estimate and calculate together the number of times they would need to run around the fitness course to cover a 1 kilometre distance. Establish a rough benchmark for 1 kilometre. (For example, 1 kilometre is 5 times around the fitness course.)

#### Session 2

This session is about recognising that rules that address fairness and safety, help to ensure that a PE activity is enjoyable. As students design a PE activity, they learn more about accurately measuring outdoor spaces.

SLOs:

• Understand that rules are designed to ensure fairness and safety.
• Pose an investigative question.
• Create and write instructions for a PE game, giving consideration to fitness, safety and enjoyment.
• Accurately measure and record the length of a given outdoor space.

Activity 1

Begin with a fitness run.

Activity 2

Explain that the class is going to play a favourite PE game (for example: Tunnel ball, Scatter Ball). Together, list the rules on the class chart.
Ask: Which of the rules are about making the game fair? Write F beside these. Discuss that fairness makes the game more enjoyable for everyone.
Ask: Which of the rules are about making the game safe? Write S beside these. Discuss any anomalies. If there are no safety rules, list at least 3.

Activity 3

Return to the class, review the enjoyment of the game and ask if playing the game will make them fit. Discuss why/why not. Refer to original “talk/sing measure” from 4 above.
(The response may be, “No, because it didn’t make me puff and I could sing.”)
Review the list of rules and confirm the fairness (F) and safety (S) decisions made earlier.
Highlight the importance of games and activities being safe and enjoyable. Ask if any other safety rules should be added and why.

Activity 4

1. List on the class chart, the words ‘enjoyment, fitness and safety’.
2. Suggest that student pairs will create their own games/activities and that the class will conduct an investigation. Through discussion, lead students to pose an investigative question. For example:
‘Can we design a game or activity that keeps us fit, is enjoyable and is safe?’
Record this on the class chart/modeling book.
3. Explain that students will be using an outdoor space with suitable boundaries, for example, the school tennis courts. Large balls and small balls will be made available. (Make available other equipment, for example hockey sticks, if skills have been taught.)
4. Clarify the task. Students will work in pairs or small groups to:
• Measure and record the size of the designated outdoor space (using skills learned in Session 1.)
• Invent a simple team game or activity that the class can play.
• Write down clear instructions and rules, checking for safety and fairness.
5. Set time limits and clarify expectations. Have students complete the task.

Activity 5

Have two student pairs swap game instructions. Have them read, critique, seek clarification and suggest refinements or improvements to the other’s game design.

Activity 6

Review pair measurements for the outdoor space. (Tennis courts are 23.77m x 8.23m) Review 1000m = 1 km. Estimate together the number of times the length of the court would need to be run to achieve the length of 1 kilometre. (eg. Round up to 25m. 25m x 40 = 1000m) Together calculate the number of lengths of the tennis court = 1km.
Students may use this as a 1 kilometre benchmark.

#### Session 3

This session is about creating a simple questionnaire to evaluate each pair’s activity, and learning about dot plots.

SLOs:

• Plan data collection.
• Collect data by trialling and evaluating an activity on its fitness, safety and enjoyment values.

Activity 1

Begin with a fitness run.

Activity 2

1. Review the investigation question recorded on the chart in Session 2.
Ask the students how they should gather the data to answer the question. Guide discussion and agree on a simple evaluation form to be completed by the class after playing each pair-lead activity. For example:
The name of the game: ________________________________
Circle for each: 1 (not so good) 2, 3, 4 or 5 (excellent),
Enjoyment: 1 2 3 4 5
Fitness: 1 2 3 4 5
Safety: 1 2 3 4 5
2. Print off the evaluation or have students copy this and practice using it by completing an evaluation for the game played in Session 2: 2 and 3.

Activity 3

Together collate and present the data using three dot plots. For example:

Discuss the dot plot features, the results, and draw conclusions.

Activity 4

For the remainder of the session, and for sessions to follow, have students participate in and evaluate each other’s games. Each pair of students will collect the data for their game to analyse and present in Session 4.

#### Session 4

This session is about student pairs sorting the data from their classmate’s evaluations of their activity and presenting the findings.

SLOs:

• Sort and display category data.

Activity 1

Begin with a fitness run.

Activity 2

Make available pencils, paper, and sets of data for each pair activity.
Have students work in pairs to sort their data and to discuss their findings. Each student should create three dot plots to present their data, record their own findings and should answer the investigative question in their own way.

As students work, have them record on a small poster, their knowledge of centimetres, metres and kilometres, the relationship between them, and explain why we have standard measures.

#### Session 5

This session is about communicating investigation findings to others and sharing their understanding of standard measures of length.

SLOs:

• Present findings.
• Review and reflect on the investigative process.
• Review and reflect upon measurement learning.
• Discuss the need for small units of length measure and introduce millimetres.

Activity 1

Begin with a fitness run.

Activity 2

Have each student present to a partner (not the person with whom they developed the game) their dot plots, their findings, and the measurement posters. Have them critique each other’s work and give feedback.

Activity 3

1. Have selected students communicate the findings about their game with the class.
2. Together, as a class:
• Summarise on the class chart conclusions about safety, enjoyment, and fitness.
• Reflect on the investigation process and suggest ways it could have been improved.

Activity 4

Arrange the length measurement tools in front of the students.
Have individual students share their learning about each of the tools.
Ask which tool would be used to measure small lengths.
Introduce the millimetre measure for tiny lengths.

Conclude by reviewing personal benchmarks for (1cm), 1m and 1km.

## Letter patterns

Purpose

In this unit we look at the number patterns we get from letters and numbers. We keep track of the numbers involved by drawing up a table of values. It’s important here to look for the pattern and see how the number of tiles changes from letter to letter.

Achievement Objectives
NA2-8: Find rules for the next member in a sequential pattern.
Specific Learning Outcomes
• Draw the next shape in a pattern sequence.
• See how the pattern continues from one shape to the next.
• Draw up a table of values.
Description of Mathematics

Patterns are an important part of mathematics. They are one of the overriding themes of the subject. It is always valuable to be able to tell the relation between two things in order to predict what will happen and understand how they interrelate.

Patterns also provide an introduction to algebra proper as the rules for simple patterns can be first discovered in words and then be written using algebraic notation.

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

Help students focus on the number patterns by discussing the tables showing the numbers of tiles used in each successive letter pattern. Look at those patterns that are made by adding a constant number of tiles onto each successive letter. Highlighting numbers on a hundreds board or using a number line may also be helpful.

Questions to develop strategic thinking:
Which number comes next in this pattern?
How do you know?
Which number will be before 36 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 did you work it out?
Could you make a letter T with 34 tiles? How do you know?

This unit can be differentiated by altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. Specifically, some students may explore the patterns and describe how the shape and number patterns are growing, but may not be ready to predict the next number in the pattern, or how many tiles would be needed to make the nth shape in the pattern.

The context of letter patterns can be adapted to recognise diversity and student interests to encourage engagement. Support students to identify and explore other growing patterns in their environment. For example, tukutuku patterns on the walls of the wharenui, or the number of seats on the bus that are occupied as students get onto the bus in pairs.

Required Resource Materials
Activity

#### Getting started

Starting with a simple pattern we build up the level of difficulty and see that it’s necessary to use a table to keep track of what is happening.

1. Build up the letter ‘I’ using coloured tiles or paper (see the diagram below).
How many tiles do we need for the first ‘I’? The second? The third?

Who can tell me how many tiles we’ll need for the fourth ‘I’?
Can someone come and show us how to make the fifth ‘I’?
How many tiles will we need for the tenth ‘I’? Make it.
What is the number pattern that we are getting?
If we had 11 tiles, which numbered ‘I’ could we make?

2. Now let’s make it a bit harder. Let’s make an ‘I’ by adding a tile to the top and the bottom each time (see diagram).

Repeat the questions from the last ‘I’ problem.
How many tiles do we need for the first ‘I’? The second? The third?
Who can tell me how many tiles we’ll need for the fourth ‘I’?
Can someone come and show us how to make the fifth ‘I’?
How many tiles will we need for the tenth ‘I’? Make it.
What is the number pattern that we are getting?
How many tiles do we add on at each step?
If we had 11 tiles, which numbered ‘I’ could we make?

3. It was easy to see what was happening in the original ‘I’ problem and to see how many tiles each ‘I’ needed. It wasn’t quite as easy with the second one we did. But what if we had a really difficult pattern? How could we keep track of what’s going on and see how many tiles we need for each letter?
Work them around to the idea of a table.
4. The original ‘I’ problem would give us an easy table. It would look like this:
 ‘I’ number 1 2 3 4 5 Number of tiles 1 2 3 4 5

What would the table look like where we added two tiles at a time?
Draw up the table with help from the students.

5. Now let the students complete the table for the letter pattern on Copymaster 1. Circulate the class while they are working and help them by asking leading questions such as:
How did you know how many tiles to use on the fourth ‘L’?
What is the pattern here?
Which ‘L’ in the sequence will use 27 tiles?
6. Bring the class back together and discuss their work.
Tell me what numbers you used to fill the table. (Check that they are correct by counting the tiles.)
What patterns can you see here?
How did you get the number of tiles for one ‘L’ from the one before?
How many tiles would you need for the 10th ‘L’?
If you had 23 tiles, what numbered ‘L’ could you make?

#### Exploring

For the next three days the students work at various stations continuing different number patterns and building up the corresponding tables. Notice that there are three series of stations here. In the first station, the students have to do a similar problem to the one that was done in ‘Getting Started’. In the second station the students have to find a missing shape in the pattern sequence. Finally in the third station they have to make their own pattern that fits the given table of values.  When the class seems to have finished most of the task, bring them back together to discuss their answers. Ask them the kind of questions that were used in ‘Getting Started’. Use the table to discuss the pattern involved and what the relation is between successive numbers in the sequence.

Day 1
The material for these stations is on Copymasters 1.1, 1.2, 1.3, 1.4. The students continue the pattern and complete the table.

Day 2
The material for these stations is on Copymasters 2.1, 2.2, 2.3, 2.4. The students find the missing element of the pattern and complete the table.

Day 3
The material for these stations is on Copymasters 3.1, 3.2, 3.3, 3.4. The students make up their own pattern to fit the values in the table.

#### Reflecting

On the final day let the students make up their own patterns using numbers instead of letters. Some students might want to leave gaps in the pictures of their shapes. Let the students share their patterns with the whole class. In the discussion, point out the importance of the table in seeing what the number pattern is.

## Voting vitality

Purpose

In this unit, which explores the context of voting, students will become familiar with and apply the five keys steps of carrying out a statistical investigation.

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
• Pose investigative questions
• Design data collection methods
• Collect and collate data.
• Display collected data in an appropriate format and make statements about the displays of data.
• Make conclusions based on a statistical investigation.
Description of Mathematics

In this unit which explores the context of voting, students will become familiar with and apply the five keys steps of carrying out a statistical investigation:

• Pose investigative questions.
• Design data collection methods.
• Collect and collate data.
• Display collected data in an appropriate format and make statements about the displays of data.
• Make conclusions by answering the investigative question based on a statistical investigation.

These five are closely linked as what data is collected can dictate the way that it is displayed and the conclusions that can be reached from the investigation. On the other hand, if some restrictions have been placed on the means of display, only certain types of data collection may be relevant.

At Level 2 you can expect students to be posing (with teacher support) a greater range of questions, including investigative questions and survey questions. They will also be helped to understand some of the issues involved in conducting surveys and learn new methods for collecting data. While at Level 1 students collected data and chose their own ways to display their findings, at Level 2 they will be introduced to pictographs, tally charts and bar charts. More emphasis here will also be placed on describing the data and the making of sensible statements from both the student’s own displays and the displays of others.

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.

Associated Achievement Objective

Social sciences AO2: understand that people make choices to meet their needs and wants.

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:

• giving students summarised data to graph rather than them having to collect it and collate it
• giving students a graph of the display and asking them to “notice” from the graph rather than having them draw the graph.

The context for this unit can be adapted to suit the interests and experiences of your students. For example, instead of class captains you could vote on a game to play at the end of the week, or a class treat for the end of term. The favourite example can be adapted to explore any favourites.

Required Resource Materials
• Paper cut into squares for voting
• Large sheets of paper and pens for recording.
Activity

This teaching sequence is designed for teachers to take students through the five key stages of a statistical investigation. The context used is voting for the class captain, however school house captains could be an alternative context.  At the end of this unit other possible statistical investigations are provided as examples of what students might further investigate in pairs or small groups. This would then allow for the unit to be extended beyond a week.

#### Session 1

It’s that time again where we need to vote for the class captains. The hunt is on to find the most suitable boy and girl to lead our class.

1. We need two class captains for our class, one girl and one boy. They are going to be in charge of meeting with the school committee and representing your ideas and thoughts to the rest of the school. If appropriate suggest a more relevant role for the class captains e.g. they may help organise the school disco or up and coming class trip.
2. Start the discussion by asking ‘What leadership qualities do you believe a person would need for this role? Brainstorm the qualities on a large sheet of paper. Ask students to think about who they might like to nominate from the class.
3. You may like students to put their hand up and nominate class members or you may like to carry out the nominations more formally by asking them to write down their nomination on a piece of paper.
4. Once the nominations have been made these need to be displayed. Students are then asked how they are going to decide "Who will be the class captains for our class?" (our investigative question). How might we collect data to answer our investigative question?
5. Examples of tally charts, lists etc could be used to exemplify ways of collecting such data. Organise students into groups of four to ask them how might this data best be collected? Reminding students of the purpose of collecting the data.
6. As a class share ideas from each group. What ideas have each group come up with? Come to a consensus of the method for collecting the data ready for the next day.  Students may decide one vote for a girl, one vote for a boy, or that each student gets two votes and they can vote for anybody.

#### Session 2

This session is about going through the voting process and collecting the data in a systematic way.

1. Recap the discussion from the previous day and encourage students to think about what we are trying to find out and why are we carrying out this investigation?
2. Have the nominated individuals’ names written on labels. Use the data collection method that was arrived at from the previous day’s discussion.  Further teaching on tally marks and how they work may be needed if this is knowledge students do not have. Why do we use tally marks when collecting large amounts of data?
3. Students are given a square piece of paper for each vote they are allowed to make (one piece of paper for one vote, two pieces of paper each if they are going to vote twice) they are asked to write down the name of the person they are voting for on the square of paper.
4. Students are given 1 minute to complete their vote and then asked to sit in a circle on the mat.
5. The teacher holds up the name of a nominated person and invites those people who voted for Sally to bring their voting square to the front. The voting squares are placed side by side as illustrated below.
6. The process is then repeated for the other nominees and the voting squares are added on to make a long strip.
7. Complete until all votes are represented in the strip.
8. Ask students to make statements about what they can see from the strip and relate this to their investigation. Be sensitive about how the information about the lowest scoring candidate is presented.

#### Session 3

In this session students will see how a strip graph can be transformed into a bar graph.

1. Break the strip graph into the votes for each nominee.  Place the name labels at the bottom of the graph and place each piece of the strip graph above the appropriate name, as illustrated below.
What would be an appropriate title for our graph?
What labels could you use for this graph and where would you write them?
3. Label the axes and give the bar graph a title so that others could make sense of the display. A good idea is to write the investigative question as the graph title.
Is this a helpful way of presenting this information?
It is easier to make statements from a bar graph or from a strip graph?
Which completed graph shows our results most clearly?

The questions could be asked in a whole class situation or students could complete a bus stop activity with the questions being posed on the top of a large piece of paper and students visiting each station to record their ideas.  Small groups would also be a valuable way for ideas and responses to the questions to be discussed and explored.
5. Summarise the responses and make recommendations about when each graph might be a useful way of presenting information
Ask the students what they notice about the information shown on the bar graph. Use the prompt “I notice…” to start the discussion. These “noticings” could be recorded as "speech" bubbles around the bar graph.
6. Conclude by revisiting the original investigative question posed: Who will be the class captains for our class?  Make statements from the results to answer the original investigative question.

#### Session 4

1. Talk about the types of things that are worth investigating. It is important that possible investigations are relevant to what is happening in the students’ lives and what is happening at school at the time.  Possible investigative questions may include:
What are the class’s favourite pastimes?
What should we spend the fundraising money on?
What class pet should we get?
2. Encourage students to review the process they went through to decide how they were going to collect and present the voting data.  List the process as questions that students can refer back to. For example
1. What is our investigative question?
2. How will we collect the data to answer our investigative question?
3. Who are we going to ask? How many people are we going to ask?
4. How are we going to display our results? In tables? What is the best graph to use?
5. What is the answer to our investigative question based on the results of our investigation?
3. Students can now work in small groups or pairs to carry out their own investigation.  This could be completed as a homelink activity or as a follow up activity.
4. Results should be shared and conclusions made based on the results. This investigation is likely to require at least three sessions of fairly intensive work; one session of planning and checking, one session of collecting and displaying data, and one of developing statements and conclusions and presenting these.

## Fraction bits and parts

Purpose

This unit explores the beginnings of proportional thinking by introducing fractions and associated language. The purpose for this unit is to make, name, and recognise wholes, halves, third parts, fourth parts and fifth parts of a variety of objects.

Achievement Objectives
NA2-1: Use simple additive strategies with whole numbers and fractions.
NA2-7: Generalise that whole numbers can be partitioned in many ways.
Specific Learning Outcomes
• count in fractions forwards and backwards to a named whole number.
• recognise the whole of an object, part of an object and equal parts and their names.
Description of Mathematics

This unit is based on the work of Richard Skemp. His ideas for teaching fractions can be used successfully with year 2 students, and also with students up to year 8 who are having difficulty understanding fractions. Skemp’s use of the word ‘parts’ is deliberate in that he uses it to refer to ‘equal parts’ whereas bits refers to non equal parts.

The use of two different physical representations, the whole and parts of a whole, are used to develop the concept of a fraction. Language also plays an important role. Being able to count in fractions helps students understand that you can have 5 thirds or 6 halves.

Using denominators that are the same, students need to know:

• a whole can be divided (partitioned) into equal parts eg one whole is equal to two half parts or 1 = 1/2 + 1/2
• each of those parts can be put back together to make a whole eg. two half parts is equal to one whole or 1/2 + 1/2 = 1
• parts can be joined to make a fraction less then 1 eg. one fourth and one fourth and one fourth is equal to three fourths or 1/4 + 1/4 +1/4 = 3/4
• parts can be joined to make a number more than 1 eg. three fourths (quarters) and one fourth and one fourth is equal to one whole and one fourth or 3/4 +1/4 +1/4 = 1 1/4

This unit supports teaching and learning activities in the Student Fractions e-ako 1 and 2 and complements the learning activities in Book 7 Teaching Fractions, Decimals and Percentages.

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

• Support students thinking by clearly and deliberately modelling how to partition a whole into equal parts, and the result of the partitioning.
• Support students to describe the process of partitioning and their understanding of equal parts in their own words. Introduce mathematical terms naturally, alongside students explanations (whole, equal parts, fractional names such as one half and one third).
• Use the terms “numerator” and “denominator” only once the underlying ideas are understood, and can be articulated in students’ own words.
• Provide additional experiences of physically partitioning materials and discussing the resultant parts, until students understand the underlying ideas.

The context of equal shares can be adapted to recognise diversity and student interests to encourage engagement. Support students to identify and explore other situations in their lives where equal sharing occurs. For example, sharing kai at home or sharing cards to play a game (the pack of cards represents one whole).

Required Resource Materials
Activity

#### Prior Experiences

• Idea of fair shares
• Know 1/2, 1/3 and 1/4 of shapes such as rectangles and circles
• Doubles and their corresponding halves

Although the unit is planned around 5 sessions it can be extended over a longer period of time.

#### Session 1

The purpose of this session is to develop students' understanding of equal parts (the denominator).

#### Resources

1. Ask the students how they would share a bale of hay (block of chocolate) between 4 sheep (4 people) fairly. Other contexts could be the sharing of pizzas but the shape of the rectangle is easier for students to cut into equal shapes. Introduce the word equal – what do you think it means?

2. Distribute copies of the eels start, action, results boards (if it is possible it would be better to have 1 board between 2 students), some play dough and a plastic knife each.

3. Have each student or pair of students make six equal sized round eels, by rolling 6 equal amounts of play dough. Put one eel in each of the outlines on the left hand side of the board. The eels are small as they have their heads and tails cut off. The eels we are making today are miniatures of the big eels. That means we are making small copies of the big ones.

4. The following story can be used to guide students through the actions as described on the board.
Hoepo and his brothers and sisters are at their Poua’s tangi and although they are sad they are looking forward to the hakari because eel is always on the menu.
Hoepo is planning to go early to the marquee because he wants one eel all to himself. He is given one whole eel. Hoepo doesn’t know it but he is going to be one sick boy!
The twins appear and they are told they have to share one eel evenly between the two of them. There are now two half parts.
The triplets come next and Aunty Wai says we will have to cut another eel into three equal parts. There are now three third parts.
Hoepo’s sister has come with her three friends. Aunty Wai says that they will have to cut the eel into four equal parts. There are now four fourth parts. Aunty Wai says they are also known as quarters.
Hoepo’s five baby cousins are only allowed to eat small portions so Aunty Wai cuts the last eel into five equal parts. There are now five fifth parts.

5. Ask the students to mark the lines and cut lightly then if they haven’t made equal parts they can smooth the play dough out and start again.

6. After cutting, the separated parts are put in the RESULT column next to their descriptions.

7. Ask the students to share with the person next to them what they can see. Hopefully someone may say "The more cuts we made the smaller the equal part" Prompt them towards that knowledge.
I want you to look at one of the third parts and one of the fifth parts. Which is bigger?
Have the students take one of each of the equal parts and put them on another blank board.
Order the equal parts from smallest to biggest.
Let’s say the names.
Students should order from 1/5 – 1
Put them back on the original board.
How many halves are equal to the whole?
How many fourths are equal to the whole?
How many thirds are equal to a whole?
How many wholes are equal to a whole?
Depending on the age of the students symbolic notation can be introduced, using the terms like 1/2 and one half part interchangeably.

8. Repeat the steps above using the biscuits start, action, results boards and fresh play dough. The first boards should, if possible, remain on view. With this second board a variety of division lines are easily found, eg fourth parts.

9. Repeat the steps above using the pies start, action, results boards. If possible, fresh play dough should be used, the other two boards remaining on view. The lines of division should be radial as shown below.

#### Session 2

The purpose of this session to develop the idea that parts of the same kind may not look alike. In Activity 1 this arose from the use of different objects. Here we see that this can be so, even with the same object.

#### Resources

• Copymaster 2
• Play dough
• Plastic knives
• Cutting boards or cut up rubbish bags on top of newspaper.

What can we remember from yesterday? Write students’ comments in your modelling book.

1. Begin with the first page. This is used in the same way as the board for Activity 1. Ask students to complete the first 3 lines (making halves in three different ways). There are three simple ways see if you can find them.
The three straightforward ways are:

2. Next, they complete the next two lines (the third parts) which offers only two straightforward ways.

3. Complete the second page (the fourth-parts). There are six ways of doing this which are fairly easy to find.

4. Some students may want to go back to the halves board and see if they can find some more.

#### Session 3

The purpose of this session is to consolidate the concepts formed in Sessions 1 and 2, moving onto a pictorial representation.

#### Resources

1. Begin by looking at some of the parts cards from Copymaster 3 together. Explain that these represent the objects which they made from play dough in the last activity, eels, chocolate bars, biscuits and some new ones. They also represent the parts into which the objects have been cut, eg third-parts, fourth-parts, halves, fifth-parts. Some have not been cut: these are wholes.

2. Shuffle the parts cards and spread them out on the table, face upwards.

3. Give each student one of the set names from Copymaster 4.

4. Each student should collect the parts cards that match the name of the set they have been given, for example, halves.

5. Students check each other’s sets and discuss if necessary.

6. Next, introduce Skemp’s mix and match game. This is a great game in that the students are consolidating what they know about denominator without being introduced to the word. Older students may have heard that word and it is important that they understand what it is. The denominator names the number of equal parts.

#### Rules of the Game

This game is best played by groups of 2-4 people.

1. Share the parts cards from Copymaster 3 evenly between all players. Each player should have their cards in front of them in a single pile, face down.
2. Place the mix and match card somewhere where all players to see it. The purpose of the mix and match card is to remind players of the directions in which they can build.
3. The first player turns over their top card and places it in the middle of the playing area.
4. Players take it in turns to turn over a card and place it alongside a card already on the playing area. When placing cards they must ensure that:
• cards in a line in the ‘match’ direction are each split into the same number of parts (eg halves, thirds…).
• cards in a line in the ‘mix’ direction are each split into different numbers of parts
5. If a player can not place their card they put it back on the bottom of their pile and it is the next player’s turn.
6. To make the game into a contest you can give a point to any player that makes ‘three in a row’ in either direction, and add a rule that says you can not have more than three cards in a row at any time. It is possible to gain 2 points by completing both a match and a mix by placing your card in the right place. The player with the most points wins.

#### Session 4

The purpose of this session is to develop students' understanding of a number of like parts (the numerator).

This is the next step towards the concept of a fraction. It is much more straightforward than that of session 1 -3 which involved (i) separating a single object into part objects (ii) of a given number (iii) all of the same amount. Here we only have to put together a given number of these parts and to recognise and name the combination

#### Resources

• 3 sets of animals (these could be models or pictures)
• Copymaster 5
• Copymaster 6
• Five food trays
• Playdough
• Plastic knives
• Cutting boards
• Five set loops

#### Warm up

1. Count in halves up to a number such as 3.
1/2, 2/2, 3/2, 4/2, 5/2, 6/2 (be prepared for students to carry on counting and not realise that 6/2 is equal to 3).

How many halves did we count (six halves = three wholes, write on the board)
How many halves do you think would equal 6? Write on the board
How many halves do you think would equal 9? Write on the board

3. Relate back to their knowledge of doubles and halves.

#### A Number of Like Parts

This is an activity for up to six students working in two teams. Its purpose is to introduce the concept described above.

1. One team acts as animal keepers, the other works in the zoo kitchen. The latter need to be more numerous, since there is more work for them to do.

2. A set of animals is chosen. Suppose that this is set 1. The kitchen staff look at the menu and set to work preparing eels, as in Making Equal parts. The animal keepers put the animals in their separate enclosures. They may choose how many of each. For example:

3. The animal keepers, one at a time, come to the kitchen and ask for food for each kind of animal in turn. The kitchen staff cut the eels as required, eg
 Animal Keepers may say: Zoo Kitchen Staff may say: Food for 2 elephants please Here it is, 2 whole bales of hay Food for 5 giraffes please 5 third parts. Tell them not to leave any scraps. Food for 3 rhino please Here you are. 3 half parts from 2 bales of hay. There is one half part left. Food for 5 zebra please Here you are 5 quarters or 5 fourth parts. Food for 6 sheep please 6 fifth parts. Lucky sheep.
4. Each time the animal keeper checks that the amounts are correct, and then gives its ration to each animal. The keepers also check each other’s work.

5. When feeding time is over, the food is returned to the kitchen for reprocessing. Steps 1 to 4 are then repeated with different animals, keepers and kitchen staff.

Note that the eels, slabs, and hay should be of standard sizes.
Note also that the eels, after their head and tails are removed, resemble the eels in a cylinder shape and the slabs of meat and hay are oblongs.

#### Session 5

So far we have covered denominator and numerator without mentioning their names. Students need to understand that the denominator names the equal parts and numerator names the number of like parts.

#### Resources

Revise some of the ideas from sessions 1-4, selecting from these activities:

• You may want to give them some unit fraction cards and ask them to order from smallest to biggest.
• Give them a piece of paper and ask them to fold the paper into equal parts that they have chosen. For example, a student may chose to fold the paper into 4 equal parts. They need to verbalise what they have done. Ask them to tell you about three of the sections.
• An activity for students if the symbolic notation has been introduced. Have a pack of cards which they can use to match pairs eg 1/4 goes with one fourth. When they have matched the cards ask each student to solve a simple addition problem for you.
I want you to give me one fourth + one fourth (write on the board). Can anyone give me cards that mean the same but are written in a different way? (1/4 + 1/4)
I want you to find one fifth and one fifth and one fifth. How many fifths are there? Can anyone give me cards that mean the same but are written in a different way? (1/5 + 1/5 + 1/5)
• Give the students three problems to solve with a partner.
What happens if we had to feed 5 giraffes and they were allowed 1/3 of a bale of hay each?
What happens if we had to feed 5 rhinos and they are allowed 1/2 a bale of hay each?
What happens if we had to feed 6 zebras and they are allowed 1/4 of a bale of hay each?

To conclude the session, ask the students to work in pairs and complete a think board (Copymaster 7 provides a blank think board and Copymaster 8 shows a completed example.) Suitable fractions for think boards include 2 halves, 3 fourth parts, 2 fifth parts, 4 third parts, 2 quarters. Show them the example of a completed think board for 3/5.

## In the garden

Purpose

This unit uses the context of a garden to explore Level 2 symmetry and transformation concepts (translation, reflection and rotation).

Achievement Objectives
GM2-7: Predict and communicate the results of translations, reflections, and rotations on plane shapes.
Specific Learning Outcomes
• Investigate shapes that tessellate.
• Make geometric patterns by translating, reflecting and rotating shapes.
Description of Mathematics

Three important ideas about symmetry are covered in this unit:

• A shape has reflective symmetry when it contains at least one line of symmetry. A line of symmetry is often described as a mirror line.
• A shape has rotational symmetry when it can be rotated by less than a full turn around a central point and look the same.
• A tessellation is the covering of a plane surface with one or more shapes, with no gaps or overlaps.

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 templates that students can use to create symmetrical butterflies. Templates could be one half of a butterfly with students needing to draw in the other half, or a full butterfly for students to colour or decorate symmetrically. Templates for bugs and flowers might also be helpful.
• Provide students with a range of A4 sheets of tessellating patterns that students can use to create garden paths.
• Have students use mirrors to help them draw symmetrical butterflies and other items.

The context in this unit can be adapted to recognise diversity and student interests to encourage engagement. For example create a classroom display of a skate park instead of a garden. Symmetrical people, dogs, and skateboards could be created, and tessellating paths and walls drawn.

Required Resource Materials
• Paper
• Scissors
• Paste
• Mosaic pattern blocks
• Coloured paper
• Rulers
• Split pins
• Blue tack
• Sticks
• Crayons
• Butterfly pictures
• School Journal Part One, Number 5, 1995, The Shapes of Leaves.
Activity

It would be helpful to introduce this work by reading a book to the students that includes a garden setting. The overall aim of the unit is to make a classroom display about a garden using the activities as starting points. Be as creative as you can!

#### Session 1: Up the garden path

In this session students will explore shapes that tessellate or repeat to cover the plane without gaps or overlaps. Although the students will only be covering a strip (path) any covering of a path can be used to tessellate the plane simply by putting paths together.

1. Explain to the students that they have the task of building a garden path. Show them examples of cobblestones and other garden paths, using online images if no real-world examples are available.
2. Give them the mosaic pattern blocks and ask them to build a path using the pattern blocks as cobblestones. All the cobblestones must fit together without any gaps. Students are to select 1 or 2 shapes to build their path. The path needs to have at least 3 or 4 rows of blocks. Ask them to experiment with the mosaic pattern shapes to design their path.
3. Students draw their path shapes and present them to the class describing the shapes that they have selected.
4. Create garden designs around the paths.

#### Session 2: Bugs, Beetles and Butterflies

In this session students will be investigating line symmetry by making butterflies out of coloured paper.

1. Show pictures of butterflies. Look at the wings and discuss reflective symmetry.
2. Ask students to make their own butterflies by folding and cutting.
3. Encourage them to cut out pieces in the wings to add detail.
4. Ask students to share their work and talk about the reflective symmetry it contains.
5. Extend the activity to making other bugs and beetles by folding and cutting.

#### Session 3: Butterfly Painting

In this session students will make symmetrical butterflies with paint.

1. Fold a piece of paper in half. On one half draw the outline of half of a butterfly. Create designs on this half of the wings with paint. Carefully fold the other half of the paper onto the wet paint. Unfold it to get a symmetrical pattern.
2. Ask students to share their work and talk about the reflective symmetry it contains.
3. Students could then make other bugs and beetles for the garden using the same technique.

#### Session 4: The Flower Garden

In this session students will be introduced to making symmetrical patterns with shape blocks. The theme for this lesson is flowers for the garden. Sunflowers would be a great example of this. Show students pictures of sunflowers or read the story "The Sunflower That Went Flop" by Joy Cowley) which can be accessed on YouTube if it is not readily available.

1.  Give students a piece of paper with a line drawn down the middle.
2. Students use mosaic pattern blocks to make half of a flower pattern on one side of the line. They give this pattern to a partner who has to then repeat the pattern on the other side of the line making sure that it is symmetrical.
3. Ask students to trace around the mosaic shapes to make the petal shapes. This can be done with coloured paper. Glue the petals onto the paper to make symmetrical flowers.
4. This activity could be extended by encouraging students to create their own symmetrical designs. They could experiment with cutting the paper shapes in half to create other pieces for their designs.
5. These could then be displayed alongside the path designs from Session 1.

#### Session 5: The Garden Wall

In this session introduce students to the idea of translation. Students will be making tiles for the garden wall. Introduce the activity by showing them examples of some wall tiles.

1. Give each student a piece of square grid paper, for example a 4x4 grid. Students are to draw a design by colouring in the squares to make a pattern.
2. They make 3 or 4 copies of this pattern.
3. Stick these in a row to make a row of tiles with repeating patterns.
4. These could then be displayed above the flowers made in the activity from Session 4.

#### Session 6: Wind Catcher in the Garden

In this session students will make a wind catcher, which illustrates rotation, as an ornament for the garden .

1. Give each student a square piece of paper.
2. Fold the square along its diagonals.
3. Make cuts along the diagonals leaving about 1 cm uncut at the centre of the square.
4. Take one of the cut ends at each corner and fold into the centre.
5. Repeat this at each corner.
6. Pin the folded pieces together with a split pin.
7. Put a little piece of blue tack onto the back of the pin to hold the pieces in place.
8. Attach the pin to a stick.
9. Blow to watch it rotate.

Cut along lines in first image

Teaching Notes:

The wind catcher has rotational symmetry but not reflective symmetry. This is because it can be rotated around onto itself but it doesn't have a line of symmetry in the plane.

#### Session 7: Making Scarecrows

Ask students to draw a scarecrow on a piece of paper. Fold the piece of paper down the middle of the scarecrow. Cut down this middle line. Students then give their scarecrow to a buddy who has to draw the other half of the scarecrow making sure that the drawing is symmetrical. Stick one half of the scarecrow onto a piece of paper. A buddy draws on the other half. Finished scarecrows could be stuck onto sticks and displayed in the garden.

Other Ideas

• Make designs for a dinner set for a picnic in the garden. Students could design a pattern for the pieces in the dinner set. The Willow Pattern story and plates could be used as motivation for this. Patterns around the edges of the plates would need to be repeating patterns.
• Paint patterns around the rim of pots. Plants could be planted in these pots.
• Make a patchwork picnic cloth with designs in each patch piece. This could be made out of paper or fabric. The patch pieces could show a tessellation or reflective symmetry.