## Early 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.

## Maps

Level Two
Geometry and Measurement
Units of Work
In this unit students are introduced to using maps. They use maps to locate landmarks, identify views from different locations, and give directions using left and right turns and distances.
• Use a map to identify views from a location.
• Use compass directions to describe the direction of landmarks.
• Describe pathways between map locations.

## Staircases

Level Two
Number and Algebra
Units of Work
In this unit students look for and describe the patterns they see in different types of staircases.
• Continue a sequential pattern.

## Data cards: Level 2

Level Two
Statistics
Units of Work
This unit introduces the students to a way of looking at information from a group of individuals, i.e. a data set. “Data cards” are used to display information about individuals and by sorting and organising a set of data cards, students can find out things or answer questions about the group.
• Pose investigative questions.
• Write data collection or survey questions to support collecting information for investigation.
• Collect information.
• Sort information into categories.
• Display information to answer investigative questions or find out things.
• Answer investigative questions by...

## Getting partial

Level Two
Number and Algebra
Units of Work
In this unit we explore fractions of regions as well as fractions of sets. We look for and develop understanding of the connection between fractions and division.
• Find fractions of regions.
• Find fractions of sets.
• Identify equivalent fractions.
• Locate fractions on a number line.

## Scavenger hunt

Level Two
Geometry and Measurement
Units of Work
In this unit students participate in a series of scavenger hunts to develop their own personal benchmarks for measures of 1cm, 10cm, 50cm and one metre. An understanding of the relationship between centimetres and metres is also developed.
• Find objects that they estimate to be a 1cm, 10cm, 50cm and one metre long.
• Measure lengths of approximately one metre to the nearest cm.
Source URL: https://nzmaths.co.nz/user/75803/planning-space/early-level-2-plan-term-4

## Maps

Purpose

In this unit students are introduced to using maps. They use maps to locate landmarks, identify views from different locations, and give directions using left and right turns and distances.

Achievement Objectives
GM2-6: Describe different views and pathways from locations on a map.
Specific Learning Outcomes
• Use a map to identify views from a location.
• Use compass directions to describe the direction of landmarks.
• Describe pathways between map locations.
Description of Mathematics

Maps provide a two dimensional representation of the real world. By looking at a map students should be able to anticipate the landmarks they will see from a given location and in which direction (N, S, E, W) those landmarks will be seen. By using maps of their school or local area students will be able to check their thinking by matching the map with the real world.

Students begin to use the map to help them follow and give directions. They start to use directions involving left and right turns and use landmarks to clarify pathways. Students also begin to use distances in whole numbers of metres.

Opportunities for Adaptation and Differentiation

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

• increase or reduce the amount of detail provided on the map template for session 1
• specific teaching around compass directions and half and quarter turns, as required.

Some of the activities in this unit can be adapted to use contexts and materials that are familiar and engaging for students. In particular, the choice of maps to use will depend on the interests of your class. Some students may respond best to maps of familiar areas, while others may be more engaged by an imagined or fantasy context. You could work as a class to create maps of a class favourite story, or the location of a television series or movie.

Required Resource Materials
Activity

#### Session 1

In this session students are introduced to using a map to locate landmarks and identify views from different locations.

1. Give the students copies of a school map with the outline of main buildings and features marked on it. Only label some of the buildings and features.
2. Work with students to label their classroom and to orientate the map.
3. Students are to label the buildings and features on the map.
4. Students then take their map and walk around the school to check their labels and to add 2 or 3 new landmarks to the map.
5. Back in the classroom ask the students to use the map to answer questions that require them to describe different views from locations on the map. For example:
Which classroom has the best view of the playground?
What building can you see from the field?
What building can you see out the library windows?

#### Session 2

In this session students use the school map to describe pathways from locations.

1. Show the students which direction to put the compass points N, S, E, W on their school map.
2. Tell the students that in today’s session they will be marking pathways on their map.
3. Ask the students to trace with their finger on their map a pathway you describe. For example, start in the school hall and walk south past Rooms 1 and 2, then walk west towards the sandpit, from the sandpit you can see the library so walk south over the lawn to the library.
4. Students work in pairs to give each other directions. Encourage the students to use the compass directions, and to use the landmarks on the map to help give directions between locations.
5. Discuss with class what they found useful when giving or following the directions.

#### Session 3

In this session students use a local map (or a fictional one) to describe different views they can see from different locations. They use compass directions to give the direction of landmarks from given locations.

1. Pose questions based on the map, which require students to describe the views from different locations. For example:
How many houses have a direct view of the playground?
What can the children see from the Playcentre?
What can the doctor see out the window?
If you sat in the doctor’s carpark what could you see?
Colour in a house that has a view of the Playcentre, the Dairy, and the Hall?
2. Pose questions based on the map which require students to use the compass directions. For example:
What building is East of the Café?
What building is North of the Hall?
What building is South of the Chemist?
What direction is the Playcentre from the Church?
What direction is the Playground from the Doctors?
How many houses are South of the Hall?
From which building can you look West to see the Church?

#### Session 4

In this session students give a set of directions between two locations using distances and quarter turns to the left and right.

1. Using the local map work out an appropriate scale for example 1cm is 50m and help the students make scale rulers with strips of card. In this example the ruler graduations will be 0, 50, 100, 150 etc.
2. On the example map the dots represent entry/exit points for buildings. Show the students how to turn the map around to orientate themselves as they follow directions and turn left and right.
3. Give the students a set of directions to follow. Focus on left and right turns, use landmarks to help provide the distances. For example, leave the Playcentre and turn right, walk along and cross the road, turn right, walk past some houses and cross the road, where are you now?
4. Students can work in pairs to give each other instructions.
5. Using their scale rulers students will be able to give directions that include distances. Give the students a set of directions to follow. For example, leave the Café and turn left. Walk 40 metres, if you turn right what will you be able to see?
6. Students can work in pairs to give each other instructions that include distances and left and right turns.

#### Session 5

In this session students learn about pathways and apply this to creating a fire escape plan for their house.

1. Using the local map, or the school ask students to draw the path from one location to another. Add conditions to the route they can take, for example draw how a class could walk from the library to the hall without walking past the office block.
2. Ask students to create a Fire Escape Plan. Before completing this activity they should draw a plan of their house and then mark the escape route out of each room. https://fireandemergency.nz/at-home/creating-an-escape-plan/
3. This activity is likely to take more than one session and can be completed as a home task.

## Staircases

Purpose

In this unit students look for and describe the patterns they see in different types of staircases.

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 sequential pattern.
Description of Mathematics

In much of students’ early pattern work, the numbers involved can be compiled in tables like the one below:

 Length of garden 1 2 3 4 5 6 Number of paving stones 8 12 16 20 24 28

Two relationships can be seen:

1. The recurrence relation allows us to calculate the next number in the pattern from the previous number. In the example above the number of paving stones increases by four each time the length of the side garden is increased by one. This pattern can be seen in the second row of the table: 8, 12, 16, 20…
2. The functional relationship allows us to calculate any number in the pattern, independent of the previous number. In the example above the number of paving stones is four times the length of the garden, plus four. We can express this relationship as an equation. If P equals the number of paving stones and L equals the length of the garden, then P = 4L + 4.

In practice, recurrence relationships are easier to identify than functional ones.

Opportunities for Adaptation and Differentiation

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

• encourage the use of physical materials as well as drawing to create patterns
• encourage students to work collaboratively with other individuals or pairs to share their discoveries
• adjust the expectations for solutions - some students may only be able to extend the pattern by one or two iterations, while others may describe a generalisation.

The materials used in this unit can be adapted to recognise diversity and student interests to encourage engagement. Instead of creating patterns with plastic classroom blocks or cubes students could be encouraged to make the patterns using environmental materials such as pebbles, shells, or daisies from the school lawn.

Required Resource Materials
• Staircase problems
• Graph paper
Activity

#### Getting Started

Today we explore up-and-down staircases to find the pattern in the number of blocks they are made from.

1. Begin the session by telling the students about up-and-down staircases:
 One block is needed to make a 1-step up-and-down staircase. It takes one step to get up and one step to get down. This is a called a 2-step staircase as it takes two steps to go up and two steps to go down.
2. Together count the steps so that the students understand why it is called a 2-step staircase.
How many blocks are in the staircase?
How many blocks do you think would be in a 3-step up-and-down staircase?
How could you work it out?
3. Give the students time to work out the number of blocks. Share the ways that they used to work it out.
4. Ask the students to guess how many cubes they think would be needed to make a 5-step staircase. Get them to build a 5-step staircase. Check their guesses.
5. Get the students to build more staircases. As they do, ask them about any patterns they see.
Some may recognise the horizontal layers as being the sequence of odd numbers. Some may see the vertical stacks.
Some may see that square numbers of blocks are always involved. This can be checked by rearranging the blocks to make square numbers (see the diagram below). This is an important discovery. Let them make it. This may require some careful scaffolding on your part.
6. Some students may wish to continue to find numbers that make larger up-and-down staircases. To keep track of the number of blocks/cubes in each staircase it might be useful to draw the staircases on graph paper. However, if the students prefer to use cubes, then they should record their results in a table.

#### Exploring

Over the next 2-3 days the students work in pairs or individually to solve the following problems. Show the students how to use grid paper to continue the patterns. As they complete the problems ask them about any patterns they see and encourage the students to record these observations with the patterns on the graph paper.

Problem 1: Straight up the stairs

How many blocks are in this 4-step-up staircase?

How many blocks would there be in a 5-step-up staircase?
How many blocks would there be in a 6-step-up staircase?
How many blocks in a 10-step-up staircase?
How many more blocks will an 11-step-up staircase need?
What is the largest up staircase that you can tell us about?

(Note: the numbers of blocks in this pattern are the triangular numbers, see Algebra Information.)

Problem 2: Climbing ladders

How many pieces of wood have we used in this 1-rung ladder?

How many pieces of wood have we used in this 2-rung ladder?

How many pieces of wood would there be in a 4-rung ladder?
How many pieces of wood would there be in a 6-rung ladder?
What is the largest ladder that you can tell us about?
How many pieces of wood will you need to add to a 7-rung ladder to get an 8-rung ladder?

(Note: the number of pieces of wood is three times the number of rungs.)

Problem 3: Small steps

Watch out! You need to take small steps to walk up and down these stairs.

 1-step 2-step 3-step

How many blocks are in the 4-step staircase?
How many blocks are in the 6-step staircase?
What is the largest staircase that you could tell us about?
Does this remind you of something you have done before?

(Note: the count here is the same as that in Problem 1.)

Problem 4: Star patterns

 This is a 1-star This is a 2-star This is a 3-star

How many blocks are in a 4-star?
How many blocks are in a 5-star?
What do you notice about the stars?
How many blocks do you need to add to a 7-star to make an 8-star?
What is the largest star that you could tell us about?

(Note: the pattern here is 1, 5, 9, 13, … At each stage you add on 4 blocks. To make a 100-star you need to have 99 lots of 4 plus one block for the centre.)

Problem 5: L-shapes

 This is a 1-L This is a 2-L This is a 3-L

How many blocks are in a 4-L?
How many blocks are in a 5-L?
What do you notice about the pattern in the L’s?
What is the largest L that you could tell us about?

(Note: to make a 100-L you need 100 + 100 – 1 = 199 blocks.)

#### Reflecting

In this session we share our solutions to the problems of the previous days. We listen carefully as the patterns are explained. We then make some block patterns of our own which we give to our classmates to continue.

1. Begin the session by asking the students to attach their solutions to the problems to a display wall. Give the students time to look at the solutions of other students. Ask for volunteers to share their solutions.
2. Give pairs of students a supply of blocks and grid paper and ask them to invent their own block pattern. Tell them to record the first three elements in the pattern on a piece of grid paper.
3. Ask the students to swap patterns with another pair. Work together to discover the pattern and then continue it.
4. Repeat with another pair’s pattern.
5. Leave the patterns on a table for the students to solve in their own time.

## Data cards: Level 2

Purpose

This unit introduces the students to a way of looking at information from a group of individuals, i.e. a data set. “Data cards” are used to display information about individuals and by sorting and organising a set of data cards, students can find out things or answer questions about the group.

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.
• Write data collection or survey questions to support collecting information for investigation.
• Collect information.
• Sort information into categories.
• Display information to answer investigative questions or find out things.
• Answer investigative questions by sorting, organising and arranging information.
• Make sensible statements about the information and be able to back up their statements with appropriate displays.
Description of Mathematics

A “data card” is simply a square piece of paper containing information about an individual person or thing. At this level the data card is divided into three areas with the same category information in the same location on each card. In this unit the terms data and information are used to mean the same thing and are interchanged throughout. Because several pieces of information about individuals are on each data card, different categories can be looked at simply by rearranging the cards.

This unit focuses on sorting and organising data sets, i.e. collections of information from a group of individuals. As the data set is looked at, questions or interesting things arise, which is different from starting with an investigative question then collecting data to answer the investigative question.

Understanding the difference between individual data and group data is central to the unit. The goal is to move students from “that is Jo’s data and that is me” to making statements about the group in general. Increasing students' ability to accurately describe aspects of a data set, including developing statistical vocabulary, is part of the unit. As students become comfortable with making statements and describing data, more precise vocabulary is to be encouraged. The meaning and usage of words like; same, similar, exactly and almost need to be explored during the unit along with the importance of using numerical descriptions, e.g. 2 more than, when describing or comparing data.

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.

Data collection or survey questions

Data collection or 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 data collection or survey question might be “What is your favourite ice cream flavour?”

As with the investigative question, data collection or survey question development is led by the teacher, and through questioning of the students, suitable data collection or survey questions are developed.

Opportunities for Adaptation and Differentiation

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:

• encouraging category within category investigative questions
• allowing for three additional data collection questions by dividing the data cards into four rather than three
• collecting data from another class and compare.

The context for this unit can be adapted to suit the interests and experiences of your students.

Required Resource Materials
Activity

#### Session One

1. Show the following data card to the class and explain what a data card is, i.e. a piece of paper contains three pieces of information about one person.
Does anyone in the class fit this data card?
Do you know someone that fits this data card that is not in this class?
How many different people could this data card be correct for?
3. Turn the data card over to reveal the name of someone familiar that fits this data card. The point to get across is that a data card could fit many people but each data card is about one person only.
4. Discuss the importance of knowing exactly what each piece of data is about.
What could “right handed” mean?
If the data card just said “brown”, instead of “brown eyes” what could it mean?
5. Explain to the students that the way to view each piece of data is to see it as the answer to a data collection or survey question. Get them to suggest the data collection questions that give these three pieces of information. Discuss how some students could answer the same data collection question differently, e.g. “Are you right handed or left handed?” could give two different answers for the person who throws a ball with one hand and writes with the other. A more specific data collection question is needed, e.g. “What hand do you write with to produce your best looking work.”
What would a data card about you look like?
6. Hand out a data card to each student to fill out (. Have each student write their name on the back of the data card hand and have a student collect these.
7. After this session the teacher needs to arrange the data cards onto pieces of paper and photocopy them. One set is made for each pair of students. Photocopying onto coloured paper is suggested to make it easy to recognise the class data set. This data set will be used during Session Three.

#### Session Two

1. Start the session by reminding the students about the data card they filled in during Session One. Select a data card one of the class filled out and read out the three pieces of data and ask the questions, “Whose data card could this one be?”, “Could it be anyone else in the class?”, “Could it be someone else in the school?”, “Could it be a teacher or other adult?” Repeat this several times.
2. Organise the students into pairs and hand out to each pair a set of Data Set One, Copymaster 1. Tell them this is a group of students from another school and get them to cut out all the data cards. Once the data cards are cut out have the students sort and organise the data cards to find out things about this data set. Remind them we are interested in the group and not individual students.
3. At a suitable time, as the pairs of students are organising the data cards,  have the class stop and look at the different ways the data cards have been arranged. Briefly discuss the different ways, along with writing up or drawing the different ways on to the board for all students to see. The question “What is good about this way?” or “When would it be good to organise the cards like this?” could be asked.
4. Ask the following investigative questions and get each pair of students to organise the data cards into one of the above arrangements to show the answer.
• Which hand do students in the class write with? Organise into rows
• What eye colour do students in the class have? Organise into columns
• Do the eye colours for boys and girls differ? Arrange into groups, i.e. groups of groups
5. Have the students suggest similar investigative questions they could explore then encourage them to look at the data cards, organising and reorganising to find out as much as they can about this group of students.
Initially encourage the students to look at one category at a time then, encourage students to look for categories within other categories, e.g. What hands do girls write with?
6. Write on a large piece of paper what the class discovers or get each pair to write up what they find out about this group. Keep this information, as it can be used later to compare with other data sets.

#### Session Three

1. Explain to the class that today they will be sorting and arranging data cards, like Session Two, except they will be using the data cards they wrote about themselves. Before the copied data cards are handed out, discuss what the students expect to find out.
What do you think we will find out about our class?
Will it be mainly different or similar to the group looked at in Session Two?
2. Hand out the copied data cards from Session One to each pair of students. The pairs are to cut out the data cards, sort them and organise them to look for other interesting things about the class.
3. The teacher is to move around getting each pair to explain and show what they have found out. The teacher is to encourage the pairs to add detail to their answers, moving students from, “Yes, there are more girls than boys” to “Yes, there are 15 girls, 5 more girls than boys.”
4. Conclude the day by considering the statements the students made at the start of the day and seeing how many where true and discussing other interesting things found out about the class.

#### Session Four

Today the students, in pairs, will design and collect their own data using data cards. Each pair of students needs to design two data collection questions to ask other students in the class. The first data collection question will be “Are you a boy or a girl?” with two new data collection questions added.

1. Discuss and brainstorm suitable data collection questions. Data collection questions for this activity need to be answered with either yes or no, or an option selected. Keep the optional answers to a maximum of three options.
Sample data collection questions:
• Do you have a pet at home?
• Do you have an older brother?
• Can you swim one length of the school pool without touching the bottom?
• If you could choose, would you sing, dance or read a book?
• Do you like fruit or meat or vegetables best?
2. Once suitable data collection questions have been developed they are to be written onto a large data card.
3. Before starting to collect data each pair of students needs to write three investigative questions they could ask of the data they will collect and to make statements about what they expect to find out about the class for these investigative questions. More able students are to be encouraged to pose investigative questions about categories within categories, leading to statements about what they will find e.g. “Most girls will select red as their favourite colour” or “About the same number of boys as girls will be youngest in their family” or “The most common data card will be boy, blue and middle”.
4. Each pair of students is to cut out enough blank data cards for the class and number them 1 to n (number in class). Once completed the pair of students are to ask half the students each their three data collection questions and fill out a data card for each student. The student’s name needs to be written on the back to make sure all students are asked. They need to remember to complete their data cards for themselves as well.
5. The importance of keeping the data in the same position needs to be stressed, i.e. the answer to the data collection question “Are you a boy or girl?” is always placed top right and the place in the family always at the bottom. This makes sorting and organising the data cards later much easier.
6. The use of abbreviation and initials could be introduced at this point if the teacher feels that class is ready for this.

#### Session Five

In pairs the students are to sort and organise their 24 data cards to look for other interesting things about the class and to see if the statements they made about the class were correct.

After a set time each pair reports what they found out about the class. This could be in the form of a written report, a conference with the teacher or a presentation to the class.

Attachments

## Getting partial

Purpose

In this unit we explore fractions of regions as well as fractions of sets. We look for and develop understanding of the connection between fractions and division.

Achievement Objectives
NA2-1: Use simple additive strategies with whole numbers and fractions.
Specific Learning Outcomes
• Find fractions of regions.
• Find fractions of sets.
• Identify equivalent fractions.
• Locate fractions on a number line.
Description of Mathematics

Fractions are probably the first departure from whole numbers that students will see. This unit introduces a number of important concepts relating to fractions. The first of these is that fractions represent parts of one whole, and can be represented in a variety of ways including regions and sets. This makes them useful in a large variety of situations where whole numbers by themselves are inadequate.

The second useful concept is that a given number can be represented as a fraction in many ways. Knowing that fractions such as ½ can be disguised as 2/4 or 3/6, etc. is important both for recognition purposes and for use in calculations.

Finally, students should know that fractions can be represented both as one whole number divided by another whole number and as points on the number line. Having a knowledge of the different representations of fractions provides connections across mathematics for students and so increases their level of understanding.

In this unit we also introduce the idea of a fraction of 100 and so lay the groundwork for the decimal representation of fractions at Level 3 and percentages at Level 4. These ideas are developed further in the units Getting the Point, Level 3 and Getting Percentible, Level 4. Facility with fractions is also an important precursor for algebra. Algebraic fractions have a wide range of uses. Without a good knowledge of how fractions work, students will be restricted in their work at higher levels of the secondary school when fractions occur in algebraic settings.

Opportunities for Adaptation and Differentiation

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:

• 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 for this unit can be adapted to recognise diversity and student interests to encourage engagement. For example, students in kapa haka teams could be used as a context for fractions of sets, and eels could be used instead of worms when considering fractions of lengths.

Required Resource Materials
• Small plastic jars
• Plastic cups and bottles
• Paper circles and strips
• Sand or oil timer
• Uni-fix cubes or multi-link cubes
• Toy money (10 cent coins)
• Plasticine, kitchen scales, ruler
• Paper clip, pencil
• Pattern blocks
• Toothpaste packets
• Copymasters onetwothree, four
Activity

Session 1

Here we look at different representations of 1/2.

1. Write the fraction 1/2 on the board. Ask students what the number is and what they think it means. Put them into groups of three or four to brainstorm ideas they have about one-half. Ensure that they record their ideas as words, numerals or diagrams to share with the whole class.
2. Get each group to report back the class on their favourite idea about one-half. Use this reporting back session to develop a class chart. Expect many of the students to have regions ideas such as cut pies and apples, half of a length, and possibly half time.
3. Set up the following quick challenges around the room as stations that each group of students must attempt. Introduce the challenges briefly. Allow the students three minutes on each station. It is critical that they record how they solved each challenge.
• The stations cards are included as Copymaster one. Ensure that the following materials are available for each challenge:
• a plastic jar with twelve beads or counters in it
• two clear plastic cups and a small plastic bottle of water
• paper circles marked with ten divisions
• a sand or oil timer
• a stack of 16 Uni-fix or multilink cubes
• twenty toy 10-cent coins in a plastic jar
• a 400 gram blob of plasticine, kitchen scales, 30 cm ruler
• spinner (Copymaster two), paper-clip, pencil
• a trapezium-shaped pattern block  and a set of blocks
• a toothpaste packet and multilink or Uni-fix cubes
4. Get the students to report their answers and the strategies they used to find them, back to the class. Highlight the equal sharing aspect of finding one half. Tell them that you want them to try each challenge again only instead of finding one-half they need to find three-quarters. Write 3/4  on the whiteboard and discuss what it means (four equal parts and three chosen). For challenge number 8 the students have to think of what will happen to the spinner three-quarters of the time.
5. Check the students recording to see how many of them have generalised three-quarters from one-half. Look for connections like two quarters make one-half so three-quarters is one-half and one-quarter.

#### Session 2

Here we look at fractions other than 1/2 and consider ways to represent these fractions that involve 100.

1. Remind the students how they found a half and three-quarters of a circle in Session 1. Discuss how many marks around the ant walked to get half way around the circle and how this could be used to divide the circle in half. Similarly the circle could be divided into quarters by marking spaces two and a half marks around and connecting the marks to the centre.
2. Give the students several circles marked with one hundred spaces around (Copymaster three). Tell them that they can use any method they like to fold one circle into quarters, one into fifths, and another into tenths. Allow them to solve this challenge in groups.
3. Share the results of their investigations. Some students will use geometry to fold the circles while others will use measurement (dividing the number of spaces around the outside). Either method is valid as one informs the other. Use the folding to make equivalent fraction statements, like 1/4 =25/100 . Challenge the students to write other equivalence statements, particularly with fractions that have other than one as their numerator (top line), e.g. 2/5 =40/100.
4. Hold onto the paper circles for Session 3.

#### Session 3

This session involves fractions in problem situations.

1. Pose this problem for the students, " It is Jo’s twelfth birthday party. Including herself there are four people at the party. If everyone gets one quarter of the cake, how many candles do they get?" Get the students to solve the problem with counters and their paper circles from the previous session.
2. Discuss the strategies that the different students used. These might include sharing twelve counters evenly onto the sections of the quarter circle, using addition (6 + 6 is 12 so 6 is a half so 3 is a quarter), or division (12 ÷ 4 = 3).
3. Give the students other sets problems (see Copymaster four). Get them to record their strategies as they solve the problems. Students who use sharing strategies should be encouraged to anticipate the result of their sharing before it is complete.
4. Students may like to write their own cake problems for others to solve. These can be made into a class book of problems for independent activity.

#### Session 4

Another way to represent numbers is the number line. Here we use the number line to show the relative positions and sizes of fractions.

1. Draw a number line from 0 to 10 on the board (about 1 metre long). Build up the number line by getting students to write where different numbers might be. Once the whole numbers are in place, ask students to think about where numbers like 1/2, 3/4, 3/2, and 4 ½  might be. This will help students to realise how fractional numbers extend the existing set of whole numbers and can be represented on a number line in the same way.
2. Give the students several paper strips of the same length cut from scrap paper. Ask them to fold one strip in half, one into quarters, and one into eighths. This is relatively easy as they can be folded by repeated halving. Ask the students to label each strip using symbols: 1/2, 1/4, and 1/8.
3. Take a full strip and use it to draw the number line from 0 to 1 by marking each end. Shift the strip to the right and mark 2 at the right-hand end, shift it again and mark 3, etc. (0 – 5 is sufficient). Ask the students if they can use their strips to show exactly where one-half would be. Expect students to align the strip folded in half to do this. Ask this for other fractions like one-third, three quarters, two-thirds, and extend it to fractions greater than one like five-halves, four-thirds, and three and seven-eighths.
4. Pose these problems for students to solve using their strips.

• Draw a number line to show 1/2, 1/4, and 1/8.  Mark where you think 1/3, 1/5, and 1/10 would go on your number line. Explain where you placed them. Why do fractions with one on the top line get smaller as the number on the bottom gets larger, e.g. one-half is larger than one-third?
• Which of these fractions is closest to one, 1/2, 2/3, or 3/4? Why?
These problems will highlight students’ knowledge of the relative size of fractions. For example, a student might find half of the distance between 0 and 1/5  to see where 1/10  should be or half the distance between 1/2  and 1 to see the location of 3/4 . The problems will also highlight their understanding of the role of the numerator (top number) as the selector of the number of parts and the role of the denominator (bottom number) as nominating how many equal parts the whole is separated into.

#### Session 5

Here we try to link the concepts of fractions in length and sets by dividing up a big worm.

1. To link the concept of fractions as they apply to lengths and sets, tell the story of the two early birds who caught a worm. Produce a stack of one hundred Uni-fix cubes or multilink cubes joined together so that sections ten cubes long are in the same colour. Tell the students that this is the worm and when the birds measured it they found that it was very long. How long? Ask, "Suppose the two birds wanted to share the worm equally. How could they do that?" Students should use the idea of half of 100 being 50. Ask, "If they caught three worms this size and shared them out, how much would each bird get?" Record their responses using equations like 1/2  of 300 is 150.
2. Extend the problem. Ask, "Suppose that it took four birds to pull this worm out of its hole. How much of the worm will each bird get? What if there were five birds, ten birds?" Record the students’ strategies using symbols and diagrams.
1. Pose a series of problems for them to solve independently, of the form:
2. There were three birds.  The worm was 18 cubes long. How much did each bird get?
3. There were four birds. Each bird got six cubes of worm. How long was the worm?
The worm was 18 cubes long. Each bird got three cubes of worm. How many birds were there?
3. Students will enjoy making up birds and worm problems for others to solve. It is vital that they record their solutions using fraction symbols. Use their responses to these problems to assess which type of strategies (sharing, adding or dividing) each student uses. Try to extend the number of strategies that each student has.

## Scavenger hunt

Purpose

In this unit students participate in a series of scavenger hunts to develop their own personal benchmarks for measures of 1cm, 10cm, 50cm and one metre. An understanding of the relationship between centimetres and metres is also developed.

Achievement Objectives
GM2-2: Partition and/or combine like measures and communicate them, using numbers and units.
Specific Learning Outcomes
• Find objects that they estimate to be a 1cm, 10cm, 50cm and one metre long.
• Measure lengths of approximately one metre to the nearest cm.
Description of Mathematics

Children need to be able to see the need to move from using non standard measures of length to standard measures of length. The motivation for this will arise out of students comparing differences in the length of their hand spans etc. From this the need for standard measurement will become evident.

Students also need to develop personal benchmarks with which to measure various objects in their daily lives. Their personal benchmarks need to gradually relate more to standard measures such as metres, 1/2 metres.

The ultimate aim is for students to be able to choose appropriately from a range of strategies including estimation, knowledge of benchmarks, and knowledge of standards measures in order approach various measuring tasks with confidence and accuracy.

Opportunities for Adaptation and Differentiation

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

• students can carry a paper strip of the target length for part of their hunt
• provide a 'hint station' with clues of suitable objects around the class.

This unit can be adapted to acknowledge student interests, encouraging engagement. The scavenger hunts could be carried out in different locations, for example, in the classroom, in the school playground, at a local park or beach.

Required Resource Materials
• Copymasters One,
• Pencils
• A selection of paper strips in a variety of lengths including 1cm, 10cm, 50 cm and 1 metre.
Activity

This unit is run as a series of stations over four days with students rotating around the stations in groups. The final session is run as a class activity with all students working on the same task in groups.

The four stations involve the students looking for objects that they estimate to be a certain length. You will need to set appropriate boundaries for their search, e.g. the classroom or the playground.

As students work, the teacher can circulate amongst the groups. Points to reinforce in your discussions with students include

• There are 100 centimetres in a metre.
How many 1 cm lengths in a metre?
How many 10 cm lengths in a metre?
Why is 50 cm sometimes called half a metre?
What is another name for a metre?
• Estimation can involve the use of personal benchmarks e.g. knowledge that your fingernail is 1cm long or the length of your stride is 1m can help you estimate these lengths more accurately.
• To measure accurately, one end of the object being measured must be aligned with zero on the ruler.
• The meaning of the unmarked gradations on the ruler may need to be considered. Measurement to the nearest cm often requires identification of the number closest to the end of the object being measured.

#### Station One

Students work in pairs or small groups to find items that they estimate to be 1cm long. They check their estimates by measuring.

Student Instructions (Copymaster One)

Go on a Scavenger Hunt!

1. Use a ruler to find out how long 1 cm is. Take a good look!
2. Find ten objects that you estimate to be 1cm long.
3. Record your objects on the table below.
4. Check your estimations using a ruler to measure the length of the objects accurately.
 Object with estimated length 1cm Measured length

How accurate were your estimates?
Were your estimates too long or too short?
What would be a good way to try and remember how long 1cm is?

#### Station Two

Students work in pairs or small groups to find items that they estimate to be 10cm long. They check their estimates by measuring.

Student Instructions (Copmaster Two)

Go on a Scavenger Hunt!

1. Use a ruler to find out how long 10cm cm is. Take a good look!
2. Find ten objects that you estimate to be 10cm long.
3. Record your objects on the table below.
4. Check your estimations using a ruler to measure the length of the objects accurately.
 Object with estimated length 10cm Measured length

How accurate were your estimates?
Were
your estimates too long or too short?
What would be a good way to try and remember how long 10cm is?

#### Station Three

Students work in pairs or small groups to find items that they estimate to be 50cm long. They check their estimates by measuring.

Student Instructions (Copymaster Three)

Go on a Scavenger Hunt!

1. Use a ruler to find out how long 50cm is. Take a good look! This length is also known as half a metre. Why?
2. Find ten objects that you estimate to be 50cm long.
3. Record your objects on the table below.
4. Check your estimations using a ruler to measure the length of the objects accurately.
 Object with estimated length 50cm Measured length Difference between estimated and measured length

How accurate were your estimates?
Were your estimates too long or too short?
What would be a good way to try and remember how long 50cm is?

#### Station Four

Students work in pairs or small groups to find items that they estimate to be 1metre long. They check their estimates by measuring.

Student Instructions (Copymaster Four)

Go on a Scavenger Hunt!

1. Use a ruler to find out how long 1 metre is. Take a good look! What is another name for this length?
2. Find ten objects that you estimate to be 1 metre long.
3. Record your objects on the table below.
4. Check your estimations using a ruler to measure the length of the objects accurately.
 Object with estimated length 50cm Measured length Difference between estimated and measured length

How accurate were your estimates?
Were your estimates too long or too short?
What would be a good way to try and remember how long 1 metre is?

#### Reflecting – Class activity

1. Before the session, set up six activity stations around the room. At each station put a selection of paper strips in a variety of lengths. Ensure that at each station there are strips with a length of 1cm, 10cm, 50 cm and 1 metre. Label the strips at each station with letters.
2. Tell the students they will be participating in the ultimate estimation challenge. Have the students rotate around the stations identifying the strips they believe to be 1cm, 10 cm, 50 cm and 1 metre long. They record their results on recording sheets (Copymaster Five).

At the conclusion of the session reveal the correct letters for the 1cm, 10cm, 50 cm and 1 metre lengths. Students check their answers and have a chance to measure the strips they chose as required.

#### Extension

Students who finish the activity early could estimate and measure the lengths of the other paper strips at the stations.

Printed from https://nzmaths.co.nz/user/75803/planning-space/early-level-2-plan-term-4 at 12:25pm on the 21st January 2022