## Early level 1 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.

## No way Jose

Level One
Statistics
Units of Work
In this unit we develop the language of probability by considering events which are likely or unlikely. We do this using the context of children's stories.
• Use everyday language to talk about chance.
• Classify events as certain, possible, or impossible.

## Numerals and expressions

Level One
Number and Algebra
Units of Work
The purpose of this unit is to develop understanding of a numeral as a symbol to represent an amount or number, and of the symbols for the operations of addition and subtraction.
• Understand that an amount or number of items can be represented with a single unique symbol.
• To correctly write numerals.
• Understand that written words and oral words can also be represented with numeral symbols.
• Recognise numbers within story contexts.
• Understand and use the addition and...

## Worms and more

Level One
Geometry and Measurement
Units of Work
This unit comprises 5 stations, which involve the students in developing an awareness of the attributes of length and area. The focus is on the development of appropriate measuring language for length and area.
• Compare lengths from the same starting point.
• Use materials to make a long or short construction.
• Use materials to compare large and small areas.

## Turns

Level One
Geometry and Measurement
Units of Work
In this unit we look at the beginning of the concept of angle. As students come to understand quarter and half turns, they also begin to see that ‘angle’ is something involving ‘an amount of turn’. These ideas are explored by using students’ bodies, toys, games and art.
• Show a quarter turn and a half turn in a number of situations.
• See that two quarter turns equal one half turn.
• Recognise the ‘corner’ of a shape that is equivalent to a quarter turn.

## Learning to count: Counting one-to-one

Level One
Number and Algebra
Units of Work
This unit develops students’ understanding of, and proficiency in, counting one-to-one.
• Understand that the number of objects in a set stays the same as changes are made to spatial layout, size, or colour.
• Understand that the count of a collection of objects can be trusted and worked from as objects are added or taken away, or the set is rearranged into parts.
Source URL: https://nzmaths.co.nz/user/75803/planning-space/early-level-1-plan-term-4

## No way Jose

Purpose

In this unit we develop the language of probability by considering events which are likely or unlikely. We do this using the context of children's stories.

Achievement Objectives
S1-3: Investigate situations that involve elements of chance, acknowledging and anticipating possible outcomes.
Specific Learning Outcomes
• Use everyday language to talk about chance.
• Classify events as certain, possible, or impossible.
Description of Mathematics

This unit is about developing the language of probability. The words that are introduced and explored in this unit are always, perhaps, no way Jose, certain, possible, impossible, will, might, won’t, will, maybe, never, yes, maybe, no. These are informal, everyday words that denote chance or probability. By using these words, that have some familiarity for the students, they should start to get a better idea of the overall concept of probability. As students progress through the primary years they will gradually learn to assign fractions or decimals to given probabilities using both a theoretical and experimental approach.

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:

• Using their own experiences of what they see as they walk or travel to school rather than the perspectives of a character in a story. Ask the students to draw or tell you about something that they “will see”, “won’t see” and “might see” on the way to school.
• Reading the selected story at the start of each of the “exploring” sessions rather than assuming that the students already know the story and can use the story packet independently.

The contexts for this unit can be adapted to suit the interests and experiences of your students. For example:

• Begin the unit by using your students’ own experiences as they walk or travel from home to school in the morning.
• Create story packets for stories that are popular with students in your class, for example, “Going on a bear/moa hunt”.
Required Resource Materials
Activity

#### Getting Started

Today we follow Red Riding Hood on her journey to Grandma’s house and look at the characters she is likely to meet.

1. Read or tell the story of Red Riding Hood.
Discuss with the students the setting of Red Riding Hood’s journey. Encourage the students to share their ideas about the objects that could be found in the forest.
2. Show a picture of a tree to the class. Ask: Do you think Red Riding Hood will see a tree in the forest?
Peg the picture of the tree beneath the word card will.
3. Show a picture of an octopus to the class. Ask: Do you think that Red Riding Hood will see an octopus in the forest?
Display the picture beneath the word card won’t.
4. Show a picture of a hedgehog to the class. Ask: Do you think that Red Riding Hood will see a hedgehog in the forest?
Display the picture beneath the word card might.
5. Show the students the rest of the pictures in the Red Riding Hood story pack. Let the students take turns placing the picture under a word card. Remember to ask the students to justify their decision.

#### Exploring

Over the next 2 to 3 days the students can look at the journeys of other fairytale characters and make decisions about who or what they might meet.

1. In pairs let the students select one of the prepared story packets. (A story packet contains 3 word cards and pictures of objects for the students to classify. It also includes 3 blank cards for the students to draw their own pictures.)
Remind the students to peg the pictures beneath the word cards.
2. As the students classify the cards, ask questions that encourage the students to explain their thinking.
Tell me why you have put that there?
Why do you think that …….. is impossible?
Could you have put it with one of the other words? Why/Why not?
3. Remind the students that between them they are to draw 3 pictures – one object for each word card in their story packet.
4. At the end of each day give the pairs an opportunity to display and discuss their word strings.
5. As the students share their strings encourage the students to use the language of probability.

#### Reflecting

1. We begin today’s session by getting the students to brainstorm a list of their favourite stories.
2. In pairs, get the students to select a story for which they can make a story pack. Discuss what the contents of a story pack are (pictures and 3 word cards).
3. Allow the pairs time to talk about the people or things that the main character will, might or won’t see.
4. Share ideas to ensure that the students understand what they are doing.
5. Let the students decide on the 3 probability words that they are going to include in their story pack. These words can either be provided on cards or written on the board for the students to copy.
6. As the students make their story packs, ask questions that focus on their use of probability words and their decisions about the likelihood of events.
7. Swap packs.

## Numerals and expressions

Purpose

The purpose of this unit is to develop understanding of a numeral as a symbol to represent an amount or number, and of the symbols for the operations of addition and subtraction.

Achievement Objectives
NA1-4: Communicate and explain counting, grouping, and equal-sharing strategies, using words, numbers, and pictures.
Specific Learning Outcomes
• Understand that an amount or number of items can be represented with a single unique symbol.
• To correctly write numerals.
• Understand that written words and oral words can also be represented with numeral symbols.
• Recognise numbers within story contexts.
• Understand and use the addition and subtraction symbols.
• Recognise and write addition and subtraction expressions from story contexts.
Description of Mathematics

This sequence of lessons lays a fundamental and important foundation for students to recognise, read and write symbols to record and communicate mathematical ideas. As the symbols become well understood, they also become tools for processing thinking. This process begins with the introduction of numerals as symbols that represent amounts or numbers of objects. Students hear and see words that are associated with the amounts, and so need to come to understand that a single symbol is representative of all forms of that number, written and spoken.

Being able to quantify and record amounts is just the beginning. We work on and with numbers. We say we operate on them, and these operations change them. As the symbols that represent the number operations of addition and subtraction are introduced, the students should ‘operate’ on items in real contexts. The language associated with addition and subtraction can be confusing. Students do not always connect addition, adding, and, plus, or subtraction, minus, takeaway, less. As many adults use this language interchangeably, students must be supported to connect these operation words with the symbol that represents them.

An expression in mathematics is a combination of symbols (e.g. 4 + 5). An equation is a statement asserting the equality of two expressions (e.g. 4 + 5 = 9). The focus in this unit of work is to have students record expressions, using symbols correctly and with confidence.

In this unit of work, subitising is given an emphasis. The early numeracy stages are defined by a student’s ability to count items, but the ability to subitise or partition an instantly recognised small group of objects into its parts is also important.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

• extending the numbers and numerals in Sessions 1 and 2 to nine
• providing additional support in session 2, activity 2 by giving students numerals to trace around as they hear them in the story.

Many of the activities in this unit suggest ways to adapt them to engage with the interests and experiences of your students. Other adaptations include:

• interchangeably using Māori words for one (tahi) to ten (tekau) throughout all activities in this unit
• replacing “Ten in the bed’ with another counting book  or story that is familiar to your students.
Required Resource Materials
Activity

This unit includes several activities for each session. Choose 2-3 activities for a single session or spread each session over more than one day.

#### Session 1

SLOs:

• To understand that an amount or number of items can be represented with a single unique symbol.
• To correctly write numerals to five.
• To understand that written words and oral words can be represented with a single symbol.
• To recognise and match words and symbols with the amounts they represent.

Activity 1

Play Number Eye Spy.
(Purpose: To identify numbers of items in groups up to five.)
Explain that in the classroom, some things we see might be in groups of two, three, four or five. Give an example, such as a group of three computers at the back of the classroom. Explain that you could give a clue about them by saying, “I spy with my little eye, three of something.”
Pose the problem for them to solve, this time changing the number in the group from three. Have students look about the classroom to identify the group of items you have seen. The person who guesses correctly has the next turn.

Activity 2

1. Play Spot the Spots, using the red and green 1 to 5 cards from Copymaster 1 pages 1-4, or the black and white 1 to 5 cards from pages 7-10.
(Purpose: To subitise different groups of dots up to five in number.)
Shuffle the cards. Hold them up one at a time, so that the students can all see them. As each card is held up, have the students show how many dots they see, by holding up the same number of fingers.
2. Shuffle the cards and distribute (at least) one to each student. Going around the mat circle, have the students take turns to hold up their card for the others to see. The other students who have a card with the same number of dots hold up their cards too.
3. Distribute numeral cards to 5, or a number fan to each student. Repeat Step 2 above, but this time the students show the numeral (card) that matches each student’s dot card as it is shown.

Activity 3

1. On the class chart revise together how to correctly write numerals 1 to 5.
Do explain that together you will be writing numerals which are symbols for numbers of things.
Emphasise the correct directionality when forming 2, 3 and 5 in particular.
2. As numerals are formed on the chart, have students practice forming them in the air, and on the mat in front of them. Have them feel and describe that correct form.
3. Have students return their dot cards from Step 2 by, one at a time, coming to the chart, showing their dot card, saying the number and forming the numeral. Watch for correct numeral formation, noting those who need further practice.

Activity 4

1. Write numerals 1 to 5 on the class chart. Ask the students what you have written (numerals). Explain that there are words for these. Show word cards, one to five, Copymaster 2.
Hold up the word cards in order and read them aloud together several times. Show them out of order, having the students quickly reading them aloud.
2. (Using Blue Tac or similar), with assistance from the students, arrange the word cards correctly beside the numerals on the chart. Together check that they match.

Activity 5

Have students work in pairs, or groups of three, to play One, Two, Three, Snap.
(Purpose: To correctly match words, numerals and images of numbers 1 to 5)
Make available to each group, two sets of word cards to five (Copymaster 2), 2 sets of numeral cards to 5 (Material Master 4-1) and one set of coloured or black dot images to 5 (Copymaster 1). Alternatively, the picture card images from Copymaster 3 can be used instead of the dots.

Each pile is shuffled and placed face down in front of the group.
Explain that students in the group each take turns to take one card from each pile, placing them face up in front of themselves as they do so. If all cards show the same amount (symbol, word and image) they say “Snap!” and keep the set.

If not, they are returned to the bottom of each pile.
The winner is the person with the most sets of three at the end of the game.

Activity 6

Conclude the session with a game of Get Together.
(Purpose: To form groups of up to five in response to hearing a number word, a written number word, or to seeing a numeral or a number word.)
Have students stand. Explain that they are to move about the room in time to music. When the music stops, the teacher will either say a number or show numeral or word cards up to five. Students are to look at the teacher, listen for a number or look for a numeral and, as quickly as possible, make a group of that number with another person, or with other people and to sit down when the group is formed.

#### Session 2

SLOs:

• To recognise and match sets of items with written and spoken words and their symbols.
• To correctly write numerals to five.
• To recognise numbers within story contexts.

Activity 1

Ensure that numerals 1 to 5 can be seen by the students. Read them together. Begin by playing Spot the Spots from Session 1, Activity 2. Instead of holding up the same number of fingers, have students write numerals in the air and on the mat with their finger.

Activity 2

1. Make paper and pencils available to the students. Explain that you are going to read them a story (Copymaster 4) and they are to listen very carefully and write down any numbers that they hear in the story. They should write these in order across the page. Alternatively, you could tell them a made up story that includes members of the class.
(Purpose: To accurately write numerals 1 to 5 in response to hearing number words within a story context.)
2. Introduce the task by reading the italicised part of the story, show the dog pictures, explain and model what to do.
4. Read the story again together, having students actively identify the number words as they are read, and writing the numerals on the class chart. Emphasise correct formation of the numerals.
5. Have students check each other’s recording.
6. Ask if students know of other stories with numbers and record their ideas. For example, The Three Little Pigs. Suggest that students listen for numbers spoken during the day.

Activity 3

1. Ensure that students can see number words and numerals.
Make a think board template (Copymaster 5) and pencil available to each student. Have the students write their favourite number (between 1 and 5) in the centre of their think board. Have them draw a picture, write their own number “story” that includes their favourite number, show any numbers by drawing dots or pictures of equipment, and write any number words.
2. Once completed, have student pairs share their thinkboards.

Activity 4

Make the cards from One, Two, Three, Snap (See Session 1, Activity 5) available to the students so they can play a Memory game as they finish their thinkboard. To play memory they should turn all the cards from three sets face down, mix them up and take turns to find matching trios.

Activity 5

Conclude the session with class sharing of thinkboards.

#### Session 3

SLOs:

• To understand and use the addition symbol.
• To recognise and use the written and spoken words for addition, with the addition symbol.

Activity 1

1. Ask the students which numerals they have been learning to write. Record these on the class chart. Have students take turns to come and write on the class chart other numerals that they know. As they do so, discuss them, highlighting their correct formation, particularly numerals 6, 7 and 9.
2. Show a selection of dot cards from 6 to 10 (Copymaster 1), asking students to say what they can see. The focus here is on seeing familiar groups of 1 to 5 dots within the larger group. Some may readily recognise immediately the larger (complete) groups, but this is not the purpose of this task.
3. Record in words what they can see. For example:
“I can see five and one.”
“I can see five (across) and two (in the corners).”
“I can see four and four.”
4. Ask, “Is there another way to write this, using numerals?”
Guide the students to shared recording on the class chart:
“I can see five and one” → 5 and 1 → 5 plus 1 → 5 + 1
“I can see five and two” → 5 and 2 → 5 plus 2 → 5 + 2
“I can see four and four” → 4 and 4 → 4 plus 4 → 4 + 4

Activity 2

Together, make a chart, or class dictionary page, about addition and its symbol, + , asking and recording what the students already know.
Included in the ideas recorded, should be:
+ is a symbol or sign, + shows that we are joining together two amounts or numbers; + is an addition symbol; when we see this sign we can read it as “and”, “plus” and “add”; + is a short way of writing “and”, “plus” and “add”.

Activity 3

1. Together write a mathematics expression about at least two other dot cards and model several ways of reading what has been written:
For example:
Write 5 + 3 and together read: ‘five plus three’, five and three’, ‘five add three’, ‘five and three more’.
2. Make paper and pencils, or, whiteboards and pens, available to the students.
Distribute to each child at least four dot cards that show numbers of dots greater than 1.
Have students write about their cards, modeling their recording on what was recorded on the class chart in Activity 1, Step 4 above.

Activity 4

Using the cards from Copymaster 6, have the students work in pairs to play Read and Draw.
(Purpose: To read a mathematical expression in at least two ways and to respond to a mathematical expression with a drawing.)
Tell the students the purpose of the Read and Draw task. Explain that they take turns to be the Reader and the Quick Draw person.
The Reader’s task is to read the mathematical expression in at least two ways to their partner. They should check their partner’s drawing before showing them the task card on which the expression is written. Their task is to check the accuracy of the drawing.
The Quick Draw person’s task is to listen carefully to the expression that is read, and to draw a diagram of what they hear, using circle dots, square boxes or triangle shapes. For example: The Reader reads, 4 + 2: “four and two, four plus two,” and the Quick Draw person draws:

Students should have at least four turns each.

Activity 5

Conclude by playing the Hands Together game.
Each Student makes a number of choice on one hand by showing that many fingers. For example:

Have them show their ‘number’ to a friend. (This is to avoid students changing the number of fingers when they see the expression.)
The teacher uses two sets of expression cards (Copymaster 6). She shows an expression, for example 5 + 3. Children who have made these numbers on their fingers must move to pair up, one student showing 5 fingers and the other showing three. The first pair to form and to show 5 + 3 collects a 5 + 3 expression card each. The game begins again. The game finishes when all cards are used up. Students take turns to read aloud to the class the cards they have ‘won’. Each student should read their cards in a different way from the student before them.
Have students bring favourite small soft toys for Session 4. Bring a small blanket.

#### Session 4

SLOs:

• To understand and use the subtraction symbol.
• To recognise and use subtraction written and spoken words, with the subtraction symbol.

Activity 1

Write this symbol on the class chart: + Have students tell you addition words and give examples of how to use them. For example, “plus”, “I have five fingers on this hand plus five fingers on this hand.” Record these.

Activity 2

1. Together, read the rhyme, “Ten in the Bed.” (Copymaster 7).
Arrange ten of the students’ soft toys in bed, using the blanket.
2. Read the rhyme a second time and have the students ‘act out’ one toy falling out each time. Have students take turns to return the toys to the bed, one at a time, and as they do so record addition expressions.
For example 1 + 1, 2 + 1, 3 + 1, reading these together. As you do so, focus on modeling the correct formation of numerals 6 to 9.
3. Write 10 and ten (symbol and word) on the class chart and discuss. Read the first verse of the rhyme once more. This time discuss how to use symbols to record what has happened. Write 10 – 1, introducing and recording this as ‘ten takeaway one.’ Continue to read the rhyme, verse by verse, writing each mathematical expression and recording the words each time. Introduce the alternative words for the subtraction symbol as you do so. For example 9 – 1, “nine minus one”; 8 – 1, “eight less one”; 7 – 1, “seven subtract one”.
4. Ask, “What was happening to the number of toys in the bed?” (The number was getting less). Discuss that subtraction symbol, brainstorm and record the student’s ideas of what it is telling us.
Read the expressions and words again and highlight the different ways we can read the subtraction symbol.

Activity 3

1. Make available to each student, paper, pencils, five plastic teddies and a tissue.
Have them put their teddies to bed under the tissue, say the rhyme to themselves and each time a teddy falls out, record the expression, for example, 5 – 1. Those who complete this quickly can write the addition expressions as they return the teddies to bed, or take more teddies and record expressions for 6 to 10.
2. Have students pair share, reading aloud their mathematical expressions. As they take turns, encourage the students to use the different language of subtraction.

Activity 4

Conclude the session with a game of Musical Chairs (or cushions).
Set out the number of chairs for students in the group. Record the number on the chart. Play a favourite piece of music. When it stops all students sit down. Have them stand and ask a student to remove one chair then come to record the expression and read aloud what they have written.
For example 10 – 1, “ten chairs minus one chair.”
Explain that one student will not have a seat this time and that this person will get to write and read the next mathematical expression on the chart.
Continue the game till no chairs remain and subtraction expressions have been written for each action.

#### Session 5

SLO:

• To recognise and write addition and subtraction expressions from story contexts.

Activity 1

1. Using class charts from sessions 3 and 4, review the symbols for addition and subtraction.
2. Either read the short scenarios from Copymaster 8, exchanging the names of students in the class for those in the scenarios, or create your own. Have the students identify if the story tells of an addition or subtraction ‘event’ and together record these on the class chart as mathematical expressions.

Activity 2

Make available to the students pencils, paper, felt pens or crayons. Have the students write at least two of their own scenarios and record the mathematical expressions that represent what is happening. Their illustrations should show what is happening in the scenario and expression.

Activity 3

Conclude the session by having the students pair share, then class share their work. Emphasise the importance of having them read their mathematical expressions in at least two ways.

## Worms and more

Purpose

This unit comprises 5 stations, which involve the students in developing an awareness of the attributes of length and area. The focus is on the development of appropriate measuring language for length and area.

Achievement Objectives
GM1-1: Order and compare objects or events by length, area, volume and capacity, weight (mass), turn (angle), temperature, and time by direct comparison and/or counting whole numbers of units.
Specific Learning Outcomes
• Compare lengths from the same starting point.
• Use materials to make a long or short construction.
• Use materials to compare large and small areas.
Description of Mathematics

Early length experiences must develop an awareness of what length is, and develop a vocabulary for discussing length. Young students usually begin by describing the size of objects as big and small. They gradually learn to discriminate in what way an object is big or small and use more specific terms. The use of words such as long, short, wide, close, near, far, deep, shallow, high, low and close, focus attention on the attribute of length. Early area experiences develop an awareness of what area is, and of the range of words that can be used to discuss it. Awareness of area as the "amount of surface" can be developed by "covering" activities such as wrapping parcels, colouring in, and covering tables with paper. The use of words such as greater, larger and smaller, focus attention on the attribute of area.

The stations may be taken as whole class activities or they may be set up as "centres" for the students to use. We expect that many students will already be aware of the attributes of length and area. For them these may be useful maintenance activities.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

• including images on the word cards used in station 2 to support beginning readers
• using “wrapping” paper with more or less shapes depending on the facility of the students in colouring-in shapes.

The contexts for activities can be adapted to suit the interests and experiences of your students. For example:

• Replace the town scenario in station 2 with, for example, a marae or farm setting.
• Replace the heart shapes in station 4 with other symbols or objects that engage your students, for example, koru, dinosaurs or ladybugs.
• Replace the bear footprint with a moa foot print.
Required Resource Materials
• Station 1: Play dough (or plasticene), picture of worm to interest the students.
• Station 2: Blocks, blank cards, toy cars.
• Station 4: Heart wrapping paper, crayons.
• Station 5: Footprints (Big Foot, Small Foot), large sheets of paper, crayons.
Activity

#### Station 1: Worms

In this activity we roll play dough to make long and short worms.

1. Give each student a ball of play dough and ask them to make a worm.
2. Get the students to bring their worms to the mat.
3. Ask the students to describe their worm.
My worm is long
My worm is fat
My worm is tiny
4. Now ask the students to make a worm that is short.
5. Look at and discuss the short worms.
How do you know that your worm is short?
Is your worm the same as everyone elses? Why?
6. Ask the students to think of other worms that could be made. (long, wiggly, thin)
7. Choose a word and ask everyone to make a worm one that fits the description.
8. Repeat with other descriptive words and create a word bank that is displayed with illustrations.
9. Conclude by asking the students to draw their favourite worm and write a descriptive word for it.

#### Station Two: Near and Far

In this activity we build a town with blocks and then "drive" our cars around it.

1. Begin by discussing the buildings that you might find in a town. Write the ideas on blank cards. (One for each student or pair of students).
2. Give a card to each student, or pair of students and ask them to build the building out of blocks.
3. Put the buildings together onto the mat (or large sheet of paper with roads drawn) and the building cards into a container.
4. Give two students a car to "drive" around the town.
5. Tell the students to stop their cars after a short time.
6. Draw a card from the container. Ask the students to identify which car is closest to the building drawn.
7. Give the cars to another two drivers and repeat.
8. Use different words, for example: furthest, nearest, far away, nearby.
9. Repeat with other descriptive words and add to the word bank that is displayed with illustrations.

#### Station Three: Shaping Ourselves

In this activity we make ourselves tall, short, wide, narrow, close and far.

1. Tell the students that we are going to play a game of sizes.
2. Ask the students to make themselves:

As tall as they can
As short as they can
As wide as they can
As close to a table as they can be
As far from the door as they can be
Take up loads of space covering the mat (lie down)

3. You could extend the activity by asking for volunteers to give instructions for "body sizes".
4. You could also link to geometry by asking the students to form different shapes with their bodies, for example, circles, triangles. The students could describe the size of their shape.
5. Repeat with other descriptive words and add to the word bank that is displayed with illustrations.

#### Station Four: Wrapping Paper

In this activity we follow directions and colour-in large and small objects on our "wrapping" paper (Copymaster 1). We then find an object to wrap in our paper.

1. Give each student a sheet of hearts wrapping paper.
2. Look at and discuss the hearts on the paper.
3. Ask the students to colour-in the small hearts.
How did you decide which were small?
4. Now ask them to colour in the large hearts.
How did you decide which were large?
Which took the longest to colour? Why?
Which were the quickest to colour? Why?
5. Ask the students to find something to wrap in their paper.
6. Bring the objects and wrapping paper to the mat.
7. Check to see if the objects will fit in the paper.
Whose didn’t?
Why not?

#### Station Five: Muddy footprints

In this activity we look at some footprints and decide who they could belong to. In our discussion we focus on the use of language associated with area.

1. Show the students the "bear" footprint from Copymaster 2.
Who could this belong to?
Why do you think that?
Is your foot as big as the bear's? How do you know? How could you check?
2. Let the students place their feet on top of the bear print.
3. Ask students to create different "prints, for example: a mouse, a dog, an albatross, a gecko, or find images of different prints online. Support the use of compartitive words in their descriptions.
4. Ask the students to draw a giant’s footprint.
5. Share and discuss giant footprints in comparison to their own.
6. Record the words used to compare the prints, collect the descriptive words and add to the word bank that is displayed with illustrations.
Attachments

## Turns

Purpose

In this unit we look at the beginning of the concept of angle. As students come to understand quarter and half turns, they also begin to see that ‘angle’ is something involving ‘an amount of turn’. These ideas are explored by using students’ bodies, toys, games and art.

Achievement Objectives
GM1-1: Order and compare objects or events by length, area, volume and capacity, weight (mass), turn (angle), temperature, and time by direct comparison and/or counting whole numbers of units.
GM1-3: Give and follow instructions for movement that involve distances, directions, and half or quarter turns.
Specific Learning Outcomes
• Show a quarter turn and a half turn in a number of situations.
• See that two quarter turns equal one half turn.
• Recognise the ‘corner’ of a shape that is equivalent to a quarter turn.
Description of Mathematics

Angle can be seen as and thought of in at least three ways. These are as:

• an amount of turning
• the spread between two rays
• the corner of a 2-dimensional figure

Angle in the New Zealand Curriculum develops over the following progression:

Level 1: quarter and half turns as angles
Level 2: quarter and half turns in either a clockwise or anti-clockwise direction; angle as an amount of turning
Level 3: sharp (acute) angles and blunt (obtuse) angles; right angles; degrees applied to simple angles – 90°, 180°, 360°, 45°, 30°, 60°
Level 4: degrees applied to all acute angles; degrees applied to all angles; angles applied in simple practical situations
Level 5: angles applied in more complex practical situations

Outside school and university, angle is something that is used regularly by surveyors and engineers both as an immediate practical tool and as a means to solve mathematics that arises from practical situations.  So angle is important in many applications in the ‘real’ world as well as an ‘abstract’ tool.  This all means that angles have a fundamental role to play in mathematics and its application.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

• In session 1, providing students more opportunities to explore the concept of turning using themselves or objects instead of recording turns on paper.
• In session 2, having students work in pairs, with one student holding the end of the string still while the other draws the arc.
• In session 5, modifying the complexity of the course students are asked to follow.

The contexts for this unit can be adapted to suit the interests and experiences of your students. For example, the contexts for identifying shapes with quarter turns could be a local playground or building of particular interest. This could involve a trip to visit it, or photos could be used.

Required Resource Materials
• Various toys that are available in the classroom
• Crayons in a variety of colours
• String
• Drawing pins
• Paint
• Cardboard
• Scissors
Activity

#### Session 1

1. Talk with the class about ‘turning’.  This can be motivated by asking them directions from their classroom to somewhere else in the school.  Emphasise ‘turning’ by asking them what they do when they get to a corner. Ask them what they have to do if they want to go left or right.  (They make a turn.) Start recording some of the vocabulary being generated by the discussion related to turns: corner, turn, spin, circle, left, right, around etc.
2. Demonstrate full, half and quarter turns. Use a big circular piece of paper or fabric, or alternatively a chalk circle drawn on carpet or concrete.  Have a student come to the centre of the circle and put their arm straight out in front. Get someone else to place a marker on the edge of the circle showing where the person is facing and their arm is pointing. Demonstrate the full turn as the person slowly turns all the way around and ends up back at their beginning point. Have everyone trace the full turn on the ground with their finger.  Choose another person to come to the centre of the circle, face the same starting point and demonstrate a half turn. How far will they need to go? Where should they stop? Stress the idea of ending up facing the opposite direction. Have someone mark where on the edge of the circle the half turn stops and the person ends up pointing. Get them to do another half turn. Where do they end up? So 2 half turns make 1 full turn? Have everyone trace the half turn on the ground with their finger. Repeat for quarter turn if the group is ready otherwise wait until they have had some practice doing full and half turns. For each demonstration, document where the pointing arm ends up, which way the person is now facing and what part of the circle the person has covered. This can also be recorded on circles on the whiteboard or modelling book.
3. Repeat the demonstration with a toy. Using a toy animal, for instance, a student could show how to move the animal through a full, half and quarter turn.
4. Give the students time to go and draw several examples of turns.  This may be done by using animal pictures, car pictures, or any other object.  Emphasise that their drawings are not to be done in any great detail.  It’s the idea of a turn that is important.
5. As you go around the class observing their drawings, check that they have the right concept and correct any misconceptions.
6. Create stories involving turns such as: forgetting something on the way to school when you would have had to turn round and go back.  This means you would have had to do a half turn. Model the turn with your toy car or stick figure on the paper.

In the three sessions that follow,  students produce artwork that they can collect in their own ‘Turns Book'.

#### Session 2

Prior to this session tie pieces of string to enough crayons for each child to have one, with a few spares to avoid arguments about colours.

1. Provide each student with a crayon with string attached. Show them how to fix one end of the string by using a drawing pin or the finger of one hand. Then show how they can make a quarter turn arc by sweeping the crayon through a quarter turn. You will need to draw their attention to the importance of keeping the end of the string still and maintaining tension on the string.
2. Ask students to make several ‘quarter turns’ in the same colour.  Check that their turns are approximately correct.
3. Having done quarter turns , students choose a new colour and create half turn arcs. Draw their attention to the relationship between quarter and half turns.
4. Have each child choose a third colour and create some full turns. Draw their attention to the relationship between full, quarter and half turns.

#### Session 3

This session is similar to that of the last session except that here the quarter turns are made using ‘combs’ the students make for themselves. As an alternative, you may prefer to have students use crayons lying flat to create the same effect.

1. To produce combs, give the students cardboard rectangles and get them to cut out ‘teeth’ to make ‘comb’ shapes similar to the diagram below.

2. By holding one end fixed, students should be able to rotate their ‘combs’ through quarter and half turns after dipping their combs in different coloured paint.
3. Give students the opportunity to make patterns with their ‘combs’ based on quarter and half turns.
4. Students could be encouraged to produce several pages of patterns. Let them choose the one that they like best for their Turns Book.
5. While they are involved in this activity check that their ‘comb’ shapes do represent quarter and half turns. There is no need to measure their work precisely but their turns should be close to the right magnitude.

#### Session 4

Corners of shapes can also be thought of as quarter and half turns. The object of this session is to find corners of shapes that are equivalent to quarter and half turns.

1. Draw a rectangle in the playground (or use a small rectangle in class). Have four students stand on the corners of the rectangle (or put four toys on the small rectangle).
2. Have one student face another one. What turn would Mike need to make in order to be looking at Jorge?
Repeat for other examples.
3. Point out that we can think of the corners of a rectangle as being made up of quarter turns. What other shapes can you think of that have corners that are quarter turns?
4. Explore right-angled and other triangles as a class.
Does this triangle (right angled) have any quarter turns? (yes)
Are all the corners quarter turns? (no)
Do all triangles have quarter turns (no, provide examples that don't)
5. Now look at shapes in the classroom that do and don't have quarter turn corners. Get them to make a class list.
6. Get them to draw two objects from the classroom (that may or may not be on the class list) that have quarter turn corners and two that don’t to add to their Turns Book.

#### Session 5

1. Have students work in pairs to guide each other around a course using instructions involving quarter turns to the left and to the right. The course could be outside, possibly following a line drawn on a netball court, or they could be in the classroom, moving around the furniture.
2. Bring the class together to talk about full, quarter and half turns. Use questions such as
How would you describe a quarter turn? A half turn?
What objects do you know that have quarter turns?
How many quarter turns make a half turn? How many half turns make a full turn?
3. Finish with a game of ‘Simon Says’ using quarter and half turns.

## Learning to count: Counting one-to-one

Purpose

This unit develops students’ understanding of, and proficiency in, counting one-to-one.

Achievement Objectives
NA1-1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
NA1-2: Know the forward and backward counting sequences of whole numbers to 100.
NA1-5: Generalise that the next counting number gives the result of adding one object to a set and that counting the number of objects in a set tells how many.
Specific Learning Outcomes
• Understand that the number of objects in a set stays the same as changes are made to spatial layout, size, or colour.
• Understand that the count of a collection of objects can be trusted and worked from as objects are added or taken away, or the set is rearranged into parts.
Description of Mathematics

Gelman and Gallistel (1978) provided five principles that young students need to generalise when learning to count. These principles are:

1. The one-to-one principle
Just like in reading when one spoken word is matched to one written word, counting involves one-to-one correspondence. One item in a collection is matched to one spoken or written word in the whole number counting sequence.
2. The stable order principle
The spoken and written names that are said and read have a fixed order. If that order is altered, e.g. “One, two, four, five,…”, the count will not work.
3. The cardinal principle
Assuming the one-to-one and stable order principles are applied then the last number in a count tells how many items are in the whole collection.

The first three principles are about how to count. The final two principles are about what can be counted.

1. The abstraction principle
Items to count can be tangible, like physical objects or pictures, or they can be imaginary, like words, sounds, or ideas, e.g. Five types of animal.
2. The order irrelevance principle
The order in which the items are counted does not alter the cardinality of the collection. This is particularly challenging for students who think that counting is about assigning number names to the items, e.g. “This counter is number three.”

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

• Provide extended opportunities to practise counting and working with the equipment.
• Support students to use counting strategies to confirm any changes made to sets, if needed (sessions 4 and 5).

The contexts for activities can be adapted to suit the interests and experiences of your students. For example, use Māori counting numbers.

Required Resource Materials
Activity

Students need to simultaneously develop proficiency with number sequences forward and backwards by one, and their capacity to apply those sequences to counting tasks. Ideally students’ ability to say word sequences develops either ahead of, or synchronous to, their need to apply it. So students who can count collections up to ten should be learning number sequences beyond ten.

#### Session 1: Hundreds board

Count by ones on a hundreds board, both forwards and backwards, and ask questions which help students connect numerals to spoken words. For example,
Can you find the number 8 without counting up or down?
What number comes after eight?
What number comes before eight?

Variations

Have students work in pairs. They take turns drawing a numeral card out of a container, then work together to find the number on a hundreds board and make a set which contains the number they have drawn, for example a set of eight counters.

First work with numbers 1–10, and extend the range as needed to increase the challenge.

#### Session 2: Slavonic abacus

Count by ones on a Slavonic abacus, both forwards and backwards. While you count, say the numbers out loud with students as you move the beads. This supports them to connect the spoken word and the quantity.
Can you find the bead for number 7?
What number comes after seven?
What number comes before seven?
Note that when practising the backward number sequence it is the amount that remains, not the bead removed, that is counted. Zero is an important number to say at the end, as the expression of the absence of quantity (no beads).

Variations
Have students work in pairs. They take turns to draw a numeral card out of a container, and challenge their partner to find the number on the Slavonic abacus. If needed, they can count the beads together to check that the right bead has been identified.

First work with numbers 1–10, and extend the range as needed to increase the challenge.

#### Session 3: Frogs in a bucket

1. Link the number after and before a given number, to adding one and subtracting one from a given collection. Use toy animals or other objects and a plastic container so there is a loud ‘plunk’ as objects go into the container. The frog animation gives an example of this with frogs in a bucket.
We had 15 frogs and one jumped out.
How many frogs are in the bucket now?
2. Increase the number of objects in the container beyond ten so that students attend to the one more/one less principle rather than image actions inside the container. Avoid putting in the objects one at a time after ten. Throw them in as imaginary groups. For example, a nice sequence is…
17 frogs and one more… 29 frogs and one more… 99 frogs and one more
7 frogs and one less… 15 frogs and one less… 27 frogs and one less…

Variations

If needed, link explicitly to the number sequence by referring to a hundreds board.

Extend the problems to two more/less, three more/less and beyond as students’ understanding and control of sequences grows.

Have students work in pairs, with one student putting objects into a container, and the other student listening and counting how many.

#### Session 4: What’s changed?

1. Begin with a set of objects in a readily subitised arrangement. For example, begin with a set of six cars.
2. Tell the students: Take a photograph of the collection and stash it in your mind.
Now close your eyes, and I’m going to change something.
Try to hold the picture in your mind.
3. Change the objects for a different kind – vary shape, colour and size to add more challenge. For example, change the cars for teddies.

4. Now open your eyes. How many things can you see now?
Look for students to recognise that the number of items is unchanged. though the items vary from the original ones. Aim for acceptance that the number is unchanged by means other than one-by-one counting, though use counting as a ‘fall back’ strategy if needed.

Variations
Students may begin to say the number is invariant because it has been the last few times, without necessarily accepting that number is conserved. Introduce the one more/one less principle with this activity. Vary the number of new objects by one while still retaining the original layout and see if the students detect the change and trust invariance enough to build on it.

Make the variations more significant such as adding or subtracting more than one and changing some parts of the arrangement.

Have the students work on the activity Odd one out in pairs, using Copymaster One.

Students work with sets of three cards, and identify one difference between each card and the other two cards in the set. For example, possible answers for the cards with teddies are: the top card is different from the other two because the teddies are sitting down; the middle card is different from the other two because there are different amounts of teddies in each row, and the bottom card is different from the other two because there are 9 teddies.

#### Session 5: Is there the same number?

1. Lay out two collections one above the other, with items aligned.
Have a look at this collection. Are there more blue teddies, more green teddies or are there the same number? Most young children will equate the length of a collection with number.
2. Tell the students: Take a photograph of this collection and stash it in your mind.
Now close your eyes, and I’m going to change something.
3. Change the spatial layout of the objects, by spreading out one of the collections.
4. Now open your eyes. Are there more blue teddies, more green teddies or are there the same number?

Variations

Vary the spatial arrangement of sets. Begin with a set of objects with some similar characteristics so that students can easily recall the objects that are present. For example, Take a photograph in your mind of the collection. Close your eyes and tell me what you see. Encourage students to pay attention to structure and the organisation of the set, with responses such as “There were two of each animal”, “The animals were on a hexagon”, and “Two on the top, two in the middle, two on the bottom.”

Now I’m going to change something. Vary the spatial layout of the objects, particularly expanding or contracting the length or area of the collection, or rearranging the layout.

Now open your eyes. Are there are still the same number of objects? How do you know?

Make minor variations to the total number as well as the spatial layout so that students are required to trust the invariance of the count and build with it or take from it.

Have students work on the activity Domino Count in pairs. This task encourages students to trust the count in the face of different spatial arrangements.

Working with at least one set of dominoes, students sort the dominoes into groups that have the same number of dots, in total. Each group can be labelled with a numeral card, or students could make their own labels. For example:

Once the dominoes have all been grouped, students can arrange the dominoes in a pattern and write the number fact for each domino. For example,

#### Session 6: Counting dots

The abstraction principle involves the idea that non-tangible items such as sounds, touches, and ideas can be counted. Developing students’ capacity to count items they cannot see and feel is important because it helps them to realise that numbers are ideas.

Messages

Play the game Messages as a class. Students sit in pairs, back to back, with one student from each pair facing the teacher. The teacher shows a pattern card from Copymaster Two for a short time. The students turn around and gently tap the number of dots they saw on their partner’s back. The partner shows how many taps they felt by holding up that many fingers. Both players turn around to check whether the number of fingers matches the number of dots on the pattern card. Alternatively, a digit card can be held up instead of a pattern card.

Pattern memory match

Students play pattern match in groups of three or four. Each group needs a set of pattern cards (Copymaster Two) and a set of digit cards.

The pattern cards are spread out individually, face down on. The digit cards are shuffled and put in a pack, face down, in the centre. Players take turns to turn over the top digit card then turn over one of the pattern cards. If the cards match the player keeps both cards and has another turn. Once all of the cards are matched the player with the most pairs wins.

After the game, talk with students about how they recognised some patterns instantly. Look for students to use combinations of smaller groupings, e.g. “I know it’s seven because four and three are seven.”