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

#### Session 1

This session is about exploring features of mosaic shapes and making a successful poster highlighting the special characteristics of one particular shape.

SLOs:

- Sort mosaic blocks by shape (and colour).
- Identify and describe the distinguishing features of mosaic block shapes, using the language of sides and corners.
- Form and express mathematics ideas in poster form, considering audience impact.

__Activity 1__

Begin by reading *The Greedy Triangle*.

__Activity 2__

Write ‘*Clever shapes*’ on the class chart. Explain that the students will be making their own small poster about a *clever* shape. Ask what the purpose of a poster is. (To capture people’s attention and to give a short clear message).

Together list the features of a good poster. It grabs the audience’s attention by using:

bold print, a simple and convincing message, interesting colours, a picture or diagram.

__Activity 3__

Make available mosaic pattern blocks, (*omit hexagons*), paper, pencils, crayons or felt pens.

- Have students in pairs take a selection of shapes, sort them into groups and explain these using the language of colour or shape.

- Have each student select one of the shape groups, working with these to come up with a reason for their ‘cleverness’. (For example: ‘they fit together with no gaps’.)

- Record on the class chart the students’ ideas as they each first describe the features of the shape they have chosen, giving the number of sides and corners, and share their creative reasons with the class: for example, ‘circles are clever because they are wheels’, ‘squares are clever because they fit together with no gaps and can be used as tiles’, ‘triangles are clever because they can stand on their heads and fit together’, etc.

**Use this time to model and record the language of the shapes**, including writing and discussing ‘tessellation’.

__Activity 4__

Explain that to make their posters, the students will need to draw around their ‘clever shape’ (as many times as necessary).

Set a time limit and have students complete their ‘clever shape’ posters.

__Activity 5__

Conclude the session by having students share their work in pairs. Display the list of criteria for a successful poster and have students self evaluate, then give partner feedback about each of the criteria.

__Activity 6__

Share clever shape posters as a class, highlighting geometric language.

#### Session 2

This session is about understanding that a 2D shape is like a footprint, a 3D shape is something you can hold or feel, and that we use different language for each.

SLOs:

- Understand the difference between 2D and 3D shapes.
- Identify distinguishing features of 2D shapes using the language of sides and corners.
- Identify distinguishing features of 3D shapes using the language of faces, edges, vertex/vertices.
- Explore hexagons, recognising that they tessellate.

__Activity 1__

- Display the posters from session one. Ask several students to describe the process of drawing around the shape and have them model this on the class chart.

- Explain that what they have drawn is like a ‘footprint’ of the shape. Write the words ‘side’ and ‘corner’ beside these features on each shape outline.

Ask, ‘Can you hold or feel an outline (‘footprint’) with your hand?’ (No. It has only 2 dimensions.). Discuss, highlighting the fact that we can say how long the outline is and how wide it is.

Record ‘width’ ‘length’ ‘dimensions’, explaining why these outlines are called two-dimensional shapes and that this is sometimes referred to as 2D.

Ask, ‘What ‘dimension’ can’t we measure?’ (How deep it is.)

- Have students write 2D beside their outlines on the class chart, explaining what this means as they do so.

__Activity 2__

Make available hexagonal mosaic blocks, pencils and paper.

Have students each draw around one block creating an outline, identify and record ‘6 sides’ and ‘6 corners’ and write a statement about the outline. For example: *“This is a 2 dimensional or 2D shape because we can only say how long it is and how wide it is.”*

__Activity 3__

- Have students discuss in pairs and decide whether the foam mosaic shape itself is a two-dimensional or three-dimensional shape.

- Have them physically take up positions in the classroom to indicate their thinking (for example: 2D on one side of the mat, 3D on the other).

Discuss, conclude and record that the mosaic block is a 3D shape because it has width, length, and thickness (depth) and we can hold it in our hand.

- Have several students draw the foam mosaic hexagon shape (hexagonal prism) on the class chart, capturing the third dimension (thickness) in their own way. Have all students complete this on their own paper.

- Write
**face, edge and vertex** on the class chart. Have students locate and identify each feature on their drawing. Write the plurals of each work beside the singular, highlighting the word *vertices*. Make the connection between the 2D language of sides and corners and the 3D terms.

Have students label their drawings using these words. Have all student touch and name those parts on their hexagonal mosaic block.

__Activity 4__

Return to the language listed in Session 1. Highlight the word **tessellate**.

Pose the task: Use your shape, a pencil and paper and show how you know whether or not a hexagon tessellates.

__Activity 5__

- Have students share their results and talk about why hexagons tessellate.

As they describe their drawings they should use the language of *side and corner*.

- Have students form groups of four, tessellating their hexagonal mosaic blocks together. Have students locate and identify faces, edges and vertices on the mosaic block, and explain to each other using this language, exactly how the tessellation is formed with 3D objects. (For example, ‘the small rectangular faces around the ‘edge’ of each block are up against each other’, ‘the edges and vertices touch’, etc.)

__Activity 6__

Conclude the session by having students each make a small poster about 2D and 3D shapes that they know, thinking about the feedback they received about their posters in Session 1.

#### Session 3

This session is about consolidating understanding of, and using language associated with, 2D and 3D shapes, understanding and making a hollow prism to create a model bees’ honeycomb.

SLOs:

- Recognise that hexagonal prisms make up bees’ honeycomb.
- Understand that the two-dimensional hexagon shape is a plane or face of the three-dimensional prism.
- Make hollow prism shapes and describe their features.

__Activity 1__

- Begin by displaying a tessellation drawing from Session 2, Activity 4.

Have students explain why the hexagons in the drawing are two-dimensional.

Ask , “What does the drawing remind you of?”

Elicit, ‘a bees’ honeycomb’.

- Display a picture of a bees honeycomb.

Have students discuss and agree whether the honeycomb is two-dimensional or three-dimensional. Have them explain their thinking.

- Write
**‘hexagonal prism’** on the class chart and show the students a model.

Have several students hold it and describe its features. Record these.

Highlight the plane (two-dimensional) shapes, the hexagon and rectangle that make up the three-dimensional prism.

- Make an inkpad and paper available to the class. Have several students make ink prints of a hexagonal face and of a rectangular face. Recognise that these prints are two-dimensional.

- Refer to the chart made in Session 2, Activity 5. Once again have students explain the connection between the 2D language of sides and corners and the 3D terms, in so doing highlighting the fact that plane shapes (2D) build (or make up) 3D shapes.

__Activity 2__

- Make available rulers, rectangular pieces of card 24cm x 15cm and cellotape.

Pose: Can you work in pairs, using this card and tape, to make a hollow hexagonal prism?

Give students time to explore and create their hexagonal tubes.

- Have students suggest how they could make a honeycomb model using their hollow prisms. Fit these together, gluing faces together, creating a model honeycomb.

Have students talk about the process using the language of faces, edges and vertices.

- Notice and discuss how important it is to be precise in the measurements and their folding, ensuring that their hexagons are
*regular*, not *irregular*. Define these.

Recognise the bee’s skill in making perfect hexagonal prisms.

If necessary, have students make their prisms with greater precision to achieve a ‘perfect’ honeycomb such as the bee produces.

__Activity 3__

- Conclude the session by listing student’s questions about bees (including why they use the hexagonal shape for their honeycombs). Have student suggest possible answers and record these.

- Suggest they could research this, perhaps with the help of parents or whānau, before the next session.

They might also like to make their own model bees to inhabit the class honeycomb.

#### Session 4

This session is about giving students opportunities to find and share information about honeybees, their honeycombs and the importance of the hexagon.

SLOs:

- Research and present information about honeybees.
- Recognise that living things have certain requirements to stay alive.
- Recognise that all living things are suited to their particular habitat.
- Recognise that bees always use hexagons because are ‘perfect in saving on labour (effort and energy) and wax’.

__Activity 1__

- Have wax or a piece of real honeycomb available. Display the class honeycomb.

Have students share the results of their research and, if any students have made their own model bees, to locate these in the honeycomb.

- Together research, read, discuss and list a summary of information about honeybees and their honeycombs. Consider online sources:

http://www.npr.org/blogs/krulwich/2013/05/13/183704091/what-is-it-about-bees-and-hexagons

http://www.nature.com/news/how-honeycombs-can-build-themselves-1.13398

__Activity 2__

Write on the class chart:

*What do honeybees need to stay alive?*

How does the honeycomb ‘suit’ the bees?

Discuss each, with reference to the research information, and record the students’ understanding of the key ideas.

Recognise that:

- It takes a bee lots of energy to make wax.
- The hexagonal structure is more compact (efficiently and tightly arranged) than any other shape (such as equilateral triangles and squares).
- The hexagonal form of the honeycomb has been shown by scientists and mathematicians to be ‘perfect in saving on energy and wax’, suiting the bees and helping them to stay alive, as their honey (food) is stored.
- It is suggested that the hexagonal cells in the honeycomb begin as circles. The angles of the hexagons a pulled into shape at the point where three cells meet.

__Activity 3__

Conclude the session by reading *A cloak for the dreamer*:

Highlight the way in which the circle shapes which did not tessellate were changed into the hexagon shapes which do tessellate and the way this is like the process in the honeycomb where the cells are thought to begin as circles.

#### Session 5

This session is about synthesising the skills and learning in Sessions 1-4.

SLOs:

- Recognise the ‘cleverness’ of the honeybee.
- Recognise that there are many more ‘clever’ shapes occurring in nature.

__Activity 1__

Review key learning over the past 4 sessions including referring to the clever shapes posters and feedback in Session 1.

__Activity 2__

Set an appropriate time limit and have students work in pairs to design and create a ‘presentation’ (poster, powerpoint, other) combining the **key ideas** about a **clever shape** and a **clever creature** (the honeybee).

__Activity 3__

Challenge students to research other clever shapes (and creatures) found in nature.

## Mathematical magic

This unit provides you with a range of opportunities to assess the entry level of achievement of your students.

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

Some of the activities in this unit can be adapted to use contexts and materials that are familiar to students. For example:

## Prior Experience

It is expected that students will present a range of prior experience of working with numbers, geometric shapes, measurement, and data. Students are expected to be able to use simple addition and subtraction in situations where sets are joined, separated, and compared.

## Session One

Talk to your students about the purpose of the unit which is to find out some information about them, so you can help them with their mathematics. In the first session students explore a ‘magic trick’ about dice and extend what they find to develop their own magic trick. Look for your students to generalise, that is, explain why the method works every time.

Dice FacesCard Sums(Acknowledgement to Jill Brown, ACU Melbourne, for the idea)In this example below only 5 is odd so the total is 20 + 1 = 21.

Note that students may find variations on a general rule:

The highest sum possible is 25, if all the odd numbers show up. Each time an odd card is turned over one is lost from the sum. The even number is always one less than the odd number. If you see how many even numbers there are you take that number from 25. For example, below there are three even cards, 0, 2, and 8, so the sum is 25 – 3 = 22.

Possible extension:

Suppose you wanted to make the trick look even more impressive by making 10 cards; 0-1, 2-3, 4-5, …,16-17, 18-19.

How could you work out the total without adding all the numbers then?## Session Two

In this session the theme of mathematical magic is continued as students look for patterns in the place value structure of 100. Begin with a Slavonic Abacus and a Hundreds Board.

Slavonic abacusDoes the student use the tens and ones structure of the abacus or attempt to count in ones?

How many beads are on the whole abacus? How do you know?If 45 beads are on the left side, how many beads are on the right side?You might revisit the fact that five tens are fifty. Fifty mean five-ty or five tens.

Do you know the nursery rhyme about Bo Peep?Read or play a YouTube clip of the Rhyme to your students if needed.

(Student A), please move some of my sheep to the left side and cover up the rest so I cannot see them (using an A4 sheet of paper).Can you work out how Bo Peep did it. How could she know 72 sheep were missing so quickly?Crosses PatternIn this task students apply place value to explain why a pattern on the hundreds board works every time.

2 + 22 = 24 (top and bottom numbers) and 11 + 13 = 24 (left and right numbers).

Possible extension:

Any square on the hundreds board is magic. The numbers along each axis have the same sum. Why?

Below 12 + 23 + 34 = 69, 13 + 23 + 33 = 69, 14 + 23 + 32 = 69, 22 + 23 + 24 = 69.

Hopefully more able students can see that this works for the same reason that the crosses pattern works. The for each line, the sum is three times the centre number, with one of the end numbers the same amount more than that number as the the other is less.

Other units that will support the development of Place Value at level 2 include:

## Session Three

In this session students explore simple two-dimensional dissections in which a shape is cut up into smaller pieces and those pieces are put together to form a different shape. They will need square shaped pieces of paper or card.

Magicians can change objects in to different shapes. We are going to see if you can be a magician.There are four challenges for you to start. You need to use all the pieces of the square and change it into each of these shapes.Copymaster 1 contains the target shapes. Either provide students copies of the Copymaster or display it on screen. Ask the students to work out how to form each shape using all the pieces from the square. Look for your students to:

Copymaster 2 has some target shapes that can be made by connecting all four pieces. Challenge your students to make each target, record the solution, and make up their own target for someone else in the class. Be aware of the need to check for uniqueness. Is the target really the same as another? This brings in ideas about reflection and rotation.

Can you use these four pieces to create a hexagon?## Session Four

In this session students consider the likelihood of magic happening. Students will consider whether the trick is really magic or if something else is involved.

I am closing my eyes now and you can move the cups around while I cannot see. Then I will guess which cup the treat is under.Am I magic or is something else going on?Their beliefs might be deterministic: "You are a teacher, so you are clever."

Some beliefs might acknowledge chance: "There are only two cups. You have a good chance of being right each time."

A student might spot that the cups are marked. If not reveal the trick to them.

What would happen if the cups were not marked?Am I magical or is it just luck?Magicians like to disguise their tricks so the two cups might be a bit simple. Let’s try the same idea but have three cups and one treat. Can you figure out a way to get a treat each time?Being magical in this situation seems bit hard. Let’s keep the three cups but add another treat.S: I always choose the position where a treat didn’t go at first.

T: Did that always work? Did anyone else try that idea? What happened? Why do you think that happened?

Other students might always choose a position where a treat was first located, or randomly select a position.

Extend the activity:

You might extend the task by varying the number of cups and treats, e.g. four cups and one, two, or three treats.

## Session Five

In this session students look for repeating patterns and connect elements in the pattern with ordinal numbers.

Mathematical magicians can think ahead. They can predict the future. Can you?Provide students with a range of materials to form sequential patterns with. The items might include bottletops, corks, blocks, toy plastic animals, pens and pencils, geometric shapes, etc. Give them a copy of the strip that can be made from Copymaster 3 after it is enlarged onto A3 size (x 1.41).

Let students create their own patterns. Look for students to:

^{th}object.Take digital photographs of the patterns to create a book, and ask students to pose problems about their patterns. Students can record the answers to their problem on the back of the page.

Discuss as a class how to predict further members of a pattern. Strategies might include:

Extend the activity:

There are many ways to increase the difficulty of repeating pattern prediction:

Dear parents and caregivers,

For the first week of school our mathematics unit is about mathematical magic. We will investigate number tricks, magical change a square into other shapes, predict the future of a pattern, explain and justify why things work.

## Building on two-digit place value

This unit supports students learning to understand the structure of two-digit numbers and how to operate with them.

Session OneSession TwoSession ThreeSession FourSession FiveOur number system is very sophisticated though it may not look it. While numbers are all around us in the environment the meaning of digits in those numbers and the quantities they represent are challenging to understand. Our number system is based on groupings of ten. Ten ones form one ten, ten tens form one hundred, ten hundreds form one thousand. And so the system continues to represent very large numbers.

To represent all the numbers we could ever want we use just ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The word for digits also comes from our fingers. We don’t need a new number to represent ten because we think of it as one hand, one group of ten. Similarly when we add one to 99 we write 100 and do not need a separate symbol for one hundred. The position of the 1 in 100 tells us about the value it represents. Zero has two uses in the number system, as the number for ‘none of something’, e.g. 6 + 0 = 6, and as a place holder, e.g. 704. Place holder means it occupies a place or places so the reader knows the values represented by of the other digits. In 500 zero is acting as a place holder in the tens and ones places.

Place value means that both the position of a digit as well as the value of that digit indicate what quantity it represents. In the number 273 the position of the 2 is in the hundreds column which means that it represents two hundred. Seven is in the tens column which means that it represents 7 units of ten, 70.

Renaming a number flexibly is important. In particular it is vital that students understand that when ten ones are created they form a unit of ten, and when ten tens are created they form a set of one hundred. For example, the answer to 210 + 390 is 6 hundreds since one ten and nine tens combine to form another hundred. Similarly when a unit of one hundred is ‘decomposed’ into tens the number looks different but still represents the same quantity. For example, 420 can be viewed as 4 hundreds and 2 tens, or 3 hundreds and 12 tens, or 2 hundreds and 22 tens, etc. Decomposing is used in subtraction problems such as 720 – 480 = □ where it is helpful to view 720 as 6 hundreds and 12 tens.

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

Some of the activities in this unit can be adapted to use contexts and materials that are familiar and engaging for students. For example:

Prior ExperienceThis unit is targeted at Level 2 so students are expected to have experience at Level 1 including:

This unit builds on the unit Two digit place value, and it would be useful for students to have worked through that unit first.

Some materials created for that unit will also be useful here.

## Session One

In this session the students analyse a game of Snakes and Ladders to determine the gains and losses from climbing a ladder or sliding down a snake.

Part OneHow many squares would a player gain by landing on the ladder on square 20?We are now going to look at some of the other snakes and ladders.How many squares would a player gain or lose by landing on the ladder or snake?A beans and bags model of 91 – 30 = 61. Putting four beans back gives 65 so the amount taken is 26.

Part TwoAsk the students to play Snakes and Ladders in pairs or threes. Use wooden or plastic cubes as the dice, using stickers on each face to write on. Copymaster One gives the playing boards. Three options of board are available to cater for different knowledge levels. Note that each board requires a different set of numbers to be on the dice. Tell your students to use efficient ways to work out where they will land with each dice roll rather than simply counting by ones. Note that the numbers are slightly different to a standard Snakes and Ladder board - the player moves from left to right across each row rather than zigzagging their way up the board.

After they have played the game for a while challenge your students to work out the gains and losses for each ladder and snake on their board. Expect them to apply the strategies they learned in Part One and record those strategies with numbers, equations or diagrams.

## Session Two

In this session students learn to represent two digit numbers using a non-proportional representation, play money (see Material Master 4-9 to make a set of play money. At this point you only need one hundred, ten, and one dollar notes.)

Part OneWhat is the amount of money that this note is representing? (Holding up $1 note)Now what happens next?How do I write this amount of money? ($10 though some students will know about the decimal point).What happens now?(Adding one $1 note gives ten $1 notes in the ones place that can be exchanged for one $10 note)14, 24, 34, 44, 54, …, 94, … What happens next?87, 77, 67, 57,…, 17, … What happens next?

95, 96, 97, 98, 99, … What happens next?

Part TwoAcknowledgement: Pay Me was invented in 1998 by Kim Budd, Kaye Griffin, and Vince Wright, mathematics facilitators in the Year 3 Development Project.

Pay Me is a task in which students make up the pay for workers. You will need a lot of envelopes, preferably recycled from the school office, and at least 30 of each note ($10 and $1) and about ten $100 notes per group. $100 notes are useful if you want to extend the activity for some students and for later work on place value. Also photocopy Copymaster Three for the students, with one challenge per group of students.

There are three different versions of Pay Me at varying degrees of complexity:

The task is worked through in groups of three students. One student becomes the banker who exchanges notes and checks the pay envelopes for accuracy. Two students work together on creating the pay envelopes. Copymaster Three has a set of instructions at the bottom of each page about what amount to start with and instructions about putting a pay slip in each envelope. Students cut out each pay slip and put it into an envelope with the correct amount. You might also ask them to write the amount in words though that can restrict the participation of some students. The Banker checks to see that the amount is correct. Each set of slips has a final question that students should answer to show they have completed the task correctly. Look for:

After the students have completed the task bring them together to share what they learned. You might like to pose other challenges for assessment purposes:

## Session Three

In this session the students gain fluency in using ‘up through ten’ and ‘back through ten’ strategies for addition and subtraction.

Part OneEach time we make one of these totals our class scores a point that might be used for some reward.Part twoI want you to think ahead about what the total will be when you add a number. Is there a way to always win or is it just good luck? Does it matter who goes first?Suppose 0, 2 and 7 did not work. How would that change your strategy?## Session Four

In this session the students explore the place value of 3 digit whole numbers, particularly building up tens to make hundreds.

Part OneYou will need about 500 groupable objects, hopefully still organised into tens and ones. You will need a way to collect ten tens into a unit of one hundred. For iceblock sticks that are bundled in tens with rubber bands, use snap lock sandwich bags. For stacks of unifix cubes, use rubber bands to collect ten tens. For small plastic bags of beans, use snap lock sandwich bags as well. You will also need a place value mat (Copymaster Two) enlarged to A3 size.

What number have I made? Make that amount with play money.If I add one more dollar what happens then? Show me with your play money.

Why does the calculator use a zero here (10)?

I am adding one, two, three,…, seven, eight, nine ones (Count them out as you add them). What is the number now? What happens if I add one? How does the number 19 change when I add one?Progress in this sequence:

Why does the calculator use two zeros to show 100?100 → 109 (Why a zero in the tens column?) → 110 (Why a zero in the ones column?) → 150 (Why does the ones digit not change?) → 198 → 200 (Two acts of rebundling or exchanging) → 204 → 214 → 294 → 300 → 305 → …etc.

Part TwoAn issue with the use of zero as a place holder is that the way numbers appear creates an impression that there are no units of a given place value in a number, e.g. 204 has no tens. A flexible understanding of place value is essential for calculation. In particular, knowing that ones are nested within tens, and tens are nested within hundreds is very important. So there are actually 20 tens in 204. This activity helps student rename three digit numbers in multiple ways.

How could I pay for this item with play money?What if the cash machine only gave out $10 and $1 notes? How could I make $247?How many $10 notes make $100? How many $10 notes make $200? How many $10 note make $240? How many extra $1 notes will you need?Could I make $247 with only one $100 note and $10 and $1 notes? How?## Session Five

In this session students extend their understanding of three digit whole numbers to include ‘breaking up’ of hundreds and tens units.

Part OneUse the ‘Modelling three digit numbers’ learning object. As with the previous day students work in pairs with a set of play money and a place value mat.

How many tens are in 378? (Students may say 7 which is correct to the place values but there are also 30 tens in 300)If I added three ones, what would happen? (two added ones would form a set of ten ones that would combine to form a ten unit and move places. Adding another one would result in 381)If I added next added three tens what would happen? (Two more tens would create ten tens which would combine to form one hundred and move places. Another ten more would result in 411)399 + 1 = 400, 400 + 7 = 407, 407 + 90 = 497, 497 + 3 = 500, …

What would happen it I took seven ones away? (Taking away six ones leaves 420 but to take another one away (click on the down arrow) would require one ten to be ‘broken up’ into ten ones and moved into the ones column. Removing one would leave 419).What would happen if I took two tens away? (Taking away 19 would leave 400. To take one more away would result to two ‘break up’ actions. One hundred would become ten tens and move columns. One ten would become ten ones and move columns. One would be removed leaving 399).183 – 4 = 179, 179 – 90 = 89, 89 – 10 = 79,…

302 – 20 = 282, 282 – 100 = 182, 182 – 80 = 102, 102 – 3 = 99,…

Part TwoBankrupt,a game in which you lose money until you have none. To play the game the students need play money, a calculator, and two dice. One dice is standard (1-6) and the other dice has these words on the faces; hundreds, tens, tens, ones, choose, choose. The game can be played in groups of four with one player being the banker. Swap roles for a new game. Encourage the students to record the transactions.## Paper planes: Level 2

This unit uses the context of making paper planes to develop understanding of metre and centimetre measures. Students investigate a variety of paper airplanes designs, experiment to see which planes fly the furthest, and decide winners by measuring and comparing results.

This unit is suitable for students who have had plenty previous experience with non-standard units and have had the concept of standard units introduced. It provides a good context for practising the use of metres and centimetres. In the second unit, Paper Planes L3, students create scatter plots of the distance their planes travel when a variable is changed.

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

Students' measurement experiences must enable them to:

The usual sequence used in primary school is to introduce the centimetre first, then the metre, followed later by the kilometre and the millimetre.

The centimetre is often introduced first because it is small enough to measure common objects. The size of the centimetre unit can be established by constructing it, for example by cutting 1-centimetre pieces of paper or straws. Most primary classrooms also have a supply of 1-cm cubes that can be used to measure objects. An appreciation of the size of the unit can be built up through lots of experience in measuring everyday objects. The students should be encouraged to develop their own reference for a centimetre, for example, a fingertip.

As the students become familiar with the size of the centimetre they should be given many opportunities to estimate before measuring. After using centimetre units to measure objects the students can be introduced to the centimetre ruler. It is a good idea to let the students develop their own ruler to begin with. For example, some classrooms have linked cubes which can be joined to form 10 cm rulers. Alternatively pieces of drinking straw could be threaded together.

The correct use of a ruler to measure objects requires specific instruction. The correct alignment of the zero on the ruler with one end of the object needs to be clarified.

Metres and millimetres are established using a similar sequence of experiences: first construct the unit and then use it to measure appropriate objects.

There are many websites that give instructions for folding paper airplanes.

This unit can be differentiated by altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. In particular, have students measure length using non-standard measures, such as hand spans or foot lengths, if they are not ready to progress to using metres and centimetres.

An alternative context for this unit is Manu tukutuku – Māori kites. Information about Manu tukutuku is readily available online, and the Te Ara website provides a useful overview. Within this context, students could design and make simplified Manu tukutuku to use for the measurement tasks, with the work culminating in a kite day, rather than an air show.

## Getting started

Make a simple paper plane. Show students your plane and ask if they have ever tried making paper planes. Discuss the different designs they have tried.

Have students work in pairs to make a simple paper plane of their own design. Alternatively they could make a plane the same as the one you have shown them.

Have students experiment with their planes to see how far they fly. Discuss:

How could we measure the distance our planes fly?What could we use to measure how far our planes have travelled?

What would we need to be careful of when measuring?

Discuss the use of non-standard measures and the need for a standard unit to allow comparison.

Show students a variety of measuring tools and discuss these.

Which of these measuring tools do you think would be best to measure the distance of our plane’s flight? Why?What other things could we use?

Emphasise the importance of an accurate starting point for the flight and accurate use of the measurement tools to the closest cm.

Have students experiment with a variety of measurement tools to measure the flights of their paper planes. As they work encourage estimation and reinforce the correct use of measurement tools to ensure measurements are accurate to the nearest metre and cm.

Students can find the dfference between their estimate and the measured length.

## Exploring

Tell the students that at the end of the week there will be an air-show. Explain that they will all participate in the show by making and flying planes and there will be a competition to see whose plane can fly the furthest.

Over the next few days have students work in pairs or small groups to try out some different designs for paper planes. As a starting point for their designs, find instructions for a selection of planes online or in books.

As students try different designs have them measure the lengths of their flights. Encourage them to record their trials in a table similar to the one below to help them keep a track of which planes fly the best. This will help them decide which plane they will use in the air-show at the end of the week.

Plane

Flight 1

Flight 2

Flight 3

Start each session with some ideas they might like to consider when making their planes. You may like to include the following points.

Weight near the bottom of the plane may increase its stability and allow it to fly further.

As work progresses you may need to set criteria for the planes. These can be discussed with students and may include the size of paper to be used and limits to the other materials that may be included, for example the number of paper clips, sellotape, glue or staples. How the planes are to be thrown may also need to be discussed.

As students work help them with their measurements and discuss these with them. Encourage estimation. Ensure that accurate starting points are used and measurements are made to the nearest cm.

Conclude each session with a discussion of the planes and how far they have flown.

How far did the plane you made today fly?How do you think you could improve your plane?

What do you think you will try tomorrow?

Reinforce the correct use of measurement tools to allow accurate measurements.

What did you use to measure the distance of your plane’s flight?What steps did you take to ensure your measurements are accurate?

## Reflecting

To conclude the work on paper planes, hold an air-show. Conduct a competition to see which of the planes flies the longest distance.

Students can work in groups to measure the distance their planes fly with one plane from each group going through to the final. This will give students maximum practise at measuring distances. Encourage the use of estimation before measurements are made.

At the conclusion of the competition reflect.

What was the difference between the first and second place getters?Which planes went the furthest?

Why do you think they flew so well?

What did we need to be careful of when we were measuring?

Which tools do you think were most useful for measuring? Why?

Family and Whānau,

This week at school we are making paper planes and measuring how far they can fly. We will be holding an air-show on Friday to see which plane can fly the furthest. Ask your child to show you the design they are using for their plane and help them experiment with things that may make the plane go further. Run some trials at home estimating how far the plane flew each time.

## Number lines and bead strings

In this unit five-based bead strings and number lines are used to solve addition and subtraction problems. The aim is to get students that use an early additive strategy to solve problems using a tidy number strategy with 10.

There are several things happening in this unit. All of them are aimed at enabling students to become more fluent in number.

The students need to realise that making a 10 is a good strategy for solving addition problems. This strategy is reinforced by the use of bead strings and the number line, so students need to understand how these representations work, and see their relevance for addition and subtraction work.

It is important that the students gradually learn to work without the bead strings and number line, so they are encouraged to ‘image’ these objects. Instead of actually using the devices they should start to think about what is happening in their heads. The next stage is for these number facts to become quickly recalled. This will take a reasonable amount of practice for most students. In the process, students are exposed to problems in context and finally they are given examples of their own to work on.

This unit is important in at least two ways for later work in mathematics in school, tertiary studies and even life itself. First, number is at the base of many ideas in mathematics so it is important to be fluent in addition and subtraction and to have strategies for carrying out these processes. Second, devices like the number line are not just useful to understand about addition and subtraction. Number lines are used extensively in co-ordinate geometry where two perpendicular number lines are used as axes. In this situation they enable us to visualise quite complicated functions. So even at this early stage in school, students are developing skills that will be useful throughout their school life as well as ideas that will grow into powerful and deep mathematics.

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

Some of the activities in this unit can be adapted to appeal to students’ interests and experiences and encourage engagement. Other contexts for number line problems include:

Note the following useful prior knowledge:

## Session 1

Sally the snail starts on number 8 and slides along 4 more spaces. Where does she end up?How can we find out where Sally will end up without counting?How many spaces will Sally need to go to get to number 10?

Now how many spaces has she got left to go?

Sally the snail is on number 9 and slides another 4 places, where will she end up?Samu the snail is on number 13 and slides backwards 5 spaces. Where does he end up?

Have the students predict where they think they will end up before getting students to come out and share their strategies on the number line.

Sally has been sliding for some time now. She is on number 27 and slides another 5 spaces. Where do you think she will end up?Ask students to talk to their partner and discuss how they would work the problem out.

Challenge students to see if they can solve the problem without counting on:

See if you can solve the problem another way?What is the nice friendly number that Sally is going to pass through?

How far is it to 30 from 27?Now how much further does she have to go?

Samu the Snail starts on 49 and slides another 8 spaces. Where does he end up?Harley the Hedgehog starts at number 87 and wanders on another 8. What number does he end up on?

## Session 2 – Marble Collections

Over the next three days the aim is to slowly remove the number lines and bead strings and encourage students to visualise what would happen on the bead string or bead frame. This is called

imaging.Begin by using a bead string 1-20 coloured in 5’s like this.

Where is number 8?Find number 11.

Where would number 16 be?

Moana has a marble collection. It starts with 9 marbles. Show me where 9 is on the bead string.Moana is on a winning streak and wins 6 more marbles. How many does she have in her collection now?Use the bead string to demonstrate putting one marble onto the 9 to make it 10 like this:

9 + 1 = 10; there was 5 left; 10 + 5 = 15.

Kate has 8 marbles and she wins 6 more. How many does she have now?George has 15 marbles and wins 6 more. How many does he have now?

Hemi has 15 marbles and loses 6. How many does he have left?

## Session 3 – Do and Hide Number line

This session is to use the number line (Copymaster 3) and bead string to solve problems and then the number lines and bead strings are taken away to encourage students to start imaging.

Freda the flea starts on 9 and hops forward 7 more spaces. Where does she end up?Ask a couple of students to take the number line and pegs away and work out the answer. Ask the students remaining to visulaise what the others will be doing on the number line. The following questions may prompt the students to image the number line.

Where did the flea start?How far does the flea have to go to get to 10?

Ask the students who took the number line away to share what they did to solve the problem.

Repeat with other problems. The following characters could be used to create similar story problems: Kev the kangaroo, Gala the grasshopper or Freckles the frog.

Encourage the students to visualise what they would do on either the number line or bead string. Extend some of the problems to numbers beyond 20.

The following types of problems will continue to challenge the students further.

Start Unknown

? + 4 = 10

Greg the grasshopper jumps 4 more spaces and ends up on 10. What number did he start on?

Change unknown

3 + ? = 8

Frances the frog starts on 3 and jumps along the number line and ends up on 8. How many spaces did she go?

## Session 4 – Problem Solving Bus Stops

In this session, problems are placed on the top of a large sheet of paper. Students move around each bus stop, solving the problem. They record their working on each sheet.

Warm up with some whole class problems like the ones that have been shared in the previous sessions. Get students to talk to their neighbour and share how they worked out the answer. Record the different ways students solved the problem by writing it on the board.

Place each of the problems from Copymaster 4 on to a large piece of paper. Place the sheets around the room. Students can either rotate around the bus stops in pairs randomly or in a sequence to solve each problem. They are to show their thinking on the large sheet of paper.

## Session 5 – Reflection

Use this session to share the solutions students came up with for each of the bus stop problems. Encourage students to act out the problems where appropriate and to remodel their answers on the number lines or bead strings.

Dear family and whānau,

This week we have been using a number line to do some addition and subtraction. Here is an example of a number line:

1

2

3

4

5

6

7

8

9

1011

12

13

14

15

16

17

18

19

20Ask your child to show you how they would solve this problem:

Kiri has 5 lollies. If she buys 8 more how many does she have altogether?

Perhaps you can make up some more problems like that and work them out together. When your child gets really quick at coming up with an answer put the number line away and ask them to try to figure out the problems by visualising the number line in their head.

## Figure it Out Links

An activity from the Figure It Out series which you may find useful is:

## Honeycomb

In this unit students sort and explore two-dimensional and three-dimensional geometric shapes, identify and describe their distinguishing features and come to appreciate the efficiency of the tessellating hexagon in meeting the needs of honeybees.

In level one, students have been learning to name some common shapes, becoming familiar with their features. When students are given opportunities, they find their own systems for sorting shapes, justifying their categories and developing the important geometric language of attributes.

As they work with three-dimensional shapes, students become aware that these are made up of flat or plane shapes that have two dimensions. It is therefore useful for students to see that two-dimensional shapes are like a print that technically cannot be ‘held’ because it has no thickness or depth. In developing the understanding of the way in which two dimensional plane shapes build three dimensional shapes, students need to have a clear understanding of the meaning and concept of ‘dimensions’ and should be able to explain in their own words what the abbreviations 2D and 3D mean.

The change in language from ‘sides and corners’ for two-dimensional shapes, to ‘faces, edges and vertex/vertices’ is not an insignificant one. The language itself conveys the shape category and should be emphasised and well understood.

As students manipulate shapes that are the same, they ‘discover’ tessellation and come to understand that this too is an identifying characteristic of a shape, albeit that they are not yet able to explain this using the precise quantification of angle.

As shapes don’t exist in isolation it is important that students have opportunities to explore shapes in structures around them. In their exploration of the remarkable structure of the honeycomb, students have opportunities to apply their new learning.

Associated Achievement ObjectivesScience

Life processesEcologyEnglish

Processes and strategiesPurposes and audiencesIdeasThis unit can be differentiated by varying the scaffolding provided and altering expectations. This will make the learning opportunities accessible to a range of learners. For example:

The context of this unit can be adapted to address diversity, and appeal to students’ interests and experiences to encourage engagement. For example:

The Greedy Triangle, by Marilyn Burns (available being read aloud on YouTube)A cloak for the dreamer, by Aileen Friedman (available being read aloud on YouTube)Whilst this unit is presented as sequence of five sessions,

more sessions than this will be required between sessions 3 and 4. It is also expected that any session may extend beyond one teaching period.## Session 1

This session is about exploring features of mosaic shapes and making a successful poster highlighting the special characteristics of one particular shape.

SLOs:

Activity 1Begin by reading

The Greedy Triangle.Activity 2Write ‘

Clever shapes’ on the class chart. Explain that the students will be making their own small poster about aclevershape. Ask what the purpose of a poster is. (To capture people’s attention and to give a short clear message).Together list the features of a good poster. It grabs the audience’s attention by using:

bold print, a simple and convincing message, interesting colours, a picture or diagram.

Activity 3Make available mosaic pattern blocks, (

omit hexagons), paper, pencils, crayons or felt pens.Use this time to model and record the language of the shapes, including writing and discussing ‘tessellation’.Activity 4Explain that to make their posters, the students will need to draw around their ‘clever shape’ (as many times as necessary).

Set a time limit and have students complete their ‘clever shape’ posters.

Activity 5Conclude the session by having students share their work in pairs. Display the list of criteria for a successful poster and have students self evaluate, then give partner feedback about each of the criteria.

Activity 6Share clever shape posters as a class, highlighting geometric language.

## Session 2

This session is about understanding that a 2D shape is like a footprint, a 3D shape is something you can hold or feel, and that we use different language for each.

SLOs:

Activity 1Ask, ‘Can you hold or feel an outline (‘footprint’) with your hand?’ (No. It has only 2 dimensions.). Discuss, highlighting the fact that we can say how long the outline is and how wide it is.

Record ‘width’ ‘length’ ‘dimensions’, explaining why these outlines are called two-dimensional shapes and that this is sometimes referred to as 2D.

Ask, ‘What ‘dimension’ can’t we measure?’ (How deep it is.)

Activity 2Make available hexagonal mosaic blocks, pencils and paper.

Have students each draw around one block creating an outline, identify and record ‘6 sides’ and ‘6 corners’ and write a statement about the outline. For example:

“This is a 2 dimensional or 2D shape because we can only say how long it is and how wide it is.”Activity 3Discuss, conclude and record that the mosaic block is a 3D shape because it has width, length, and thickness (depth) and we can hold it in our hand.

on the class chart. Have students locate and identify each feature on their drawing. Write the plurals of each work beside the singular, highlighting the wordface, edge and vertex. Make the connection between the 2D language of sides and corners and the 3D terms.verticesHave students label their drawings using these words. Have all student touch and name those parts on their hexagonal mosaic block.

Activity 4Return to the language listed in Session 1. Highlight the word

tessellate.Pose the task: Use your shape, a pencil and paper and show how you know whether or not a hexagon tessellates.

Activity 5As they describe their drawings they should use the language of

.side and cornerActivity 6Conclude the session by having students each make a small poster about 2D and 3D shapes that they know, thinking about the feedback they received about their posters in Session 1.

## Session 3

This session is about consolidating understanding of, and using language associated with, 2D and 3D shapes, understanding and making a hollow prism to create a model bees’ honeycomb.

SLOs:

Activity 1Have students explain why the hexagons in the drawing are two-dimensional.

Ask , “What does the drawing remind you of?”

Elicit, ‘a bees’ honeycomb’.

Have students discuss and agree whether the honeycomb is two-dimensional or three-dimensional. Have them explain their thinking.

on the class chart and show the students a model.‘hexagonal prism’Have several students hold it and describe its features. Record these.

Highlight the plane (two-dimensional) shapes, the hexagon and rectangle that make up the three-dimensional prism.

Activity 2Pose: Can you work in pairs, using this card and tape, to make a hollow hexagonal prism?

Give students time to explore and create their hexagonal tubes.

Have students talk about the process using the language of faces, edges and vertices.

regular, notirregular. Define these.Recognise the bee’s skill in making perfect hexagonal prisms.

If necessary, have students make their prisms with greater precision to achieve a ‘perfect’ honeycomb such as the bee produces.

Activity 3They might also like to make their own model bees to inhabit the class honeycomb.

## Session 4

This session is about giving students opportunities to find and share information about honeybees, their honeycombs and the importance of the hexagon.

SLOs:

Activity 1Have students share the results of their research and, if any students have made their own model bees, to locate these in the honeycomb.

http://www.npr.org/blogs/krulwich/2013/05/13/183704091/what-is-it-about-bees-and-hexagons

http://www.nature.com/news/how-honeycombs-can-build-themselves-1.13398

Activity 2Write on the class chart:

What do honeybees need to stay alive?How does the honeycomb ‘suit’ the bees?

Discuss each, with reference to the research information, and record the students’ understanding of the key ideas.

Recognise that:

Activity 3Conclude the session by reading

A cloak for the dreamer:Highlight the way in which the circle shapes which did not tessellate were changed into the hexagon shapes which do tessellate and the way this is like the process in the honeycomb where the cells are thought to begin as circles.

## Session 5

This session is about synthesising the skills and learning in Sessions 1-4.

SLOs:

Activity 1Review key learning over the past 4 sessions including referring to the clever shapes posters and feedback in Session 1.

Activity 2Set an appropriate time limit and have students work in pairs to design and create a ‘presentation’ (poster, powerpoint, other) combining the

key ideasabout aclever shapeand aclever creature(the honeybee).Activity 3Challenge students to research other clever shapes (and creatures) found in nature.

Dear parents and whānau,

We have been learning that a two dimensional shape is a bit like a footprint, with width and length as the two dimensions, and that other shapes are three dimensional.

Your child would like your help to find out more about the way in which honeybees create hexagonal honeycombs, and to learn more about other ‘clever’ shapes in nature.

Thank you.