Prior Experience

It is expected that students will have a range of prior experiences 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 Faces

1. With the whole class, demonstrate a dice magic trick. Shake two dice in your hands and then present them to the class with two sides held together so you can't see the numbers on them. Explain that you can predict the total of the two joined faces. Act out the same trick a couple of times inviting individual students to join the dice for you to prove that you are not cheating.
   
    
2. The key to the trick is that opposite faces of a die add to seven. For any pair of joined dice look at the end faces. The opposite faces that are hidden will be the complements of seven. For example, if three dots are at one end the opposite meeting face will have four dots (3 (toru) + 4 (wha) = 7 (whitu)). If one dot is at the other end, then the opposite meeting face will have six dots. The total number of dots meeting will be four (wha) plus six (ono) equals ten (tekau) dots.
    
3. After several examples, put the students into pairs with two dice and encourage them to discover how the trick works. After a suitable period, bring the class back together to discuss students’ ideas.
    
4. Some students may say that you figure out the missing face on each dice by looking at the five you can see, then add those dot numbers together. That works but it is quite hard to do in your head and seems to take a lot of time. Others may say that there are 21 dots on each dice, so the total is 42 dots. By adding up the dots that are showing you can find out how many dots are hidden. This also works but is very slow and requires a lot of work.
    
5. The ‘adds to seven’ feature of opposite faces on a dice is the key bit of noticing. You may need to bring this to students’ attention. Challenge them to consider three dice joined together. Is it still possible to work out the dot total of the hidden faces? (There will be four faces to consider) Ask your students to work out a rule for three dice.
   
    
6. Whatever way the centre dice is orientated the two hidden faces add to seven. The two hidden faces of the end dice can also be worked out using the ‘opposite faces add to seven’ rule. Therefore, the dot total will be just like that for two dice plus seven.

Card Sums

1. Tell your students that you are going to continue the theme of mathematical magic. While students are watching, create five cards. The image below shows the numbers to go on the front and back of each card. You can use square pieces of paper if you want, such as those found in a memo pad.
   
    
2. Toss the five cards on the ground so they land randomly. Tell students that you know the total of the five numbers without needing to add them up. Do not tell the students how you are doing it. Look at the number of odd numbered cards. Add that number to 20. Say there are four odd numbers. Add four to 20. The total is 24. Here is an example, 1, 3, 5, and 9 are all odd.
   
   In this example below only 5 is odd so the total is 20 + 1 = 21.
   
    
3. Get the students to make their own set of cards and ask them to work in pairs to figure out how you know the total without adding the numbers. Look for students to:
   • carry out some trials of tossing the cards to get an idea of how the activity works
   • systematically record the sums (totals) that come up. What sums are possible? What is the lowest possible sum? What is the highest possible sum?
   • classify the numbers on the cards as odd and even numbers
   • consider the effect on the total of turning over one card, two cards, three cards… Is the effect different if the number showing on the card is odd or even?
      
4. Can the students develop a way to know the sum without adding all five numbers?
    
5. After a suitable time of exploration, talk about the questions above. Do students generalise a strategy that works every time?
   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 abacus

1. Choose a two-digit number on the hundreds board and ask a student to make the number on the left side of the abacus. For example, suppose you choose 45:
   
   Does the student use the tens and ones structure of the abacus or attempt to count in ones?
    
2. You may need to revisit the meaning of ‘forty’ as four tens, and ‘forty-five’ as four tens and five.
   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?
    
3. Do your students use the place value structure of ten and ones, even if counting by tens and ones?
   You might revisit the fact that five tens are fifty. Fifty mean five-ty or five tens.
    
4. Record the result as an equation 45 + 55 = 100. Talk through two more examples, like 29 + 71 = 100 and 84 + 16 = 100.
    
5. Ask students: Do you know where kiwi live? 
    
6. Tell the students that there were 100 kiwi living in a burrow in the local reserve. The kiwi were very inquisitive and got lost all the time. The carer for the kiwi made up some mathematical magic to tell straight away how many kiwi were missing. Act out being the kiwi carer.
   (Student A), please move some of my kiwi to the left side and cover up the rest so I cannot see them (using an A4 sheet of paper).
   
    
7. Role play working out the number of kiwi you can see, then recording the number. For example, “Two tens, that’s twenty, five and three, that’s eight. I can see 28 kiwi.”
    
8. Write 28 + 72 = 100 on the board, pausing a little at the 72 to show a bit of working out. Confirm that 72 is correct on the Slavonic Abacus.
    
9. Have the students work in pairs. Can you work out how the kiwi carer did it? How could they know 72 kiwi were missing so quickly?
    
10. Give the students time to work on the task. Students may use a Slavonic abacus to support them if needed and, later, to explain and justify their strategy. Listen to the discussions of your students:
    • Do they use the tens and ones structure of two digit numbers?
    • Are they aware that ten tens are 100?
    • Do they look for relationships in the digits of the two-digit numbers that make 100?
       
11. After a suitable time, bring the class together. Look for ways to capture what students say in ways that support other students to understand. For example:
    
     
12. Invite the students to justify why the method works and any exceptions to it. Look for responses like, “Three and seven makes the other ten. Then you have ten tens and that is 100,” and “It is different for numbers like 40 + 60 because they already make ten tens.” If your students prove to be competent with two digits you might consider extending the task to two addends that make 1000.

Crosses Pattern

In this task students apply place value to explain why a pattern on the hundreds board works every time.

1. Play Video 1, pausing at appropriate points to allow students to calculate the sums of the top and bottom and left and right numbers separately. For example:
   
   2 + 22 = 24 (top and bottom numbers) and 11 + 13 = 24 (left and right numbers).
    
2. See if students notice that the two sums are always equal and are the double of the middle number. You might invite students to use a hundreds board to try crosses of their own to see if the trick works. Ask your students to try to explain why the trick works every time.
    
3. After some discussion time, bring the students together to share their ideas. Look for students to apply the place value structure of the numbers in the cross. Attendance to place value can be supported by using materials to model each number in the cross. Any proportional place value representation will work. 
    
4. Look for ideas of balance like:
   • “The ones digits are one more and one less. Two is one less than three and four is one more. So the numbers balance to make the same as three plus three.”
   • “Both sums have six tens. Two tens and four tens equal six tens and three tens and three tens equals six tens.”
   • “Look at the middle number. The top number is ten less and the bottom number is ten more than that. The left number is one less and the right number is one more than the middle number.”

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 cross pattern works. 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 other is less.

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

• Place value with two digit numbers
• Building on two digit place value
• Place Value People
• Place Value to four digits

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.

1. Tell students: Magicians can change objects into different shapes. We are going to see if you can be a magician.
    
2. Ask your students to cut their square into three pieces as shown below. You may need to discuss the meaning of diagonal. When they are finished ask the students to put the square back together.
   
    
3. Now tell students: 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:
   • attend to properties of the target shapes, in particular, angles and side lengths?
   • align sides that are of equal length?
   • visualise pieces within the target shapes?
4. After a suitable time, bring the class together to discuss the strategies they used. Ask them how they might record a solution. Usually students suggest drawing the pieces in the completed position. 
    
5. Extend the task by cutting the largest triangle in half to form two right angled triangles that are the same size as the other two. The resulting pieces are quarters of the original square.
   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.
    
6. Many dissection puzzles were created by magic mathematicians. Ask your students to find other ways to dissect a square then create target puzzles. Here is a simple example:
   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.

1. Begin with two plastic cups, one of which is marked in a barely discernible way (e.g. pencil mark or a smile sticker). Tell a student to hide a treat under one cup while you are watching. The treat might be a small toy or a packaged food item. Small kete and shells could be used here instead of plastic cups and treats.
    
2. Tell the student: 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.
    
3. After the student has moved the cups around, choose the correct cup knowing from the subtle marking. Simulate this trick three times choosing the opposite position to the one before. You might record wins and losses, i.e. 3-0.
   Am I magic or is something else going on?
    
4. Expect students to express their beliefs about the trick.
   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.
    
5. Ask students: What would happen if the cups were not marked?
    
6. Repeat the simulation with unmarked cups. Choose the cup that is in the other position to where the treat was originally located. In most cases the student will randomly allocate the treat position and you will ‘magically’ choose the right cup only some of the time. Repeat the simulation three times and record the wins-losses, e.g. 2-1.
    
7. Ask the students: Am I magical or is it just luck?
    
8. After a brief discussion you could get your students to trial the two cup one treat situation. You might gather data about the number of students who are magical (correct) or not magical (incorrect) and graph the data quickly, possibly using a spreadsheet. It is interesting to compare bar chart and pie representations.
   
    
9. Expect your students to consider that the chances of being correct by luck are 50:50.
    
10. Extend the problem:
    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?
     
11. Am I Magical 1, Am I Magical 2, and Am I Magical 3 can be used to put students in the position of magician. Students close their eyes as the cups are moved then guess where the treat is hidden. Later in the video the treat cup position is revealed. You might ask the students each time, who were magicians and guessed correctly (with a show of hands)?
     
12. Tell the students: Being magical in this situation seems a bit hard. Let’s keep the three cups but add another treat. 
     
13. Let students trial the three cups, two treats situations. Ask them to gather data about the times they were magic, chose a treat cup, and the time when their magic deserted them.
     
14. After a suitable time of exploring bring the class back to discuss their conjectures:
    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.
     
15. Discuss with your students:
    • Is there a best cup to choose? Why?
    • What are the chances of being magical by luck?
       
16. Encourage students to create models of the situation, such as, “There are two ways of being magical and only one way of not being magical.”
    
     
17. Can your students compare the two cup and three cup situations? Do they assign descriptive words to the likelihoods, such as more likely, less chance, etc.?
    • Am I more likely to be magical in this game compared to the two-cup game? Why?
    • What if there was only one rabbit in the three-cup game?
    • Can we change the game so it is impossible to be magical? (no treats)
    • Can we change the game so you are certain to be magical? (treats in every cup)

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.

1. Tell your students: Mathematical magicians can think ahead. They can predict the future. Can you?
    
2. PowerPoint 1, PowerPoint 2, and PowerPoint 3 relate to repeating patterns of increasing sophistication. The animations guide you with prompting questions for your students to discuss.
    
3. Look at the way your students anticipate further members of each pattern.
   • Do they fail to see any repeating element? In that case reading the pattern like a poem or chant can help.
   • Do they recite the repeating element one after the other and try to track the ordinal counting? For example, “kiwi (One), tuatara (two), kiwi (three), tuatara (four), ….”
   • Do they use skip counting to anticipate which animal will be in given positions? For example, “The weta comes every three animals. 3, 6, 9, 12… so the weta will be in number 12.”
      
4. Provide students with a range of materials to form sequential patterns with. The items should be locally sourced and might include shells, leaves, pebbles, 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).
    

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

   • create and extend an element of repeat
   • use one or more variables in their pattern
   • predict ahead what objects will be for given ordinal numbers, e.g. the 16^th object.

    

6. Take 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.
    
7. Discuss as a class how to predict further members of a pattern. Strategies might include:
   • Create a word sequence for each variable, e.g. blue (kikorangi), yellow (kōwhai), red (whero), blue, yellow, red, …
   • Use skip counting sequences to predict further members, e.g. If the colour sequence is blue, yellow, blue, yellow,… then every block in the ordinal sequence 2, 4, 6, … etc. is yellow.
8. Some students may be able to apply simple multiplication knowledge to the patterns. For example, if the element that is repeated is made of five objects, e.g. kiwi, tuatara, ruru, weta, piwakawaka,… then the five times tables might be used. For each position, 5, 10, 15, 20, … the animal is a weta. Other ordinal positions can be worked out by adding and subtracting from multiples of five. For example, position 23 must be a tuatara since 25 was a weta.

Extend the activity:

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

• Use a longer unit of repeat, especially a number of objects that produce a difficult sequence of multiples. For example, ○, □, →, ∆, ○, □, →, ∆, ○, □, →, ∆, … has four shapes in the unit of repeat so multiple of four will be needed for prediction.
• Use more than one variable in the unit of repeat, such as colour, position and size.
• Leave missing shapes or objects in the repeating pattern, e.g. ○, □, ­_, ∆, ○, ­_, →, ∆, …                

Session One

Begin the session by acting out the following scene with your class (mahi tahi model).

Characters:
Captain Kaiwhakaako - teacher
Crew - ākonga

Props:
Treasure - a small box
Crooked palm tree - desk

Captain Kaiwhakaako, the pirate, decided to bury their treasure.
They started from the crooked palm tree and carefully counted 12 steps, (heel, toe) and then stopped and placed the treasure on the ground.
To make sure that they remembered where they left it, they wrote down on their map - 12 steps.
He wanted to make really sure that he had measured correctly before digging the hole so he asked a cabin boy or girl to check.
Captain Kaiwhakaako was puzzled. How could the crew member have a different number of steps?
Had they made a mistake?

1. Discuss with ākonga the reasons for the differences.
   Can you think of a measure that Captain Kaiwhakaako could use that is the same for everyone?
   If ākonga come up with the suggestion of a metre, ask:
   How long is it?
   When and where is used as a measurement?
2. Tell the ākonga that Captain Kaiwhakaako is really interested in using metres on their map but they're not sure how long, wide or high a metre is. Captain Kaiwhakaako wants their crew to go around the island (classroom), and make a list of all the things that are less than one metre, about one metre and more than one metre and then share it with them so that they can learn about a metre.
3. Provide ākonga with a metre stick or a one metre length cut from ribbon/string/wool or cardboard.
4. At the conclusion the ākonga can share their findings with the crew and Captain Kaiwhakaako and find out if they had similar measurements for objects in the room.
5. Finally they could measure from the crooked palm tree to the treasure and record the answer in metres. The letter m could be introduced as a means of recording. Suggestions on how to record incomplete metres could also be discussed.

Session two

1. Tell ākonga that Captain Kaiwhakaako has decided that now they know what a metre is, they want to start drawing up plans for their new pirate ship and that they would like the crew to help. 
   Discuss with ākonga the type of boats that pirates sailed in. This could include discussion about waka and waka ama (outrigger canoe).
   Provide them with chalk and a metre measure and take them outside to draw the boat to Captain Kaiwhakaako requirements.

   Measurements of Captain Kaiwhakaako's new pirate ship:

   • Length: 10 metres
   • Middle mast: 5 metres
   • Front/back mast: 4 metres
   • Plank: 1 metre

    

2. Ākonga might like to add extras like flags, anchor ropes, and cannons and add them to the measurement list. Encourage ākonga to estimate before drawing.
3. Ask ākonga to stand and show how high they think a metre would be from the floor. Check with their metre measure and reference it to their body.
   A metre is as high as …………….(my ribs).
   How wide is a metre? A metre is from my fingertips to ……………

Session three

1. Ask ākonga to estimate how many of their handspans would be the closest to a metre. 
2. Trace an outline of their handspan on to paper and then cut it out and use it to measure along the metre. Record results. Have ākonga estimate how many of their footprints would be closest to a metre. Make outlines by removing their shoe and tracing around their foot.

3. Check how ākonga position the shapes when measuring.
   Do they begin from the same baseline?
   Do they use the measuring unit consistently without gaps or overlapping?
   Ākonga can show their results by pasting their outlines on to paper and recording the number beside it.

   To measure 1 metre it takes:
   ____ of my handspans
   _____ of my footprints

4. To finish, pose this problem for the crew,
   Captain Kaiwhakaako has gone to a boat shop to buy some new canvas for sails. They want two metres. Can you show me using a body measurement how long two metres would be?

Session four

1. Provide ākonga with a standard metre ruler to explore. Look at the markings on it and discuss what they can see.
2. Talk about where you begin measuring from. (Ākonga can have difficulties identifying the starting points on calibrated rulers. They start from the edge rather than the markings.)
3. If ākonga haven’t offered the word centimetre in the discussion, explain to them that the space between the numbers is one centimetre and place centimetre cubes along the ruler.
4. Ask: How many centimetre cubes might fit along the metre?
   If 1cm cubes that connect are available join 100 using two different colours to distinguish the decades. Place the line of cubes on top of the metre ruler and count in tens to 100.
5. What are the advantages and disadvantages of using standard measures?

6. Provide ākonga with string, scissors and glue and let them investigate the different ways of creating patterns with 1 metre of string. Ākonga can first measure a metre, and then make a pattern.

   e.g. spirals
   zig zags
   straight lines
   curves

7. Glue their discoveries to cardboard and display the one-metre patterns.
   Discuss that different patterns look as though they have different lengths.

Session five

Captain Kaiwhakaako has decided to have a sports day for the pirate crew. The events for the day are:

• Toss the cannon ball (1kg of clay or a shot put from the PE shed): Copymaster 1: Toss the Cannon Ball
• Jump from the plank (A standing jump from a rectangular piece of cardboard): Copymaster 2: Jump from the Plank
• Metre kick (A rugby ball): Copymaster 3: Metre Kick
• A graphic organiser (e.g. table) for ākonga to record their measurements in

You could adapt this session to include games you have played as a class that involve throwing, kicking, jumping, and tossing. The key learning is estimating and measuring in metres. At each station, ākonga need to estimate how far they will kick/jump/throw/toss in metres, and then measure the actual distance covered.

1. Set up the activities in the three stations and provide each student with a one metre long piece of string or metre ruler. Model how to complete the activity at each station. With ākonga, come up with a criteria for how to measure the different tasks properly (e.g. the string must be straight, no gaps between the measuring tools). Set a time limit at each station (approximately 10 minutes).
2. When ākonga share their results at the end, talk about the half metre, or the extra bit and the need to have a smaller unit of measure.
3. Ākonga will need to work with a partner who can stand where the rugby ball lands after the kick, a tuakana/teina model could work well here. As above, have ākonga record their estimation prior to measuring. After tossing the cannon ball, ākonga estimate how many metres, and then measure. 

Whilst this unit is presented as a sequence of five sessions, more sessions than this may be required. Any session may extend beyond one teaching period. This unit is written to focus on the work of Pablo Picasso, who co-founded the cubist movement. You may prefer to focus the activities on similar works by New Zealand artists.

Session 1

This session is about naming and describing plane (2D) shapes, and using these to create a picture. Note: By drawing around the shape ākonga are creating a two-dimensional shape. Limit the colour selection, as this is relevant to the work in a later lesson.

SLOs:

• Sort geometric blocks and explain their groupings.
• Use geometric language to describe plane shapes.
• Understand the features of a two-dimensional shape.
• Create an artwork using plane shapes.

Activity 1

Make geometric blocks available to pairs of ākonga (tuakana/teina).
Begin by having individual ākonga sort a selection of the geometric blocks into groups, and explain to their partner the groupings they have made. Have them repeat the sorting task, this time categorising them differently.
Encourage and affirm appropriate geometric language, including the correct use of shape names and descriptions of their features.

Activity 2

As a class (mahi tahi), brainstorm and record on the class chart, all shape and attribute language associated with the task in Activity 1.

Activity 3

Make paper, pencils, and pastels or crayons available, but limit the colour selection.
Challenge ākonga to make a picture of a person, object or place that is important to them. Explain that they are to make their picture using the shape blocks to help them. 
Demonstrate how to begin the picture by drawing around several shapes and then colouring in the outline. For example:

Explain why your picture is important to you. (For example: ‘My Dad used to drive an old blue car a bit like this one.’)

Activity 4

Have ākonga make and complete their own pictures. When pictures are complete, have each ākonga name their picture, write a short story about it, using words from the brainstorm list in Activity 3. Their story should explain why the subject of the picture is important to them and how they made their picture.  This activity could be integrated with explicit writing instruction (e.g. explanation writing).

Activity 5

Refer to the example picture made in Activity 3 above and to the artworks they have just completed.
Ask: “Are the shapes two-dimensional shapes or three-dimensional shapes?” Discuss ideas.
Write the word ‘dimension’ and 2D below the picture.
Ask ākonga to discuss in pairs the meaning of what has been written.
Through discussion, develop understanding of the meaning of the word ‘dimension’, of two dimensions and of the abbreviation, 2D.
Highlight the shapes that they have drawn are like (foot) prints only. They are wide and long, but not deep. Explain that two-dimensional shapes have no depth or thickness.

Activity 6

1. Have ākonga discuss in pairs and decide whether the geometric blocks themselves are two-dimensional or three-dimensional shapes.
2. 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 geometric blocks are 3D shapes because they have width, length, and thickness (depth), and we can feel these. Recognise that the geometric block shapes have different thickness or depth.
3. Write face, edge and vertex on the class chart. Have ākonga locate and identify each feature on several of the geometric blocks. Write the plurals of each word beside the singular, highlighting the word vertices. Make the connection between the 2D language of sides and corners, and the 3D terms edges and vertices.

Activity 7

Invite ākonga to share their art works and stories. Conclude by writing on the class chart, ‘We used two-dimensional shapes to make our artworks today.’

Session 2

This session is about choosing and responding to a piece of Picasso’s artwork.

SLOs:

• Recognise how shape is an important feature of Picasso’s artworks.

Activity 1

Have several ākonga share with the class their art stories from Session 1, Activity 3.
Acknowledge ākonga as artists.

Activity 2

Explain that you have a true story to tell about another artist. Read Attachment 1: Picasso. (Omit the quote in the box).
Ask ākonga what they found most interesting in the story.
Record their ideas on the class chart, summarising their learning about Picasso.

Activity 3

Write on the class chart: “Art is a lie that makes us realise the truth.” Explain this is something Picasso said. Have ākonga discuss what he might mean by this.
Elicit ideas: for example, art does not always show us how things really are (“they lie”) but we recognise this by comparing art with how things are ("the truth").’

Activity 4

1. Locate around the classroom, several individual small copies of each Picasso’s pictures from Attachment 2. There should be enough for each ākonga to have a picture of their own choice.
   Explain what you have done. You do not need to elaborate on cubist art at this stage, it will be discussed in Session 3 and 4.
2. Have ākonga silently complete an ‘art-walk’ once around the ‘gallery of artworks’. Have them make a second rotation, this time choosing and taking an artwork of their choice and returning to their place.
3. Make available paper, pencils and glue sticks.
   Have ākonga glue their chosen picture onto their paper, leaving sufficient space to write about it.
   Remind the ākonga that in Session 1 they wrote about their own artworks.
   Explain that each ākonga is to write about the Picasso picture they have chosen.
   Their writing should:
   • explain their feelings about the picture
   • explain how they think Picasso made the picture
   • include a short story about why the picture might be important to Picasso
4. Clarify the task with the ākonga, list on the class chart any special words that they might need and set a time limit.
5. Some ākonga may want to explain how their picture fits with Picasso’s statement in Step 3 above.

Activity 5

1. When the task is complete, have ākonga who chose the same picture, form a group. Have the ākonga share their responses in their groups, comparing their ideas.
2. Ask, “Who wrote something about “shapes” in their writing?”
   Ākonga take turns to talk about the way Picasso uses shapes in his pictures, including identifying the features of those shapes. For example: In Picture 1, triangles with lots of corners (angles) have been used.

Activity 6

Conclude the session by encouraging ākonga to share their writing about Picasso’s artwork.

Session 3 and 4

This session is about exploring the features of a three-dimensional shape. Ākonga recognise that a 3D shape is comprised of plane shapes, and represent 3D shapes in an artwork.

SLOs:

• Understand and describe the features of a cube.
• Create an artwork, developing a practical understanding of the relationship between 2D and 3D shapes.

Activity 1

Begin by having more ākonga read their Picasso artwork stories from session 2.

Activity 2

1. Make paper and pencils available.
   Have ākonga form pairs with their Picasso artworks. Partners should have different pictures.
   Write on the class chart the headings: Colour      Shape      Other
   Have ākonga write these headings on one shared piece of paper. Set a time limit.
   Have ākonga look at both Picasso pictures and record on the chart under the three headings the things they notice about both art works.
2. Have partners share their findings with another pair.
   Discuss as a class, highlighting (in most instances) the narrow range of colours, light/dark contrasts, different angles of geometric shapes.

Activity 3

1. Show the class a wooden cube.
   On the class chart, write cube and list its features, including the number of faces, edges and vertices. Highlight that we can view a cube from different angles.
    

2. Write cubism on the class chart. Explain that it is a name for an art style that Picasso is famous for. Have ākonga suggest what this might be and record their ideas/definitions on the class chart.
   If required, complement ideas with these points:
   Cubism:

   • uses simple geometric shapes
   • shows things from different viewpoints in any one artwork
   • sometimes breaks up (or fragments) 3D shapes into parts
   • shows the plane (2D) shapes that make up 3D shapes.

   Talk about each of these, having ākonga find and discuss examples in their artworks in front of them.

Activity 4

1. Explain that ākonga will make their own cubist artwork about one thing that is important to themselves. Refer to Picasso’s use of music/musical instruments or parts of these.
   Make available at least one copy of Attachment 3 per ākonga, scissors, glue, A4 paper, pencils, crayons/pastels.
   Explain that their completed artwork should:
   • include parts of something that are important to them personally
   • fill the A4 page
   • use the shapes or parts of shapes from Attachment 3
   • show the shapes connected or touching in some way
   • include lines they have drawn
   • include their own (limited) choice of colour in empty spaces.
2. Model the beginning of this process.
   Cut out one cube shape from Attachment 3. Discuss this with reference to the wooden cube, highlighting that this is a way of capturing a 3D shape in art.
   Cut the cube (or cuboid) picture into its component parts. It is important for ākonga to understand the differences and similarities between cuboids and cubes - a cube is a cuboid with all edges the same length. 
   Recognise and discuss the squares (and rectangles) that result. These are the 2D shapes that make up the 3D shape.
   Look at the parallelogram shapes. Discuss that this is what happens to the square and rectangular faces when they are shown in 2 dimensions.
3. Show the developing process.
   For example: a plant (leaf) may be something important to the artist.
   
   This can be cut and arranged alongside some shapes to produce the artwork.
   
    
4. Refer to the features noted in the headings: colour, shape and other, in Activity 2, Step 1 above. Are these Picasso style features reflected here?
    
5. Refer to the cubism features list in Activity 3, Step 2 above. Are these features evident here?
    
6. Identify next steps to complete the artwork. (Fill the page, include more cuboid shapes or parts of these, fill in any white paper spaces remaining with appropriate colours.)

Activity 5

Ākonga can now begin their artworks. Ask them to stop and review progress throughout, reflecting on their own work and giving feedback to others.

Activity 6

Finish artworks with a title.

Session 5

This session is about reflecting upon and consolidating the key learning about 2D and 3D shapes and about one artist.

SLOs:

• Identify and articulate key learning about geometric shapes.

Activity 1

Ākonga can display their cubist art (including titles) on their desks. Explain that ākonga will undertake a slow and silent art-walk in which they are to notice works they particularly like. They should look closely at these and decide what it is that makes them appealing to them personally.  The two stars and a wish feedback structure could be used here. That is, ākonga should give two positive comments and a suggestion for improvement to another ākonga. 

Activity 2

Have several ākonga share their ideas and feedback on the artwork they have noticed, explaining what they like about it and why. Have them refer to the artwork criteria when making their comments.

Activity 3

Make available poster paper large enough to accommodate ākonga artworks from Session 1 and Session 4, Activity 5.
Have each ākonga place (and glue) both artworks onto the poster paper, leaving sufficient space to attach a reflective comment.

Activity 4

On writing paper, have ākonga:
a. Write which of their own artworks they prefer, writing  2-3 reasons for their preference.
b. Explain what they have learned about geometric shapes through their exploration of Picasso’s art and of cubism.
c. Attach their reflections to their poster paper beneath their artworks.

Activity 5

1. Ākonga can share and display their reflections. Discuss.
2. Reflect on Picasso’s statement: “Art is a lie that makes us realise the truth.”
   Recognise that the artworks do not show things as they are, but they helped us to see some things that are true.
   On the class chart list the ‘true’ things (truth) that ākonga have learned about art and about mathematics (geometry).

Activity 6

Conclude the session by sharing some of your own favourite Picasso artworks. Discuss the fact that shape is a feature of much of his work.

Getting Started

Here the concept of a 5-mat is introduced. It is constructed from combinations of Cuisenaire rods that all have the same length as the yellow rod (5). The 5-mat is a device to help ākonga explore equality of combinations of numbers. It also helps them to see that '=' means 'is equal to'.

1. Give each pair of ākonga a set of Cuisenaire rods. Allow time for free play if the rods are new to the ākonga. During the free play, encourage building activities that lead to comparison of the length of the rods and activities that fit them together tightly. A tuakana/teina model could work well here. 
2. Conduct a class discussion about the lengths of the rods (mahi tahi model). Begin by making a staircase of the rods in increasing length. Then by covering the rods with the unit (white rods), establish the lengths as 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 times the length of the white rod.
   When Cuisenaire rods are exactly made in units of 1 cm, some ākonga may also be able to check the length by measuring.
   Draw a clearly labelled diagram on the whiteboard or as a poster, for reference.
   
3. Introduce the idea of a 5-mat.  First, ākonga take a 5-rod and put together one other combination of rods that makes 5.  For example, ākonga might make 5 as 1 + 3 + 1 (white, green, white), 4 + 1 (pink, white), 2 + 3 (red + green), or as 5 whites. Then these different combinations can be put together as shown in the diagram to make a 5-mat. Of course there are many different 5-mats, but they are all rectangles with the yellow rod (5) as one side.
   

4. Ākonga can then suggest the number combinations demonstrated by the 5-mat on the board. The 5-mat above has:

   1 + 3 + 1 4 + 1 2 + 3 1 + 1 + 1 + 1 + 1 or 5 x 1

   = 5 = 5 = 5 = 5 = 5

5. Note that it is important to keep the numbers in the order that they appear on the mat: 2 + 3 and 3 + 2 would be different rows of the 5-mat.  Note also that there are two relationships shown by the row of 5 whites; one is an addition and one a multiplication.
6. Next, demonstrate links between the relationships. Explain that the 5-mat also shows other relationships that are true. For example, 1 + 3 + 1 = 4 + 1  and 5 x 1 = 2 + 3 etc. This use of equality may seem strange to some ākonga because it does not give an answer on the right hand side. Explain that '=' means 'is equal to'.
7. Ākonga write down relationships from the 5-mat and share some with the class.
8. Connect the relationships between the mats and the real world. For example, ‘e rua ika, a toru ika’.Then another could be ‘e wha ika a tahi ika’. Number sentences could be written below. Other examples could be different coloured poi or shirts of people in the waka.

Exploring

Here the concept of 'equal' is explored further using mats of different sizes.

1. Ākonga choose a mat to make of a given size. Controlled choice of the size of the mat can allow for individual differences. The diagram below shows four rows of a 12-mat.
   
2. Ask ākonga to record the relationships shown on their mat. The mat in the diagram illustrates many relationships. For instance, 3 + 3 + 3 + 3 = 12 and 4 x 3 = 12.
   Note that 4 x 3 is interpreted as 4 groups of 3 here and not 3 groups of 4.
3. Ākonga should also record some of the relationships between rows on the mat. For example, 6 + 1 + 5 = 3 + 3 + 3 + 3.
4. Initiate a class discussion on interesting examples: for example, 5 + 7 = 7 + 5.
   Did anyone else find something like this?
   Is 4 + 8 = 8 + 4? Why?
   Did anyone else find something like this that did NOT work?
5. Other interesting examples that are worth discussing are things like 4 x 3 = 3 x 4.
   Did anyone else find something like this?
   Is 2 x 6 = 6 x 2? Why?
   Did anyone else find something like this that did NOT work?
   Rows that show a strong visual pattern may also show interesting number patterns.
6. The activity can be repeated using a mat of a different size.
7. Turn the situation around.
   Make me a mat that shows that 4 + 7 = 2 + 9.
   What other equalities can a mat like this show?
   Make me a mat that shows that 2 x 5 = 3 + 7.
   What other equalities can a mat like this show?
   Let ākonga pursue this aspect of the problem in pairs, independently or in small groups. Rove and support ākonga as necessary.

Reflecting

This section brings together what ākonga have discovered so far. 

1. Ākonga can make a poster or digital presentation (e.g. using Google Slides) of their work on a large piece of paper individually or in a pair. This could involve taking photos of some of the number mats they have made. Some ākonga can report their most interesting findings to the class.
2. Highlight the important points. This will include observations about;
   • addition (for example, 8 + 1 = 7 + 2 = 6 + 3, 8 + 1 = 1 + 8,  and 7 + 2 = 2 + 7)
   • multiplication (3 x 4 = 4 x 3)
   • and the meaning of equality.
3. Is it true that 4 + any number = that same number + 4? Why? Why not?
   Is it true that 2 x any number = that same number x 2? Why? Why not?

Possible extensions for Levels 3 and 4
This unit can be extended for ākonga working at Level 3 or Level 4.

1. Ākonga can be challenged by changing the value of the white rod from 1 to, say 2, or even 0.1.
2. Carefully removing a rod from a number mat leads to a natural setting for equation solving. For example, removing the dark green rod from the mat above, leads to equations such as
   ? + 1 + 5 = 12; and
   ? + 1 + 5 = 3 x 4.
3. A variety of other questions can be asked within the context of the number mats and checked visually. For example, I am making a 16-mat: Can I make a row just out of the light green rods (3-rod)?
   Answering this could lead to a statement such as 5 x 3 + 1 = 16.
   Use a mat to check whether 2 x 5 + 4 = 6 + 1 + 7 or 5 + 3 x 4 = 7 + 9.

Session 1: Balloons investigation

Today we will make a pictograph of our favourite balloon shapes. We are going to answer the investigative question “What different balloon shapes do the ākonga in our class like?”

1. Take a bag of balloons and spread out. Discuss shapes. Suggest the investigative question “What shape balloons do the ākonga in our class like?”
2. Ākonga choose favourite shape (or colour if different shaped balloons are not available) and draw it on a piece of paper (one eighth of an A4).
3. Ākonga work together to discuss ways to display the data. If matching pictures in 1:1 lines (pictograph) is not suggested, direct them to this.
4. Ākonga attach their drawing to the class chart.
5. Ask the ākonga what they notice about the information shown on the pictograph. Use the prompt “I notice…” to start the discussion. These “noticings” could be recorded as "speech" bubbles or on post it notes around the chart.
6. Talk about the need to label the axes and give the chart a title so that others could make sense of the display. The investigative question could be written as the chart title.
7. Ask analysis questions to extend the noticing about the results that require ākonga to combine sets:
   How many ākonga liked long wiggly balloons?
   How many ākonga liked long straight balloons?
   How many ākonga liked long balloons altogether?
   How can you add the numbers together?
   How many ākonga liked balloons that were not long?
   How many more ākonga liked long wiggly balloons than long straight balloons? All ākonga counting methods should be valused in this activity. However, it may be appropriate for you to (model and reinforce the use of subtraction or addition, rather than counting on or back.)
   Try to find analysis questions that will allow ākonga to use strategies such as near doubles and adding to make 10s.

Session 2: Birthday Party investigation

This birthday party investigation is described in full as a possible model for teaching and developing ideas for each of the stages of the statistical enquiry cycle at Level 2.  In New Zealand we use the PPDAC cycle (problem, plan, data, analysis, conclusion) for the statistical enquiry cycle.  You can find out more about the PPDAC cycle at Census At School New Zealand.

If the birthday party context is not suitable for your ākonga, choose another context (e.g. Diwali, matariki). The process described here will work for other contexts.

PROBLEM: Generating ideas for statistical investigation and developing investigative questions

1. Ask ākonga to think about the topic of birthday parties.  Explain that we are collecting some information to answer different investigative questions about birthday parties that we are going to pose.
   Using the starter “I wonder…” Ask the ākonga what they wonder about birthday parties.  Record their ideas. For example:
   I wonder…
   1. What are our favourite types of kai to eat at birthday parties?
   2. What games do we like to play at birthday parties?
   3. How many birthday parties have ākonga gone to? Consider whether this question might be inappropriate or emotionally distressing for your ākonga.
   4. Where do ākonga like to have their birthday parties?
   5. What birthday presents we want?
   6. What types of people go to birthday parties or school events?
   7. Using the “I wonder” prompt helps with generating investigative questions, questions we ask of the data. Record all questions and ideas on a brainstorming document (e.g. large chart, PowerPoint). 
       

2. The amount of work needed to tidy up the investigative questions will depend on the responses of your ākonga in the brainstorming session. New Zealand based research has identified six criteria to support the development of and/or critiquing of investigative questions. These criteria are used in the example below.  The teacher asks questions of ākonga to identify the information needed e.g. variable, group and with this information develops the investigative question. 

   • The intent of the investigative question is clear – we need to pose summary investigative questions (about category or whole number data)
   • The variable of interest is clear e.g. favourite kai, number of birthday parties/school events ākonga have been to already.
   • The group we are interested in is clear e.g. our class, Room 30, Kauri class
   • We can collect data to answer our investigative question
   • The investigative question considers the whole group e.g. What is Room 30's favourite kai to have at a celebration? – considers the whole group; whereas 'What is the most popular kai that ākonga in Room 30 like to eat at a birthday party – does not and is not an investigative question (it is an example of an analysis question and asks about an individual)
   • The investigative question is interesting and/or purposeful – in this case if the ideas are generated by ākonga then we would expect them to be interesting to ākonga.
      

   For the favourite kai at a birthday party example some possible questions are:

   • What will we want to find out about? (Favourite kai at a birthday party)
   • Who are we going to ask? (Our class)
   • Do you think we could find this out by asking our class? (Yes)
   • Are you interested in knowing about people's favourite kai at a birthday party? (Yes (if no, then change the question).)
      

   For each of the ideas generated in part 1, possible investigative questions are:

   1. What are favourite kai at birthday parties for ākonga in Room 30?
   2. What are Room 30’s favourite birthday games?
   3. How many birthday parties have the children in Room 30 been to before?
   4. Where do Room 30 children want to have their birthday parties?
   5. What presents do Room 30 children want for their birthday?
   6. How many people do Room 30 children want at their birthday parties?
       

3. Each group selects one of the investigative questions to explore.

    

PLAN: Planning to collect data to answer our investigative questions

4. Explain to ākonga that they need to think about what question or questions they will ask to collect the information they need to answer their investigative question.
    

5. Explain that these questions are called survey questions and they are the questions we ask to get the data. Work with groups to generate survey questions. For example: 

   • If the investigative question is: “What are the favourite birthday cakes for children in Room 30?”, ask ākonga how they could collect the data. 
   • A possible response is to ask the other ākonga “What is your favourite birthday cake?”
   • This might lead to a discussion about whether they mean the flavour and/or the style (a possible way to extend this for some ākonga), e.g. I might respond my favourite birthday cake is a dinosaur, when they might have been meaning the flavour e.g. chocolate, banana etc.
   • This might also mean that the investigative question needs adjusting to: What are favourite birthday cake flavours for children in Room 30?
   • Also, the ākonga might want to ask, “What is your favourite birthday cake out of chocolate, banana, vanilla and carrot?” You could challenge them as to if this would really answer the investigative question and suggest that possibly they might change the survey question to allow for other answers.
      

   Possible survey questions are:

   • What is your favourite birthday cake flavour?
   • What is your favourite game to play on your birthday?
   • How many birthday parties have you been to before?
   • Where do you want to have your birthday party?
   • What presents do you want for your birthday? (this could give multiple answers, may want to change to what is the present you most want…)
   • What is your favourite kai at birthday parties?
   • How many people do you want to have at your birthday party?

    

6. In these examples you can see that the survey question and investigative question are very similar, but there are key differences that make it an investigative question (What are favourite birthday cakes for the ākonga in Room 30? – overall about the class data) rather than a survey question (What is your favourite birthday cake flavour? – asking the individual).
    
7. Ask ākonga to think about how they will record the information they get. Options may include:
   • Tally chart
   • Writing down names and choices
   • Using predetermined options
   • Using a class list to record responses
      
8. Let ākonga try any of the options they suggest.  They are likely to encounter problems, but this will provide further learning opportunities as they reflect on the difficulties and how they can improve them.

DATA: Collecting and organising data

1. Ākonga collect data from the rest of the class using their planned method. You might provide a graphic organiser for your ākonga to use to organise their data. Consider also whether your ākonga need explicit instruction in how to record tally marks. Expect a bit of chaos. Possible issues that lead to useful teaching opportunities include:

• Predetermined options
• What happens for ākonga whose choice is not in the predetermined options?
• What if nobody likes the options given and they end up with a whole lot of people choosing "other"? They only have tally marks so they cannot regroup to new categories.
• Using tally marks only
  • The discussed issue above about the “other” category
  • Have fewer tally marks than the number of ākonga in the class 
    • and they think they have surveyed everyone
    • or they do not know who they have not surveyed yet
  • Have more tally marks than the number of ākonga in the class
• Possible solutions to the above issues could be (generated by the ākonga if possible)
  • Recording the name of the ākonga and their response and then tallying from the list
  • Giving everyone a piece of paper to write their response on, then collecting all the papers in and tallying from the papers

2. Regardless of the process of data collection we are aiming for a collated summary of the results.

ANALYSIS: Making and describing displays

1. Taking their summarised information, ākonga make a pictograph to help to answer their investigative question. As for the balloon activity we want to have uniform pieces. Provide:
   • Squares of paper all of the same size for ākonga to create their own pictures
   • Chart paper
2. Ākonga give the chart a title – a good option is the investigative question.
3. Ākonga make the pictograph by glueing enough pictures to represent the data they collected. This could be done by hand, or with an online tool.
4. Teacher roams, questioning for understanding and ensuring that ākonga can correctly construct a pictograph.
5. Once ākonga have completed their pictograph they should write or share 2-3 "I notice…” statements about their pictograph. These could be written on post it notes and stuck onto the chart. This could also be filmed and presented as a mini movie or set of slides with voiceover accompaniment.
6. Teachers can prompt further statements by asking questions such:
   • What do you notice about how many ākonga liked cakes that were not chocolate?
   • What do you notice about the number of birthday parties attended? Did you notice the greatest number of birthday parties? The least number of birthday parties?
   • Emphasise questions that require ākonga to operate with the numbers in their displays.
7. Check the “I notice…” statements for the variable and reference to the class.  For example: “I notice that the most favourite birthday cake flavour for Room 30 children is chocolate cake.”This statement includes the variable (favourite birthday cake flavour) and the class (Room 30 children). Support ākonga to write statements that include the variable and the group.
    
8. Ask ākonga to leave their charts on their desks.  Hand out post-it notes to the ākonga and ask them to wander around the class and to look at all the other graphs.  Encourage them to add “I notice…” statements to the graphs of others by using post-it notes.
    

CONCLUSION: Answering the investigative question

At the end of the session get each group to share their chart. They should state their investigative question and then the answer to the investigative question. The answer should draw on the evidence from their graph and their “I notice…” statements.

For example: What are some favourite birthday cake flavours for children in Room 30?

Answer: The most popular birthday cake flavour for Room 30 is chocolate cake. 15 ākonga in our class had chocolate as their choice. The other flavours that were liked included carrot cake, banana cake and ice-cream cake.  Carrot cake was the least popular cake flavour for Room 30.

Extending: If I (the teacher) was to make a cake for the class what flavour should I make?

Session 3: Popcorn

The previous session involved the full PPDAC cycle.  In this session today we are going to look at using tally marks to record the number of pieces of popcorn in a small cup and a bar graph to display the data.  We are focusing on the data collection and analysis phases.

1. Display a small cup (or bag) of popcorn and ask ākonga to guess how many pieces of popcorn they think are in the cup.
2. Pose the investigative question: How many pieces of popcorn are in the small cups?
3. We are going to collect data to answer our investigative question by counting how many pieces of popcorn are in each of the cups I have here (count the number of pieces of popcorn in the small cup).
4. How should we do that? Elicit ideas including counting them all. Ask how we could count them and keep a track? Accept all ideas including using tally marks to keep a track.
5. Teacher models using tally marks to track how many pieces of popcorns she/he counts. Ensure hands are washed and tables are clean.
6. Distribute individual cups of popcorn to small groups. You may wish to provide one cup of popcorn per ākonga, or use the tuakana/teina model and provide one between two.
7. Ākonga count the pieces of popcorn and use tally marks to record the number of pieces of popcorn in each cup. They should add together the total of the tally marks each ākonga in the group recorded. (Record the number of pieces of popcorn in the cup, but don’t combine, later we will use each ākonga cup count as a data point).
8. Gather the total tallies on the board or a chart.
9. Using a prepared bar graph outline, the teacher constructs a bar graph with the information from the individual total tallies.
10. Discuss features of the graph and summarise the information shown.
    What was the most common number of pieces of popcorn?
    What was the least common number of pieces of popcorn?
    How many more pieces of popcorn were there in the cup with the most, than the one with the least?
11. As a class challenge, try to work out how many pieces of popcorn the class counted altogether.
    How many pieces of popcorn did each table group count?
    Discuss strategies for adding the numbers together (for example: combine the numbers that add to 'tidy' numbers; use place value; use doubles or near doubles).

Session 4. Favourites

In this session we will undertake a statistical investigation using the idea of favourites as our starting point.  The big ideas for the investigation are detailed in session 2.  Ideas to support the specific context are given here.

PROBLEM

Brainstorm with ākonga different things that they have a favourite of. You might use the starter “I wonder what are favourite _________ for our class?”

Using the ideas developed previously, identify 10-15 favourites to be explored and develop investigative questions for pairs of ākonga to explore. A tuakana/teina model could be used here.

Investigative questions might be:

• What are favourite sports that the children in our class play?
• What are our class’s favourite waiata?
• What are Room 30’s favourite kai?

PLAN

As ākonga have had some practice with planning previously, allow them some freedom, as appropriate, to plan their data collection. Check in on the survey questions they are planning to ask. Encourage ākonga to use the tuakana/teina model to support their learning journey.

DATA

Ākonga collect the data that they need to answer their investigative question. Be prepared for some potentially inefficient methods. Use any resulting errors or problems to improve their data collection methods.

ANALYSIS

Ākonga can display the data to answer their investigative question.  They may use a pictograph or a bar graph.  Remind them to label using the investigative question and to write “I notice…” statements about what the data shows.

CONCLUSION

Allow time for pairs to present their findings by giving their investigative question and then answering it using evidence from their displays and noticings. 

This series of lessons provides different contexts to explore multiplication concepts using arrays such as the one below. This array has 5 rows and 10 columns.

Session One: Getting started

1. We begin the week with the ‘Orchard Problem’. A picture book about gardens, such as Nana's Veggie Garden - Te Māra Kai a Kui by Marie Munro, could be used to ignite interest in this context.
   Jack the apple tree grower has to prune his apple trees in the Autumn. He has 6 rows of apple trees and in every row there are 6 trees. How many apple trees does Jack have to prune altogether?
   

The start of PowerPoint 1 shows the whole array. Show the complete array. Ask your students to open their eyes and take a mind picture of what they see. Click once to remove all the trees and ask your students to draw what their mind picture looks like. One child could draw their picture on the whiteboard. This could then be referred back to throughout the rest of the lesson.

Look to see if they attend to the rows and columns layout even if the numbers of trees have errors. Discuss the layout.

1. Have a pile of counters in the middle of the mat. Ask a volunteer to come and show what the first row of trees might look like. Or get 6 individuals to come forward and act like trees and organise themselves into what they think a row is.
   Alternatively click again in the PowerPoint so it’s easy for all to see what the first row of apple trees will look like. Ask your students to improve their picture if they can.
   What will the second row look like? 
   It’s important for students to understand what a row is so they can make sense of the problem. It is also important for them to notice that all rows have the same number of trees.
2. Arrange the class into small mixed ability groups with 3 or 4 students in each. Give each group a large sheet of paper. Ask them to fold their piece of paper so it makes 4 boxes (fold in half one way and then in half the other way).
3. Allow some time for each group to see if they can come up with different ways to solve the Orchard Problem and record their methods in the four boxes. Tell them that you are looking for efficient strategies, those that take the least work.
   Allow students to use equipment if they think it will help them solve the problem.
   Rove around the class and challenge their thinking with questions like:
   • How could you count the trees in groups rather than one at a time?
   • What facts do you know that might help you?
   • What sets of numbers do you know that might help you?
   • What is the most efficient way of working out the total number of trees?
4.  Ask the groups to cut up the 4 boxes on their large sheet of paper and then come to the mat. Gather the class in a circle and ask the groups to share what they think is their most interesting strategy. Place each group’s strategy in the middle of the circle as they are being shared. Once each group has contributed, ask the students to offer strategies that no one has shared yet. 

5. Likely strategies	Possible teacher responses
   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 …. Tahi, rua, toru, whā, rima, ono, whitu, waru, iwa, tekau, tekau mā tahi…	Can you think of a more efficient way to work out how many trees there are? How many trees are there in one row?
   6, 12, 18, 24, 30, 36	Do you know what 6 + 6 =? Or 3 + 3 = ?  Can that knowledge help you solve this problem more efficiently?
   6 x 6 = 36	What if Jake had 6 rows of trees and there were 7 trees in each row?
   6 + 6 = 12; 12 + 12 = 24; 24 + 12 = 36	You used addition to work that out.  Do you know any multiplication facts that could help?
   2 x 6 = 12; 12 + 12 + 12 = 36	If 2 x 6 = 12, what does 4 x 6 =? How could you work out 6 x 6 from this?
   3 x 6 = 18 and then doubled it	That is very efficient. Could you work out 9 rows of 6 for me using 6 x 6 = 36?
   5 x 6 = 30; and 6 more = 36

   The shared strategies can be put into similar groups.
   Who used a strategy like this one?

6. Show students PowerPoint 2. The PowerPoint encourages students to disembed a given smaller array of trees from within a larger array. They are also asked to use their knowledge of the smaller array to work out the total number of trees in the larger array. This is a significant ability for finding the totals of arrays using the distributive property of multiplication.
7. Provide your students with Copymaster 1. The challenge is to find the total number of trees in each orchard. Challenge your students to find efficient strategies that do not involve counting by ones.
8. As a class, share the different ways that students used to solve the Orchard Problems. You might model on the Copymaster to show how various students partitioned the arrays.

Sessions Two and Three: Exploring through work stations

The picture book Hooray! Arrays! by Jason Powe could be used to ignite interest in this learning. In the next two sessions students work in pairs or threes to solve the problems on Copymaster 2. Consider choosing these pairs to encourage tuakana teina through the pairing of more knowledgeable and less knowledgeable students. Enlarge the problem cards and place them at each station. Provide students with access to copies of Copymaster 3 and Copymaster 4 (arrays students can draw on), and physical equipment such as counters, cubes, and the Slavonic Abacus.

Read the problems from Copymaster 2 to the class one at a time to clarify the wording. You may need to revisit the meaning of rows and columns by creating simple examples.
As students work on a station activity, ask them to create a record of their thinking and solutions. The record might be a recording sheet or in their workbook. Note that Part 2 of each problem is open and requires a longer period of investigation.

As the students work watch for the following:

• Can they interpret the problem wording either as a physical representation or as symbolic equations?
• Do they create arrays of equal rows and columns?
• Are they able to use skip counting, additive or multiplicative strategies to find the total number of trees?
• Do they begin to see properties of whole numbers under multiplication? (for example, Apple Orchard Part 2 deals with the commutative property)

At times during both sessions you might bring the class together to discuss confusions or misconceptions, clarify language and share efficient strategies and ways of representing the problems.

Below are specific details related to each problem set.

Orange Orchard

Orange Orchard (Part 1) involves 6 x 8 (or 8 x 6). Students might use their knowledge of 6 x 6 = 36 and add on 12 more (two columns of six). That would indicate a strong understanding of the multiplicative structure of arrays.

Most students will use strategies that involve visualising the array and partitioning the array into manageable chunks (dis-embedding). For example, they might split rows of eight into two fours (6 x 8 = 6 x 4 + 6 x 4), or into fives and threes (6 x 8 = 6 x 5 + 6 x 3). Other students will use less sophisticated strategies such as counting in twos and fives, or a combination of skip counting and counting by ones.

Part 2 is an open task which requires students to identify the factor pairs of 24.

Encourage capable students to be systematic in finding all the possibilities (1 x 24, 2 x 12, 3 x 8, 4 x 6).

Kiwifruit Orchard

Part 1 requires students to coordinate three factors as the problem can be written as 3 x (4 x 5). Multiplication is a binary operation so only two factors can be multiplied at once. Do your student recognise the structure of a single orchard (4 x 5) and realise that the total is consists of three arrays of that size?

Similarly, in Part 2 students must restructure 36 plants into two sets. Do they partition 36 into two numbers, preferably that have many factors? The problem does not say that the two orchards must contain the same number of plants though 18 and 18 is a nice first solution. Once the two sets of plants are formed can your students find appropriate numbers of rows and columns that equal the parts of 36?

Strawberry Patch

Part 1 is a single array (5 x 12). Students might use the distributive property and solve the problem or 5 x 10 + 5 x 2 (partitioning 12) or 5 x 6 + 5 x 6. Some may re-unitise two fives as ten to create 6 x 10. These strategies are strongly multiplicative. Most students will use smaller units such as fives or two and apply a combination of repeated addition (5 + 5 = 10, 10 + 10 = 20, etc.) or skip counting (2, 4, 6, 8, …).

Part 2 is about factors that have the same product (24). This gives students a chance to recognise that some numbers have many factors and the expressions of those factors have patterns. For example, 6 x 4 and 3 x 8 are related by doubling and halving. The logic behind the relationship may be accessible for some students. If the rows are halved in length, then twice as many rows can be made with the same number of plants.

Apple Orchard

Part 1 gives students a chance to ‘discover’ the commutative property, the order of factors does not affect the product. In this case 5 x 10 = 10 x 5.

Part 2 applies the distributive property of multiplication though many students will physically solve the problem with objects. Look for students to notice that 12 extra trees shared among six rows results in two extra per row. So, the number of rows stays the same, but the rows increase in length to six trees. Similarly, if more rows are made the 12 trees are formed into three rows of four. The number of rows would then be 9. 6 x 6 and 9 x 4 are the possible options.

Sessions Four and Five

Sessions Four and Five give students an opportunity to recognise the application of arrays in other contexts.

The chocolate block problem involves visualising the total number of pieces in a block even though the wrapping is only partially removed. PowerPoint 3 provides some examples of partially revealed chocolate blocks. For each block ask:

• How many pieces are in this block?
• How do you know?

Look for students to apply two types of strategies, both of which are important in measurement:

Iteration: That is when they take one column or row and see how many times it maps into the whole block.

Partitioning: That is when they imagine the lines that cut up the block, particularly halving lines. They look to find a partitioning that fits the row or column that is given.

Copymaster 5 provides students with further examples of visualising the masked array.

The Kapa Haka problem is designed around the array structure of seating arrangements for Kapa Haka performances at school.

Begin by role playing the Kapa Haka problem. Use chairs to make a simulated arrangement of seats. You might like to include grid references used to locate specific seats.

Try questions like:

• How many rows are there? How many columns are there?
• How many audience members could be seated altogether?
• If the performance needed 24 seats what could they do?

Use different arrangements of columns and rows.

Give the students counters, cubes or square grid paper to design possible seat layouts with 40 seats. Encourage them to be systematic and to look for patterns in the arrangements. Some students will find efficient ways to record the arrangements such as:

2 rows of 20 seats                4 rows of 10 seats                5 rows of 8 seats

Record these possibilities as multiplication expressions on rectangles of card. Put pairs of cards together to see if students notice patterns, like doubling and halving.

It is important to also note what length rows do not work.

• Could we make rows of 11 sets? 9 seats? Why not? (40 is not divisible by 11 or 9 as there would be remaining seats left over)

If students show competence with finding factors, you could challenge them to find seating arrangements with a prime number of seats such as 17 or 23. They should find that only one arrangement works; 1 x 17 and 1 x 23 respectively.

Reflecting

As a final task for the unit, ask the students to make up their own array-based multiplication problems for their partner to solve.

1. Tell the students that they are to pretend to be kūmara growers. They decide how many rows of kūmara plants they want in each row and how many rows they will have altogether. As part of this learning, you could look into how early Maori people grew kūmara. This plant arrived in New Zealand with Polynesian settlers in the 13th Century. However, the climate here was much colder than the Polynesian islands. As a result, the kūmara had to be stored until the weather was warm enough for it to grow. The kūmara plant became even more important once settlers discovered that some of their other food plants would not grow at all in New Zealand’s climate. These kūmara were different to the ones we eat today - which came to us from North America. The books Haumia and his Kumara: A Story of Manukau by Ron Bacon, and Kumara Mash Forever by Calico McClintock could be used to engage students in this context.
2. Then they challenge their partner to see if the partner can work out how many kūmara plants they will have altogether.
3. Tell the students to create a record of their problem with the solution on the back. The problems could be made into a book and other students could write other solution strategies on the back of each problem page.
4. Conclude the session by talking about the types of problems we have explored and solved over the week. Tell them that the problems were based on arrays. Let them know that there are many ways of solving these problems, tough multiplication is the most efficient method. Ask students where else in daily life they might find arrays.

Getting Started

Today we look at the number patterns in a tower of tins (tini).

1. Tell the students that today we will stack tins for a supermarket (hokomaha) display.

2. Show the students the arrangement:

   

   How many tins are in this arrangement?
   How many tins will be in the next row (kapa)?
   Then how many tins will there be altogether?
   How did you work that out?

3. Encourage the students to share the strategy they used to work out the number of tins. “I can see 4 tins and know that you need 5 more on the bottom. 4 + 5 = 9”

   “I know that 1 + 3 + 5 = 9 because 5+3= 8 and 1 more is 9.”
   [These strategies illustrate the student’s knowledge of basic addition facts.]

4. Show the students the next arrangement of tins. They can check that their predictions were correct.

   

   How many tins will be in the next row? 
   Then how many tins will there be altogether?
   How did you work that out?

5. Encourage the students to share the strategies they used to work out the number of tins.
   “I know that we need to add 7 to 9 which is 16.” [knowledge of basic facts]
   “I know that 7+ 9 = 16 because 7 + 10 = 17 and this is one less." [early part-whole reasoning]
   “I know that we are adding on odd numbers each time. 1+3+5+7 = 16 because 7+3 is 10 + 5 + 1 = 16."

6. Add seven tins to the arrangement and ask the same questions. As the numbers are becoming larger expect the range of strategies used to be more varied.

   

   “16 + 9 = 25. I counted on from 16.” [advanced counting strategy]
   “16 + 10 = 26 so it is one less which is 25.” [part-whole strategy]

7. Tell the students that the supermarket has asked for the display to be 10 rows high.
   How many tins will you need altogether?
8. Ask the students to work in small groups to find out how many tins are needed. As the students work circulate asking:
   How are you keeping track of the numbers?
   Do you know how many tins will be on the bottom row? How do you know?
9. Gather the students back together as a class to share solutions.
10. Discuss the methods that the groups have used to keep track of the number of tins.
11. Work with students to make a table showing the number of rows and total number of tins. Complete the first couple of rows together.
12. Ask the small groups to complete their own copy of the table on Copymaster 1. As they complete the chart ask:
    Can you spot any patterns?
    Write down what you notice?
    Can you predict how many tins would be needed when there are 15 in the bottom row?
13. Encourage the students to explain their strategies for “counting” the numbers of tins.
14. As a class, share the patterns noted.

Exploring

Over the next 2-3 sessions the students work with a partner to investigate the patterns in other stacking problems. Consider pairing together students with mixed mathematical abilities (tuakana/teina). We suggest the following introduction to each problem.

1. Pose the problem to the class and ask the students to think about how they might solve it. In particular encourage them to think about the table of values that they would construct to keep track of the numbers.
2. Share tables.
3. Ask the students to work with their partner to construct and complete their own table.

4. Write the following questions on the board for the students to consider as they solve the problem.

   How many tins are in the first row? 
   How many are in the second row?
   By how much is the number of tins changing as the rows increase?
   What patterns do you notice?
   Can you predict how many tins would be needed for the bottom row if the stack was 15 rows high?
   Explain the strategy you are using to count the tins to your partner?
   Did you use the same strategy?
   Which strategy do you find the easiest?

5. As the students complete the tables and solve the problem, circulate and ask them to explain the strategies that they are using to “count” the numbers of tins in the design.
6. Share solutions as a class.

Problem 1:

Copymaster 1

A supermarket assistant was asked to make a display of sauce tins. The display has to be 10 rows high.
How many tins are needed altogether?
What patterns do you notice?

Problem 2:

Copymaster 2

A supermarket assistant was asked to make a display of sauce tins. The display has to be 10 rows high.
How many tins are needed altogether?
What patterns do you notice?

Problem 3:

Copymaster 3

A food demonstrator likes her products displayed using a cross pattern. The display has to be 10 products wide.
How many products are needed altogether?
What patterns do you notice?

Reflecting

In this session the students create their own “growth” pattern for others to solve.

1. Display the growth patterns investigated over the previous sessions.
2. Gather the students as a class and tell them that their task for the day is to invent a pattern for the supermarket to use to display objects.
3. Ask the students in small groups to decide on a pattern and the way that it will grow. (A supply of counters may be helpful for some students.)
4. Direct students to construct a table to keep track of their pattern (up to the 10^th model). Model how to construct and use this. Alternatively, you could provide a graphic organiser for students to use.
5. Once they have constructed the table ask them to record the any patterns that they spot in the numbers. Ask them also to make predictions about the 15^th and 20^th model. 
6. Direct students to swap problems with another group. When the problem has been solved, they should compare solutions with each other.

Session 1

We start this unit with a guessing game which introduces the idea of estimation. Consider how the mystery object you choose might reflect the learning interests and cultural make-up of your class. 

1. Show ākonga the outline of an object, for example; a small book, a shell or a rākau stick.
   What do you think that this could be the outline of?
   How many cubes do you think I would need to cover this shape?

2. Give each student a 1cm cube and ask them to write their guess on a piece of paper. Introduce the idea that an estimate is a thoughtful guess. 
    

   I think the area of the mystery object is ........... cubes Lillie-Moana

3. Show the class a shape made with 5cm cubes, for example a rākau stick

   Ask ākonga to record the shape on cm squared paper.
   What is the area of this shape?

4. If ākonga say 5 squares tell them that the unit square is called a square centimetre.
   Why do you think it is called a square centimetre?
5. Ask a volunteer to make a different shape with the 5 cubes. Tell them that the shape must be flat and the whole sides of the squares must touch.
   What is the area of this shape? (5 square centimetres or 5 square cm)
6. Give each student 5 cm cubes and challenge them to find other shapes that can be made with the cubes. Ask them to record the shapes on the cm grid paper.
   
   (These shapes are called pentominoes and there are 12 distinct shapes that can be made. Some ākonga may wish to explore this concept further).
7. Share shapes. Check again that the ākonga understand that each has an area of 5 square cm. You may wish to introduce the recording device of 5cm², although this is not the purpose of this unit. 
8. As an extension, ākonga could make larger shapes in response to a prompt (e.g. remember when we looked at tukutuku panels yesterday, can you make a new pattern that would fit on a tukutuku panel?) and could estimate and measure the area of these larger shapes. Encourage ākonga to use effective counting methods (e.g. skip counting in fives, repeated adding). This could be further adapted by changing the size of the shapes ākonga make (e.g. 2cm2, 3cm2.)

Session 2

1. Look at the outline of the mystery object from yesterday.
 How can we work out whose guess was closest to the area of the object? 
2. Give each pair of ākonga an outline of the mystery object and ask them to work out its area in square centimetres. These pairs could be based on the tuakana/teina model to encourage shared learning. Have centimetre cubes and squared paper available and support ākonga to make decisions about how they will measure the area. Share areas and approaches used.
3. Talk about how to handle part squares. Within this, draw on the understandings of halves and quarters that is demonstrated by your ākonga.
4. Ask ākonga to write what they think the object is, and their measurement for its area, on the object’s outline. Display the outlines on a Mystery Object chart.

Session 3

1. Pose the question: What objects do you think have about the same area as our Mystery Object? Note that ākonga will need to use their estimation skills to accurately identify objects of a similar area and discuss possible estimation strategies. 
2. Brainstorm ideas for objects that have about the same area as the Mystery Object. Write the names of these objects on strips of paper and put them into a hat.
3. Working with a partner, ākonga take a strip, and find the object it names. They then make an outline of the object, calculate its area, and write the name of the object and its area on the outline. Ākonga could use cm² to record their area.
4. At the end of the session work together (mahi tahi) to order the objects measured from smallest area to largest area, and identify objects with a similar area to the Mystery Object.

Session 4

1. Establish a challenge: Today we’re going to challenge ourselves to identify objects with a specific area. We’ll need to use our estimation skills.  
2. Before the session, fill the hat with strips of paper. Each strip needs to have the measurement of an area written on it. Include several strips of the same measurement.
3. Ākonga work in pairs to take a strip with the measurement of an area, and draw or find five objects with that area. Encourage tuakana/teina by pairing more knowledgeable and less knowledgeable ākonga together. You might provide a poster or digital presentation of objects that match the given measurements, to prompt ākonga in their thinking. Consider how these objects could support links to the cultural make-up and learning interests of your class (e.g. if learning about traditional Māori games and activities, objects could include poi, manu tukutuku (kites), and ruru (knucklebones).
4. As a class, review the task together (mahi tahi) and find out how successful ākonga were at estimating the area. Discuss useful estimation strategies.
5. Ākonga who have been working with the same measurement compare results and discuss any differences, checking each other’s measurements.

Session 5

Today we use the measurement skills we've been working on to find out who has the smallest and largest footprint in our class.

How could we find out?
About how many square centimetres do you think it would be? Why do you think that?

1. Ask small groups of ākonga to think about a way of measuring footprints to find out who has the smallest and who has the largest.
2. Support ākonga to draw outlines of their feet. If a variety of measuring ideas have emerged, you could model a few (or all) of the ideas and choose one idea to follow as a class.
3. When the outline is made the ākonga need to work out the area of their footprint.
4. Share outlines and measurements. Display from smallest to largest.

Prior Experience

The activities are mostly open ended so they cater for a range of achievement levels. It is expected that students have some experience with naming and classifying basic geometric shapes, with measuring weight in kilograms, and with translating, reflecting and rotating shapes. They should also have place value knowledge to at least 200.

Session One

1. The Māori New Year is celebrated at a different time each year. That is because the date depends on two events, the rising of the star cluster Matariki and the arrival of a new moon. In June, Matariki and the other six or eight stars of the cluster become visible in the Eastern Sky about 30 minutes before dawn. This is known as the rising of Matariki as, for the month prior to Matariki, it is below the horizon. After the rising of Matariki, Māori look for the next new moon to signal the New Year. The week prior to the new moon, excluding the night of no moon, is when Matariki is celebrated. Slide 1 of the PowerPoint shows the best known seven stars of Matariki. Slide 2 shows how to find Matariki should you want to organise a pre-dawn star spotting expedition. 
2. The Tūwharetoa legend of Tamarereti has connections to Matariki. Versions of the story vary but all name him as responsible for creating the stars in the night sky. Slides 3-8 tell the legend in abbreviated form. You may like to show an animated version of the story.
3. Tamarereti cast the shining stones into the heavens on his journey across Lake Taupō. The stones stuck in the dark night sky to become the stars. Ranganui, the sky father, put Tamarereti’s waka into the sky in honour of his deeds, and the waka appears today as the Milky Way. The Southern Cross and Pointers make up the anchor and rope of this great canoe (see slide eight).
4. Show the students slide 9 which shows a satellite picture of Lake Taupō. Click to discuss where Tamarereti drifted to while asleep, cooked his fish, then set off from to return to his village.
5. Select a student to act out the next part of the lesson. 
   What the legend does not tell you is that Tamarereti collected 200 bright shiny stones and put them at the bottom of the waka.
   Have a ‘waka’ with 200 counters ready for the student to act out the story. Any narrow container will make a good waka.
6. Tell the student to start rowing, then grab a small handful of stones to throw into the sky. Remember that the stones have to last the whole journey so Tamarereti cannot use them up all at once. Ask the student to cast the counters onto a sheet of art paper so the whole class can see.
7. Ask: Is there a way to group these stones to count them easily?
   Look for students to suggest ways to group the counters. Combinations that add to ten are especially useful. 
8. Tell students: Tamarereti is being careful not to use all of his stones up because the Taniwha will eat him if he is unable to see. He wants to know how many stones he has left. How could he work that out?
9. Let the students work out the remaining number in pairs. Then share the different ways the answer could be achieved. Look for part-whole strategies rather than counting back. For example if 15 stones are thrown into the sky subtracting ten then five is a good strategy. If 19 stones are thrown then subtracting 20 and adding one is effective. 
10. Ask: What might Tamarereti scratch into the side of his waka to keep track of his number of stones?
11. Invite suggestions from the students about how to record the number of stones. Write on a map of Tamarereti’s journey the first stone toss or use the animation on Slide 5 to show how the journey might be recorded. Let the students work in pairs to act out and record Tamarereti’s journey across Lake Taupō. Each pair will need a container, 200 counters and a copy of Copymaster 1. The number of counters can be lessened or increased to vary the challenge. Expect students to manage the distance to go on the map and the number of stones left. They should record the results of their calculations as each handful of stones is cast into the heavens. The grid system on Copymaster 1 could be used to create coordinates so students can indicate the position of Tamarereti’s waka each time he casts out shining stones.
12. After a suitable period, bring the class back together to discuss the strategies they used to calculate the remaining stones. Use an empty number line or two connected hundred bead strings to illustrate strategies as students suggest them. 
    • Back through ten (173 – 16 = 157)
      
    • Tidy numbers and compensation (145 – 29 = 116)
      
    • Standard place value (156 – 34 = 122)
      
13. You might use the work samples students produce as evidence of their additive thinking.

Session Two

1. Slide 11 of the PowerPoint shows the phases of the moon. Ask the students why the moon (marama) changes appearance. Some may know that the change is caused by the moon’s orbit around the Earth and the extent to which the half of the moon lightened by the sun’s rays is visible. You can demonstrate this with a ball and a lamp.
2. The phases of the moon are important to Māori as they indicate which days are best for traditional food gathering, particularly fishing. Slide 12 shows a page from Mathematics Across Cultures (1992). Ask the students to interpret the calendar.
   • Why does the month only have 30 days? That is the length of one moon orbit of the Earth (actually 29.5 days).
   • When are the best days to fish in the lunar month? (The red days which are days 18, 24, and 25)
   • When are the worst days of the month to fish? (The first 2 days of the new moon, days 6-7, 10, 16, 20-22, and the last two days of the old moon). 
   • So what are the best days to fish during Matariki? Matariki is celebrated in the last quarter of the lunar cycle but not on the day of the new moon. 
3. Use the timeanddate.com website to capture the lunar calendar for the current month. Give each student a copy of the calendar and ask them to make a puzzle for a classmate. They do that by cutting the calendar up into jigsaw pieces. Set the maximum number of pieces to eight and tell the students to use the straight lines of the calendar to cut along. They can cut vertically or horizontally so shapes like an L or a Z are encouraged.
4. Once they have cut up their calendar, students give their pieces to a partner to reassemble. Look to students to attend to the progression of days at the top of the calendar, the maximum of seven days in each row, and the sequence of whole numbers to put the puzzle back together. Have the students glue their completed calendar into their mathematics book.
5. With their calendar intact students can answer these questions:
   How do we find out the date of the full moon from this calendar? 
   So when will the last quarter start? 
   When are the good days for fishing? 
   When will the new moon appear?
   So when does the New Year start? 
6. Tell the students that, in honour of Matariki, they are going fishing. If the day is not a good fishing day, wish them luck. If it is a good day for fishing, say you are expecting a lot of success. The fishing game can be played in two ways:
   • Cut the fish out (see Copymaster 2) and attach a paper clip to each fish. Make a fishing rod using a stick, a piece of string and a magnet (magnetic strip is a relatively cheap way to do this). Students capture a fish by getting it to stick to the magnet.
   • Cut out the fish cards and turn them upside down. Players take turns to choose a fish.
7. The game can be played in pairs or threes. The object of the game is for each player to gather fish that add to 100. They do that as often as they can. At any time players can trade fish with each other to make 100.
8. Once the students have played the game on Copymaster 2, gather the class to share another legend. Māui was known as a trickster. It was Matariki, the New Year, and it was very cold outside. Māui’s brothers were getting bored (again) so he decided to play a trick on them. He made up the second set of fishing cards (see page 2 of Copymaster 2). The brothers tried for a long time to make 100 with the fish. They could not. Can you?
9. Let the students try the second fishing game to see if they can do better than Māui’s brothers. It is actually impossible to make 100 with the cards but see if your students can figure out why. They may need to take the game home to their whānau to see if anyone can explain how Māui’s clever trick works. All of the numbers on the fish are answers to the nine times table so the totals must always be in the nine times table (multiples of nine). 100 is not a multiple of nine.

Session Three

Matariki is a time of cultural pursuits and feasting to celebrate the New Year ahead. The hāngī or earth oven has particular significance at the time of the new moon after the rise of Matariki in the eastern pre-dawn sky. Matariki is the star at the bow of Te Waka o Rangi and her travels around the sky for eleven months of the year are exhausting. It is said the steam of the first hāngī in the New Year rises into the sky and replenishes the strength of Matariki. From the offerings she gathers strength to lead the giant canoe for another year. Without Matariki at the bow the canoe cannot travel and Taramainuku cannot cast his net to gather the souls of the departed. At the New Year the names of the dead are called out so the souls of the departed may be cast into the heavens as stars.

There are many resources already available about hāngī.

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

The notes below are an adaptation more suitable for Level 2 students.

1. Tell your students about the types of food that are usually cooked in a hāngī. Chicken, pork and lamb are the most common meats used and the vegetables tend to be root crops like kūmara, potato and pumpkin. Stuffing is also popular. Before your class can plan the hāngī you will need to find out what people like to eat.
2. Your investigation question is “What hāngī foods do people in our class like to eat?”
3. Copymaster 3 has a photocopy sheet of ‘choice squares’. Put a container such as a shoebox or 2L plastic ice-cream container in the centre of the room. That is where the data will be placed. Show the students the first page of the Copymaster.
4. Ask: If you want to eat any of these foods at our hāngī you need to cut out that square and put it into the box. Should there be some restrictions on what you can eat?
   Students might mention that people should not eat every meat and every vegetable. Agree on some restrictions like one or two meats and up to three vegetables. Point out that stuffing is a yes or no choice.
5. Explain that the data will be used to order the food. “If someone chooses two meats while another person chooses only one meat, how will we deal with that?” Students might suggest that a person choosing two meats can put in one half of each square while a person choosing one meat might put in the whole square. 
6. Give the students time to make their choices and put the squares of the food they choose into the container. It is important that they cut out squares rather than the food within the squares as scale is important for possible data displays. Once you have brought the class together in a circle on the mat, empty the container of squares.
7. Ask: How might we organise these data so we can order food for the hāngī?
8. Students should suggest putting the squares into categories so get a few students to sort the data into piles. Ask, “How might we show the data so the number of squares for each food is easier to see?” After some discussion you should end up with a picture graph made with the squares. Managing the half squares should provoke a discussion about how large fractions such as five halves are. You might glue the squares in place on a large sheet of paper and add labels and scale to the axes. The graph might also be given a title. If relevant, you could use a digital graphing tool (e.g. Microsoft Excel, Google Sheets) to create a spreadsheet and bar graph.
   
9. Once the data display is complete, put the students into small groups to discuss “How might we use this data to order food for the hāngī?” After a suitable time, gather the class to share ideas. Expect students to consider the idea of a portion, that is how much of a food is reasonable as part of a meal. For example, one pumpkin is too much for a single portion so a fraction such as one eighth or one tenth is more sensible.
10. Share the information about meat (see PowerPoint slide 13) for a poster about this information). The poster has some questions for the children to consider. Have a set of kitchen scales available to identify objects around the room that weigh the same as a lamb chop or a size 14 chicken. You might use the scales to count in lots of 100 grams to find out how many portions are in one kilogram of meat.
11. Ask the students to work with a partner to decide how much of each food to buy. Look for them to consider the data on preferences you have collected, the information about portions of meat and their estimates of how much of each vegetable is required for each portion. You may decide to pool the data across several classes to make the task more challenging and avoid having a lot of pork left over! The students should produce a shopping list with clear working about how they decided on each amount.
12. Share the shopping lists and agree on suitable amounts of each food. The amounts of vegetables are likely to be expressed as numbers of whole vegetables, e.g. two pumpkins, which will add interest to the next part of the lesson – working out the cost per person. Copymaster 3 has a fictitious flyer from the local butcher and fruit and vegetable shop so that the students can create a budget for the hāngī food (see also Slide 13 of the PowerPoint). Allow students to use a calculator if they need to. Some may like to use a spreadsheet to keep track of their budget. Students will need to convert from numbers of vegetables into kilograms by estimating. For example, four or five good sized potatoes weigh 1 kilogram. Students may realise that they need a recipe for stuffing so they can calculate how much bread to order. Let them search for a stuffing recipe. Onions are an important ingredient in stuffing.
13. The final part of the budget is to work out a cost per person. This is a sharing context. The total cost, say $75, is divided equally among all the people in the class. Look for students to realise that the operation needed is division. You may need to link to simpler sharing problems so they connect the equal sharing to division and can write an equation for the solution, e.g. 75 ÷ 25 = 3. Talk about the meaning of the numbers in the equation, e.g. 3 represents $3 per person.

Session Four

Matariki was a time when food was already stored, and it was cold outside. So whānau (families) spent time together engaging in cultural pursuits such as storytelling, arts and games. Whai (string games) were popular with tamariki (children) and adults alike, especially when they involved co-operation. Whai has a long history and is common to many indigenous cultures around the world, including the indigenous tribes of North America. Traditionally whai was played with twine made from flax. The best man-made fibre to use for whai is nylon since it slides and flexes, and is soft on your hands. It is commonly used to form lines for brickwork so is available at most hardware stores in a variety of colours. Nylon string is usually available in craft shops.

1. Ask your students to make a tau waru (number 8) loop by wrapping the string loosely around their palms eight times, cutting the string, knotting it with minimal wastage, and trimming any loose ends. 
2. Whai relies on algorithms that are standard procedures. Algorithms are an important part of mathematics. Processes that initially take some time to master become standardised routines. The more complicated whai rely on the some basic moves being well known by the maker. That is where you should start with your students.
3. Dasha Emery has created an excellent series of videos that give clear instructions about making well known whai. A good first move for students to learn is called “Opening A” which is a standard algorithm. This opening is the start of many whai, particularly those that end in diamond shapes. Play the video at the link below which talks students through Opening A. Dasha refers to this pattern as kotahi taimana (one diamond). Let students practise the opening until they have it mastered: 
   • Kotahi taimana - One diamond (YouTube)
4. Next follow the instructions in this video to create te kapu me te hoeha (cup and saucer): 
   • Cup and saucer (YouTube)
     At 1:08 it is easier to think of going over two strings and ‘picking up the third string’ in that move. Note that the move where you use your mouth to shift the bottom of two strings over your thumbs (2:00 - ) is called ‘Navajoing your thumbs’ and is another algorithm common in whai. 
5. The next whai your students might make is ngā taimana e rua (two diamonds) which builds on Opening A. Tell your students to make the standard opening before the video starts. Alert them that somewhere in the video they will need to Navajo their thumbs. Work through the video, with students supporting each other to create the ngā taimana e rua pattern. Alternatively give the students copies of Copymaster 4 that has the instructions in graphic form. The written form will be much harder to interpret. 
   • Two of diamonds
6. Put the students into small groups of about three or four. Provide them with opportunity to become class experts in a particular whai pattern. Their job will be to teach the rest of the class how to make that pattern. Another YouTube chanel with many examples of string patterns is available at the link below:
   • Mom's Minivan YouTube channel
7. Once your students have practised their whai in small groups invite them to teach others how to make their pattern. You could do this as a whole class or create expert-novice pairs.
8. For an interesting geometry challenge consider looking for different shapes within a whai pattern. For example the gate pattern looks like this
   
9. Ask your students what shapes they can see in the figure. Here are a few possibilities:
   
10. Some interesting points might arise such as:
    • A three sided polygon is called a triangle, irrespective of the length of the sides, size of angles or orientation. The same is true of all five sided polygons being called pentagons and all six sided polygons being called hexagons.
    • Non-regular means that all the sides and angles are not equal so a regular polygon, such as a square, must have equal sides and angles.
    • The prefix ‘tapa’ means sides and the number name tells how many of those sides are in the shape, e.g. tapatoru means three sides (very helpful).

Session Five

In this session students learn to play the traditional Māori game Mū Tōrere which is like a form of draughts. The original game is sometimes referred to as the wheke (octopus) game or the whetū (star) game due to the shape of the board. It is appropriate that students learn to play the game at the time of Matariki, since the Māori New Year is a time of engaging in cultural pastimes. The board (see Copymaster 5) has been altered to include the nine or seven stars of Matariki, depending on the version of the game that is played. A digital version of this game is available online - search for “Mū Tōrere - HEIHEI Games”.

1. Introduce the Mū Tōrere Ngāwari (easy) version first. The game is played in pairs with each player needing three counters for the easy game and four counters for the original game. Their counters should be of one colour. The rules are included on the game boards. 
2. Let the students explore the easy game in pairs. Tell them that they need to record the winning position if one of them wins. Using black and grey for the counter colours can help to identify the arrangements that create wins. After they have played awhile bring the students together to share the winning positions. Create a set of diagrams. In these examples grey wins.
   
3. While they may look different the winning arrangements are actually the same, and are just rotations or reflections of one another. That can be demonstrated by putting the patterns on cards and turning them.
4. Ask: What must be true for a player to win in the easy game?
   The winner must occupy the centre circle, the opponent’s stones must be clustered together around the hexagon and the winner must have the ends of the cluster blocked off. You might try to find a winning arrangement by separating the loser’s stones into a group of two and one but there is no way for the other player to stop them moving.
5. Transfer to the original game that has the same set of moves but more winning arrangements. Ask the students to create winning arrangements on their board prior to playing the game. Create a gallery so the students can look for similarities and differences. Here are winning positions for black. Notice how all four, three and one, and two and two configurations of grey can all result in a victory to black but the winner must always occupy the centre. Discuss the similarity of winning arrangements created by students as the diagrams are reflected or rotated.
   
6. Is it possible to trap a player that has four ones, or a two and two ones (as shown below)? Try colouring in four circles grey to achieve a trap. It is not possible.
   
7. Once winning positions have been analysed let the students play the game. Competitive games go for over thirty moves so tell your students to be patient and think ahead. An interesting idea is that players can always create a draw if they know what they are doing. Is that true?

Session 1

Begin by asking students to bring in their school bag.  Pose the question Who has the heaviest bag to carry to school and who carries the lightest bag to school?  Several kete filled with rocks or books could be used for the same purpose.

1. Begin by selecting 5 or 6 bags (or kete) from around the class – it is important to select bags of various sizes and shapes to discuss that biggest doesn’t necessarily mean heaviest to further explore the idea of conservation.
2. Discuss what it means to “weigh” something. Look for students to mention how “heavy” an object is. Students might make connections with weighing ingredients when cooking, or weighing luggage when travelling. As a class, compile a list of contexts in which weighing is an important act of measurement. Display this for the students to reflect on.
3. Ask students how they think early Māori people measured different things? Although there is little research to say how early Māori measured weight or mass, it is thought that they measured length by using their body parts (e.g. one arm could measure the length of a fish). Ways of measuring, that used the body as a measuring tool, were used for building houses, whakairo (carving), raranga (weaving), and tā (tattooing). To make sure their measurements were consistent, one person (often a high ranking chief) was chosen within a tribe (iwi) and was given the task of being the “standard measure”.  This person was considered to be a taonga, and was remembered throughout generations. Encourage students to share their thoughts - what objects do you think early Māori might have used to measure weight (e.g. stones, tools they made).
4. How are we going to go about ordering these bags from the lightest to the heaviest bag to answer our question?
5. Gather suggested solution strategies then trial strategies to establish an effective way to order the bags by weight.
6. Group students in groups of 5-6 with their bags (or kete).  Ask each group to order their bags from least to most heavy.
7. Share the techniques and strategies used by each group to order the bags.
8. Ask 2 groups to pair up to combine their bags on one continuum of least to most heavy. 

Session 2

The following activities are to provide students with experiences to compare weights of different objects and to create a benchmark of what a kilogram feels like.

1. Make available a 1kg weight for students to use to give them the ‘feel’ of a kilogram.
2. Seat the class in a circle around a variety of items from around the room, from your kitchen, environment etc.  Ensure items like 1kg bag of sugar or a 1 litre container of milk or water are included in the items as such items will become useful benchmarks.
3. On large sheets of paper draw and label the following buckets. 
   
4. Ask individuals to select an item and place it in the most appropriate bucket. Before each item is placed in the bucket it would be a good idea to pass the object around the circle for students to feel the mass of each object. This activity could be carried out in smaller groups if necessary to give individuals more hands-on experience.

In preparation for Session 3 ask students to locate items from around their home that they believe would make good benchmarks for 1kg. Ask them to bring along an object that they think has a mass of one kilogram.

Session 3

1. Ask individuals to bring their 1 kg benchmark items to the mat. 
2. Using scales check the actual measurement of each of the items to see how close they are to 1kg. Record the weight of each item in a large table or on separate pieces of paper.
3. Give students 5-10 minutes to rove around the circle and hold one another’s item.
4. Ask students to discuss which of the benchmarks are the most useful.  For example, objects which you don’t usually pick up are not particularly good benchmarks as you will not be familiar with their mass.
5. Either individually or in small groups give students a reusable bag and ask them to put one kilogram of something in it.  You may prefer to do this activity outside in the sand area (using sand to make a kilogram), in the local environment (use rocks to make a kilogram) or you may do it inside and suggest a range of items that could be used to make one kilogram.
6. Weigh the bags and discuss why they are not all exactly one kilogram.  Answers could include that different items have been collected of different shapes and sizes, or that people have not collect enough, or have collected too many items. Compare them to the benchmarks.
7. Class discussion needs to now focus on how many grams are there in one kilogram?  The following types of questions can be asked to explore the connections between grams and kilograms. Students can record their ideas on video to be shared with others. 
   What does kilo stand for? 
   How many kilograms is 2000g?
   How many grams in 1.5 kg?

Session 4

1. Organise students into groups of 2-4 and ask each group to select one of the near 1kg items that were identified from the previous day. This will be used as the group’s benchmark to measure various other items around the room.
2. Group members take turns to be blindfolded. In one hand they hold the bench mark and in the other hand they are given another item. The task is to estimate the mass of the mystery item by comparing its mass with the benchmark item.
   The blindfolded individual verbally announces their estimate and a non-blindfolded recorder records the estimation. The non-blindfolded individuals can also estimate the mass of the mystery object.
3. After each estimate students then use scales to measure the item’s mass. The comparison can then be made between the actual mass and the estimated mass. 
   The process can be repeated for each group member.
4. This could be turned into a game in which the individuals who estimate within 100 grams earn themselves a point. The first group member to earn four points is the winner. As an extension, you could ask students to figure out how many grams are in each of the objects, or all of the objects together. 

Session 5

1. Bring this unit to a conclusion by asking students to share the benchmarks they are going to use for 1kg. 
2. List the various benchmarks on a large sheet of paper or digital device to be displayed and shared as a reference.
3. Share the various strategies and techniques students have developed to establish near estimates for objects they are asked to weigh.
4. Create a class display or powerpoint of benchmarks, strategies, and techniques.

Prior Experience

This unit is targeted at Level 2 so students should have experience of the following skills from Level 1:

• Forward and backward number sequences to 100 at least
• Counting and forming sets of objects to 100 at least
• Reading and writing numbers to 100 at least

If your students have not yet developed proficiency in these skills, consider revising them prior to, or alongside, this unit. 

Session One

In this session the students explore how groupings of ten can be used to aid counting and to perform calculations. They create the sets of countable objects that will be used in later lessons. Consider framing the creation of these countable objects in relation to other relevant learning (e.g. creating painted stones for the school garden, collecting cans for a school-wide food drive).

Acknowledgment: The game 60 second challenge was created by Ann Downton from Monash University, Melbourne.

1. Play the game “Sixty Second Challenge”. You need a measure for one minute (e.g. stopwatch, egg timer) and a lot of countable items such as beans, cubes, ice block sticks, or counters. You will use these countable objects as your place value material for the week at least. Set the timer for one minute. During that time the students work in pairs in this way. One partner rolls the dice and the other partner takes that many objects and adds them to their collection. Then the partner rolls the dice again and their partner collects that many objects. Rolling the dice cannot occur until the previous collection is made. When sixty seconds is up the students count how many objects they have collected in total. Each total can be recorded on a post-it note or a scrap of paper. Each pair needs to bring their number to the mat. Put a skipping rope on the ground (1m +) and label the ends 0 and 100 by pegging a card to each end. This activity can also be done outside with a chalk number line. 
2. Ask the students, one pair at a time, to place their score on the line where they think it belongs and peg the label to the rope. Discuss the placement in terms of proximity to benchmarks like 50, 75, 25, etc. After five numbers are placed, ask the remaining groups to put their score where they think it belongs.
3. Discuss how students counted their collections. Look for different ways of grouping. Discuss which way is easiest. Counting in twos and fives is relatively easy but there are a lot of counts. Counting in threes is difficult because there is no pattern to the sequence to help you. Grouping in tens and ones makes writing the number easy to say and write. For example four groups of ten and six is ‘forty-six’ and is written as 46. You could revise these counting strategies with a video or song. Value all methods of counting, and consider pairing up students with similar times-table knowledge. This will help them to feel confident in their ability to share their knowledge with a peer.
4. Play the “Sixty Second Challenge” with the players swapping roles of dice thrower and object gatherer. Look for students to group systematically to find the totals. Add the second attempt numbers to the number line. If there are many notes for a number the notes can be allocated to the same position on the line beneath one another. An interesting question is, “Is the class getting better at this game?” Ask students to justify their answers.
5. Pose two questions related to the final game for students to solve. Discuss key words like total and difference.
   What is the middle score? How can you tell?
   What is the difference between the highest and the lowest score?
6. For the latter question, make the highest and lowest collections using groupings of tens and ones. “I am using tens and ones because I think that will be easier.” Organise the two collections on the place value mat (Copymaster 1) to create a useful image of difference. Below packets of beans are used and the lowest and highest totals are 24 and 87 respectively.
   
7. Send the students off to solve the difference problem in pairs. Tell them to organise their counting objects in packets/groupings of ten. These groupings will be useful for the remainder of the week. So iceblock sticks might be bundled with rubber bands, cubes might be connected into lengths, and beans might be bagged in small see-through bags or white film canisters. Look for:
   • Do your students use the groupings of ten to simplify the task rather than count in ones?
   • Do they recognise that basic facts can be applied to groups of ten as well as ones, e.g. 8 tens is 6 more tens than 2 tens (20 + 60 = 80)?
   • Do they combine tens and ones and name the quantity, i.e. 6 tens and 3 ones makes 63?
   • Discuss these important ideas when you gather the class together to sum up what they have learned.

Session Two 

In this session students learn to match quantities with two-digit numbers and vice versa.

Part One

1. Have your students sit on the mat in a circle so they can all see an A3 sized place value mat you have put in the centre. Give each student a copy of a hundreds board (Copymaster 2) and one counter.
2. Tell them you are going to say a number and they are to put their counter on it. Stress the importance of distinguishing “teens” from “tys”, for example sixteen from sixty. Some students may need more practice counting these types of numbers - they could be supported with the use of a song or by using flashcards with a partner.
3. Once students are confident at listening and placing the counter on the correct number, introduce the counting objects (cubes, beans, iceblock sticks, something relevant to your classroom context etc.) packaged in tens from the previous day. Discuss how the objects are organised in sets of ten. Note that the A3 sheet is divided into two columns for tens (left) and ones (right).
4. Play a game where you create a collection of objects (whole tens and ones) and the students indicate how many objects are in the collection by placing the counter on that number on the hundreds board. Focus on the language connection of “ty” as tens and “teen” as one ten. Some important ideas to think about when developing your sequences:
   • teen and ‘ty’ numbers, e.g. 13, 17 compared to 30 and 70
   • iterating by tens, e.g. 13, 23, 33, 43, … (Note that the ones digit does not change)
   • re-unitising ten ones to form a ten unit, e.g. 67, 68, 69, 70.
   • re-unitising ten tens to form a hundred unit, e.g. 97, 98, 99, 100. (Note that students must visualise the continuation of the hundreds board pattern to show numbers greater than 100.
   • go backwards in sequences as well, e.g. 87, 77, 67, ...
5. Move to masking where you create a hidden collection under a shield of some kind (card or plastic container) and tell the students information like:
   There are four tens and seven ones.
   It is not quite eight tens. There are three beans missing.
6. Repeat the process of iterating by tens.  Start with a collection, say 24, then add ten repeatedly to the collection. Get the students to show the total each time by moving their counter. Go through 100 forwards and down to zero backwards. See if the students can “shortcut” several iterations of ten by using fact knowledge, e.g. 40 + 30 is just 4 + 3 in units of ten. Move to more complicated examples like:
   I have 26 (put the collection under the shield). I am putting 60 more beans in. What does sixty look like (6 bags of ten). How many beans are there now? How do you know?
7. Compare the starting number and the final number using an empty number line:
   
   Is there an easy way to know it will be 86 without going 26, 36, 46, ... 86?
8. Get the students to work in pairs with one student creating and building with physical collections and the other moving their counter to show the quantity on a hundreds board. Letting the counter mover see the collection or masking the collection is an important variable. More capable students could work with the hundreds board that is missing many numbers (See Copymaster 3) or use a different century from the Thousands Book (see Copymaster 4) using place value blocks to build the collections.

Part Two

1. Play Close to 100 using individual hundreds boards (See Copymaster 2) to keep track of each person’s score. Collections of counting objects in packages of ten and individual ‘ones’ might also be used if needed to support some students.
   Close to 100 is played in pairs with a dice (1-6). Players take turns to roll the dice and decide if the digit that comes up represents tens or ones. For example, if 5 is rolled it may be used as 5 or 50.
   The player adds whatever they chose to their running total. That total is recorded each roll. Players have a total of seven rolls and must use all of these rolls. The player with the total closest to 100 after seven rolls wins. Players’ totals may go over 100. Here is an example:
   
   If appropriate, organise students into pairs or small groups to play this game. It can be adapted to the different knowledge in your class by changing the “close to” number (i.e. close to 10, close to 1000).
    
2. After a few games, discuss winning strategies with the students. Some may suggest getting to 80 as fast as possible then choosing ones, or letting small digits represent tens and large digits ones. Discuss how the winner is decided if one score is less than 100 and the other is greater, e.g. 93 and 106. In that case 93 is ‘seven away’ from 100 and 106 is ‘six away’ so 106 wins.
3. To cater for student differences, vary the game to backwards to zero from 100 (integers may arise) or using 3-digit numbers with each digit representing hundreds, tens or ones, aiming for 1000 with ten rolls.

Session Three

In this session students investigate how 100 can be partitioned to form ‘number buddies’ like 20 + 80 and 1 + 99.

Part One

1. Introduce the students to the Slavonic abacus. An internet search for "Slavonic Abacus online" leads to many different interactive tools that could be used in the absence of a physical abacus. Each student should have their copy of the hundreds board and a transparent counter. Below is a Slavonic abacus showing 25 (left) and 75 (right). 
   
   How many beads are in each row? How do you know?
   How many beads are on the whole abacus (100)? How do you know?
2. Make blocks of beads by moving across whole rows of tens and some ones. Students put their counter on the matching number on their hundreds board. Discuss the structure of the number (tens and ones) and the way the number is written. To move beyond counting tens and ones verbalise the structure of the block of beads as you create it.
   Eight tens and six ones, No tens and nine ones, five tens and two ones.
3. Develop imaging by masking the Slavonic abacus (turning it around or hiding it) and just saying the structure while creating the block of beads.
   I’ve got seven tens and four ones. Move over 100 to see how the students react. Look for them imaging where the counter might go below the hundreds board. 
   I have 12 tens and five ones (125)
4. Next, work on different names for 100. Make 100 with place value material and then model various ways to split the 100. Each time a new split of 100 is made, invite students to show that split on the abacus.
5. Ask another student to record an equation for the split and a number story to go with it. Model this, making explicit links to cultural contexts and learning from other curriculum areas that are relevant to your learners. See below for an example. See below for an example.
   

Number story:

On day 1 the community planted 48 trees that had been delivered by the council. On day 2 the local marae offered 52 native trees for planting. Altogether 100 trees were planted.

Or:              36 + 17 + 26 + 21 = 100 (36 + 64)

At the local beach, when the tide was down, Aniwa and her cousin collected 36 cockles. The next time they went collecting cockles Aniwa found 17, Rei found 26 and Kori managed to get 21. Over two days they had collected 100 cockles.

Part Two

1. Play the game Number Buddies to 100 with calculators and the sets of tens and ones groupable objects students created in lesson one. Start with the Slavonic abacus. Move across a block of complete tens and ones. For example:
   I have seven tens and two ones. How many beads is that? (72)
2. Point to the remaining collection of beads (not moved).
   How many are left over here? How do you know? (28 - There are two tens and eight ones)
3. Try to highlight non-counting strategies, e.g. There are eight because 2 + 8 =10.
4. Show the students how Number Buddies to 100 is played in pairs with a calculator. One player enters a two-digit number. Get a student to do this on the IWB calculator or with a simple online calculator. Make a block on the Slavonic abacus of that many beads. Say the first player keyed in 46.
   The other player must add on the correct number to that number to make the total 100. Highlight that this involves thinking about what is left from 100 using the abacus. So if correct the other player would key in + 54 = and the calculator would display 100.
   The game can be played competitively in pairs with a point awarded for each correct answer (best of ten tries). Get the students to record their work with equations, e.g. 46 + 54 = 100.
5. After playing the game for a while, ask the students to think about patterns that allow them to work out the “number buddies”. They might note that numbers where the ones digit is zero are easy, e.g. 40 + 60 = 100, as the tens digits need to add to ten (ten tens make 100). The other examples are more difficult, e.g. 39 + 61 = 100, as the tens digits add to nine and the other ten comes from collecting ones.
6. Note: More advanced students might play Number Buddies to 1000. Students who are still developing their knowledge of numbers to 100 might benefit from playing Number Buddies to 10, 20, or 50. Consider the addition-facts knowledge of your students, and adapt the game in reflection of this. Alternatively, you could play Number Buddies to 100 with a small group of students, whilst others play Number Buddies to 1000 etc. Ultimately, you should support all students towards working with 100.

Session Four

In this session the students explore different names for the same two-digit number.

Part One

In the last lesson the class explored how 100 can be renamed in lots of ways. In this lesson, we explore the same concept with other numbers.

1. Use place value equipment to model how 75 can be renamed as 6 tens and 15 ones or as 5 tens, and 25 ones. Joke about the funny names that are created, ‘Sixty-fifteen’ and ‘Fifty-twenty-five’. Record the pattern of names:
   • 7 tens and 15 ones
   • 6 tens and 25 ones
2. Ask what would come next in the pattern.
   What would happen to the beans to get the next name for 75? (A bag of ten would be shifted into the ones place)
   If we kept the pattern going, when would it end? (75 ones)
3. Model 2 tens and 43 ones so the number showing is 63. Replace 10 ones with 1 ten, and repeat. This leads to a pattern of:
   • 2 tens and 43 ones
   • 3 tens and 33 ones
   • 4 tens and 23 ones
4. At each new movement of beans ask the students to predict the outcome and the new name for 63.
5. Send the students off to practise renaming a two-digit number in pairs. Some may need their materials for Session One and some will be able to rename numbers without support. After a short period of practice, bring the class together to see if the students have understood the idea. Create a model of 5 tens, and 28 ones in the centre of the mat using your place value mat from Session Two.
   What is this number? ‘fifty– twenty-eight’ or 78
   What other names for that number can you find? ‘sixty-eighteen’, ‘seventy-eight’ but don’t forget ‘forty-thirty eight’, ‘thirty-forty-eight’.

Part Two

1. Introduce the game of Cover Cathy Crocodile. This game is an adaptation of a game created by Joan Paske who was a prominent figure in New Zealand Mathematics education in the 1970s and 1980s. The game is in honour of her extensive contribution.
   In the game students choose cards to cover the crocodile numbers of their board. The cards provide many options for covering numbers but most options involve renaming. For example, if a student wants to cover 72 they could do so by nominating:
   • 7 tens and 2 ones from the □ tens and 2 ones card
   • 6 tens and 12 ones from the 6 tens and □ ones card
   • 3 tens and 42 ones from the 3 tens and □ ones card
2. There are many other options. Some students may need support with materials though it is hoped they can do the renaming mentally. Look for the following:
   • Do your students fluently work between tens and ones?
   • Can your students check that a card nominated by another player is viable?, e.g. 2 tens and 36 ones is a name for 56
   • Do your students write numerals to support them?, e.g. Checking 74 by writing 40 + 34
3. Let the students play the game until winners emerge. Gather your students together to discuss the thinking involved in the game.
   When might renaming two-digit numbers be useful? Examples might include having ten dollar notes and one dollar coins and trying to find out how much money you have in total.
4. Pose this problem. Build 63 on the place value mat with groupable materials where all the students can see.
   Suppose I have 63 toys. I give 26 of them to my brother or sister. How many do I have left?
5. Let students suggest ways to solve the problem. It is easy to allocate 20 packets of ten leaving 43 toys.
   How many toys do we still need to take away? Can we rename 43 so it is easy to take away the other six toys?
6. Hopefully students suggest that a packet of ten toys can be broken up to change 43 into 30 and 13. Then the six toys can be removed leaving 37 toys. This task is a good indicator of whether students can apply renaming in a more difficult context of subtraction. You might change the numbers in the toy problem to give them a chance to practise renaming and subtracting.

Session Five

In this session students apply the place value structure of two-digit numbers to change a given number into a different number either mentally or with support of materials.

Part One

1. Make the number 30 with grouped materials from Lesson One (3 tens) using the place value mat (Copymaster 1). Show the online calculator with 30 in the display.
   Suppose I am set the challenge of changing from 30 to 80. What could I do? Is there a single operation I could key in?
2. Act on suggestions by students. Key in their suggestions and have a student change the object model to match. Students might suggest adding 50 to 30, like 50, 60, 70, 80 (skip counting in tens). Remind them that they can use basic facts with tens, just like they do with ones. If 3 + 5 = 8, then 30 + 50 = 80 since the units added are tens. They could also use the Slavonic Abacus to show this.
   Suppose I am set the challenge of changing 80 to 10 in a single operation. What might I do?
3. Again mirror the calculations on the calculator with physical manipulation of the objects model. Look for students to recognise that if 8 – 7 = 1 then 80 - 70 = 10 since the units are tens.
4. Next form 15 with the materials and enter 15 into the calculator.
   This time I have to change 15 into 46 with one operation. What can I do?
5. This problem is more complex since it involves working with tens and ones. Invite suggestions from students and manipulate the materials as well as keying in the operations. Look to see if the students recognise the changes in the digits from 15 to 47 and what that means for the quantity to be added.
   If I write 15 + □ = 47 does that help you work out what to do?
6. Some students may see that the tens digit has increased by three and the ones digit has increased by two. So the number added is 3 tens and 2 ones (32).
7. Pose similar problems but increase the difficulty by following the sequence below. Watch to see how students cope with the change in re-unitising demands of the problems. Record each problem as a change unknown equation, e.g. 78 - □ = 45, to see if students link the value of the digits to the solution.
   • Change 78 into 45
   • Change 43 into 81
   • Change 62 into 26

Part Two

1. Ask the students to play the change game in pairs. Each pair will need a calculator and possibly a set of groupable objects from Lesson One if they need more support.
   Players take turns to enter a starting number, say 34, and pass over the calculator with a change instruction, say “Change 34 into 88 with one operation.” Restrict the numbers to two places though moving to 3 digits is a significant extension for more competent students. A player gets a point for every correct change they give. The asker gets a point if the suggested change is incorrect. Look for the following:
   • Do the students use the place value of the digits in deciding what change to try?
   • Do they recognise what digits need to change?
   • Do they notice when a ten needs to be created or partitioned? e.g. 58 + □ = 92 or 73 - □ = 29
2. After a suitable period of playing, bring the students together on the mat.
   What did you do to make the problems harder for your partner?
3. Look for students to explain that the hardest challenges required both digits to change and that problems were hard if renaming was involved, particularly subtraction.
4. For assessment of students’ understanding of place value, ask them to solve the problems on Copymaster 5. That will give you valuable data about their control over re-unitising tens and ones. 

We introduce the unit by rolling dice and investigating the numbers that come up.

1. Begin the session by showing the students the large die and asking them which number they think will come up if you roll it.
   What number do you think I will roll?
   Why do you think that?
   Roll the die and see whether students' predictions were correct. Repeat a couple of times.
2. What are the possible numbers that I can roll?
   List these on the board and tell the students that this list of all the possible outcomes is called the sample space.
3. What if I rolled the die twenty times. What do you think will happen? Why?
   List these predictions on the board or on chart paper.

4. With the class, roll the die twenty times keeping track with a tally chart. Summarise onto a class chart.

   1	2	3	4	5	6
   lll	llll	l	lll	llll l	lll

5. Give pairs of students a die and ask them to work together to roll it 20 times. As they finish, ask them to record their results on the class chart.

   Pairs	1	2	3	4	5	6
   Mr Tihi	3	4	1	3	6	3
   Ben & Tane	2	5	3	2	4	4

6. Discuss the results with the class. Look back at their earlier predictions.
   Why are all our results different?
   If you rolled the die another twenty times what do you think would happen? Why?

7. Now let's add our results together.
   What do you think that we will find?
   Use a calculator to sum down each of the columns

   
   Number rolled

   Pairs	1	2	3	4	5	6
   Mr Tihi	3	4	1	3	6	3
   Ben & Tane	2	5	3	2	4	4
   Jay & Sarah	5	3	3	2	5	2
   Class totals 240 rolls	45	36	42	31	39	47

   At this level it is likely that the students may attribute uneven results to "special luck". It is through continued experience that students come to appreciate the mathematics of probability.

Exploring

Over the next 3 days the students play a number of probability games. Copymasters for each game are attached to this unit.  They are encouraged to think about the sample space and to make predictions as they play the games. They will also begin to think about whether the games are fair.

Tell the students that they are going to play a number of games in pairs over the next 3 days and there are some general things they need to do with each game:

• as they play each game they are to write down the possible outcomes (the sample space). They are also to write a prediction about what they think will happen in the game
• play the game, recording the results
• compare what happens with their prediction.

Note: At this level do not expect the students to make mathematically sound predictions or systematically identify all possible outcomes. It is likely that they will make incomplete lists of possible outcomes. In future work, as they have similar experiences, their thinking will become more systematic and mathematically sound. The main aim of this unit is to start the students thinking about possible outcomes and notions of fairness.
Probability games to select from:

Bunny hop (Copymaster 1)
Sample space = {Heads, Tails}
As there is an equal chance of getting a head or a tail you would expect the bunny to be close to 0 after a number of turns.

Doubles (Copymaster 2)
Sample space

There are 6 ways of getting a double or 6 out of 36.

It is unlikely that the students will be this systematic about identifying the sample space. However they should identify that you are more likely to get non-doubles.

Pūkeko racing (Copymaster 3)

Sample space = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
The table shows the ways of getting each sum. 7 (6 ways) is the most likely followed by 6 and 8 (5 ways).

Odds or evens (Copymaster 4)

Sample space = {1 3, 5 (odd) and 2, 4, 6 (even)}
For this game the probability of each player winning is equal.

Sums (Copymaster 5)
From the table for Pūkeko racing you can see that there are 24 ways of rolling a 5, 6, 7, 8 or 9 which is a probability of 24 out of 36 or 2/3. The probability of rolling a 2, 3, 4, 10, 11, or 12 out of 35 or 1/3.

Up or down (Copymaster 6)
Sample space = {heads, tails}
This is very similar to the bunny hop game. You expect the climber to be close to 0 after a number of turns.

At the end of each session have a class sharing time to discuss a couple of the games.

• Tell us about one of the games you played today
• What were the possible outcomes?
• What did you think would happen?
• What happened when you played the game?
• Did anyone else play the same game?
• Did you get the same results?
• Do you think that the game was fair? Why? Why not?

Reflecting

On the final day of the unit ask the students to invent their own games using either coins or dice. Share the games with others in the class.

Which number came up the most? Why do you think it did?
If we rolled the die again do you think we would get the same results? Why?

As ākonga work through these activities the teacher may need to bring the class or small groups of ākonga together from time to time to discuss and model. Make sure an understanding of what line symmetry is and the names and attributes of common two-dimensional mathematical shapes is developing, alongside appropriate vocabulary.

Teachers may also like to generate a class display of the names and attributes of the shapes to be used over the course of the unit.

Session 1

1. Take a square piece of paper and fold it in half in front of the class. 
   
2. Using scissors cut out the shape as shown below.  Before opening the paper ask the class:
   Will the other half be exactly the same?
   How do you know the other half will be exactly the same?
   When I open this piece of paper, what shape will the hole in the middle be?
   
3. Open the paper and open up the piece that was cut out. Talk about the attributes of the shape.
4. Repeat this process cutting out the following shapes.
   
5. Discuss the shapes when the paper was folded in half and when it was unfolded.  The aim of this discussion is to find out what ākonga know and notice. Questions like the following could be used:
   Why did it work like that?
   How many sides and how many angles?
   What do you notice about the length of the sides?
   Are any angles the same?
   Does anyone know the name of this shape?
6. Challenge ākonga:  What other shapes could be made by folding a square piece of paper in half and cutting? What shapes do you think are impossible to make?
7. Hand out square pieces of paper and get the class to experiment and try to make some new shapes. Some ākonga could record their thinking about the relative attributes that go with the shapes they make. Some ākonga may need shape blocks available to try to replicate with folding and cutting. 

Session 2 - Straight Line Shapes

How many different straight line shapes can be made by folding a square piece of paper in half and cutting?

For most of this unit the focus is on straight line shapes. Using a ruler to draw the straight lines onto the folded paper before cutting is encouraged. Working in small groups, the ākonga are to make as many of the following as they can. A tuakana/teina model could work well here. 

Make . . .

• 4 different looking shapes with 3 straight sides
• 4 different looking shapes with 4 straight sides
• 4 different looking shapes with more than 4 straight sides

Place these shapes into three piles.

1. Shapes with 3 straight sides
2. Shapes with 4 straight sides
3. Shapes with more than 4 straight sides

Once as many different shapes as possible have been made, assign a category of shapes to pairs of ākonga, e.g. shapes with 3 straight sides.  The pairs sort their shapes according to the way they look.  A tuakana/teina model could work well here. The ākonga then share with the rest of the class why they sorted their shapes as they did. Ākonga could use known shapes or reference posters to help with this.

Session 3 – Make the Shapes

How many of the following shapes can you make by folding and cutting?

Ask ākonga to fold a square piece of paper in half and cut out a shape so that when they unfold it the hole will be one of the shapes below.

Model doing one shape in front of everyone. Emphasise that you are looking for a line in the shape that you could fold on, so both halves would be the same (this is called a line of symmetry or reflective symmetry).

Get the ākonga to predict which shapes will be the easiest to make, the hardest to make and whether any will be impossible. Ask why they think they will be easy, hard or impossible.

Make some more challenges like the ones above for others in your class. You could provide photographs of things in nature that have lines of symmetry that ākonga could replicate. Search 'reflective symmetry' in a google image search for some ideas.

Session 4 - Alphabet Shapes

Make as many letters of the alphabet as you can by folding and cutting. This activity could also be adapted to use two-dimensional shapes that are relevant to your current context of learning (e.g. the shape of a marae, a koru, and the shape of a wave; sports gear). Some ākonga may benefit from working in pairs, and/or with the teacher in this session - at least initially.

As a reference point, here are the alphabet letters that do and do not have lines of symmetry.

Session 5 - Reflecting

Ask ākonga (mahi tahi model) to think about the things they have learnt this week, the names of shapes and about their reflective or line symmetry.

Give pairs of ākonga 3-4 of the shapes from the list below that ākonga have become familiar with while completing this unit. Ask them to describe them using their own words and the words they have been learning this week. Also ask them to identify which shapes have line or reflective symmetry. A tuakana/teina model could work well here.

Equilateral triangle, Right angle triangle, Isosceles triangle, Scalene triangle, Square, Rectangle, Trapezium, Rhombus, Parallelogram, regular and non regular Pentagon, Hexagon and Octagon.

Session 1

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

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

Session 2

In this session ākonga use the kura map to describe pathways from locations.

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

Session 3

In this session ākonga use a local or imaginative map to describe different views they can see from different locations. They use compass directions to give the direction of landmarks from given locations. The map below is available as Copymaster 1.

1. Pose questions based on the map, which require ākonga to describe the views from different locations. For example:
   How many whare have a direct view of the marae?
   What can the children see from the playcentre?
   What can the doctor see out the window?
   If you sat in the doctor’s carpark what could you see?
   Colour in a whare that has a view of the playcentre, the dairy, and the hall?
2. Pose questions based on the map which require ākonga to use the compass directions. For example:
   What building is east of the café?
   What building is north of the hall?
   What building is south of the chemist?
   What direction is the playcentre from the church?
   What direction is the marae from the doctors?
   How many whare are south of the hall?
   From which building can you look west to see the church?

Session 4

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

1. Select a map to use for this session with your ākonga, it could be the kura map you used in Session 1, Copymaster 1 or a different map that ākonga are already familiar with. Work out an appropriate scale, for example 1cm is 50m, and help ākonga make scale rulers with strips of card. In Copymaster 1, the ruler graduations will be 0, 50, 100, 150 etc.
2. On the Copymaster 1 map, the dots represent entry/exit points for buildings. Show ākonga how to turn the map around to orientate themselves as they follow directions and turn left and right.
3. Give ākonga a set of directions to follow. Focus on left and right turns, and using landmarks to help provide the distances. For example, leave the playcentre and turn right, walk along and cross the road, turn right, walk past some whare and cross the road, where are you now?
4. Ākonga can work in pairs to give each other instructions. These pairs could be a tuakana/teina model.
5. Using their scale rulers, ākonga will be able to give directions that include distances. Give ākonga a set of directions to follow. For example, leave the café and turn left. Walk 40 metres, if you turn right what will you be able to see?
6. Ākonga can work in pairs to give each other instructions that include distances and left and right turns. The tuakana/teina model could be appropriate for this learning also. 

Session 5

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

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

Getting Started

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

1. Begin the session by telling ākonga about up-and-down staircases. This type of staircase can be likened to traditional lattice poutama which can be found on tukutuku panels in many marae.

   	One block is needed to make a 1-step up-and-down staircase. It takes one step to get up and one step to get down.
   	This is called a 2-step staircase, as it takes two steps to go up and two steps to go down.

2. Together count the steps so that ākonga understand why it is called a 2-step staircase. If possible, observe staircases in your school, and count how many steps it has.
   How many blocks are in the staircase?
   How many blocks do you think would be in a 3-step up-and-down staircase?
   How could you work it out?
3. Give ākonga time to work out the number of blocks. Ākonga could use blocks, draw or think about the result. Share the ways that they used to work it out.
4. Ask ākonga to guess how many cubes they think would be needed to make a 5-step staircase. Get them to build a 5-step staircase. Check their guesses.

5. Get ākonga to build more staircases. As they do, ask them about any patterns they see.
   Some may recognise the horizontal layers as being the sequence of odd numbers. Some may see the vertical stacks.
   Some may see that square numbers of blocks are always involved. This can be checked by rearranging the blocks to make square numbers (see the diagram below). This is an important discovery. Let them make it. This may require some careful scaffolding on your part. Ask ākonga to make several squares and look for patterns. The first square is 1 block, then the next square is 1 + 3 followed by 4 + 5. The square term is a number raised to the second power (n2  e.g., 2 x 2 = 4). Using blocks to build some squares and then asking ākonga to draw three more builds a good understanding of square numbers.

6. Some ākonga may wish to continue to find numbers that make larger up-and-down staircases. To keep track of the number of blocks/cubes in each staircase it might be useful to draw the staircases on graph paper. Other ākonga will prefer to use cubes. Demonstrate and show ākonga how they can record their results in a table.
7. There could also be an opportunity for some ākonga to create their own staircase patterns to help challenge their thinking at this early stage of the unit.

Exploring

Over the next 2-3 days, ākonga work in pairs or individually to solve the following problems (Copymaster of problems). A tuakana/teina model could work well here. Show ākonga how to use grid paper to draw the patterns and continue them. They could also use materials. As ākonga complete the problems, ask them about any patterns they see and encourage ākonga to record these observations with the patterns on the graph paper or by building the patterns with materials. Ākonga can also record their patterns using a table. The teacher will need to demonstrate how to do this and potentially provide blank tables for ākonga to use. For example:

Problem 1: Straight up the stairs

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

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

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

Problem 2: Climbing ladders

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

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

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

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

Ice-block sticks could be used to create ladders. 

Problem 3: Small steps

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

1-step	2-step	3-step

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

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

Problem 4: Star patterns

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

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

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

Problem 5: L-shapes

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

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

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

Reflecting

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

1. Begin the session by asking ākonga to attach and display their solutions to the problems on a display wall or table. These solutions could be constructions out of materials, drawings on grid paper, tables or oral/written explanations. Give ākonga time to look at the solutions of other ākonga. Pairs of ākonga could share with other pairs of students. Encourage ākonga to share their solutions with the class.
2. Give pairs of ākonga a supply of blocks (or other objects) and grid paper and ask them to invent their own pattern. A tuakana=teina model could work well here. Ask them to record the first three elements in the pattern on a piece of grid paper. They could also make a table to explain their pattern.
3. Ask ākonga to swap patterns with another pair. Work together to discover the pattern and then continue it.
4. Repeat with another pair’s pattern.
5. Leave the patterns on a table for all ākonga to solve in their own time.

Session One

1. Show the following data card to the class and explain what a data card is, i.e. a piece of paper contains three pieces of information about one person.
   
2. Discuss the importance of knowing exactly what each piece of data is about. 
   What could “tūī” mean? What could “reading” mean? What could “Even date” mean?
3. Ask the class to tell you something about this student. 
   Does anyone in the class fit this data card?
   Do you know someone that fits this data card that is not in this class?
   How many different people could this data card be correct for?
4. Turn the data card over to reveal the name of someone familiar that fits this data card. The point to get across is that a data card could fit many people but each data card is about one person only.
5. Explain to the students that the way to view each piece of data is to see it as the answer to a data collection or survey question/pātai. Get them to suggest the data collection questions that give these three pieces of information. Discuss how some students could answer the same data collection question differently, e.g. “What type of native bird do you like?” A more specific data collection question is needed, e.g. “What type of native bird do you like best- pīwakawaka, tūī or kererū?
   What would a data card about you look like?
6. Hand out a data card to each student to fill out (Copymaster 2). Have each student write their name on the back of the data card hand and have a student collect these.
7. After this session the teacher needs to arrange the data cards onto pieces of paper and photocopy them. One set is made for each pair of students. Photocopying onto coloured paper is suggested to make it easy to recognise the class data set. The names of the students on the back of the data cards are not needed. This data set will be used during Session Three.

Session Two

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

Session Three

1. Explain to the class that today they will be sorting and arranging data cards, like Session Two, except they will be using the data cards they wrote about themselves. Before the copied data cards are handed out, discuss what the students expect to find out. 
   What do you think we will find out about our class?
   Will it be mainly different or similar to the group looked at in Session Two?
2. Hand out the copied data cards from Session One to each pair of students. The pairs are to cut out the data cards, sort them and organise them to look for other interesting things about the class.
3. The teacher is to move around getting each pair to explain and show what they have found out. The teacher is to encourage the pairs to add detail to their answers, moving students from, “Yes, there are more students who like tūī than pīwakawaka or kererū.” to “Yes, there are 10 more students who like tūī compared to the total of 8 students who like pīwakawaka and kererū.”
4. Conclude by considering the statements the students made at the start of the day and seeing how many were true and discussing other interesting things they found out about the class.

Session Four

Today the students, in pairs (tuakana/teina model could work well here), will design and collect their own data using data cards. Each pair of students needs to design three data collection questions to ask other students in the class. 

1. Discuss and brainstorm suitable data collection questions. Data collection questions for this activity need to be answered with either yes or no, or an option selected. Keep the optional answers to a maximum of three options.
   Sample data collection questions:
   • What is your favourite kai - pizza or burgers?
   • Have you ever caught a fish?
   • If you could choose, would you sing, dance or read a book?
   • Do you prefer to play at the beach or the river?
2. Once suitable data collection questions have been developed they are to be written onto a large data card.
   
3. Before starting to collect data each pair of students needs to write three investigative questions they could ask of the data they will collect and to make statements about what they expect to find out about the class for these investigative questions. Students should be encouraged to pose investigative questions about categories within categories, leading to statements about what they will find e.g. “People who like to read will select burgers as their favourite kai ” or “Most people surveyed will like  kererū”.
4. Each pair of students is to cut out enough blank data cards for the class and number them 1 to n (number in class). Once completed the pair of students are to ask half the students each, their three data collection questions and fill out a data card for each student. The student’s name needs to be written on the back to make sure all students are asked. They need to remember to complete their data cards for themselves as well.

Session Five

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

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

Session 1

Here we look at different representations of 1/2.

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

Session 2

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

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

Session 3

This session involves fractions in problem situations.

1. Pose this problem for the students, There are 12 kūmara in the hangi. There are four people wanting kūmara on their plate. If everyone gets one quarter of the kūmara, how many kūmara do they get?" Get the students to solve the problem with counters and their paper circles from the previous session.
2. Discuss the strategies that the different students used. These might include sharing twelve counters evenly onto the sections of the quarter circle, using addition (6 + 6 is 12 so 6 is a half so 3 is a quarter), or division (12 ÷ 4 = 3).
3. Give the students other set problems (see Copymaster 4). Get them to record their strategies as they solve the problems. Students who use sharing strategies should be encouraged to anticipate the result of their sharing before it is complete.
4. Students may like to write their own problems for others to solve. These can be made into a class book or digital resource (e.g. Google Classroom post, Padlet Board) of problems for independent activity.

Session 4

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

1. Draw a number line from 0 to 10 on the board (about 1 metre long). Build up the number line by getting students to write where different numbers might be. Once the whole numbers are in place, ask students to think about where numbers like 1/2, 3/4, 3/2, and 4 ½  might be. This will help students to realise how fractional numbers extend the existing set of whole numbers and can be represented on a number line in the same way.
2. Give the students several paper strips of the same length cut from scrap paper. Ask them to fold one strip in half, one into quarters, and one into eighths. This is relatively easy as they can be folded by repeated halving. Ask the students to label each strip using symbols: 1/2, 1/4, and 1/8.
3. Take a full strip and use it to draw the number line from 0 to 1 by marking each end. Shift the strip to the right and mark 2 at the right-hand end, shift it again and mark 3, etc. (0 – 5 is sufficient). Ask the students if they can use their strips to show exactly where one-half would be. Expect students to align the strip folded in half to do this. Ask this for other fractions like one-third, three quarters, two-thirds, and extend it to fractions greater than one like five-halves, four-thirds, and three and seven-eighths.
4. Pose these problems for students to solve using their strips:
   • Draw a number line to show 1/2, 1/4, and 1/8. Mark where you think 1/3, 1/5, and 1/10 would go on your number line. Explain where you placed them. Why do fractions with one on the top line get smaller as the number on the bottom gets larger, e.g. one-half is larger than one-third?
   • Which of these fractions is closest to one, 1/2, 2/3, or 3/4? Why?
     
     These problems will highlight students’ knowledge of the relative size of fractions. For example, a student might find half of the distance between 0 and 1/5  to see where 1/10  should be or half the distance between 1/2  and 1 to see the location of 3/4 . The problems will also highlight their understanding of the role of the numerator (top number) as the selector of the number of parts and the role of the denominator (bottom number) as nominating how many equal parts the whole is separated into.

Session 5

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

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

This unit is run as a series of stations over four days with students rotating around the stations in groups. The final session is run as a class activity with all students working on the same task in groups. Consider grouping together students with mixed mathematical abilities in order to encourage collaboration (mahi tahi) and tuakana-teina (peer supported learning).

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

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

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

Introduce the concept of a scavenger hunt, and model how to complete the tasks at each station. Depending on the needs of your students, it may also be appropriate to model how to accurately measure items with a ruler. This modelling could be used to create a class chart or set of guidelines for measuring. In turn, this could be used to support students in practising accurate modelling skills throughout the session.

Station One

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

Student Instructions (Copymaster 1)

Go on a Scavenger Hunt!

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

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

Station Two

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

Student Instructions (Copymaster 2)

Go on a Scavenger Hunt!

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

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

Station Three

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

Student Instructions (Copymaster 3)

Go on a Scavenger Hunt!

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

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

Station Four

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

Student Instructions (Copymaster 4)

Go on a Scavenger Hunt!

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

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

Reflecting – Class activity

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

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

Extension

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