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.