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. ○, □, ­_, ∆, ○, ­_, →, ∆, …                

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

• 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

This unit builds on the unit Place value with two digit numbers, and it would be useful for students to have worked through that unit first. Some materials created for that unit will also be useful here.

Activity

Session One

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

1. Show the students the first Snakes and Ladders board (page one) of Copymaster 1. Make sure students know how the game works, then ask.
   How many squares would a player gain by landing on the ladder on square 20?
2. Look for students to use place value to work out the answer of 19 by counting squares. You may need to draw students’ attention to the rows of ten squares. Make the connection to a hundreds board to support their thinking if needed.
   We are now going to look at some of the other snakes and ladders.
3. For each ladder climb or snake slide ask the students to work out:
   How many squares would a player gain or lose by landing on the ladder or snake?
4. Look for your students to use the differences between digits rather than counting by tens and ones to find their answers. For example, the snake at the top left has the player sliding from 91 to 65. A slide of 30 would take them from 91 to 61, since the tens digit changes from 9 to 6 and 9 tens – 3 tens = 6 tens. Since 65 is four more than 61 the actual drop is 26 (four less than 30). You may need to use groupable materials to model the operations.
   
   A beans and bags model of 91 – 30 = 61. Putting four beans back gives 65 so the amount taken is 26.
5. Challenge the students to work in pairs to work out the number of squares gained or lost by other ladders and snakes. Start with an expectation that students can solve the problems using numbers but allow students to use groupable materials if they need support. Look for:
   • Do the students record the beginning and end numbers for a snake or ladder?
   • Do they look for changes to the digits to work out the gains or losses?
   • Do they apply basic facts to find the gains and losses, rather than rely on counting?
   • Do they use strategies like tidy numbers, as with 91 - □ = 65?
6. Gather the class together to discuss their strategies, using symbols, diagrams and groupable materials to represent the quantities involved. For example, here is a tidy number strategy illustrated on an empty number line.
   

Ask the students to play Snakes and Ladders in pairs or threes - a tuakana/teina model could work well here. Use wooden or whiteboard cubes as the dice, using stickers or whiteboard pens on each face to write on. Copymaster 1 gives the playing boards. Three options of board are available for differentiation. Note that each board requires a different set of numbers to be on the dice. Tell your students to use efficient ways to work out where they will land with each dice roll rather than simply counting by ones. Note that the numbers are slightly different to a standard Snakes and Ladder board - the player moves from left to right across each row rather than zigzagging their way up the board.

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

Session Two

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

Part One

1. Students need to be seated so they can all see an A3 sized place value mat (Copymaster 2). Each pair of students should have their groupable materials behind their backs for when they need them.
   What is the amount of money that this note is representing? (Holding up $1 note) 
2. Ask the students to count as you repeatedly place $1 notes in the ones column of the place value mat until you reach $9.
   Now what happens next?
3. See if students realise you could just leave ten $1 notes in the ones place or you could exchange those notes for one $10 note. Let a student act as ‘the bank’ as you exchange the money and place a single $10 note in the tens column.
   How do I write this amount of money? ($10 though some students will know about the decimal point).
4. Build up the teen numbers by adding $1 notes to the ones place until $19 is reached.
   What happens now? (Adding one $1 note gives ten $1 notes in the ones place that can be exchanged for one $10 note)
5. Tell the students that you are going to show them amounts of money on the place value mat. Their job is to represent the same number of groupable objects as there are dollars. The purpose of doing this is to help students recognise that within a ten dollar note are ten ‘objects’, dollars. You might use examples like; $17, $28, $53, $67, $95. Below $95 is represented by nine $10 notes and five $1 notes with the matching representation with beans in bags.
                  
6. Use counting to represent the link between changes in the symbols and changes in quantity, both with money and with the grouped objects. Good sequences might be:
   14, 24, 34, 44, 54, …, 94, … What happens next?
   87, 77, 67, 57,…, 17, … What happens next?
   95, 96, 97, 98, 99, … What happens next?
7. In the first sequence ten $10 notes are exchanged for one $100 note and a new column on the place value mat is needed (see Copymaster 2 – enlarge to A3). The second sequence is interesting because ten dollars can only by subtracted until $7 is reached though some students might propose negative $3, like a debt. This is an opportunity for differentiation if it occurs.
8. In the third sequence two exchanges are needed. Ten $1 notes can be exchanged for one $10 note then there are ten $10 notes. These ten notes can be exchanged for one $100 note. Then counting by ones can continue; 100, 101, 102, 103,…
9. Note that students might recognise that their groupable object representation is different to the money in that no exchanges are needed. They simply need to repackage ten tens as a unit of one hundred. You may find suitable containers to house ten tens, and use that as a representation of one hundred. For example, large kete, ice cream containers or big rubber bands.

Pay Me is a task in which students make up the pay for employees. You will need:

• a lot of envelopes, preferably recycled from the school office
• at least 30 of each note ($10 and $1) and about ten $100 notes per group. $100 notes are useful if you want to extend the activity for some students and for later work on place value.
• Copymaster 3 for the students, with one challenge per group of students.

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

• Making up two digit whole numbers with money, though exchanging with the bank is challenging
• Combining two digit amounts of money before paying, with exchange as well

The task is worked through in groups of three students. One student becomes the banker who exchanges notes and checks the pay envelopes for accuracy. Two students work together on creating the pay envelopes. Copymaster 3 has a set of instructions at the bottom of each page about what amount to start with and instructions about putting a pay slip in each envelope. Students cut out each pay slip and put it into an envelope with the correct amount. You might also ask them to write the amount in words though that can restrict the participation of some students. The Banker checks to see that the amount is correct. Each set of slips has a final question that students should answer to show they have completed the task correctly. Look for:

• Do the students represent each amount correctly?
• Do they exchange notes correctly with an expectation of how much money they will have after the exchange?
• Do they use place value to add and subtract amounts where required?

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

• Melissa earns $72 and has to pay $20 in tax. How much money should go in her pay envelope?
• Hone earns $35 on one job and $47 on another job. How much money should go in his pay envelope?

Session Three

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

1. Either use a set of playing cards with the picture cards and joker removed, or create a set of digit cards using Copymaster 4. Each pair of students will need groupable objects or play money behind them. Allow them to choose which representation they prefer.
2. Write these two digit numbers of the board: 12, 20, 34, 45, 51, 67, 73, 86, 90, 100
3. Tell the students that the first activity is a class challenge.
   Each time we make one of these totals our class scores 10 points.
4. The game is simple. A deck of digit cards is shuffled and placed in the middle beside the place value mat (Copymaster 2). Someone turns over the top card and that number of objects is added to what is already there. If a zero (or ten with playing cards) comes up someone gets to choose a single digit number to add.
5. Draw the first card, say 3 comes up. The students get three objects or three $1 notes.
6. Draw the next card, say 9 comes up. The students get nine objects or nine $1 notes and add them to their collection. Expect the students to form a ten with ten of the objects, or exchange ten $1 notes for a $10 note. The total in this case is 12 so the class gets a point as 12 is a target number.
7. Draw the third card, say 0 comes up. A student is chosen to decide what will be added on. They choose 8 so the total is 20, also a target number. Again ten ones will need to be regrouped as one ten or exchanged for one $10 note.
8. Play continues like that with the class getting ten points for every target number scored and ten tens being exchanged for one hundred. It is important during the game that students anticipate the result of adding one before physically doing the addition. Anticipation will promote ‘up over ten’ strategies.
9. Look to see that students are anticipating the result of adding the card number, carrying out correct exchanges with their money or groupable materials, and thinking ahead to the next target number. You might also choose to capture the additions using the empty number line learning object. For example:
   
10. The target game can be played backwards by starting with 100 and taking away the card number. This will help practice ‘back through ten’ strategies, such as 83 – 7 = □ as 83 – 3 = 80, 80 – 4 = 76.

1. Play the game Race to 100 in pairs. Students need a calculator to share. The first player enters a single digit number other than zero, say 4. The second player adds a single digit number other than zero to get a new total, say adds 5 to get 9. The first player goes next, adding a single digit number, then player two adds a single digit number, etc.
2. Play continues like that until either of two things happen:
   • A player gets the total to exactly 100 (They win).
   • A player gets the total over 100 (They lose).
3. Before the students go away, challenge them to find a strategy to win the game. It is also important that they take turns to start the game.
   I want you to think ahead about what the total will be when you add a number. Is there a way to always win or is it just good luck? Does it matter who goes first?
4. Let the students play while you watch for:
   • Do the students anticipate the total before they add a number?
   • Are they looking for a winning strategy?
5. A winning strategy may not be found which means the game can remain a challenge for a few days. Students can also play racing down to zero, starting at 100, and taking away single digit numbers. A winning strategy is to be the second player and ‘cover your opponent’ to the next decade.
6. Consider this sequence of moves.
   • When Player One enters three, Player Two adds seven to make ten.
   • When Player One adds six to get 16, Player Two adds four to make the next decade, 20.
   • When Player One adds eight to get 28, Player Two adds two to make the next decade, 30.
   • If Player Two responds by adding the number needed to make the next decade they must always win, since 100 is a multiple of ten.
7. If students do discover a winning strategy, change the target number to see how they respond. Having a target like 123 is a good extension. You might also ‘disable’ keys other than zero.
   Suppose 0, 2 and 7 did not work. How would that change your strategy?

Getting started

1. Make a simple paper plane. Show students your plane and ask if they have ever tried making paper planes. Discuss the different designs they have tried.
2. Have students work in pairs to make a simple paper plane of their own design. Alternatively they could make a plane the same as the one you have shown them.
3. Have students experiment with their planes to see how far they fly. Discuss:
   How could we measure the distance our planes fly?
   What could we use to measure how far our planes have travelled?
   What would we need to be careful of when measuring?
4. Discuss the use of non-standard measures and the need for a standard unit to allow comparison.
5. Show students a variety of measuring tools and discuss these.
   Which of these measuring tools do you think would be best to measure the distance of our plane’s flight? Why?
   What other things could we use?
6. Emphasise the importance of an accurate starting point for the flight and accurate use of the measurement tools to the closest cm.
7. Have students experiment with a variety of measurement tools to measure the flights of their paper planes. As they work encourage estimation and reinforce the correct use of measurement tools to ensure measurements are accurate to the nearest metre and cm.
8. Students can find the dfference between their estimate and the measured length.

Exploring

1. Tell the students that at the end of the week there will be an air-show. Explain that they will all participate in the show by making and flying planes and there will be a competition to see whose plane can fly the furthest.
2. Over the next few days, have students work in pairs or small groups to try out some different designs for paper planes. Consider grouping students with different levels of confidence and understanding together to encourage tuakana-teina and mahi tahi. If required, provide instructions for a selection of planes online or in books.
3. As students try different designs, have them measure the lengths of their flights. Explicitly model how to correctly measure centimetres using a ruler. Create a class set of measurement guidelines for students to refer back to throughout the following sessions.

4. Encourage them to record their trials in a table similar to the one below, on paper or a device, to help them keep a track of which planes fly the best. This will help them decide which plane they will use in the air-show at the end of the week. Support students as they work through this process as needed. Students may wish to video their week’s investigation to share with whānau and classmates.

   Plane	Flight 1	Flight 2	Flight 3

5. Start each session with a discussion about what they have noticed they might need to think about when making planes. They may suggest the following points.
   • Planes with longer wing spans and larger surface areas for their wings will tend to fly further than planes with shorter wingspans and smaller surface areas. As paper is not very strong it can be difficult to lengthen the wingspan.
   • For planes to fly a long way they need to be stable in flight. A symmetrical plane is more likely to be stable.
   • Weight near the bottom of the plane may increase its stability and allow it to fly further. 
6. As work progresses, you may need to set criteria for the planes. These can be decided on through discussion. They may include the size of paper to be used and limits to the other materials (e.g. the number of paper clips, sellotape, glue or staples available to each group). How the planes are to be thrown may also need to be discussed.
7. As students work, help them with their measurements and discuss these with them. Encourage estimation. Ensure that accurate starting points are used and measurements are made to the nearest cm. Encourage discussion with each other about the maths concepts being explored. 
8. Conclude each session with a discussion of the planes and how far they have flown.
   How far did the plane you made today fly?
   How do you think you could improve your plane?
   What do you think you will try tomorrow?
9. Reinforce the correct use of measurement tools to allow accurate measurements.
   What did you use to measure the distance of your plane’s flight?  
   What steps did you take to ensure your measurements are accurate? Encourage use of the language of measurement when students are discussing their measurements.

Reflecting

1. To conclude the work on paper planes, hold an air-show. Conduct a competition to see which of the planes flies the longest distance.
2. Students can work in groups to measure the distance their planes fly with one plane from each group going through to the final. This will give students maximum practice at measuring distances. Encourage the use of estimation before measurements are made.
3. At the conclusion of the competition reflect.
   What was the difference between the first and second place getters?
   Which planes went the furthest?
   Why do you think they flew so well?
   What did we need to be careful of when we were measuring?
   Which tools do you think were most useful for measuring? Why?

Note the following useful prior knowledge:

• Students have had experience making two types of facts with materials: combinations to 10 (e.g. 6 + 4 = 10, 3 + 7 = 10) and facts with a 10 (e.g. 10 + 6 = 16, 10 + 8 =18).
• Students can recall these two types of facts.

Session 1

1. Begin the session by reminding the class what a number line is. Show an example of what a number line is, and encourage students to share where they have seen them before/what they have used them for. The picture books Ten on a Twig by Lo Cole and Hello Numbers! What Can You Do? By Edmund Harriss and Houston Hughes could be used to engage your students in number thinking. Pose the following problem using either English or Māori numbers. 
   Kahu the kiwi starts on number eight and walks along four more spaces. Where does she end up? or
   Kahu the kiwi starts on number waru and walks along whā more spaces. Where does she end up?
2. Ask a student to come forward and place a peg on the number line where Kahu started.
   How can we find out where Kahu will end up without counting?
   How many spaces will Kahu need to go to get to number 10?
   Now how many spaces has she got left to go?
   
3. Ask similar types of problems such as;
   Kahu the kiwi is on number 9 and walks another 4 places, where will she end up?
   Kiri the kea is on number 13 and flies backwards 5 spaces.  Where does he end up?
   Have the students predict where they think they will end up before getting students to come out and share their strategies on the number line.
4. Now increase the size of the starting number.  For example:
   Kahu has been walking for some time now.  She is on number 27 and walks another 5 spaces.  Where do you think she will end up?
   Ask students to talk to their partner and discuss how they would work the problem out.
   Challenge students to see if they can solve the problem without counting on:
   See if you can solve the problem another way?
   What is the nice friendly number that Kahu is going to pass through?
   How far is it from 27 to 30?
   Now how much further does she have to go?
5. Pose a few more problems that start with a larger number. Continue to model on the number line with pegs. Possible problems are:
   Kiri the kea starts on 49 and slides another 8 spaces.  Where does he end up?
   Tāne the takahe starts at number 87 and wanders on another 8.  What number does he end up on?
6. Send those students who have got the idea off with Copymaster 1.  Give students the option of remaining on the mat with you to go over some more problems.

Session 2 – Marble Collections

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

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

1. Warm up.  Build up students’ knowledge of the bead string so that they know such things as bead 6 is after the first set of yellow beads.  We want students to be able to find these beads without counting each single bead.
   Where is number 8?
   Find number 11.
   Where would number 16 be?
2. Encourage students to explain how they found where each bead was by using groupings, that is by using non-counting strategies. E.g. I knew that 11 was after 10. 
3. Now pose some story problems.
   Moana has a marble collection.  It starts with 9 marbles. Show me where 9 is on the bead string.
   Moana is on a winning streak and wins 6 more marbles. How many does she have in her collection now?
   Use the bead string to demonstrate putting one marble onto the 9 to make it 10 like this:
   
4. Record together on the board:
   9 + 1 = 10; there were 5 left; 10 + 5 = 15.
5. Continue to pose similar problems:
   Kauri has 8 marbles and she wins 6 more. How many does she have now?
   George has 15 marbles and wins 6 more. How many does he have now?
   Hemi has 15 marbles and loses 6.  How many does he have left?
6. Give students Copymaster 2.  Show them a couple of examples of how you would show your working.  Students complete the activity in pairs.

Session 3 – Do and Hide Number line

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

1. Talia the Tūi starts on 9 and flies forward 7 more spaces.  Where does she end up?
   Ask a couple of students to take the number line and pegs away and work out the answer.  Ask the students remaining to visualise what the others will be doing on the number line. The following questions may prompt the students to image the number line.
   Where did the Tūi start?
   How far does the Tūi have to fly to get to 10?
2. Ask the students who took the number line away to share what they did to solve the problem.
3. Repeat with other problems. The following characters could be used to create similar story problems: Kākāriki the kererū, Giana the giant wētā or Tama the tuatara.
   Encourage the students to visualise what they would do on either the number line or bead string.  Extend some of the problems to numbers beyond 20.
4. The following types of problems will continue to challenge the students further.

Session 4 – Problem Solving Bus Stops

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

1. Warm up with some whole class problems like the ones that have been shared in the previous sessions.  Get students to talk to their neighbour and share how they worked out the answer.  Record the different ways students solved the problem by writing it on the board.
2. Place each of the problems from Copymaster 4 on to a large piece of paper.  Place the sheets around the room.  Students can either rotate around the bus stops in pairs randomly or in a sequence to solve each problem.  They are to show their thinking on the large sheet of paper.

Session 5 – Reflection

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

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

Session 1

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

SLOs:  

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

Activity 1

Begin by reading The Greedy Triangle.

Activity 2

Write ‘Clever shapes’ on a class chart. Explain that the students will be making their own small poster about a clever shape. Ask what the purpose of a poster is (To capture people’s attention and to give a short clear message). This could be linked to writing instruction.
Together list the features of a good poster. It grabs the audience’s attention by using:
bold print, a simple and convincing message, interesting colours, a picture or diagram.

Activity 3

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

1. Have students in pairs take a selection of shapes, sort them into groups and explain these using the language of colour or shape. Consider modelling this first, e.g. explaining the colour and number of sides of a triangle as a “think-aloud”.  If necessary, revise the shape names using a song, chart, or game. Consider integrating māori kupu for colours and 2D shape names.
2. Have each student select one of the shape groups, working with these to come up with a reason for their ‘cleverness’. (For example: ‘they fit together with no gaps’.)
3. Record on the class chart the students’ ideas as they each first describe the features of the shape they have chosen, giving the number of sides and corners, and share their creative reasons with the class: for example, ‘circles are clever because they are wheels’, ‘squares are clever because they fit together with no gaps and can be used as tiles’, ‘triangles are clever because they can stand on their heads and fit together’, etc.
   Use this time to model and record the language of the shapes, including writing and discussing ‘tessellation’.

Activity 4

Explain that to make their posters, the students will need to draw around their ‘clever shape’ (as many times as necessary).
Set a time limit and have students complete their ‘clever shape’ posters.

Activity 5

Conclude the session by having students share their work in pairs. Display the list of criteria for a successful poster and have students self evaluate, then give partner feedback about each of the criteria. Model how to give and receive feedback with a partner, making explicit reference to the criteria listed.

Activity 6

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

Session 2

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

SLOs:

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

Activity 1

1. Display the posters from session one. Ask several students to describe the process of drawing around the shape and have them model this on the class chart.
2. Explain that what they have drawn is like a ‘footprint’ of the shape. Write the words ‘side’ and ‘corner’ beside these features on each shape outline.
   Ask, ‘Can you hold or feel an outline (‘footprint’) with your hand?’ (No. It has only 2 dimensions.). Discuss, highlighting the fact that we can say how long the outline is and how wide it is.
   Record ‘width’ ‘length’ ‘dimensions’, explaining why these outlines are called two-dimensional shapes and that this is sometimes referred to as 2D (i.e. because the two-dimensions are width and length).
   Ask, ‘What ‘dimension’ can’t we measure?’ (How deep it is.)
3. Have students write 2D beside their outlines on the class chart, explaining what this means as they do so.

Activity 2

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

Activity 3

1. Have students discuss in pairs and decide whether the foam mosaic shape itself is a two-dimensional or three-dimensional shape.
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 mosaic block is a 3D shape because it has width, length, and thickness (depth) and we can hold it in our hand.
3. Have several students draw the foam mosaic hexagon shape (hexagonal prism) on the class chart, capturing the third dimension (thickness) in their own way. Have all students complete this on their own paper.
4. Write face, edge and vertex on the class chart. Have students locate and identify each feature on their drawing. 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.
   Have students label their drawings using these words. Have all students touch and name those parts on their hexagonal mosaic block.

Activity 4

Return to the language listed in Session 1. Highlight the word tessellate.
Pose the task: Use your shape, a pencil and paper and show how you know whether or not a hexagon tessellates.

Activity 5

1. Ask students to share their results and talk about why hexagons tessellate.
   As they describe their drawings they should use the language of side and corner.
2. Students form groups of four, tessellating their hexagonal mosaic blocks together. Ask them to locate and identify faces, edges and vertices on the mosaic block, and explain to each other using this language, exactly how the tessellation is formed with 3D objects. (For example, ‘the small rectangular faces around the ‘edge’ of each block are up against each other’, ‘the edges and vertices touch’, etc.) These ideas could be recorded on digital devices for sharing.

Activity 6

Conclude the session by having students each make a small poster, or spoken or digital presentation about 2D and 3D shapes that they know. Encourage them to think about the feedback they received about their posters in Session 1. Model this explicitly.

Session 3

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

SLOs:

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

Activity 1

1. Begin by displaying a tessellation drawing from Session 2, Activity 4.
   
   Have students explain why the hexagons in the drawing are two-dimensional.
   Ask , “What does the drawing remind you of?”
   Elicit, ‘a bees’ honeycomb’.
    
2. Display a picture of bees' honeycomb. You could also watch a video about honeycomb, draw on local community expertise, or read a picture book (e.g. The Beeman by Laurie Krebs) to engage students in this context.
   
   Have students discuss and agree whether the honeycomb is two-dimensional or three-dimensional. Have them explain their thinking.
3. Write ‘hexagonal prism’ on the class chart and show the students a model.
   
   Have several students hold it and describe its features. Record these.
   Highlight the plane (two-dimensional) shapes, the hexagon and rectangle that make up the three-dimensional prism.
4. Have several students make art prints of a hexagonal face and of a rectangular face. You could use ink or paint, and create 3D cardboard stamps. Recognise that these prints are two-dimensional.
5. Refer to the chart made in Session 2, Activity 5. Once again have students explain the connection between the 2D language of sides and corners and the 3D terms, in so doing highlighting the fact that plane shapes (2D) build (or make up) 3D shapes.

Activity 2

1. Make available rulers, rectangular pieces of card 24cm x 15cm and cellotape.
   Pose: Can you work in pairs, using this card and tape, to make a hollow hexagonal prism?
   Give students time to explore and create their hexagonal tubes.
2. Have students suggest how they could make a honeycomb model using their hollow prisms. Fit the cubes together and glue the faces together to create a model honeycomb.
   Have students talk about the process using the language of faces, edges and vertices.
3. Notice and discuss how important it is to be precise in the measurements and their folding, ensuring that their hexagons are regular, not irregular. Define these (e.g. a regular hexagon has six sides that are all the same length, an irregular hexagon has six sides that are of different lengths). You could investigate this further by drawing up a T chart on the board and getting students to draw regular hexagons (by tracing around their hexagonal prisms) and irregular hexagons (by drawing a closed, 2D shape with six sides of different lengths). Alternatively, you could create the T chart and 5 regular and irregular hexagons on a Powerpoint. The students could direct you to drag the different hexagons into the different columns, depending on whether they were a regular or irregular hexagon.
   Recognise the bee’s skill in making perfect hexagonal prisms.
   If necessary, have students make their prisms with greater precision to achieve a ‘perfect’ honeycomb such as the bee produces.

Activity 3

1. Conclude the session by listing student’s questions about bees (including why they use the hexagonal shape for their honeycombs). Ask students to suggest possible answers and record these.
2. Suggest they could research this, perhaps with the help of parents or whānau, before the next session.
   They might also like to make their own model bees to inhabit the class honeycomb.

Session 4

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

SLOs:

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

Activity 1

1. Have wax or a piece of real honeycomb available. Display the class honeycomb.
   Have students share the results of their research and, if any students have made their own model bees, to locate these in the honeycomb.
2. Together research, read, discuss and list a summary of information about honeybees and their honeycombs. Consider online sources, journal and early reading resources, and videos:
   http://www.npr.org/blogs/krulwich/2013/05/13/183704091/what-is-it-about-bees-and-hexagons
   http://www.nature.com/news/how-honeycombs-can-build-themselves-1.13398
   
   You may need to structure this task to ensure the engagement and success of your students (e.g. pair up students who would benefit from a tuakana-teina relationship and direct them to research one thing - such as where honey bees live). It may be effective for you to provide the different resources for the students to use in their research, to ensure the language and content is age appropriate and accessible.

Activity 2

Write on the class chart:
What do honeybees need to stay alive?
How does the honeycomb ‘suit’ the bees? 
Discuss each, with reference to the research information, and record the students’ understanding of the key ideas.
Recognise that:

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

Activity 3

Conclude the session by reading A cloak for the dreamer:

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

Session 5

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

SLOs:

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

Activity 1

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

Activity 2

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

Activity 3

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