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.

Session 1-2 

In these sessions we investigate the volume of corn kernels before and after popping. The amount of popcorn to be made is based on the batch size using a popcorn maker, which is usually about 1/2 a cup of kernels. We use non-standard units to measure the volume of both kernels and popped corn. We think about why the measurements vary (because the units being used vary) and what could be done to improve the consistency of the measurements (use a standard unit).

1. Present the students with a small container (about 1/2 cup size). Have the students measure the volume of the container using spoonfuls of kernels. Use a variety of spoons from teaspoons to large salad and serving spoons.
2. Record the different measurements of volume on a chart with illustrations of the spoons used.
   Ask: Why are we coming up with different numbers of spoonfuls needed to fill the container? 
   Record students’ responses, encouraging them to compare the sizes of spoons.
3. Make the popcorn – popcorn makers are the easiest to use in the classroom setting, but other methods provide the same results.
4. As the popcorn pops ask students to make predictions about how much popcorn there will be. Explore some containers including ice-cream containers, bowls and boxes and ask the students to identify a container that they think will be the same volume as the popped corn. (Note that 1/2 cup of kernels makes about 4 litres of popcorn in a popcorn maker.)
5. Once the popcorn is popped, tip it into the selected container to check the students’ prediction, then find a container that is a good fit for all of the popped corn.
6. Get the students to consider the original measure used for the kernels, i.e. 1/2 cup, and to think about how many of these would be filled by the popped corn. Have the students measure the popped corn with the measure used for the kernels. For example, they might find that a half-cup scoop of corn kernels produces 32 half-cup scoops of popped corn.

Session 3

In this investigation the students revisit the results from the previous investigation, and the idea of measuring the popcorn using a litre measure is explored.

1. Revisit the results of the previous volume investigations and talk about the volume of kernels used, and  how much popcorn was made. Focus on the fact that the volume of kernels used was different when measured with different sized spoons, and introduce the idea of a standard measure. Note that standard measures are useful because they enable accurate communication, suggesting that if you were going to send instructions for making popcorn to another class, spoonfuls of corn kernels would not be a useful way to measure, because they might be using different sized spoons than you.
2. Introduce the litre as a standard measure of volume, and ask students to identify things they know that use litres (for example, 2L milk bottles, 2L ice-cream tubs, 1L bottles of juice). Show some litre containers to the class (soft drink bottles could be cut down to a litre or half litre quite easily).
3. Have the students measure the volume of the container which the popped corn fitted into using the litre measure (water, rice, wheat or sand could be used for this task, it doesn’t have to be popcorn.)
4. Get the students to explore the other containers available and measure them in the same way using the litre container and counting how many fit into each bowl, ice cream container and box. Have the students label and order the containers and identify any that would have fitted the batch of popcorn.

Session 4

In this investigation the students think about a standard serving size for the popcorn. The students will find out how many litres of popcorn will be needed for everyone in the class to get one serving.

1. Discuss the question: How much popcorn do you like to eat when you go to the movies? 
2. Have lots of small containers available so that students can choose the size of container which represents the amount of popcorn they would like. Try to come to a consensus about the size of a share of popcorn.
3. Choose one container, which represents the size of a serving of popcorn (having some containers which are 1/2 or 1/4 litre size and directing towards those will make the measurement easier, but this isn’t necessary).
4. Use the serving size container and talk about how many servings you would need for the class. Students may like to consider making some popcorn for the principal, or a neighbouring class. Use water to measure out the right number of serves – a large plastic bucket or container will be needed.
5. Use the litre containers to find out how many litres of popcorn will be needed for everyone to get a serving. Measure by filling the litre container and counting.
6. Look back at the results of Session 3 and have the students work out how many batches of popcorn the class will need to make. Some students may be able to calculate this easily, others will have to use the container that the popped corn fitted into to help them work it out.

Session 5

On the final day of the unit the students make cones to fit one serving of popcorn into. The batches of popcorn will be made and the students will be able to measure out their serving to eat.

1. Provide materials to make cones to hold the popcorn. Ask the students to construct a cone that will hold the agreed serving size of popcorn, then check the volume of their cone by measuring. The volume of the cone can easily be adjusted by making the cone wider or narrower or by cutting from the top.
2. Make the batches of popcorn as the students work on the containers. Have the students fill their cone using the serving size container as the measure.
3. When all the containers are filled, ask the students to record some facts about making the popcorn to share. 
   What volume of corn kernels have been used?
   How many batches of popcorn have been made?
   How many servings were made?
   How many litres of popcorn have been made?
   How much is left over?
   If you were making some popcorn for your whānau how many batches would you need to make?
4. Discuss responses together.

Getting Started

1. Introduce the session by asking the students to work through several equal group (set) problems first and then ask them to pose their own problems. For example:

   • There are 8 cars. Each one can take 2 people on the school trip. How many people are there altogether?
   • There are 6 bags of shellfish (kaimoana). Each bag contains 4 pipi. How many pipi are there altogether?
   • There are 7 waka in the race. Each waka holds 3 students. How many students are in the race?
      

   When writing these problems, consider what times tables your students are confident in applying to word problems. Also consider how students might benefit from working in pairs (tuakana-teina).
   
   The students can represent these and similar ‘equal sets’ problems with:

   • towers of interlocking cubes 
   • threading beads 
   • jumps on the number line 
   • interlocking cubes on a number track 
   • drawing a picture to show the number of waka and the corresponding number of students. 
      

   Note: It is important to link the examples (where possible) to the structure of repeated addition of equivalent sets as multiplication.  For example: 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28 or 7 x 4 = 28. Discuss what the numbers 4, 7, and 28 refer to and what the operations symbols + and x refer to. The multiplication symbols can be thought of as meaning ‘of’. For example, 7 x 4 = 20 means seven sets of four.
    

2. Now ask the students to make up word problems using the problem structure above with different answers. For example, “Write a multiplication problem with an answer of 24”.
    

3. Use several sets of ice-cream containers (all with the same number of items in them) with the contents of each covered except for one.   Ask the students to write story problems for each example.

   • What strategy do the students use to solve the problems?
   • Do they try to count the contents of each ice-cream container by ones; that is, those that are visible and those that are concealed?
   • Do they use skip counting, e.g. 3, 6, 9, …, or repeated addition, e.g. 3 + 3 = 6, 6 + 3 = 9, …?
   • Do they apply multiplication facts, e.g. 5 x 3 = 15 so 6 x 3 = 18 (3 more)?
      

   Be aware that students’ choice of strategy depends on the connection between the conditions of each problem and the number resources that they have available. Expect that the strategies used by individual students will vary.

Exploring

Over the next three days the students are exposed to a variety of different types of story problems. They are encouraged to model the problems using different equipment and explain their answers to others. They think about the most efficient ways of solving the problems. It is important that students are provided with opportunities to build up multiplication facts to 10 and then to 20. Some students may solve these problems without equipment, using the number knowledge they have available.

Rate problems

1. On the first day, work through several measurement rate problems. Draw on the multiplicative strategies students used previoulsy, and model how to work out the answer when necessary. It is good if students notice that the situations are structurally similar to the ‘equal sets’ problems from the previous session. Measurement quantities, especially time, are more intangible than ‘bags of’ or ‘packets of’ in equal sets situations. Acting out problems can support students to see the common structure.
    
2. Use these measurement rate situations:
   [Name] can write their name in 10 seconds. How long does it take to write their name four times?
   You might select a student to role play writing their name and use an analogue clock (less battery) and/or stacks of cubes to track the time in seconds
   
   [Name] drinks four cups of water each day. How many cups does he/she drink in one week?
   Use plastic cups to build up the equal sets of four cups that are involved in this problem. Use another material to track the number of days.
   
   [Name] puts five pieces of harakeke into every bundle. How many pieces of harakeke does he/she need for eight bundles?
   
   [Name] puts three spoons of Milo in each cup. How many spoons of Milo does he/she need for 10 cups?
   
   These problems are more accessible than the time related tasks. Shirts can be cut out of paper and buttons represented by counters. Cups and plastic spoons can be used to model the Milo problem. Both quantities in each rate are tangible.
    
3. Copymaster 1 has some rate problems for students to solve. The problems contain a mixture of tangible and intangible units.
   The students can create similar types of problems with pictures and pose the problems to each other. Encourage them to explain their strategies to each other.
    
4. After time solving the problems, gather the class.
   Discuss what is the same about all the problems you have just worked on.
   Do students express the idea of a rate as being a “for every” relationship?
   Now ask the students to make up similar word problems and to pose their problems to each other. Encourage them to explain their answers to each other. 

Multiplicative comparison

1. On the second day of exploration use PowerPoint 1 to expose your students to comparison situations. The first preference of students may be to look for additive relationships. For example, here is a correct additive response to the question on Slide One.
   How much taller is Jill’s apartment block than Jack’s apartment block?
   S: Jill’s apartment block has 12 floors and Jack’s has four floors. Jill’s block is eight floors higher.
   Students may not offer a ‘times as many’ multiplicative answer. If that occurs pose this problem:
   Jill says that her apartment block is three times higher than Jack’s block. I wonder what she means?
    
2. Slides Two and Three show the additive and multiplicative comparisons that can be made.
   Look for students to note the inverse relationships:
   S: Jack’s apartment block is eight floors less (shorter) than Jill’s.
   S: Jack’s apartment block is one third of the height of Jill’s.
3. Work through the other slides of PowerPoint 1 looking to see if students identify the common structure of finding difference (additive) and finding the scale factor (multiplicative).
    
4. Copymaster 2 contains many multiplicative comparison problems. Some problems are in the form where the relationship is required while others require application of a given scale factor. Students are also encouraged to write equations to represent the situations. Let the students solve the problems, with the support of materials like counters if they need it. After a suitable time, share the answers as a class.
   
   Do your students:
   • Recognise the meaning of “times as many.”?
   • Represent the situations correctly with materials?
   • Identify the scale factor and the set to be scaled?
   • Record the equations correctly?

Arrays

1. On the third day of exploration work through several array problems based on situations in which there are equal groups. When modelling arrays, it may be helpful to talk about the lines across as ‘rows’ and the lines up and down as ‘columns’.  Teams with the same number of members in each are often used during the school day. 
   Pose problems such as:
   The students are lined up in 3 teams for sport. Each team has 6 members. How many students are there altogether?
   Encourage the students to draw representations of problems like this using three rows (one for each team) and six columns (one for each team member). Alternatively, use the students as the objects in the problem. If students draw the situation as three columns of six it opens discussion of the commutative property since 6 x 3 = 3 x 6.
    
2. Other problems you might use include:
   • A tray of eggs has five rows and six columns. How many eggs are in the tray altogether?
   • The kapa haka group is arranged in four rows of children. There are ten children in each row. How many children are there altogether?
   • A chocolate block has six columns, with four pieces in each column. How many pieces are in the whole block?
   • A tray of rēwana bread for the school whānau day has seven columns, with six loaves of rēwana bread in each column. How many loaves of rēwana bread can fit on the whole tray?
      
3. The students can model these and other array problems with:
   • pegboards:  pegs on a pegboard can be used to illustrate arrays in multiplication. 
     For the problem above this could be talked about as 3 rows of 6 pegs or 3 sixes, 
     or 3 rows of 6, or 3 x 6 = 18
     By turning the pegboard a quarter turn, the array still has a total of 18 pegs.
     This could be talked about as 6 columns of 3 pegs or 3 rows of 6 pegs or 6 threes or 6 columns of 3 or 6 X 3 + 18.
   • interlocking cubes
   • colouring grid paper
   • diagrams.
4. Copymaster 3 contains some problems related to arrays. Encourage your students to record an equation for each array, to show how they found the total number of objects. 

Reflecting

On the final day of the unit we play a game called 'Array Trap' in which the students use graph paper to plot arrays.

For this activity you will need:

• a sheet of graph paper for each player (at least 30 x 20 squares);
• 3 x ten-sided dice - each side has a different digit on it (you can use 3 x standard 1-6 dice, but the facts are limited)
• different coloured felt pens.

Players take turns to:

1. Roll the dice and choose two numbers, for example 8 and 2;
2. Mark out a rectangle of that size, for example 8 rows of 2, or 2 columns of 8; or a different pair of factors with the same area, for example, four columns of four.
3. Write the multiplication basic fact in the rectangle, for example 8 x 2 = 16 or 2 x 8 = 16;
4. Players take turns until one player is trapped. That is, they are unable to find space for the array they have rolled.

Discuss strategies for the game such as:

• placing large arrays near the edge to maximise the space available for more arrays
• renaming the factors so the array fits a given space.

Although this unit is set out as five sessions, to cover the topic of statistical investigations in depth will likely take longer. Some of the sessions, especially sessions 4 and 5 could easily be extended as a unit in themselves. Alternatively, this unit could follow on from a unit on data presentation to give students an appreciation of practical applications of data display.

Session 1

Session 1 provides an introduction to statistical investigations. The class will work together to answer the investigative question – How many brothers and sisters do people in our class have? Be sensitive to the needs of your class - if this context is inappropriate for your students , then it may need to be altered.

1. Explain to the class that their job for maths this week will be to gather information or data on the class, summarise the information, or data, collected and then present this as a report which will be sent home to parents and displayed in the class. 
2. Ask students whether they can explain what the word statistics means.
   Explain that statistics concerns the collection, organisation, analysis and presentation of data in a way that other people can understand what it shows.
3. Explain that the class will work in small groups, each of them with the job of finding out information about the class.
4. First we will work as a whole class to answer the investigative question:
   How many brothers and sisters do people in our class have?
5. Ask the students what information we need to get from everyone in the class to answer our investigative question.  
   Students might suggest that we ask how many siblings they have, or they might suggest we ask how many brothers, how many sisters and how siblings they have altogether. 
   The idea of asking about brothers and sisters separately allows for a deeper exploration of the data and a more in depth answer to our investigative question. 
6. Agree as a class to ask about the three pieces of information. See if anyone can suggest how we could collect the data.
7. Working with student ideas, move towards a solution whereby each student records their information on a piece of paper.  
   Sticky notes could be a good way to collect this information from the students as it will allow rearrangements of the data quickly.
8. Suggest that the students divide their paper into three as shown in the diagram below to answer three survey questions. 
   • How many brothers do you have?
   • How many sisters do you have?
   • How many siblings do you have (or total number of brothers and sisters)?
     
9. Get the students to fill in their responses for their brothers and sisters. Check what a response of zero means – in one or all the sections (no brothers/no sisters, no siblings/only child)
10. Work with a partner to check that the information is correct and in the correct place. A good way to do this is for the partner to take the piece of paper and describe to another student the number of brothers and sisters the student has. 
    For example: 
    Pip records the following information about her brothers and sisters.  She gives it to her partner.  Her partner, Kaycee shares this information with another student.  Kaycee says that Pip has three brothers and one sister.  Altogether Pip has four siblings.
    
11. Collect all the pieces of paper (or sticky notes) and ask:
    How can we use the pieces of paper (or sticky notes) to show someone else how many brothers and sisters people in the class have?
    How can we show the information so that people can easily understand what it is showing?
    Hopefully, someone will suggest a more organised list, or counting the number of 0s, the number of 1s etc and writing sentences to explain how many there are of each.
12. Carry out these suggestions to show how much clearer they make the information.
13. Ask for suggestions for other ways to show the information. If nobody suggests it, introduce the idea of using a dot plot.
14. Demonstrate how to draw a dot plot of the information, ensuring that you highlight important features of dot plots; axis, scale and labels on the axis, title (use the investigative question), and accurately plotted points.
15. Students could draw their own versions as a practice exercise. It may be useful to provide a template with an appropriate scale for students to use.
16. Encourage students to draw separate graphs showing just brothers and just sisters as well.
17. Now that we have made a display of the data, in this case a dot plot, we need to describe the dot plot.
18. Ask the students what things they notice about the data. Record these ideas on the board. Write the words “I notice…” on the board or chart paper and capture ideas under this. They might notice:
    • What is the most common number of siblings/brothers/sisters?
    • The largest number of siblings, the smallest number of siblings
    • Where most of the data lies e.g. most of our class have 2-4 siblings.
19. Work with the students to tidy up their statements to ensure that they include the variable and reference the class. For example:
    • The most common number of siblings for people in our class is 2.
    • The largest number of siblings in our class is 7.
    • Most of the people in our class have between 2 and 4 siblings.
20. Explain that over the next few days students will be investigating some other ideas about the class, making their own graphs to show the information and describing what the information shows.

Session 2

This session is ultimately about choosing an appropriate topic to investigate about the class. There will be a real need for discussion about measurable data and realistic topics that can be investigated in the given time frame. It would be a good idea to provide the students with a list of topics if they get stuck, but they should be encouraged to try and come up with something original where possible.

1. Recap the previous session’s work, discussing how the information was collected, how it was presented, and how it was discussed.

PROBLEM: Generating ideas for statistical investigation and developing investigative questions

2. Explain that in this session students will work in small groups to come up with three topics to explore about the class.  The topics need to be ones that they can collect information from the class about and therefore complete the investigation.
3. Discuss criteria that the topics must meet.
   • Is this a topic that the students in our class would be happy to share information with everyone?
   • Would the topic apply to everyone in the class?
   • Is the topic interesting or purposeful?
4. Put students into small groups and give them a few minutes to come up with some ideas that they think they might use. Try to group together students with different levels of competence and encourage tuakana-teina. Encourage them to think of topics that use categories and topics that use counts (e.g. number of siblings). Ideally they should have at least one of each across their three topics.
   If groups are having trouble thinking of ideas, you could try writing a list of suggestions on the board but limiting groups to using one of your ideas only, to encourage them to think of their own. Some ideas could be:
   • Favourite flavour of ice cream/pizza/soft drink etc.
   • Favourite pet
   • Number of pets
   • Colour of eyes
   • Shoe size
   • Favourite native bird
   • Favourite beach
   • Cultural identity
   • Birthday month
   • Home language
   • Number of skips (using a skipping rope) in 30 seconds
   • Number of hops in 30 seconds
   • How they travel to school
   • Number of (whole) hours sleep the previous night
   • Number of languages students speak
   • Number of letters in their first name
   • Number of letters in their first and last names
   • Number of items in their school bag
5. Once groups have decided on their topics, work with them to pose investigative questions.  Model examples of these to help the students pose their own. 
   • How many brothers and sisters do people in our class have?
   • What are Room 30’s favourite pets?
   • What are Room 30's favourite native birds?
   • What eye colours do the people in our class have?
   • What cultures are present within Room 30?
   • How did Room 30 students get to school today?
   • How long are our class’s first names?
   • When are Room 30’s birthdays?
6. Once they have posed their investigative questions, share them as a class, and ensure that they are all appropriate, checking in on the criteria specified in 3.
7. If groups need to change any of their investigative questions, give them time to do so now.

PLAN: Planning to collect data to answer our investigative questions

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

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

   • If the investigative question is: “What are Room 30’s favourite pets?”, ask the students how they could collect the data. 
   • A possible response is to ask the other students “What is your favourite native bird?”
   • Also, the students might want to ask, “What is your favourite native bird out of Tūi, Kiwi, Kerēru, or Kea?” You could challenge them as to if this would really answer the investigative question and suggest that possibly they might change the survey question to allow for other answers.

   Possible survey questions are:

   • What is the colour of your eyes?
   • How did you travel to school today?
   • What language do you speak at home?
   • What culture do you mostly identify with?
   • What month is your birthday?

   In these examples you can see that the survey question and investigative question are very similar, but there are key differences that make it an investigative question (What are Room 30’s favourite native birds? – overall about the class data) rather than a survey question (What is your favourite native bird? – asking the individual).

10. Ensure that all groups record their investigative and survey questions for the next session.

Session 3

Data collection is a vital part of the investigation process. In this session students will plan for their data collection, collect their data and record their data and summarise using a tally chart or similar for analysis in the following sessions.

PLAN continued: Planning to collect data to answer our investigative questions

1. Get the students to think about how they will record the information they get. Options may include:
   • Tally chart
   • Writing down names and choices
   • Using predetermined options
   • Using a class list to record responses
2. Let them try any of the options they suggest.  They are likely to encounter problems, but this will provide further learning opportunities as they reflect on the difficulties and how they can improve them.

DATA: Collecting and organising data

3. Students collect data from the rest of the class using their planned method.  Expect a bit of chaos. Possible issues aka teaching opportunities include:
   • Predetermined options
     • What happens for students whose choice is not in the predetermined options?
     • What if nobody likes the options given and they end up with a whole lot of people choosing the 'other' category and only have tally marks so they cannot regroup to new categories?
   • Using tally marks only
     • The discussed issue about the “other” category
     • Have less tally marks than the number of students in the class 
       • and they think they have surveyed everyone 
       • or they do not know who they have not surveyed yet
     • Have more tally marks than the number of students in the class
   • Possible solutions to the above issues could be (generated by the students)
     • Recording the name of the student and their response and then tallying from the list
     • Giving everyone a piece of paper to write their response on, then collecting all the papers in and tallying from the papers
4. Regardless of the process of data collection we are aiming for a collated summary of the results. 
   

Session 4

In this session the students will work on creating data displays of the data collected in the previous session.

ANALYSIS: Making and describing displays

Numerical data – displaying count data e.g. “How many…” investigative questions

1. Show the dot plot created in Session 1 of numbers of siblings.
2. Discuss how it was made and what needs to be included on it.
3. Get students to identify which one (or more) of their investigative questions involves count data. Choose one of these to work on first if they have more than one.
4. Give students time to work on their first graph, providing support as required. Providing pre-drawn axes may be useful, but students may still need help selecting an appropriate scale to use and placing the “dots”.
5. After all students have completed one of their graphs, bring the class together to share what students have done.
6. Discuss and compare graphs between groups.

Categorical data – displaying data that has categories e.g. “What…” investigative questions

7. Get students to identify which one (or more) of their investigative questions involves category data.  Choose one of these to work on first if they have more than one.
8. Ask students if they can remember how to graph categorical data (you may have already done some work on using pictographs or bar graphs e.g. Parties and favourites). Reference back to this previous work and discuss how it was made and what needs to be included.
9. Give students time to work on their categorical data graph, either a pictograph or a bar graph. You may want to encourage a bar graph depending on how much statistics you have already done prior to this unit.
10. After all students have completed one of their categorical data graphs bring the class together to share what the students have done.
11. Discuss and compare graphs between groups.
12. Send students to work on the last graph of their three.

Session 5

Session 5 is a finishing off session. Students should be given time to complete their graphs if they have not already, and to write statements about what the graphs show.

1. Give groups time to finish graphs as required.
2. Students should also write statements under each graph telling what the graph shows. Ideas for describing graphs were discussed in session 1.  Refer to these ideas.  In addition, starters for these statements could be given:
   • The most popular…
   • The most common…
   • The least popular…
   • The least common…
   • Most students in our class…
   • The largest number of…
   • The smallest number of…
3. Check their descriptive statements for the variable and the group. For example, favourite pets and our class; travel to school and Room 30.
4. Discuss with the students whether there is any action we should take as a result of any of the information we have found out in our investigations. 
5. Ask if there are any conclusions we can make from the investigations we have done.
6. Students could compile their displays as a booklet to take home to their families entitled "About our class" or similar. Alternatively, create a class display of the findings, or share them with another class.

As each pede is developed, help ākonga focus on the number patterns involved by creating tables as below. Similar tables can be drawn for each type of pede.

Use of a hundreds board may help ākonga visualise the number patterns more easily and help them to predict which numbers will be part of the patterns.

The conclusion of each session is an ideal time to focus on the number patterns involved. Questions to develop number knowledge include:

Which number comes next in the number pattern for this pede? How do you know?
Which number will be before 20 in this pattern? (or another number as appropriate)
How do you know?
What is the largest number you can think of in this pattern? How do you know? 
Could a pede with 20 squares be a Spotted Pede? Why / Why not?
Could a pede with 32 squares be a 2-pede? Why / Why not?
Are there any numbers that could be Spotted Pedes and Humped Back Pedes? What are they? How did you work that out?

Session 1

Here we explore number patterns related to the mythical creatures that live on the planet Elsinore. The patterns involve skip counting by 2s.

Introduce the idea of a mythical creature with a picture book like Taniwha by Robyn Kahukiwa, or Zog by Julia Donaldson. Consider whether a local iwi member could come to talk to your class about the history of taniwha in your local area. On planet Elsinore there lives a strange collection of creatures. There is the Humped-Back Pede. The Humped Back 1-pede looks like this. Can you see their eye? And the Humped Back 2-pede looks like this. They have an eye too. (Show your ākonga the pictures below.) Ask ākonga to work individually or in pairs (tuakana/teina model) to make a Humped Back 3-pede with the green tiles, or draw a picture.
Can you work out how many squares a Humped Back 4-pede has?

Gather ākonga together to talk about the creatures that they drew. Explore the number pattern of counting in 2s that comes from the Humped-Back Pedes. Also ask questions like:
Can you tell me how many green squares a Humped Back 5-pede will have?
Can you tell me how many green squares a Humped Back 7-pede will have?
Can you tell me how many green squares a Humped Back 10-pede will have?
How many feet has a Humped-Back Pede with 12 squares?
How many feet has a Humped-Back Pede with 18 squares?
How many feet has a Humped-Back Pede with 20 squares?
Can you tell me how to get the number of squares that a Humped-Back Pede with a particular number of feet has?
Can you tell me how to get the number of feet that a Humped-Back Pede with a particular number of squares has?

Session 2

Here we investigate some more mythical creatures that live on planet Elsinore. The patterns here involve skip counting by 3s.

1. There are other creatures on the planet Elsinore. They look as if they have been made up from squares. The ones with one foot are called 1-pedes. The ones with two feet are called 2-pedes and the ones with 3 feet are called 3-pedes. (Show ākonga the picture below.)
   
2. Did you know that ‘pede’ means ‘foot’?
   How many feet would a 4-pede have? What about a 5-pede?
   Can you tell me how many squares a 1-pede has?
   How many squares does a 2-pede have?
   What about a 3-pede?
   (Put the numbers of squares beside the creatures as ākonga answer the questions, or create a table)
3. Can someone tell me what a 4-pede looks like?
   How many feet will it have?
   How many squares will it have?
   Can someone make one for me with these square tiles?
   Does everyone agree with that?
   (Write 11 under the 4-pede.)
4. Let’s have a look at the number of squares that these Pedes have. Count them out. 2, 5, 8, 11.
   I wonder what sort of Pede comes next? (5-pede.)
   How many squares does a 5-pede have? (14)
   I wonder what sort of Pede comes next? (6-pede.)
   How many squares does a 6-pede have? (17)
   Is there any pattern in the number of squares that Pedes have? (Add on 3 for each extra foot. Talk about skip counting by 3s.)

Session 3

Here we explore patterns further. Where we are particularly interested in linking the number of squares on a creature and the number of feet it has.

1. Here we are going to explore the Spotted Pedes. This is what they look like. Talk about the number of squares they have, and the number of blue and red squares.  Record these beside each of the Spotted Pedes.
   
2. Ask ākonga to draw or make the Spotted 4-pedes? As they work ask them the following questions:
   How many red squares does a Spotted 4-pede have?
   How many blue squares does a Spotted 4-pede have?
   How many squares are all together? How did you work that out?
   Why are there more blue squares than red squares? How many more?
3. Repeat with Spotted 5-pedes and Spotted 6-pedes.
   How many red squares does a Spotted 5-pede have?
   Can you tell me how many blue squares a Spotted 5-pede has? Draw it.
   How many red squares does a Spotted 6-pede have?
   How many more blue squares does a Spotted 6-pede have? Draw it.
4. For those ākonga who need challenging, ask them about the Spotted 10-pede?
   These ākonga could also be encouraged to make a table of there mathematical thinking.
   How many red squares does a Spotted 10-pede have?
   How many blue squares does a Spotted 10-pede have?
5. At the end of the session spend time sharing findings as a class (mahi tahi). Ask ākonga:
   What did you find out about the Spotted Pedes?
   What patterns did you find?
   (Try to get them to see that they have as many red squares as they have feet. This means that it is very easy to find out how many red squares they have.)

Session 4

1. We are going to find out about the Big-Headed Pedes. This is what they look like.
   
2. Ask ākonga to work with a partner (tuakana/teina model could work well here) to draw the next three Big-Headed Pedes (4, 5, 6). As they work, ask ākonga the following questions:
   How many yellow squares does a Big-Headed Pede 4-pede have?
   How many yellow squares does a Big-Headed Pede 5-pede have?
   How many yellow squares does a Big-Headed Pede 6-pede have?
   Do you need to draw the creatures to work out how many squares they have? Why or why not?
   Could you work out how many yellow squares a Big-Headed Pede 10-pede would have?
3. At the end of the session spend time sharing findings as a class (mahi tahi). Ask them:
   What did you find out about the Big-Headed Pedes?
   What patterns did you find?
   (Try to get them to see that they have three more yellow squares than they have feet. This means that it is very easy to find out how many yellow squares they have.)
   I saw a Big-Headed Pede with 16 yellow squares. How many feet did she have?

Session 5

1. Can you tell me what kinds of Pedes we have been talking about this week?
   Discuss the Pedes, the Humped-Back Pedes, the Spotted Pedes and the Big-Headed Pedes.
2. Ask ākonga to work with their partners (tuakana/teina model) to invent a Pede of their own. Ask ākonga to record the first three pedes on one piece of paper and the other three on a second sheet. Ask them to also invent a name for their creature.
3. Pairs could then swap the first three Pedes to see if they can work out the next three Pedes for each other’s creature. They can then check with each other to see if they arrived at the same Pedes for the 4, 5 and 6 creatures.
4. As time allows, ākonga could swap pedes with other pairs.
5. As the pairs work, ask them to discuss the various patterns that they have produced. Ask them questions such as:
   How many squares does one of your 5-pedes have?
   Can you tell me the number of squares an X Pede with 10 feet (or some other relatively large number) has?
   If I had 15 squares, what is the Pede with the largest number of feet that I could draw?
6. Collate the classes Pedes into a book of Pede problems that can be worked on by ākonga during any choosing time.

Prior Experience

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

Session One

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

Session Two

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

Session Three

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

There are many resources already available about hāngī.

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

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

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

Session Four

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

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

Session Five

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

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

Getting Started

1. This unit of work starts with a class discussion about robots. Find out what the students know about robots and their uses. The main point of the discussion is that the students start to understand that robots do not think for themselves. They move because someone has given the robot a set of instructions. Without these instructions the robot can not do anything. The following questions could be used:
   What is a robot?
   What are they used for?
   Why are they used?
   Can robots think?
   Do they have a brain like a human brain?
   If not, how do they know what to do?
   What sort of instructions do robots need to follow?
   What is the name of a person that writes instructions for robots?  (programmer)
2. Explain to the students that they are going to be writing some instructions to see if they can get a robot to do things for them. Show the students the instruction cards and explain what each card instructs the robot to do.
   
3. Have the class sit around the outside of an 8 x 8 grid. This could be marked in chalk on concrete outside, in marker on a large piece of fabric, or with masking tape on the carpet. This is called the walk-on grid later in the unit. This grid will be used throughout the unit.
   F means move forward (haere whakamua) one square.
   B means move backwards (haere whakamuri) one square.
   R means turn to the right (huri whakamatau) 1/4 turn (90°).
   L means turn to the left (huri whakamaui) 1/4 turn (90°).
   End (kua mutu) means the set of instructions is finished.
   If the robot had the cards, F  F  F  F end,  what would it do?
   What would the instructions, R  L  R  L  end, do?
4. Explain to the students that we will be getting the robot to move and do things for us on an 8 x 8 grid. The squares in the grid need to be large enough for a student to stand in them, e.g. 50cm x 50cm. Also explain that they will sometimes be the programmer, writing the instructions, and sometimes the robot, following the instructions.
5. Place the following set of instruction cards, so the students can see them. Pegging the instruction cards onto a line would be a good way to display them. This way the teacher can change the order easily and the students can clearly see the instructions are a set of individual instructions.
   F  F  F  F  R  F  F  F  F  end
6. Get one student to be the robot while the teacher calls out the instructions. Before starting, have the students predict where the robot will end up.
7. Work through the instruction cards, one at a time, to see where the robot ends up. Reinforce the idea of “One card, one action” with the robot only doing what the instruction cards say, nothing more, nothing less. L means 1/4 turn to the left staying within the square the robot is in, it does not mean turn left then move into the next square.
8. Continue this discussion exploring how the robot works and the meaning of each instruction card. The impact of changing the cards, the order of the cards and the starting point of the robot need to be considered. The following questions could be used to facilitate this discussion.
   What would happen if we changed the L to an R in the set of instructions above?
   Where would the robot end up?
   Would we end up at the same place if we used the same set of instruction cards but the robot started in a different place?
   Where would the robot end up if we started here, but kept the same set of instruction cards?
   Starting here, where do you think the robot would end up with this set of instruction cards?
   F  F  F  F  F  B  B  B  B  B  end
   Starting here, where do you think the robot would end up with this set of instruction cards?
   Which direction will the robot be facing at the end?
   B  B  B  L  B  B  B  L  B  B  B  L   B  B  B  end

Exploring

Over the next 2 or 3 days the students will work in pairs, using the instruction cards to programme the robot to do a series of tasks. The number of tasks and the choice of tasks, need to be worked out by the teacher to ensure all students are challenged and engaged. The tasks do not need to be completed in order, although they do get progressively more difficult going down the page.

The tasks are designed for an 8 x 8 grid with the following headings.

An important part of this unit is developing the students' ability to debug. “Debugging” is the process of finding a mistake, identifying the cause and fixing it. To help students think through the mistake they will make, a counter with an arrow on it to show direction moved over a paper grid may help. Others may need to walk through their instruction cards on the walk-on grid. Working out the set of instructions away from the walk-on grid is important.

Getting the students to place their set of instruction cards onto a ring, a length of string or a pipe cleaner will keep their cards in order when dropped.

Task 1

Start the robot at 7, facing into the grid. Move around the grid and leave at D.

Task 2

Start the robot at 5, facing into the grid. Move around the grid and leave at G. There must be more than one direction cards used, i.e. more than one L or R.

Task 3

Start the robot at F, facing into the grid. Move around the grid and leave at 8. Each type of instruction card, L, R, F, B must be used at least two times.

Task 4

What is the least number of instruction cards needed to start the robot at E, facing into the grid and move around the grid and leave at 6?

Task 5

Start the robot at F, facing into the grid. Move around the grid and leave at 8. Each type of instruction card, L, R, F, B must be used at least two times. For the next two tasks two new instruction cards need to be introduced.

PD means to put down the object the robot is carrying in the square the robot is in. PU means to pick up the object in the square the robot is in. The robot doesn’t move or change directions as it picks up the object.

Task 6

Start the robot at F, facing into the grid. Move around the grid and pick up the object from the diamond and place it at the cross. Leave at F

Task 7

What is the least number of instruction cards needed to start the robot at A, facing into the grid and move around the grid, picking up an object at the star and putting it down at the circle, then leaving the grid at 1?

Task 8

Challenge each other by designing a task for others in your class.

Reflecting

Ask students to think about the things they have learnt this week and discuss. Giving each student a blank piece of paper, ask them to write down one or two important things they have learned, as well as to write down their favourite set of instruction cards. Conclude the week with each team selecting one set of instruction cards, i.e. a completed task, the teacher reads out the instruction task as a student, not the developers, follows the instructions.

Session 1

SLOs:

• Recognise three numbers that are related through the operations of addition and subtraction.
• Recognise that there are two addition and two subtraction members of a ‘family of facts’.

Activity 1

1. Make available to the students, digit cards, cards with addition, subtraction and equals symbols, tens frames and counters.
   Ask students to think of three numbers that they like, between and including 1 and 10. Accept all groups of three numbers, and record them on a class chart.
   For example:
   1, 2, 3
   3, 4, 5
   2, 4, 6
   4, 7, 10
   1, 5, 9
   3, 5, 8
   Ensure, without pointing this out to the students, that there are some sets that include ‘family of fact’ numbers, For example. eg. 1, 2 ,3, or 3, 5, 8 or 4, 6 10.
   Have each pair of students select a set of three numbers for their investigation.
   Students take at least four digit cards for each of their chosen numbers, symbol cards, an empty tens frame and counters of two colours.
   Pose, “What equations can you make with your numbers? Make a display or record digitally.” (The students with an ‘unrelated’ set of numbers, for example 3, 4, 5 will quickly discover that no equations can be made using all three digits at the same time.)
2. When some have completed the task, have all the students move to look at the sets of equations that have been made using all three digits. For example: 1 + 2 = 3, 2 + 1 = 3, 3 – 1 = 2, 3 – 2 = 1. Tell them to be prepared to explain what they notice. 
   Discuss their observations, eliciting observations such as, “there are four equations”, “there are two subtraction equations and two addition equations”, “the addition equations are just the other way around (commutative property).”
3. Record one set of four related equations and write Family of Facts on the class chart (this could be digital). Ask whether this is a good name and why. Record student ideas, highlighting the fact that these numbers are related through the operations of addition and subtraction. Families are related.

Activity 2

1. Return to the list created in Activity 1, Step 1. Identify the sets of numbers that have been found to be related by addition and subtraction. Record four equations for some of these. For example:
   1, 2, 3
   3, 4, 5
   2, 4, 6 (2 + 4 = 6, 4 + 2 = 6, 6 – 4 = 2, 6 – 2 = 4)
   4, 7, 10
   1, 5, 9
   3, 5, 8 (3 + 5 = 8, 5 + 3 = 8, 8 – 5 = 3, 8 – 3 = 5)
   Ask, “Can you see what we could do to the other groups of numbers to make each of them into a family?” (Change one of the numbers) Accept suggestions and explore ideas.
   Take one of these groups, for example 3, 4, 5. Have a student model with counters on a tens frame, 3 + 4. This will show that the third number is 7. 
   
   Conclude that 5 can be changed to 7. Explore the other options of changing one of the numbers: 4 to 2, or 3 to 1.
   Explore one more example, for example 4, 7, 10. Discuss that instead of 10 this number should be 11. Alternatively, model seven
   
   and highlight the fact that the 4 could be changed to 3. Ask, is there a third thing we could do? (Change 7 to 6).
2. Continue to make available to the students, digit cards, cards with addition, subtraction and equals symbols, tens frames and counters, paper and pencils.
   Have students choose to work in pairs or on their own, to explore the other sets of ‘unrelated numbers’ on the list. They should make a change and write the four equations (+-) which result.
3. Have students (pair) share their recordings. Discuss what is the same or different about them (this depends on which number they change). They should draw a box around sets of four equations that they have written that are the same as a partner has recorded.

Activity 3

Conclude this session by summarising on the class chart, the features of a family of related facts: three number members of the family, and four equations, two of addition and two of subtraction.

Session 2

SLOs:

• Recognise that there are two addition and two subtraction members of a ‘family of facts’.
• Write and read sets of related addition and subtraction equations.
• Explain, in their own words, the inverse relationship between addition and subtraction.
• Recognise and understand the additive inverse, a (+) - a = 0. (It is not necessary for students to know the name for this.)

Activity 1

1. Begin by reading together the concluding notes from Session 1.
2. Distribute sets of Family Shuffle cards (Copymaster 1) to students.
   (Purpose: To recognise related addition and subtraction equations)
   Explain how to play.
   Each student has one set of 16 shuffled cards. These are dealt out, face up, in a four-by-four array. Two cards (any) are removed from the array and set aside, creating two empty spaces in the array. Individual cards can be slid across or up and down within the array space, but not lifted, till the array shows one complete ‘family’ in each line or a column. The two cards that were set aside are replaced to complete the array.
   To increase the challenge of the task, remove one card only, and/or place each ‘family’ in the same order. (eg. two addition equations then two subtraction equations). Students can swap sets and explore other ‘families’.
   Students could write their own sets to create alternative puzzles.

Activity 2

1. Write a ‘family’ of teen number equations on the class chart. For example:
   14 + 5 = 19, 5 + 14 = 19, 19 – 14 = 5, 19 – 5 = 14.
   Ask students what they notice about what is happening with the numbers.
   Record observations such as, “you can add the numbers both ways without changing the result”, “when you subtract you write the biggest number first”, “when you take one of the numbers away you get the other number.”
2. Pose the question, “How are addition and subtraction related to each other?” Record student’s ideas.
3. Read to the class (or have written on the class chart) this scenario.
   “Sam helped his Gran with lots of jobs. He earned $5. He helped Grandpa too and he gave Sam $1. Unfortunately Sam lost the $1 on his way home.”
   Record, or ask a student to write, the equations that express the scenario on the class chart. (5 + 1 – 1 = 5 and 5 + 1 = 6, 6 – 1 = 5).
   Discuss what is happening in these equations. Elicit the observation that subtraction is ‘undoing’ addition.
4. Have student pairs discuss, create, and agree upon, a parallel scenario in which subtraction “undoes” addition.
   Record several of these in words and in equations on the class chart.
5. Read to the class (or have written on the class chart) this scenario or a similar one relevant for the class.
   “Sam had $6 and lost $1 on his way home. He did some extra jobs for his Mum and she gave him $1.”
   Record, or ask a student to write, the equations that express the scenario on the class chart. (6 -1 + 1 = 6 and 6 - 1 = 5, 5 + 1 = 6).
   Once again, discuss what is happening in these equations. Elicit the observation that addition is ‘undoing’ subtraction.
   Explain that we say this relationship between addition and subtraction is known as an inverse relationship. Record this in answer to the question posed in Step 2 above. Have students suggest a meaning for inverse, then confirm this with a dictionary.
6. Explain that each student is to create a small creative A4 poster or slideshow showing what they have learned so far about number ‘families’ and about the relationship between addition and subtraction. Make paper, pencils and felt pens available to the students.
   Each student could write their own scenario, including a picture or diagram to show what is happening, writing related equations, and an explanation in words of ‘inverse relationship.’

Activity 3

Conclude by sharing and discussing student work.

Session 3

SLOs:

• Recognise that addition is commutative but that subtraction is not.
• Recognise how knowing about number families is helpful for solving problems.
• Solve number problems that involve application of the additive inverse.

Activity 1

1. Begin by sharing the student work from Session 2, Activity 2, Step 6.
   Ask “Why is knowing about families of related facts useful?”
   List student suggestions. In particular highlight the commutative property of addition. For example: If you know 17 + 5 = 22, you will also know 5 + 17 = 22. Also highlight the related subtraction facts.
2. Distribute the addition and subtraction grids (Copymaster 2) to each student. Use the larger class copies to model how to complete each grid. In particular, show how to complete the subtraction grid, subtracting the numbers down the side from those along the top row, and putting a dot for those that ‘cannot be subtracted’, rather than discussing negative numbers at this point.
   Highlight the importance of the students writing their observations about each grid once they are completed. These observations should include number patterns and the fact that the subtraction grid cannot be fully completed.
3. Once completed, have students share what they notice and record their observations.
4. On the class addition grid look for the same sums for both addends.
   For example, 2 + 1 = 3 and 1 + 2 = 3, 2 + 3 = 5 and 3 + 2 = 5.
   
   (Discuss the pattern and also notice the pattern of doubles)
   Write this statement on the class chart and read it with the students:
   We can carry out addition of two numbers in any order and this does not affect the result. Introduce the word commutative.
5. Ask why they can’t complete the subtraction grid in the same way they have the addition grid. Record a student statement that states, in their words, that addition is commutative, subtraction is not.

Activity 2

1. Have students complete the number problems on Copymaster 3. Nana’s party. In giving instructions highlight the importance of the students recording equations and on explaining what is happening with the numbers in the problems.
2. Have students share their work. Discussion should focus on highlighting the relationships between addition and subtraction. Buddy students to support each other if needed.

Activity 3

1. Introduce the game Families on Board. Model how it is played.
   (Purpose: to identify three fact family members and to record four related equations.)
   The game is played in pairs. The players have one Hundreds Board, 25 counters of one colour each, pencil and paper.
   Tens frames showing ten, blank tens frames, and extra counters should be available to the students to model or work out equations if appropriate.
   
   Round one: The Hundreds Board is screened so that numbers 1 – 20 only are visible. Students take turns to place 3 counters on three related numbers. Counters cannot all be placed in the same row. 
   For example: students cannot cover 3, 4 and 7 because they are all in the same row.
   As they place their counters they say and write the four related equations. Students can use tens frames, if needed, to work out or demonstrate the equations and their relationship.
   For example:
   Player 1 
   
   Player 2
   
   Turns continue.
   The challenge is to complete the task between them, leaving only two numbers uncovered. If they have more than two uncovered on the first try, they try again with different combinations.
   
   Round two:
   The Hundreds Board is screened so that numbers 1 – 30 only are visible. Students take turns to place 3 counters on three related numbers. Counters cannot all be placed in the same row. As they place their counters they say and write the four related equations, using tens frames if needed.
   The challenge is to complete the task between them, leaving no more than three numbers uncovered by counters.
   
   Round three:
   Numbers 1-50 are made visible. The task is completed. The challenge is to leave just two numbers uncovered by counters.
2. Conclude this lesson and series of lessons, by recording students’ reflections on their learning about addition and subtraction, and the relationship between them. They can share these reflections with a buddy or whānau.

Getting Started

1. Begin a discussion about activities that ākonga are involved in every week.
2. Brainstorm a list of things such as sleeping, playing, and eating.
3. Once the list is compiled, discuss how often the activities occur. Are they daily, just on school days or a few times a week?
4. Explain that this week they are going to look more carefully at some of these activities and work out how long they spend on them over the course of a week.
5. Have ākonga work in pairs (a tuakana/teina model could work well here) to choose one of the activities from the list, and calculate how long they spend on this activity each week. Copymaster 1 can be used to help guide this process. Ākonga could present their findings digitally, in writing, or verbally. You can support ākonga to select which activity to focus on to ensure the process is not too complicated. Alternatively, ākonga who are ready for extension could be encouraged to think about multiple activities. 
6. As ākonga work, help them with the numbers involved and discuss the strategies they are using.
   How could we work out what seven lots of 2 are all together?
   How do you know 5 lots of 5 minutes is 25 minutes altogether?
   How could you check?
7. Encourage ākonga to discuss the methods they are using. If ākonga are familiar with the + and - symbols and their meaning, use the calculator as a way for ākonga to confirm their answers. Encourage students to check the answers the calculator gives them are reasonable.
   If we know 2 lots of 5 are ten is it reasonable for 5 lots of 5 to be 8 which is less than that?
8. At the conclusion of the session bring all ākonga together to report and discuss their findings.
   How long do you think you would spend brushing your teeth in a week? What did your group find out?
   From our calculations which activity takes the most of our time over the course of a week? Which takes the least?
9. Emphasise that the calculations they have done are based on estimates and are not statements of fact.

Exploring

1. Show ākonga Copymaster 2.
2. Over the course of the next few days look at the statements individually and assess whether or not they are reasonable. You need not look at all the statements, but work at the pace of your ākonga and cover as many as appropriate.
3. For each statement have ākonga work in pairs to do their own calculations to check the statements. Copymaster 3 can be used to guide this process.
4. Encourage ākonga to compare their ideas about what times they think are reasonable for the various activities as well as their methods for calculating to check the statements.
5. A calculator could be used as a final check. Ākonga should be encouraged to confirm the reasonableness of the answers provided with mental calculations.
6. At the end of each session have ākonga share their findings. The types of questions you might use to help develop their understanding include:
   How many minutes in an hour? In half an hour?
   How did you work out how long she spends each day?
   How could you check your calculations?
   Is it reasonable to spend that long to brush your teeth? How long do you think it takes you?
7. Show ākonga Copymaster 4. Use the same process as above to decide which statements are reasonable and which ones are unreasonable.
8. Compare the times taken by Sally for the various activities, with those taken by Hone.
   Who takes the longest?
   How much longer do they take?
   How does this compare with how long you would take to do that?
   Copymaster 5 can help in these comparisons.

Reflecting

1. Have ākonga work in pairs to write their own set of statements about the time spent on different activities in a week.
2. Get ākonga to write three statements but make only two of them reasonable and one unreasonable. Copymaster 6 can be used to record their statements.
3. Once the statements are written, have the pairs of ākonga swap statements and work out which one of the other group’s statements is unreasonable.
4. Can they describe to the other group how they found it and why they think it is unreasonable? Does the other group agree?
5. As a conclusion to the session, have ākonga share their experiences.
   Did you find the statement that was unreasonable? How?
   What made you think it was the unreasonable one? How did you check?
6. Discuss both the strategies used in calculations to check the reasonableness of the answers and the lengths of time taken for various activities. Discussion may cover debate about what is a reasonable length of time taken for a particular activity if required.

The first four sessions of this unit are structured around a number of challenges. The cube challenges involve randomly taking one cube from a bag of coloured cubes. To win the challenge you need to take a cube of a particular colour from the bag. Similarly, the spinner challenges involve one spin on a spinner and are won by landing on a particular colour.  

For each challenge:

1. Introduce the challenge and discuss with ākonga ideas about whether the challenge is fair, and why.
2. Have ākonga play the challenge in pairs, recording how many games they play, and how many of these they win (a tuakana/teina model could work well here).
3. Discuss how ākonga ideas about whether the challenge is fair have changed now that they have tried it.
4. If the challenge is unfair, ask ākonga to suggest how the rules could be changed to make it fair, and then try the challenge with some of the rules suggested.
5. Discuss experiences of playing with the changed rules, and whether ākonga think the challenge is now fair.

When discussing whether each challenge is fair, support ākonga to consider the probability of all possible events in a challenge, ordering them from most likely to least likely and identifying events that have the same likelihood of occurring. Ākonga do not need to know the theoretical probabilities involved. However, they should be able to explain their reasoning.

Session one challenges

Cube Challenge I:

Bag contents: one red and one blue multi-link cube

Choose one cube
To win the challenge: take a red cube

This challenge is fair, because there is an equal likelihood of winning (by selecting a red cube) or losing (by selecting a blue cube).

Cube Challenge II:

Bag contents: one red and two blue multi-link cubes
Choose one cube
To win the challenge: take a red cube

This is not a fair challenge because it is more likely that a blue cube will be taken than a red cube. In fact, players are twice as likely to lose the challenge as to win it.

The challenge will be fair if there are an equal number of red cubes and blue cubes. The easiest way to change the challenge so that you win more often, is to add more red cubes. The more red cubes you add, the more likely you are to win the challenge.

Session two

Cube Challenge III:

Bag contents: one red, one blue and one green multi-link cube
Choose one cube
To win the challenge: take a red cube

This is not a fair challenge. There are three equally likely events: take a red, take a blue, or take a green. In terms of the challenge, players are more likely to lose by taking a blue or a green cube, than they are to win by taking a red cube.

The challenge will be fair if there are an equal number of red cubes and cubes that are not red. The easiest way to change the challenge so that you win more often, is to add more red cubes. The more red cubes you add, the more likely you are to win the challenge.

Cube Challenge IV:

Bag contents: three red and two blue multi-link cubes
Choose one cube
To win the challenge: take a red cube.

This is not a fair challenge. There are two events: take a red, or take a blue, and taking a red is more likely than taking a blue. As far as the challenge is concerned players are more likely to win by taking a red (three out of five times) than they are to lose by taking a blue (two out of five times).

The challenge will be fair if there are an equal number of red cubes and cubes that are not red, so the easiest way to change this into a fair challenge is to add one blue cube.

Session three

Spinner Challenge I:

Spinner:

Spin the spinner once

To win the challenge: spinner lands on green

This a fair game as there is an equal likelihood of winning by landing on a green segment, and losing by landing on a red segment.

Spinner Challenge II:

Spinner:

Spin the spinner once

To win the challenge: spinner lands on green

This is not a fair game. There are three equally likely events: land on green, land on red, or land on blue. In terms of the challenge, players are more likely to lose by landing on red or blue, than they are to win by landing on green.

The challenge will be fair if there are an equal number of green segments and segments that are not green. One way to make the challenge fair is to divide the blue segment in half, and colour half of it red, and half of it green.

Session four

Work with Spinner Challenge III and Spinner Challenge IV. For each challenge have ākonga play the game, suggest adaptations to the rules to make the game more fair, and try the new rules out. Discuss their ideas about whether the game is fair and why, throughout.

Spinner Challenge III:

Spinner:

Spin the spinner once.

To win the challenge: spinner lands on green.

This is not a fair challenge. There are two events: land on red, or land on green, and landing on green is less likely than landing on red. As far as the challenge is concerned players are more likely to lose by landing on green (two out of five times) than they are to lose by landing on red (three out of five times).

The challenge will be fair if there are an equal number of green segments and segments that are not green. The easiest way to change this into a fair challenge is to divide one of the red segments  in half, and colour half of it green.

Spinner Challenge IV:

Spinner:

Spin the spinner once.

To win the challenge: spinner lands on green.

This is a fair challenge because there is an equal likelihood of winning (by landing on green) or losing (by landing on a colour other than green).

Session five

1. Review a few of the challenges from the week. Discuss the idea of “fairness”. Students might make connections to events that have happened in the playground or during sports games. The key idea to emphasise is that a fair game is a game in which there is an equal chance, or probability, of winning or losing. If the rules of a game change, then the chance of winning or losing (and therefore the fairness of the game) might also change. The picture books Pigs at Odds by Amy Axelrod, It’s Probably Penny by Loreen Leedy, or Bad Luck Brad by Gail Herman could be used to engage students in this discussion.
2. Ask ākonga to think about the probability of all possible events in a challenge, ordering them from most likely to least likely and identifying events that have the same likelihood of occurring. Ākonga need not know the theoretical probabilities involved but should be able to explain their reasoning. Words like tōkeke (fair) and tōkeke-kore (unfair) could be introduced here.
3. Ask ākonga to work in pairs (a tuakana/teina model could work well here) to make a new challenge using cubes, spinners, or something else that they select. Make specific links to learning from other curriculum areas to support ākonga - the more support they have in an independent task such as this, the more likely they are to succeed. Some ākonga will benefit from working in a small group with the teacher, before going to work independently. Ultimately, ākonga should develop an idea of whether their challenge is fair or not. For extension, ākonga could select fewer or more cubes/segments in their games.
4. Have ākonga swap challenges and play them.
5. Discuss the challenges faced by ākonga, and have ākonga explain and justify whether or not they think particular challenges are fair.

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 playing a familiar game and evaluating its health benefits and safety considerations. 
Students of this age may be challenged to accurately measure their pulse rate. Therefore these lessons use an alternative ‘indicator’ of the effects of exercise: that is that the intensity of worthwhile exercise should prevent you from singing, but should not prevent you from talking.

SLOs:

• Understand how running a distance contributes to fitness and wellbeing.
• Create a personal benchmark for 1 metre.
• Accurately use three measuring devices to measure a distance of more than 3 metres.
• Correctly record length measurements using abbreviations.
• Understand how many metres are in one kilometre.
• Establish a person benchmark for 1 kilometre.

Activity 1

Begin the lesson by singing a favourite waiata. The aim of the lesson is not to learn a new waiata, so consider using one your ākonga are already familiar with. 

Activity 2

1. Explain to the students that they will be undertaking their regular fitness run (a distance of up to half a kilometre). Discuss the possible effects/benefits of this, and elicit specific statements. Possible responses could include “we get puffed”, “our heart beat/pulse speeds up”, and “it’s good for us”. In response, explain that the increase of beats per minute (bpm) is because their body physically needs to circulate oxygen more quickly as they exercise. Science has shown this is good for us.
   Explain that today, when they each return from their run they should (individually) immediately sing the waiata from activity 1 (above), and then talk to a classmate about their run.
2. Have students complete their run and this task.
3. Ask: ‘Who was able to sing the waiata immediately?’
   ‘Who was able to talk to their classmate?’ 
   Count the responses for each and record these on the class chart. Discuss the results, explaining that being unable to sing immediately shows that they exercised well and their bodies will benefit.

Activity 3

Ask, “How far did you run today?” and “How can we find out?”
Make available 1 centimetre cubes, meter rulers, a 10 metre + tape measure and a measuring wheel.
Have the students share what they know about the metre ruler. Establish that it is called a metre ruler. It is 1 metre long. If centimetres have already been introduced and used, have students line up 1 cm cubes along the ruler to confirm that 1 metre = 100 centimetres.

Activity 4

Develop a personal benchmark by asking: “Who can jump 1 metre?”
Have student pairs measure a 1 metre length on the carpet/floor, marking this with chalk.
Highlight that the measure begins at 0 and ends at 100. Discuss the ‘extra’ space at each end of the ruler.
In their pairs, have students check if each person can jump 1 metre. A tuakana/teina model could work well here. 
Agree that when we think about how big 1 metre is, we can think of it as one big personal jump.

Activity 5

1. Introduce the tape measure and measuring wheel, explaining and showing how each measures 1 metre and multiples of 1 metre.
   Highlight the 1 metre personal benchmark by asking:
   If the tape measure is 10 metres, about how many of your jumps is that?
   If we measured 100 metres with the wheel, about how many of your jumps is that?
2. Write ‘standard measure’ on the class chart and ask what it means. Elicit responses and point out that standard units have been created to allow consistency and communication of measures. We understand each other’s measurements if we use the same measures.
   Explain that the standard units used in New Zealand, and in most countries in the world, are metric units. Some students may be familiar with the use of feet and inches, and could share their knowledge at this point. Discuss possible situations in which a standard measure might be useful (e.g. travel, building). You might be able to make links to community members (e.g. builders) or favourite hobbies (e.g. sprinting). Consider also how links might be made to your cultural context.
3. Write centimetre, metre and kilometre on the class chart. Explain that when we write them often, we want a quick way to record them. Model cm, m and km abbreviations. Students may mention that people often refer to "ks" when talking about kilometres. 

Activity 6

Have a student model both the correct and an incorrect way to measure using a metre ruler. This could be completed in pairs. Highlight how to mark the beginning and end of the measure and how to correctly replace the metre ruler, when measuring a distance greater than 1 metre.
Have several students measure a length that is more than a metre, read the measure aloud, and record this on the class chart. 
Model examples of parts of a metre as well, for example 2 ½ m.

Activity 7

Explain that students will pair up (tuakana/teina) and participate in two measuring tasks to become familiar with the measuring tools. Emphasise that their recording should use the correct abbreviations.

1. Show and have students make a recording sheet, as demonstrated below. Alternatively, you may feel it would be more effective to provide some, or all, of your students with a graphic organiser to be used in this activity.
    

   Measurement from ... to	Metre ruler	Measuring tape

   
   Have students measure at least three different lengths around the classroom, hall, or other designated area, using a metre ruler and a measuring tape. They should that they get the same measure using each tool.

2. Clarify the exact fitness course route, the start and end points, and set relevant boundaries. Have student pairs take turns using the measuring wheel to measure the distance around the course and to then record the result.

Activity 8

Conclude the session by sharing measurement results and reviewing the fitness course distance. Discuss how many metres in 1 kilometre. Estimate and calculate together the number of times they would need to run around the fitness course to cover a 1 kilometre distance. Establish a rough benchmark for 1 kilometre. (For example, 1 kilometre is 5 times around the fitness course.)

Session 2

This session is about recognising that rules that address fairness and safety, help to ensure that a PE/fitness activity is enjoyable. As students design a PE/fitness activity, they learn more about accurately measuring outdoor spaces.

SLOs:

• Understand that rules are designed to ensure fairness and safety.
• Pose an investigative question.
• Create and write instructions for a PE/fitness game, giving consideration to fitness, safety and enjoyment.
• Accurately measure and record the length of a given outdoor space.

Activity 1

Begin with a fitness run.

Activity 2

Explain that the class is going to play a favourite PE game (for example: Ki-o-Rahi, Tunnel ball, Scatter Ball). Together, list the rules on the class chart.
Ask: Which of the rules are about making the game fair? Write F beside these. Discuss that fairness makes the game more enjoyable for everyone.
Ask: Which of the rules are about making the game safe? Write S beside these. Discuss any anomalies. If there are no specific safety rules, list some generic ones.

Activity 3

Return to the class, review the enjoyment of the game and ask if playing the game will make them fit. Discuss why/why not. Refer to the “talk/sing measure” from session 1.
(The response may be, “No, because it didn’t make me puff and I could sing.”)
Review the list of rules and confirm the fairness (F) and safety (S) decisions made earlier.
Highlight the importance of games and activities being safe and enjoyable. Ask if any other safety rules should be added and why.

Activity 4

1. List on the class chart, the words ‘enjoyment, fitness and safety’.
2. Suggest that students will work in pairs or small groups to create their own PE/fitness games. Through discussion, lead students to pose an investigative question. For example:
   ‘Can we design a game or activity that keeps us fit, is enjoyable and is safe?’
   Record this on the class chart/modeling book.
3. Explain that students will be using an outdoor space with suitable boundaries, for example, the school tennis courts. Large balls and small balls will be made available. (Make other equipment available, as appropriate.)
4. Clarify the task. Students will work in pairs or small groups to:
   • Measure and record the size of the designated outdoor space (using skills learned in Session 1.)
   • Invent a simple game that the class can play.
   • Write down clear instructions and rules, checking for safety and fairness.
5. Set time limits and clarify expectations. Have students complete the task.

Activity 5

Have groups swap game instructions with another group. Have them read, critique, seek clarification and suggest refinements or improvements to the other’s game design. You may wish to come up with guidelines and/or a rubric for students to use during this. The groups may wish to play each other’s games as well.
Give time for these adjustments to be made.

Activity 6

Review pair measurements for the outdoor space. Tennis courts are about 23.8m x 8.2m. Remind students that 1000m = 1 km. Estimate together the number of times the length of the court would need to be run to achieve the length of 1 kilometre. (eg. Round up to 25m. 25m x 40 = 1000m) Together calculate the number of lengths of the tennis court needed to run 1km.
Students may use this as a 1 kilometre benchmark. You could also consider other known lengths such as from the school gate to the walking pou, or other local community locations.

Session 3

This session is about creating a simple questionnaire to evaluate each group's activity, and learning about dot plots.

SLOs:

• Plan data collection.
• Collect data by trialing and evaluating an activity on its fitness, safety and enjoyment values.

Activity 1

Begin with a fitness run.

Activity 2

1. Review the investigation question recorded on the chart in Session 2.
   Ask the students how they should gather the data to answer the question. Guide discussion and agree on a simple evaluation form to be completed by the class after playing each game. For example:
   The name of the game: ________________________________
   Circle for each: 1 (not so good) 2, 3, 4 or 5 (excellent),
   Enjoyment: 1 2 3 4 5
   Fitness: 1 2 3 4 5
   Safety: 1 2 3 4 5
2. Print off the evaluation or have students copy this and practice using it by completing an evaluation for the game played at the start of Session 2.

Activity 3

Working together (mahi tahi model), collate and present the data using three dot plots. For example:

Discuss the dot plot features, the results, and draw conclusions.

Activity 4

For the remainder of the session, and for sessions to follow, have students participate in and evaluate each other’s games. Each group of students will collect the data for their game to analyse and present in Session 4.

Session 4

This session is about student groups sorting the data from their classmate’s evaluations of their activity and presenting the findings.

SLOs:

• Sort and display category data.
• Answer an investigative question.

Activity 1

Begin with a fitness run.

Activity 2

Make available pencils, paper, and sets of data for each pair activity.
Have students work in their groups to sort their data and to discuss their findings. Each student should create three dot plots to present their data, record their own findings and should answer the investigative question in their own way.

As students work, have them record on a small poster, their knowledge of centimetres, metres and kilometres, the relationship between them, and explain why we have standard measures.

Session 5

This session is about communicating investigation findings to others and sharing their understanding of standard measures of length.

SLOs:

• Present findings.
• Review and reflect on the investigative process.
• Review and reflect upon measurement learning.
• Discuss the need for small units of length measure and introduce millimetres.

Activity 1

Begin with a fitness run.

Activity 2

1. Have each group present their findings about their game to the class. Allow time for other students to provide feedback.
2. Together, as a class:
   • Summarise on the class chart conclusions about safety, enjoyment, and fitness.
   • Reflect on the investigation process and suggest ways it could have been improved.

Activity 4

Arrange the length measurement tools in front of the students.
Have individual students share their learning about each of the tools.
Ask which tool would be used to measure small lengths.
Introduce the millimetre measure for tiny lengths.

Conclude by reviewing personal benchmarks for (1cm), 1m and 1km.

Getting started

Starting with a simple pattern, we build up the level of difficulty and see that it’s necessary to use a table to record what is happening.

1. Build up the letter ‘I’ using coloured tiles or paper (see the diagram below).
   How many tiles do we need for the first ‘I’? The second? The third?
   

   Who can tell me how many tiles we’ll need for the fourth ‘I’?
   Can someone come and show us how to make the fifth ‘I’?
   How many tiles will we need for the tenth ‘I’? Make it.
   What is the number pattern that we are getting?
   If we had 11 tiles, which numbered ‘I’ could we make?

2. Now let’s make it a bit harder. Let’s make an ‘I’ by adding a tile to the top and the bottom each time (see diagram).

   Repeat the questions from the last ‘I’ problem.
   How many tiles do we need for the first ‘I’? The second? The third?
   Who can tell me how many tiles we’ll need for the fourth ‘I’?
   Can someone come and show us how to make the fifth ‘I’?
   How many tiles will we need for the tenth ‘I’? Make it.
   What is the number pattern that we are getting?
   How many tiles do we add on at each step?
   If we had 11 tiles, which numbered ‘I’ could we make?

3. It was easy to see what was happening in the original ‘I’ problem and to see how many tiles each ‘I’ needed. It wasn’t quite as easy with the second one we did. But what if we had a really difficult pattern? How could we keep track of what’s going on and see how many tiles we need for each letter (reta)?
   After korero, suggest the idea of a table.

4. The original ‘I’ problem would give us an easy table. It would look like this:

   ‘I’ number	1	2	3	4	5
   Number of tiles	1	2	3	4	5

   What would the table look like where we added two tiles at a time?
   Draw up the table with help from the ākonga

5. Now let the ākonga complete the table for the letter pattern on Copymaster 1. Support ākonga the while they are working and help them by asking leading questions such as:
   How did you know how many tiles to use on the fourth ‘L’?
   What is the pattern (tauira) here?
   Which ‘L’ in the sequence will use 27 tiles?
6. Bring the class back together and discuss their work.
   Tell me what numbers you used to fill the table. (Check that they are correct by counting the tiles.)
   What patterns can you see here?
   How did you get the number of tiles for one ‘L’ from the one before?
   How many tiles would you need for the 10th ‘L’?
   If you had 23 tiles, what numbered ‘L’ could you make?

Exploring

For the next three days the ākonga work at three stations continuing different number patterns and building up the corresponding tables. In the first station, the ākonga complete a similar problem to the one in ‘Getting Started’. In the second station the ākonga find a missing shape in the pattern sequence. Finally in the third station ākonga make their own pattern that fits the given table of values.  Ākonga could work in three groups that provide tuakana/teina support. At the end of each day, bring them back together to discuss their thinking. Ask them the kind of questions that were used in ‘Getting Started’. Use the tables to discuss the patterns involved and the relation between successive numbers in the sequence.

Day 1
The material for these stations is on Copymasters 1.1, 1.2, 1.3, 1.4. The ākonga continue the pattern and complete the table. Ākonga could continue to use tiles to support their learning.

Day 2
The material for these stations is on Copymasters 2.1, 2.2, 2.3, 2.4. The ākonga find the missing element of the pattern and complete the table. Tiles could be provided for some ākonga who may need to construct the missing element before drawing it.

Day 3
The material for these stations is on Copymasters 3.1, 3.2, 3.3, 3.4. The ākonga make up their own pattern to fit the values in the table. Ākonga could use the tiles to create patterns and count to check that they match the numbers in the table before drawing them on the sheet. 

Reflecting

On the final day let the ākonga make up their own patterns using numbers or shapes, instead of letters. They could construct these with tiles first, or by drawing. Encourage ākonga to think of patterns in their environment, for example, tukutuku patterns in the local wharenui or museum. Ākonga should also provide a table to show the number pattern of their number or shape.  Some ākonga might want to leave gaps in the patterns of their numbers or shapes. Other ākonga could fill this in when they share their pattern with the class.

When ākonga are sharing their patterns with the class, point out the importance of the table in seeing what the number pattern is.

The teaching sequence is designed for teachers to guide students through the five key stages of a statistical investigation. The context used is voting for a native tree which is to be planted in the school. You may choose to conclude the unit by buying and planting the chosen tree, or you may prefer to simply vote on students’ favourite tree. At the end of this unit other possible statistical investigations are provided as examples of what students might further investigate in pairs or small groups. This would allow for the unit to be extended beyond a week.

Session 1

This session is about framing the context for an investigation, and deciding on an investigative question, a sample, and a method of investigation.

1. Introduce the topic by explaining that we are going to choose a favourite tree for the school. Four native trees have been selected as the most suitable and our class is responsible for organising the senior classes for the final voting. The four trees at the top of the list are: totara, kauri, kowhai and ti kouka. We need only one of these trees and it will be planted where there is plenty of space.  We want this tree to have special significance and to represent the pupils who attended our school as well as future students. Some of the meanings associated with the trees selected include:
   • Totara - life and growth, 
   • kauri - strength, 
   • kowhai - personal growth and moving from the past with renewed adventure and 
   • ti kouka - independence.
2. Start the discussion by asking “What are some of the special features of these four New Zealand trees?” Brainstorm the features on a large sheet of paper. Ask students to think about which tree they would like and why. “Which tree will we choose to plant in our school?” (a. our investigative question).
3. Examples of tally charts or lists or specific criteria could be used to exemplify ways of collecting such data. Organise students into groups of four to ask them how might this data best be collected? Remind students of the purpose of collecting data. (b. How might we design and collect data to answer our investigative question?).
4. As a class, share ideas from each group. What ideas have each group come up with? Come to a consensus of the method for collecting the data ready for the next day. 
5. Decide who the participants should be. (c. Who are we going to ask? How many classes? If the school population is too large limit to two or three classes).

Session 2

This session is about going through the voting process and collecting the data in a systematic way. 

1. Recap the discussion from the previous day and encourage students to think about what we are trying to find out and why are we carrying out this investigation?
2. Have the nominated tree names written on labels. Use the data collection method that was decided upon in the previous day’s discussion.  Further teaching on tally marks and how they work may be needed depending on the needs of your students. Why do we use tally marks when collecting large amounts of data? 
3. Give students two square pieces of paper: one for the votes of other classes, and one for their own vote.
4. Everyone will be asked to write down the name of the tree they are voting for on the square piece of paper.
5. On returning to the classroom, ask everyone to sort their votes in groups. Ask them to sit in a circle with their data cards in front of them. Hold up the name of a nominated tree and invite those people who voted for kōwhai to bring their voting square to the front. The voting squares are placed side by side as illustrated below.
   
   
    
6. The process is then repeated for the other nominees and the voting squares are added on to make a long  strip.
7. Complete until all votes are represented in the strip. 
8. Ask students to make statements about what they can see from the strip and relate this to their investigation.

Session 3

In this session students will see how a strip graph (kauwhata tāhei) can be transformed into a bar graph (e.g. How are we going to display our results? In tables? What is the best graph to use?).

1. Break the strip graph into the votes for each tree. Place the name labels at the bottom of the graph and place each piece of the strip graph above the appropriate name, as illustrated below.
   
2. Ask the following questions:
   What would be an appropriate title for our graph?
   What labels could you use for this graph and where would you write them?
3. Label the axes and give the bar graph (kauwhata pou) a title so that others could make sense of the display. A good idea is to write the investigative question as the graph title.
4. Ask the following questions:
   Is this a helpful way of presenting this information? 
   It is easier to make statements from a bar graph or from a strip graph?
   Which completed graph shows our results most clearly? 
   The questions could be asked in a whole class situation or students could complete a bus stop activity with the questions being posed on the top of a large piece of paper and students visiting each station to record their ideas.  Small groups would also be a valuable way for ideas and responses to the questions to be discussed and explored.
5. Summarise the responses and make recommendations about when each graph might be a useful way of presenting information
   Ask the students what they notice about the information shown on the bar graph. Use the prompt “I notice…” to start the discussion. These “noticings” could be recorded as speech bubbles around the bar graph.
6. Conclude by revisiting the original investigative question posed: "Which tree will we choose to plant in our school?" Make statements from the results to answer the original investigative question (e. What is the answer to our investigative question based on the results of our investigation?). 

Session 4

In this session students discuss the types of things that are worth investigating and carry out their own investigation.

1. Talk about the types of things that are worth investigating. It is important that possible investigations are relevant to what is happening in the students’ lives and what is happening at school at the time. Possible investigative questions may include:
   What game should we play at the end of the week? 
   What should we spend the fundraising money on? 
   What should we plant in the school/community garden?
2. Encourage students to review the process they went through to decide how they were going to collect and present the voting data.  List the process as questions that students can refer back to. For example
   1. What is our investigative question?
   2. How will we collect the data to answer our investigative question?
   3. Who are we going to ask? How many people are we going to ask?
   4. How are we going to display our results? In tables? What is the best graph to use?  
   5. What is the answer to our investigative question based on the results of our investigation?
3. Students can now work in small groups or pairs to carry out their own investigation.  This could be completed as a homelink activity or as a follow up activity.
4. Results should be shared and conclusions made based on the results. This investigation is likely to require at least three sessions of fairly intensive work; one session of planning and checking, one session of collecting and displaying data, and one of developing statements and conclusions and presenting these. Digital links could be made by directing students to display their graph and findings as a PowerPoint or set of Google slides.

Prior Experiences

• Idea of fair shares
• Know 1/2, 1/3 and 1/4 of shapes such as rectangles and circles
• Doubles and their corresponding halves

Although the unit is planned around 5 sessions it can be extended over a longer period of time.

Session 1 

The purpose of this session is to develop students' understanding of equal parts (the denominator).

Resources:

• Copymaster 1
• Playdough (see recipe) 
• Plastic knives
• Cutting boards

1. Ask the students how they would share a bale of hay (block of chocolate) between 4 sheep (4 people) fairly. Other contexts could be the sharing of pizzas but the shape of the rectangle is easier for students to cut into equal shapes. Introduce the word equal – what do you think it means?
    
2. Distribute copies of the eels start, action, results boards (Copymaster 1), some playdough, a cutting board and a plastic knife each. If it is possible it would be better to have 1 board between 2 students.
    
3. Have each student or pair of students make six equal sized round eels, by rolling 6 equal amounts of play dough. Put one eel in each of the outlines on the left hand side of the board. The eels are small as they have their heads and tails cut off. The eels we are making today are miniatures of the big eels. That means we are making small copies of the big ones.
    
4. The following story can be used to guide students through the actions as described on the board. The story could be adapted to a different context, such as the sharing of rēwena bread, to suit your students.
   Hoepo and his brothers and sisters are at their Poua’s tangi and although they are sad they are looking forward to the hakari because shortfin eel is always on the menu.
   Hoepo is planning to go early to the marquee because he wants one eel all to himself. He is given one whole eel. Hoepo doesn’t know it but he is going to be one sick boy!
   The twins appear and they are told they have to share one eel evenly between the two of them. There are now two half parts.
   The triplets come next and Aunty Wai says we will have to cut another eel into three equal parts. There are now three third parts.
   Hoepo’s sister has come with her three friends. Aunty Wai says that they will have to cut the eel into four equal parts. There are now four fourth parts. Aunty Wai says they are also known as quarters.
   Hoepo’s five baby cousins are only allowed to eat small portions so Aunty Wai cuts the last eel into five equal parts. There are now five fifth parts.
    
5. Ask the students to mark the lines and cut lightly then if they haven’t made equal parts they can smooth the playdough out and start again.
    
6. After cutting, the separated parts are put in the RESULT column next to their descriptions.
    
7. Ask the students to share with the person next to them what they can see. Hopefully someone may say, "The more cuts we made, the smaller the equal part." Prompt them towards that knowledge.
   I want you to look at one of the third parts and one of the fifth parts. Which is bigger? 
   Have the students take one of each of the equal parts and put them on another blank board.
   Order the equal parts from smallest to biggest. 
   Let’s say the names. 
   Students should order from 1/5 – 1
   Put them back on the original board.
   How many halves are equal to the whole? 
   How many fourths are equal to the whole? 
   How many thirds are equal to a whole? 
   How many wholes are equal to a whole? 
   Depending on the age of the students symbolic notation can be introduced, using the terms like 1/2 and one half part interchangeably. 
    
8. Repeat the steps above using the biscuits start, action, results boards and fresh playdough. The first boards should, if possible, remain on view. With this second board a variety of division lines are easily found, eg fourth parts. Rēwena bread could be used as another context instead of biscuits or pies (as in number 9). 
   
9. Repeat the steps above using the pies start, action, results boards. If possible, fresh play dough should be used, the other two boards remaining on view. The lines of division should be radial as shown below.
   

Session 2

The purpose of this session is to develop the idea that parts of the same kind may not look alike. In Activity 1 this arose from the use of different objects. Here we see that this can be so, even with the same object.

Resources:

• Copymaster 2
• Playdough 
• Plastic knives
• Cutting boards

Revise knowledge about equal parts.
What can we remember from yesterday? Write students’ comments in your modeling book.

1. Begin with the first page. This is used in the same way as the board for Activity 1. Ask students to complete the first 3 lines (making halves in three different ways). There are three simple ways, see if you can find them.
   The three straightforward ways are:
   
    
2. Next, they complete the next two lines (the third parts) which offers only two straightforward ways.
    
3. Complete the second page (the fourth-parts). There are six ways of doing this which are fairly easy to find.
    
4. Students may want to go back to the halves, thirds or fourths/quarters boards and see if they can find some more.

Session 3

The purpose of this session is to consolidate the concepts formed in Sessions 1 and 2, moving onto a pictorial representation.

Resources:

• Copymaster 3
• Copymaster 4

1. Begin by looking at some of the parts cards from Copymaster 3 together. Explain that these represent the objects which they made from playdough in the last activity, eels, chocolate bars, biscuits and some new ones. They also represent the parts into which the objects have been cut, e.g. third-parts, fourth-parts, halves, fifth-parts. Some have not been cut: these are wholes.
    
2. Shuffle the parts cards and spread them out on the table, face upwards.
    
3. Give each student one of the set names from Copymaster 4.
    
4. Each student should collect the parts cards that match the name of the set they have been given, for example, halves.
    
5. Students check each other’s sets and discuss if necessary.
    
6. Next, introduce Skemp’s mix and match game. This is a great game in that the students are consolidating what they know about denominator without being introduced to the word. Older students may have heard that word and it is important that they understand what it is. The denominator names the number of equal parts.

Mix and Match: Rules of the Game

This game is best played by groups of 2-4 people (a tuakana/teina model could be used here).

1. Share the parts cards from Copymaster 3 evenly between all players. Each player should have their cards in front of them in a single pile, face down.
2. Place the mix and match card somewhere where all players can see it. The purpose of the mix and match card is to remind players of the directions in which they can build.
3. The first player turns over their top card and places it in the middle of the playing area.
4. Players take it in turns to turn over a card and place it alongside a card already on the playing area. When placing cards they must ensure that:
   • cards in a line in the ‘match’ direction are each split into the same number of parts (e.g. halves, thirds…).
   • cards in a line in the ‘mix’ direction are each split into different numbers of parts
     
5. If a player cannot place their card they put it back on the bottom of their pile and it is the next player’s turn.
6. To make the game into a contest you can give a point to any player that makes ‘three in a row’ in either direction, and add a rule that says you can not have more than three cards in a row at any time. It is possible to gain 2 points by completing both a match and a mix by placing your card in the right place. The player with the most points wins.

Session 4 

The purpose of this session is to develop students' understanding of a number of like parts (the numerator).

This is the next step towards the concept of a fraction. It is much more straightforward than that of session 1 -3 which involved (i) separating a single object into part objects (ii) of a given number (iii) all of the same amount. Here we only have to put together a given number of these parts and to recognise and name the combination.

Resources:

• 3 sets of animals (these could be models or pictures)
• Copymaster 5 
• Copymaster 6 
• Five trays or plates
• Playdough
• Plastic knives
• Cutting boards
• Five set loops

Warm up

1. Count in halves up to a number such as 3.
   1/2, 2/2, 3/2, 4/2, 5/2, 6/2 (be prepared for students to carry on counting and not realise that 6/2 is equal to 3).
    
2. Ask the students:
   How many halves did we count (six halves = three wholes, write on the board)
   How many halves do you think would equal 6? Write on the board
   How many halves do you think would equal 9? Write on the board
    
3. Relate back to their knowledge of doubles and halves.

A Number of Like Parts

This is an activity for up to six students working in two teams (mahi tahi). Its purpose is to introduce the concept described above.

Set the scene for your students:

Yesterday we went to the zoo and saw the zoo keepers feeding some of the animals. We are going to pretend that some of us are the animal keepers feeding the animals and some of us are the zoo kitchen staff, preparing and cutting up all the food.

1. One team acts as animal keepers, the other works in the zoo kitchen. The latter need to be more numerous, since there is more work for them to do.
    
2. A set of animals is chosen. Suppose that this is set 1. The kitchen staff look at the menu and set to work preparing eels, as in Making Equal parts. The animal keepers put the animals in their separate enclosures. They may choose how many of each. For example:
   
    

3. The animal keepers, one at a time, come to the kitchen and ask for food for each kind of animal in turn. The kitchen staff cut the hay as required (using the shaped playdough, plastic knives and cutting boards), e.g.

   Animal Keepers may say:	Zoo Kitchen Staff may say:
   Food for 2 elephants please	Here it is, 2 whole bales of hay.
   Food for 5 giraffes please	5 third parts. Tell them not to leave any scraps.
   Food for 3 rhino please	Here you are. 3 half parts from 2 bales of hay. There is one half part left.
   Food for 5 zebra please	Here you are, 5 quarters or 5 fourth parts.
   Food for 6 sheep please	6 fifth parts. Lucky sheep.

4. Each time the animal keeper checks that the amounts are correct, and then gives its ration to each animal. The keepers check each other’s work.
    
5. When feeding time is over, the food is returned to the kitchen for reprocessing. Steps 1 to 4 are then repeated with different animals, keepers and kitchen staff.

Note that the eels, slabs, and hay should be of a standard size.
Note also that the eels, after their head and tails are removed, resemble the eels in a cylinder shape and the slabs of meat and hay are oblongs.

Session 5

So far we have covered denominator and numerator without mentioning their names. Students need to understand that the denominator names the equal parts and numerator names the number of like parts. 

Resources:

• Copymaster 7
• Copymaster 8

Revise some of the ideas from sessions 1-4, selecting from these activities:

• You may want to give them some unit fraction cards and ask them to order from smallest to biggest.
• Give simple fraction addition problems to add below 1.
• Give them a piece of paper and ask them to fold the paper into equal parts that they have chosen. For example, a student may choose to fold the paper into 4 equal parts. They need to verbalise what they have done. Ask them to tell you about three of the sections.
• An activity for students if the symbolic notation has been introduced could be to have a pack of cards which they can use to match pairs, e.g. 1/4 goes with one fourth. When they have matched the cards, ask each student to solve a simple addition problem for you.
  I want you to give me one fourth + one fourth (write on the board). Can anyone give me cards that mean the same but are written in a different way? (1/4 + 1/4)
  I want you to find one fifth and one fifth and one fifth. How many fifths are there? Can anyone give me cards that mean the same but are written in a different way? (1/5 + 1/5 + 1/5)
• Give the students three problems to solve with a partner (tuakana-teina model could be used).
  What happens if we had to feed 5 giraffes and they were allowed 1/3 of a bale of hay each?
  What happens if we had to feed 5 rhinos and they are allowed 1/2 a bale of hay each?
  What happens if we had to feed 6 zebras and they are allowed 1/4 of a bale of hay each?

To conclude the session, ask the students to work in pairs and complete a think board (Copymaster 7 provides a blank think board and Copymaster 8 shows a completed example.) Suitable fractions for think boards include 2 halves, 3 fourth parts, 2 fifth parts, 4 third parts, 2 quarters. Show and discuss the example of a completed think board for 3/5.

Ignite children’s prior knowledge by discussing home or local community gardens that they are familiar with. It may also be helpful to introduce this unit by reading a book about garden settings or viewing images of garden settings online. The overall aim of the unit is to create a classroom display of a garden using the activities as starting points. Be as creative as you can! 

Session 1: Up the garden path

In this session students will explore shapes that tessellate or repeat to cover the plane without gaps or overlaps. Although the students will only be covering a strip (path) any covering of a path can be used to tessellate the plane simply by putting paths together.

1. Explain to the students that they have the task of building a garden path. If possible, show them examples of garden paths in the local area. Use online images if no real-world examples are available. 
2. Ask students to build a path using shape blocks. All the shapes must fit together without any gaps. Students are to select 1 or 2 shapes to build their path. The path needs to have at least 3 or 4 rows of blocks. 
3. Students draw the paths they have created (or take digital photos) and present them to the class, describing the shapes that they have selected.
4. Create garden designs around the paths. 
   

Session 2: Bugs, Beetles and Butterflies

In this session students will be investigating line symmetry by making butterflies out of coloured paper.

1. Show pictures of native butterflies such as the Red Admiral (Kahukura) or Rauparaha's Copper. Look at the wings and discuss reflective symmetry.
2. Ask students to make their own butterflies by folding and cutting.
   
3. Encourage them to cut out pieces in the wings to add detail.
4. Ask students to share their work and talk about the reflective symmetry it contains.
5. Extend the activity to making other native bugs and beetles such as the Huhu Beetle or Puriri Moth by folding and cutting.
6. If adapting using the marae as the context, symmetrical tekoteko could be made in the same way. If using a skate park as the context, symmetrical people and dogs could be created.

Session 3: Butterfly Painting

In this session students will make symmetrical butterflies with paint. Refer to the pictures of native butterflies from the previous session as inspiration.

1. Fold a piece of paper in half. On one half draw the outline of half of a butterfly. Create designs on this half of the wings with paint. Carefully fold the other half of the paper onto the wet paint. Unfold it to get a symmetrical pattern.
2. Ask students to share their work and talk about the reflective symmetry it contains.
3. Students could then make other bugs and beetles for the garden using the same technique.

Session 4: The Flower Garden

In this session students will be introduced to making symmetrical patterns with shape blocks. The theme for this lesson is flowers for the garden, so showing the students images of flowers and reading or viewing a story about flowers would be beneficial. Sunflowers would be a great example of a flower to use in this session.

1.  Give students a piece of paper with a line drawn down the middle.
2. Students use shape blocks to make half of a flower pattern on one side of the line. They give this pattern to a partner who has to then repeat the pattern on the other side of the line making sure that it is symmetrical.
   
3. Ask students to trace around the shape blocks to make the petal shapes. Coloured paper could be used to cut out the petals. Glue the petals onto the paper to make symmetrical flowers.
4. This activity could be extended by encouraging students to create their own symmetrical designs. They could experiment with cutting the paper shapes in half to create other pieces for their designs.
5. These could then be displayed alongside the path designs from Session 1.

Session 5: The Garden Wall

In this session introduce students to the idea of translation. Students will be making tiles for the garden wall. Introduce the activity by showing them examples of some wall tiles from the local area.

1. Give each student a piece of square grid paper, for example a 4x4 grid. Students are to draw a design by colouring in the squares to make a pattern.
2. They make 3 or 4 copies of this pattern.
3. Stick these in a row to make a row of tiles with repeating patterns.
   
4. These could then be displayed above the flowers made in the activity from Session 4.
5. This session could be extended by encouraging students to use more grid squares or by creating more complex designs within each grid.
6. If adapting using the marae as the context, students could make tukutuku panels for a wharenui by showing them some examples and then having them copy and translate some of the patterns they have seen.

Session 6: Wind Catcher in the Garden

In this session students will make a wind catcher, which illustrates rotation, as an ornament for the garden .

1. Give each student a square piece of paper.
2. Fold the square along its diagonals.
3. Make cuts along the diagonals leaving about 1 cm uncut at the centre of the square.
4. Take one of the cut ends at each corner and fold into the centre.
5. Repeat this at each corner.
6. Pin the folded pieces together with a split pin.
7. Put a little piece of blue tack onto the back of the pin to hold the pieces in place.
8. Attach the pin to a stick or paper straw.
9. Blow to watch it rotate.
   
   Cut along lines in first image

Note: The wind catcher has rotational symmetry but not reflective symmetry. This is because it can be rotated around onto itself but it doesn't have a line of symmetry in the plane.

Other Ideas

• Make designs for a dinner set for a picnic in the garden. Students could design a pattern for the pieces in the dinner set. The Willow Pattern story and plates could be used as motivation for this. Patterns around the edges of the plates would need to be repeating patterns. This could also be adapted to include Māori or Pasifika desgins.
• Paint patterns around the rim of pots. These designs could include Māori and Pasifika aspects. Plants could be planted in these pots.
• Make a patchwork picnic tapa cloth with designs in each patch piece. This could be made out of paper or fabric. The patch pieces could show a tessellation or reflective symmetry of Māori or Pasifika designs.
• Have a touch table in your classroom of items from nature that show symmetry and transformation (for example, leaves, flowers, insects). These could be added to by the students in your class (see whānau link). Encourage students to bring fallen things rather than harming our environment. 
• Go to your school or community kāri/garden and notice any natural symmetry and transformation. Or use online images of kāri/gardens from around Aotearoa. Draw pictures of what you see and label any symmetry and/or transformation.