## Early level 2 plan (term 1)

Planning notes
This plan is a starting point for planning a mathematics and statistics teaching programme for a term. The resources listed cover about 50% of your teaching time. Further resources need to be added to meet the specific learning needs of your class, to support your local curriculum and to provide sufficient teaching for a full term.

## Mathematical magic

Level Two
Integrated
Units of Work
This unit provides you with a range of opportunities to assess the entry level of achievement of your students.
• Classify whole numbers as even or odd and generalise the nature of sums when even and odd numbers are added.
• Recognise that sums remain the same if the same amount is added and subtracted to the two addends, e.g. 17 + 19 = 27 + 9.

## Pirate plays

Level Two
Geometry and Measurement
Units of Work
In this unit we explore the size of a metre and develop our own ways to estimate a metre length.
• Recognise the need for a standard unit of length.
• Recognise a metre length.
• Estimate and measure to the nearest metre.

## Picasso

Level Two
Geometry and Measurement
Units of Work
This unit uses the context of Picasso's art to explore two and three-dimensional shapes, to recognise their features, and to develop in the students appropriate language to discuss Picasso's and their own artworks.
• Sort attribute blocks and explain their groupings.
• Understand and use geometric language to describe the features of two-dimensional (plane) shapes.
• Create an artwork using plane shapes.
• Recognise how shape is an important feature of Picasso’s artworks.
• Understand and describe the...

## Cuisenaire mats

Level Two
Number and Algebra
Units of Work
In this unit students use Cuisenaire rods (or other equivalent material that fits together precisely) to make “number mats” that illustrate a variety of numerical patterns and can be visually appealing. They formally record the number relationships in the mat.
• Use addition and multiplication to find number combinations that 'make' a given result.
Source URL: https://nzmaths.co.nz/user/75803/planning-space/early-level-2-plan-term-1

## Mathematical magic

Purpose

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

Specific Learning Outcomes
• Classify whole numbers as even or odd and generalise the nature of sums when even and odd numbers are added.
• Recognise that sums remain the same if the same amount is added and subtracted to the two addends, e.g. 17 + 19 = 27 + 9.
• Create and follow instructions to make a model made with shapes.
• Recombine parts of one shape to form another shape.
• Extend a repeating pattern to predict further members, preferably using repeated addition, skip counting or multiplication.
• Order the chance of simple events by looking at models of all the outcome.

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

• in session one, students can predict the total number of hidden dots on the dice, and check by counting
• In session two, students could work with a total of 10 or 20 on the hundreds board, rather than the full 100.

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

• in session two, instead of Little Bo Peep and her 100 sheep work with 100 kiwi and have some of them hiding in their burrows
• in session four, instead of cups and treats use kete and shells for the magic trick
• in session five, create repeating patterns like the ones shown with environmental materials such as leaves, shells, and sticks, or items that are currently of interest to students, such as rugby cards.
Required Resource Materials
• Digital camera to record students’ work.
• Session One – Two large dice, standard 1-6 dice, squares of paper or card for students to construct cards (file cards are ideal)
• Session Two – Hundreds Board and Slavonic Abacus (physical or virtual versions), Video 1.
• Session Three – Squares of paper, scissors, Copymaster 1, Copymaster 2.
• Session Four – Plastic cups, objects to act as ‘treats’, Am I Magical 1, Am I Magical 2, Am I Magical 3.
• Session Five – Objects to form patterns, e.g. natural materials like acorns, shells, stones, or toy animals, geometric shapes, blocks, Copymaster 3, PowerPoint 1, PowerPoint 2, PowerPoint 3.
Activity

#### Prior Experience

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

#### Session One

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

Dice Faces

1. Begin with the whole class, demonstrating 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 + 4 = 7). 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 plus six equals ten 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. What ever 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 (Acknowledgement to Jill Brown, ACU Melbourne, for the idea)

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 they 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 the nursery rhyme about Bo Peep?

6. Tell the students that Bo Peep had 100 sheep. Her sheep were very naughty and hid all the time. She made up some mathematical magic to tell straight away how many sheep were missing. Act out being Bo Peep.
(Student A), please move some of my sheep to the left side and cover up the rest so I cannot see them (using an A4 sheet of paper).

7. Role play working out the number of sheep you can see, then recording the number. For example, “Two tens, that’s twenty, five and three, that’s eight. I can see 28 sheep.”

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 Bo Peep did it. How could she know 72 sheep 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 the 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 crosses pattern works. The for each line, the sum is three times the centre number, with one of the end numbers the same amount more than that number as the the other is less.

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

#### Session Three

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

1. Tell students: Magicians can change objects in to 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 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 food item.

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 1Am 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 bit hard. Let’s keep the three cups but add another treat.

13. Let students trial the three cup, two treats situation. 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.

• 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, “Lion (One), bear (two), lion (three), bear (four), ….”
• Do they use skip counting to anticipate which animal will be in given positions? For example, “The giraffe comes every three animals. 3, 6, 9, 12… so the giraffe will be in number 12.”

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

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 16th object.

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

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, yellow, red, 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 of repeat is made of five objects, e.g. bear, tiger, giraffe, elephant, hippopotamus,… then the five times tables might be used. For each position, 5, 10, 15, 20, … the animal is a hippopotamus. Other ordinal positions can be worked out by adding and subtracting from multiples of five. For example, position 23 must be a giraffe since 25 was a hippopotamus.

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 a 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. ○, □, ­_, ∆, ○, ­_, →, ∆, …

## Pirate plays

Purpose

In this unit we explore the size of a metre and develop our own ways to estimate a metre length.

Achievement Objectives
GM2-1: Create and use appropriate units and devices to measure length, area, volume and capacity, weight (mass), turn (angle), temperature, and time.
Specific Learning Outcomes
• Recognise the need for a standard unit of length.
• Recognise a metre length.
• Estimate and measure to the nearest metre.
Description of Mathematics

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

Students measurement experiences must enable them to:

• develop an understanding of the size of the standard unit
• estimate and measure using the unit.

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

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

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

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

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

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

• Students could continue to use non-standard units of heel toe steps or hand spans to measure if they are not ready for standard units.
• Clearly and deliberately model the correct use of a metre ruler, ensuring that the start of the scale is used as the starting point rather than the end of the ruler, and there are no gaps or overlaps between measures.
• Work directly with small groups of students to measure accurately, reinforcing the correct use of the metre ruler.

The context for this unit can be adapted to recognise diversity and student interests to encourage engagement. For example, the unit could be focused around the voyage from Hawaiki to Aotearoa with activities including measurements for a new waka, and challenges between rival navigators.

Required Resource Materials
• A metre strip of card, ribbon or a metre ruler
• Chalk for drawing on asphalt
• Light card for handspans and footprints
• 1kg weight, clay wrapped in plastic wrap
• Rugby ball
• 1cm blocks
• String
• Copymasters for the "sports day" Toss the Cannon BallJump from the PlankMetre Kick
Activity

#### Session one

Begin the session by acting out the following scene with the students.

Characters:
Captain Blood - teacher
Crew / cabin boys and girls - students

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

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

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

#### Session two

1. Tell the students that Captain Blood has decided that now he knows what a metre is he wants to start drawing up plans for his new pirate ship and that he would like the crew to help.
Discuss with the students the type of boats that pirates sailed in.
Provide them with chalk and a metre measure and take them outside to draw the boat to Captain Blood’s requirements.

Measurements of Captain Blood’s New Pirate Ship

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

The students might like to add extras like flags, anchor ropes, and cannons and add them to the measurement list. Encourage students to estimate before drawing.

2. Ask the students to stand and show how high they think a metre would be from the floor. Check with their metre measure and reference it to their body.
A metre is as high as …………….(my ribs).
How wide is a metre? A metre is from my fingertips to ……………

#### Session three

1. Ask the students to estimate how many of their handspans would be the closest to a metre.
2. Trace an outline of their handspan on to paper and then cut it out and use it to measure along the metre. Record results. Have the students estimate how many of their footprints would be closest to a metre. Make outlines by removing their shoe and tracing around their foot.
3. Check how students position the shapes when measuring. Do they begin from the same baseline?
Do they use the measuring unit consistently without gaps or overlapping?
The students can show their results by pasting their outlines on to paper and recording the number beside it.
 To measure 1 metre it takes: ____ of my handspans _____ of my footprints
4. To finish pose this problem for the crew,
Captain Blood has gone to a boat shop to buy some new canvas for sails. He wants two metres. Can you show me using a body measurement how long two metres would be?

#### Session four

1. Provide the students with a standard metre ruler to explore. Look at the markings on it and discuss what they can see.
Talk about where you begin measuring from. (Students can have difficulties identifying the starting points on calibrated rulers. They start from the edge rather than the markings.)
2. If the students haven’t offered the word centimetre in discussion explain to them that the space between the numbers is one centimetre and place centimetre cubes along the ruler.
3. Ask: How many centimetre cubes might fit along the metre?
If 1cm cubes that connect are available join 100 using two different colours to distinguish the decades. Place the line of cubes on top of the metre ruler and count in tens to 100.
4. Provide the students with string, scissors and PVA glue and let them investigate the different ways of creating patterns with 1 metre of string. The students first measure a metre and then make a pattern.
 e.g. spirals zig zags straight lines curves
5. Glue their discoveries to cardboard and display the one-metre patterns.
Discuss that different patterns look as though they have different lengths.

#### Session five

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

1. Set up the activities in the three stations and provide each student with a one metre long piece of string or metre ruler.
2. When the students share their results at the end, talk about the half metre, or the extra bit and the need to have a smaller unit of measure.
3. The students will need to work with a partner who can stand where the rugby ball lands after the kick. As above, have the students record their estimation prior to measuring. After tossing the cannon ball the students estimate how many metres and then measure.
Attachments

## Picasso

Purpose

This unit uses the context of Picasso's art to explore two and three-dimensional shapes, to recognise their features, and to develop in the students appropriate language to discuss Picasso's and their own artworks.

Achievement Objectives
GM2-3: Sort objects by their spatial features, with justification.
GM2-4: Identify and describe the plane shapes found in objects.
GM2-7: Predict and communicate the results of translations, reflections, and rotations on plane shapes.
Specific Learning Outcomes
• Sort attribute blocks and explain their groupings.
• Understand and use geometric language to describe the features of two-dimensional (plane) shapes.
• Create an artwork using plane shapes.
• Recognise how shape is an important feature of Picasso’s artworks.
• Understand and describe the features of a cube.
• Create an artwork, developing a practical understanding of the relationship between 2D and 3D shapes.
• Identify and articulate key learning about geometric shapes.
Description of Mathematics

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

Initially, students come to understand two-dimensional shapes as flat or plane shapes that have two dimensions. They have length and width but no depth, and therefore, technically, cannot be ‘held’. As they work with three-dimensional shapes, they come to understand the way in which two-dimensional plane shapes build three-dimensional shapes. Students need to have a clear understanding of the meaning and concept of ‘dimensions’ and should be able to explain in their own words what the abbreviations 2D and 3D mean. The change in language from ‘sides and corners’ for two-dimensional shapes, to ‘faces, edges and vertex/vertices’ is not an insignificant one. The language itself conveys the shape category and should be emphasised and well understood.

As students work with physical shapes they have opportunities to come to understand their defining characteristics. Having them talk about and explain these within a particular learning context, consolidates conceptual understanding. The representation of three-dimensional shapes in the two-dimensional medium that a painted artwork is, creates its own challenge and interest. It should involve deconstructing 3D shapes, recognising that they are comprised of 2D plane shapes. This is a key understanding to be developed at this level.

In levels 3 and 4 students are challenged to explore and represent objects from different viewpoints and perspectives. The work in this unit of work is a useful precursor to developing these concepts.

Picasso’s art provides a useful context for the exploration of two and three dimensions as well as developing recognition in the students of the importance of shape as fundamental structure of art itself.

Associated Achievement Objectives

Art
Visual Arts

• Share ideas about how and why their work and others’ works are made, and their purpose, value and context.
• Investigate and develop visual ideas in response to a variety of motivations, observation or imagination.
• Share the ideas, feelings, and stories communicated by their own and others’ objects and images.

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

• Provide templates for students to use to create their art works from 2D shapes (session 1). They can either copy or colour the template as appropriate.
• Provide students with a glossary of terms and phrases that they can use to write about Picasso’s art (session 3).
• Minimise the number of shapes from Attachment 3 that students work with to create their own cubist art work (session 4)

The contexts in this unit can be adapted to recognise cultural diversity and encourage engagement. For example, in addition to the works of Picasso students could respond to cubist works from New Zealand artists. Charles Tole, John Weeks, and Colin McCahon all include elements of cubism in some of their work.

Required Resource Materials
• Attribute blocks
• Pencils
• Crayons, or pastels
• Glue sticks
• Scissors
• Art paper
Activity

Learning activities
Whilst this unit is presented as sequence of five sessions, more sessions than this may be required. It is also expected that any session may extend beyond one teaching period.

Session 1

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

SLOs:

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

Activity 1

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

Activity 2

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

Activity 3

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

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

Activity 4

Have students make and complete their own pictures. When pictures are complete, have each student name their picture, write a short story about it, using words from the brainstorm list in Activity 3. Their story should explain why the subject of the picture is important to them and how they made their picture.

Activity 5

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

Activity 6

1. Have students discuss in pairs and decide whether the attribute blocks themselves are two-dimensional or three-dimensional shapes.

2. Have them physically take up positions in the classroom to indicate their thinking (for example: 2D on one side of the mat, 3D on the other).
Discuss, conclude and record that the attributes blocks are 3D shapes because they have width, length, and thickness (depth), and we can feel these. Recognise that the attribute block shapes have different thickness or depth.

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

Activity 7

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

Session 2

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

SLOs:

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

Activity 1

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

Activity 2

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

Activity 3

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

Activity 4

1. Locate around the classroom, several individual small copies of each Picasso’s pictures from Attachment 2, enough for each student to have a picture of their own choice.
Explain what you have done.

2. Have students silently complete an ‘art-walk’ once around the ‘gallery of artworks’. Have them make a second rotation, this time choosing and taking an artwork of their choice and returning to their place.

3. Make available paper, pencils and glue sticks.
Have students glue their chosen picture onto their paper, leaving sufficient space to write about it.
Remind the students that in Session 1 they wrote about their own artworks.
Explain that each student is to write about the Picasso picture they have chosen.
Their writing should:
• explain their feelings about the picture
• explain how they think Picasso made the picture
• include a short story about why the picture might be important to Picasso.
Clarify the task with the students, list on the class chart any special words that they might need and set a time limit.

4. Extend the task for those who finish quickly, by having them explain how their picture fits with Picasso’s statement in Step 3 above.

Activity 5

1. When the task is complete, have students who chose the same picture, form a group. Have the students share their responses in their groups, comparing their ideas.

Have several students take turns to talk about the way Picasso uses shapes in his pictures, including identifying the features of those shapes. For example: In Picture 1, triangles with lots of corners (angles) have been used.

Activity 6

Conclude the session by having several students share their writing about Picasso’s artwork.

Session 3 and 4

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

SLOs:

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

Activity 1

Begin by having several more students read their Picasso artwork stories.

Activity 2

1. Make paper and pencils available.
Have students form pairs with their Picasso artworks. Partners should have different pictures.
Write on the class chart the headings: Colour      Shape      Other
Have students write these headings on one shared piece of paper. Set a time limit.
Have students look at both Picasso pictures and record on the chart under the three headings the things they notice about both art works.

2. Have student partners share their findings with another pair.
Discuss as a class, highlighting (in most instances) the narrow range of colours, light/dark contrasts, different angles of geometric shapes.

Activity 3

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

2. Write cubism on the class chart. Explain that it is a name for an art style that Picasso is famous for. Have students suggest what this might be and record their ideas/definitions on the class chart.
If required, complement the student’s ideas with these points:
Cubism:
• uses simple geometric shapes
• shows things from different viewpoints in any one artwork
• sometimes breaks up (or fragments) 3D shapes into parts
• shows the plane (2D) shapes that make up 3D shapes.
Talk about each of these, having students find and discuss examples in their artworks in front of them.

Activity 4

1. Explain that the students will make their own cubist artwork about one thing that is important to themselves. Refer to Picasso’s use of music/musical instruments or parts of these.
Make available at least one copy of Attachment 3 per student, scissors, glue, A4 paper, pencils, crayons/pastels.
Explain that their completed artwork should:
• Include parts of something that are important to them personally
• Fill the A4 page
• Use the shapes or parts of shapes from Attachment 3
• Show the shapes connected or touching in some way
• Include lines they have drawn
• Include their own (limited) choice of colour in empty spaces.
2. Model the beginning of this process.
Cut out one cube shape from Attachment 3. Discuss this with reference to the wooden cube, highlighting that this is a way of capturing a 3D shape in art.
Cut the cube (or cuboid) picture into its component parts.
Recognise and discuss the squares (and rectangles) that result. These are the 2D shapes that make up the 3D shape.
Look at the parallelogram shapes. Discuss that this is what happens to the square and rectangular faces when they are shown in 2 dimensions.

3. Show the developing process.
For example: a plant (leaf) may be something important to the artist.

This can be cut and arranged alongside some shapes to produce the artwork.

4. Refer to the features noted in the headings: colour, shape and other, in Activity 2, Step 1 above. Are these Picasso style features reflected here?

5. Refer to the cubism features list in Activity 3, Step 2 above. Are these features evident here?

6. Identify next steps to complete the artwork. (Fill the page, include more cube shapes of parts of these, fill in any white paper spaces remaining with appropriate colours.)

Activity 5

Have students begin their artworks. Have them stop and review progress throughout, reflecting on their own work and giving feedback to others.

Activity 6

Finish artworks with a title.

Session 5

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

SLOs:

• Identify and articulate key learning about geometric shapes.

Activity 1

Have students display on desks, their cubism art with titles. Explain that the students will undertake a slow and silent art-walk in which they are to notice works they particularly like. They should look closely at these and decide what it is that makes them appealing to them personally.

Activity 2

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

Activity 3

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

Activity 4

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

Activity 5

1. Have student share and display their reflections. Discuss.

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

Activity 6

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

Attachments
picasso-1.pdf191.83 KB
picasso-2.pdf372.16 KB
picasso-3.pdf161.52 KB

## Cuisenaire mats

Purpose

In this unit students use Cuisenaire rods (or other equivalent material that fits together precisely) to make “number mats” that illustrate a variety of numerical patterns and can be visually appealing. They formally record the number relationships in the mat.

Achievement Objectives
NA2-6: Communicate and interpret simple additive strategies, using words, diagrams (pictures), and symbols.
Specific Learning Outcomes
• Use addition and multiplication to find number combinations that 'make' a given result.
Description of Mathematics

It is important for students to know the meaning of the equality sign. Realising that “=” indicates that the two expressions on either side of it are equal is a key step on the road to algebra. These two expressions have the same status - one does not have to be the “answer” of the other.

This very important idea is fundamental to a sound platform on which to proceed to algebra proper at a later level. It is introduced here by way of a geometric technique that links numbers via Cuisenaire rods.

This unit could be repeated at a higher level by assigning a value other than 1 to the white rod.

This unit provides an opportunity to develop students’ number knowledge in the area of Grouping and Place Value. It also provides a way develop early part-whole thinking in the addition and subtraction domain as it allows students to clearly see the different ways a number can be partitioned.

To develop students knowledge of groupings within 5 and ten focus their attention while they are working with the 5-mat and the 10-mat.

Can you find 2 numbers that join together to make 10 on the 10-mat?

Can you find two different numbers?

How many different combinations can you find?

List the combinations as they are identified. Encourage students to see the relationships between the two addends: as one increases, the other decreases. This may also be illustrated using the Cuisenaire mats:

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

• Simplify the task by working with smaller numbers, such as 5-mats, and make the task more complex by working with larger number mats such as 12-mats.
• Some students may be able to move from using the structured physical representation of Cuisenaire rods to drawing diagrams to support their thinking.

The context in this unit can be adapted to recognise cultural diversity and student interests to encourage engagement. For example, in place of a Cuisenaire mat students could work with images of a waka ama with a defined number of seats, with coloured shirts to represent number patterns. For example in a 5-waka, there can be 3 red and 2 blue shirts (3 + 2), 4 red and 1 blue shirt (4 + 1), or 3 red, 1 blue and 1 green shirt (3 + 1 + 1).

Required Resource Materials
• Cuisenaire rods for each pair of students.
• If they are available Magnetic Cuisenaire rods which stick onto a blackboard are very useful.
Activity

#### Getting Started

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

1. Give each pair of students a set of Cuisenaire rods and allow time for free play if they are new to the students. During the free play, encourage building activities that lead to comparison of the length of the rods and activities that fit them together tightly.
2. Conduct a class discussion about the lengths of the rods. Begin by making a staircase of the rods in increasing length. Then by covering the rods with the unit (white rods), establish the lengths as 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 times the length of the white rod.
(When Cuisenaire rods are exactly made in units of 1 cm, some students may also be able to check the length by measuring.)
Draw a clearly labelled diagram on the blackboard, for reference.
3. Introduce the idea of a 5-mat.  First: students take a 5-rod and put together one other combination of rods that makes 5.  For example, different students might make 5 as 1 + 3 + 1 (white, green, white), 4 + 1 (pink, white), 2 + 3 (red + green), or as 5 whites. Then these different combinations can be put together as shown in the diagram to make a 5-mat. Of course there are many different 5-mats, but they are all rectangles with the yellow rod (5) as one side.
4. Number combinations in the 5-mat.  Students suggest the number combinations demonstrated by the  5-mat on the board. The mat above has:
 1 + 3 + 1 4 + 1 2 + 3 1 + 1 + 1 + 1 + 1 or 5 x 1 = 5 = 5 = 5 = 5 = 5

Note that it is important to keep the numbers in the order that they appear on the mat: 2 + 3 and 3 + 2 would be different rows of the 5-mat.  Note also that there are two relationships shown by the row of 5 whites; one is an addition and one a multiplication.

Explain that the 5-mat also shows other relationships that are true. For example, 1 + 3 + 1 = 4 + 1  and 5 x 1 = 2 + 3 etc. This use of equality may seem strange to students because it does not give an answer on the right hand side. Explain that “=” means “is equal to”.
6. Students write down relationships from the 5-mat and read out their favourites to the class.

#### Exploring

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

1. Students pick a mat to make of a given size. Controlled choice of the size of the mat can allow for individual differences. The more able students should be challenged by being given larger numbers. The diagram below shows the first four rows of a 12-mat.
2. Ask the students to record the relationships shown on their mat. The mat in the diagram illustrates many relationships. For instance, 3 + 3 + 3 + 3 = 12 and 4 x 3 = 12. (Note that 4 x 3 is interpreted as 4 groups of 3 here and not 3 groups of 4.)
3. Students should also record some of the relationships between rows of the mat. For example, 6 + 1 + 5 = 3 + 3 + 3 + 3.
4. Initiate a class discussion on interesting examples: for example, 5 + 7 = 7 + 5.
Did anyone else find something like this?
Is 4 + 8 = 8 + 4? Why?

Did anyone else find something like this that did NOT work?
5. Other interesting examples that are worth discussing are things like 4 x 3 = 3 x 4.
Did anyone else find something like this?
Is 2 x 6 = 6 x 2? Why?
Did anyone else find something like this that did NOT work?

Rows that show a strong visual pattern may also show interesting number patterns.
6. The activity can be repeated using a mat of a different size.
7. Turn the situation around.
Make me a mat that shows that 4 + 7 = 2 + 9.
What other equalities can a mat like this show?
Make me a mat that shows that 2 x 5 = 3 + 7.
What other equalities can a mat like this show?

Let the students pursue this aspect of the problem in pairs.

#### Reflecting

This section brings together all that the students have discovered so far.

1. The students can make a poster of their work on a large piece of paper individually or in a pair. Selected students can report their most interesting findings to the class.
2. Highlight the important points.  This will include observations about addition (for example, 8 + 1 = 7 + 2 = 6 + 3, 8 + 1 = 1 + 8,  and 7 + 2 = 2 + 7) multiplication (3 x 4 = 4 x 3) and the meaning of equality.
3. Is it true that 4 + any number = that same number + 4? Why? Why not?
Is it true that 2 x any number = that same number x 2? Why? Why not?

Possible extensions
The ideas above can be extended for more able students and older students. So this unit could be used with students at Levels 3 or even 4.

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