## Early level 2 plan (term 3)

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.

## Making benchmarks: Mass

Level Two
Geometry and Measurement
Units of Work
In this unit we will explore the idea of having benchmarks of 1 kilogram and 1/2 kilogram or 500 grams to aid in estimating the mass of given objects.
• Use objects of 1kg mass to estimate the mass of other objects.
• Discuss the need for having and using standard measures of mass.
• Make sensible estimates about the mass of given objects.
• Explain the meaning of metric prefix terminology (e.g kilo).

## Place value with two-digit numbers

Level Two
Number and Algebra
Units of Work
This unit supports students learning to understand the structure of two-digit numbers and how to operate with them.

Session One

• Group a set of objects in tens to aid in counting the objects.

Session Two

• Recognise how many tens and ones are in a given collection.
• Read two-digit numbers used to represent quantities.

Session Three

• Partition (break up) 100 into two numbers using tens and...

## That's not fair!

Level Two
Statistics
Units of Work
In this unit we play probability games and learn about sample space and a sense of fairness.
• Use dice etc to assign roles and discuss the fairness of games.
• Play probability games and identify all possible outcomes.
• Compare and order the likelihood of simple events.

## Fold and cut

Level Two
Geometry and Measurement
Units of Work
In this unit students explore line or reflective symmetry and the names and attributes of two-dimensional mathematical shapes. They fold and cut out shapes to make shapes that have line symmetry.
• Explain in their own language what line symmetry is.
• Describe the process of making shapes with line symmetry.
• Name common two-dimensional mathematical shapes.
• Describe the differences between common two-dimensional mathematical shapes in relation to number of sides.
Source URL: https://nzmaths.co.nz/user/75803/planning-space/early-level-2-plan-term-3

## Making benchmarks: Mass

Purpose

In this unit we will explore the idea of having benchmarks of 1 kilogram and 1/2 kilogram or 500 grams to aid in estimating the mass of given objects.

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.
GM2-2: Partition and/or combine like measures and communicate them, using numbers and units.
Specific Learning Outcomes
• Use objects of 1kg mass to estimate the mass of other objects.
• Discuss the need for having and using standard measures of mass.
• Make sensible estimates about the mass of given objects.
• Explain the meaning of metric prefix terminology (e.g kilo).
Description of Mathematics

It is difficult to estimate the mass of individual items.  Try picking up a schoolbag and estimating it’s mass.  It is something most people aren’t that good at because we haven’t had much practice or we don’t have the same ‘onboard’ means in which to benchmark like in length e.g fingertip to shoulder – 1 metre. Students need to develop personal benchmarks with which to measure various objects in their daily lives.  Their personal benchmarks need to gradually relate more to standard measures such as 1 kilogram or 500 grams.

Students also need to be provided with opportunities and experiences to explore the connections between kilograms and grams. To support the understandings of these connections students will explore the language of measurement including prefixes such as kilo.  The ultimate aim is for students to be able to choose appropriately from a range of strategies including estimation, knowledge of benchmarks, and knowledge of standards measures in order to approach various measuring tasks with confidence and accuracy.

It is of note that mass and weight are not the same thing. The mass of an object is a measure of the amount of matter in it. Weight is the force that gravity exerts on an object and so can vary from place to place. The terms mass and weight are used loosely, and inaccurately, in everyday speech to mean the same thing.

This unit can be differentiated by varying the scaffolding of the tasks to make the learning opportunities accessible to a range of learners. For example, provide a range of items with a weight of 1 kg for students to use throughout the activities. Items such as a 1kg bag of icing sugar or 1 litre of juice would be ideal.

The context in this unit can be adapted to recognise diversity and student interests to encourage engagement. Support students to measure the mass of familiar items or items of interest, and encourage students to develop benchmarks for mass using items of importance to them. For example, how many marbles or LEGO bricks are in 1kg? How heavy is your favourite book? Can you find a book that weighs 1kg? How many rugby balls in 1 kg? (two, the weight of a regulation rugby ball is 460g).

Required Resource Materials
• Scales
• Various 1kg weights
• Plastic bags
Activity

#### Session 1

Begin by asking students to bring in their school bag.  Pose the question Who has the heaviest bag to carry to school and who carries the lightest bag to school?

1. Begin by selecting 5 or 6 bags from around the class – it is important to select bags of various sizes and shapes to discuss that biggest doesn’t necessarily mean heaviest to further explore the idea of conservation.
2. How are we going to go about ordering these bags from the lightest to the heaviest bag to answer our question?
3. Gather suggested solution strategies then trial strategies to establish an effective way to order the bags by weight.
4. Group students in groups of 5-6 with their bags.  Ask each group to order their bags from least to most heavy.
5. Share the techniques and strategies used by each group to order the bags.
6. Ask 2 groups to pair up to combine their bags on one continuum of least to most heavy.

#### Session 2

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

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

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

#### Session 3

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

#### Session 4

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

#### Session 5

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

## Place value with two-digit numbers

Purpose

This unit supports students learning to understand the structure of two-digit numbers and how to operate with them.

Achievement Objectives
NA2-4: Know how many ones, tens, and hundreds are in whole numbers to at least 1000.
Specific Learning Outcomes

Session One

• Group a set of objects in tens to aid in counting the objects.

Session Two

• Recognise how many tens and ones are in a given collection.
• Read two-digit numbers used to represent quantities.

Session Three

• Partition (break up) 100 into two numbers using tens and ones place value.

Session Four

• Rename two-digit numbers in many ways.

Session Five

• Change a two-digit number to a target number with one operation.
Description of Mathematics

Our number system is very sophisticated though it may not look it. While numbers are all around us in the environment the meaning of digits in those numbers and the quantities they represent are challenging to understand. Our number system is based on groupings of ten. So, of all the ways we could collect single objects into groups, we choose ten as our preferred group. Using ten is so common around the world because humans have ten fingers. In fact the part of our brain that controls our fingers is also associated with counting.

To represent all the numbers we could ever want we use just ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The word for digits also comes from our fingers. We don’t need a new number to represent ten because we think of it as one hand, one group of ten. When we write 10 for ten we are using place value for the first time. So the place of the digit 1 tells us the size of the quantity it represents 1 ten. Zero has two uses in the number system, as the number for ‘none of something’ and as a place holder. That means it occupies a place or places so the reader knows the values represented by of the other digits. In 10 zero is acting as a place holder in the ones place.

Place value means that both the position of a digit as well as the value of that digit indicate what quantity it represents. In the number 73 the position of the 7 is in the tens column which means that it represents 7 units of ten, 70. Renaming a number flexibly is important. In particular it is vital that students understand that when ten ones are created they form a unit of ten. For example, the answer to 25 + 35 is 6 tens since 5 ones and 5 ones combine to form another ten. Similarly when a unit of ten is ‘decomposed’ into ones the number looks different but still represents the same quantity. For example, 42 can be viewed as 4 tens and 2 ones, or 3 tens and 12 ones, or 2 tens and 22 ones, etc. Decomposing is used in subtraction problems such as 72 – 48 = □ where it is helpful to view 72 as 6 tens and 12 ones.

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

• Provide place value materials, play money or a calculator for students to use if they need to check their answers. Work first with materials where ones can be combined to make tens (such as bundles of sticks or unifix cubes) and progress to materials which represent tens and ones differently (such as place value blocks or money).
• Provide additional opportunities for students to practise modelling two-digit numbers on place value materials.

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

• use Māori names for counting numbers to reinforce the tens-based structure of numbers
• use environmental materials to model the tens and ones structure of two-digit numbers, for bundles of flax.
Required Resource Materials
• Groupable materials such as: Iceblock sticks and rubber bands, Lima beans and small plastic bags, Unifix cubes, BeaNZ and film cannisters, Counters and bags (only one type of material is needed)
• Calculators
• Slavonic Abacus
• Copymasters One to Five (see end of unit for links).
• Games of Cover Cathy Crocodile Tens and Ones (5 students per game)
Activity

Prior Experience

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

#### Session One

In this session the students explore how groupings of ten can be used to aid counting and to perform calculations. They also create the sets of countable objects that will be used in later lessons.

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

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

Session Two

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

Part One

1. Have your students sit on the mat in a circle so they can all see an A3 sized place value mat you have put in the centre. Give each student a copy of a hundreds board (Copymaster Two) and one counter.
2. Tell them you are going to say a number and they are to put their counter on it. Stress the importance of distinguishing “teens” from “tys”, for example sixteen from sixty.
3. Once students are confident at listening and placing the counter on the correct number introduce the counting objects (cubes, beans, iceblock sticks, etc.) packaged in tens from the previous day. Discuss how the objects are organised in sets of ten. Note that the A3 sheet is divided into two columns for tens (left) and ones (right).
4. Play a game where you create a collection of objects (whole tens and ones) and the students indicate how many objects are in the collection by placing the counter on that number on the hundreds board. Focus on the language connection of “ty” as tens and “teen” as one ten. Some important ideas to think about when developing your sequences:
• Teen and ‘ty’ numbers, e.g. 13, 17 compared to 30 and 70
• Iterating by tens, e.g. 13, 23, 33, 43, … (Note that the ones digit does not change)
• Re-unitising of ten ones to form a ten unit, e.g. 67, 68, 69, 70.
• Re-unitising of ten tens to form a hundred unit, e.g. 97, 98, 99, 100. (Note that students must visualise the continuation of the hundreds board pattern to show numbers greater than 100.
• Go backwards in sequences as well, e.g. 87, 77, 67, ...
5. Move to masking where you create a hidden collection under a shield of some kind (Card or plastic container) and tell the students information like:
There are four tens and seven ones.
It is not quite eight tens. There are three beans missing.
6. Iterate by tens by starting with a collection, say 24, then adding ten repeatedly to the collection. Get the students to show the total each time by moving their counter. Go through 100 forwards and down to zero backwards. See if the students can “shortcut” several iterations of ten by using fact knowledge, e.g. 40 + 30 is just 4 + 3 in units of ten. Move to more complicated examples like:
I have 26 (putting collection under the shield). I am putting 60 more beans in. What does sixty look like (6 bags of ten). How many beans are there now? How do you know?
7. Compare the starting number and the final number using an empty number line:

Is there an easy way to know it will be 86 without going 26, 36, 46, ... 86?
8. Get the students to work in pairs with one student creating and building with physical collections and the other moving their counter to show the quantity on a hundreds board. Letting the counter mover see the collection or masking the collection is an important variable. More capable students could work with the hundreds board that is missing many numbers (See Copymaster Three) or use a different century from the Thousands Book (see Copymaster Four) using place value blocks to build the collections.

Part Two

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

Session Three

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

Part One

1. Introduce the students to the Slavonic abacus. Each student should have their copy of the hundreds board and a transparent counter. Below is a Slavonic abacus showing 25 (left) and 75 (right).

How many beads are in each row? How do you know?
How many beads are on the whole abacus (100)? How do you know?
2. Make blocks of beads by moving across whole rows of tens and some ones. Students put their counter on the matching number on their hundreds board. Discuss the structure of the number (tens and ones) and the way the number is written. To move beyond counting tens and ones verbalise the structure of the block of beads as you create it.
Eight tens and six ones, No tens and nine ones, five tens and two ones.
3. Develop imaging by masking the Slavonic abacus (turning it around or hiding it) and just saying the structure while creating the block of beads.
I’ve got seven tens and four ones. Move over 100 to see how the students react. Look for them imaging where the counter might go below the hundreds board.
I have 12 tens and five ones (125)
4. Next, work on different names for 100. Make 100 with place value material and then model various ways to split the 100. Each time a new split of 100 is made invite students to show that spilt on the abacus.
5. Ask another student to record an equation for the split. Look below for an example:

Part Two

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

Session Four

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

Part One

In the last lesson the class explored how 100 can be renamed in lots of ways. Now is time to do the same thing with other numbers.

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

Part Two

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

Session Five

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

Part One

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

Part Two

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

## That's not fair!

Purpose

In this unit we play probability games and learn about sample space and a sense of fairness.

Achievement Objectives
S2-3: Investigate simple situations that involve elements of chance, recognising equal and different likelihoods and acknowledging uncertainty.
Specific Learning Outcomes
• Use dice etc to assign roles and discuss the fairness of games.
• Play probability games and identify all possible outcomes.
• Compare and order the likelihood of simple events.
Description of Mathematics

Three important ideas underpin this unit:

• The set of all possible outcomes of a random phenomenon is called the sample space.
• An event is any outcome, or set of outcomes of a random phenomenon.
• A fair game is a game in which there is an equal chance of winning or losing.

Students should be given lots of experience with spinners, coins, dice and other equipment that generates outcomes at random, such as drawing a name from a hat. The equipment can be used to play games, which should lead to a discussion of fairness (or otherwise) of the equipment and to finding the possible outcomes of using it. As they play games, record results and use the results to make predictions they find out that with probability they can never know exactly what will happen next, but they get an idea about what to expect.

Students at this Level will begin to explore the concept of equally likely events, such as getting a head or tail from the toss of a coin, or the spin of a spinner with two equal sized regions. Students can handle simple fractions at Level 2, and assigning simple probabilities provides them with an interesting and useful application of these numbers. Students can understand that the probability of getting a head when tossing a coin is 1/2. Given a spinner that is marked off equally in three colours, students can also understand that the probability of getting any one of the colours is 1/3 because there are three equally likely events and one of them has to happen.

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

• Work directly with students as they work through the probability games. Guide them to think through all possible outcomes, predict outcomes, record outcomes and reflect on results.
• Encourage students to work at their own pace taking as long as they need to work through each game. Students do not need to complete all of the games listed.
• Expect students to share their thinking about the fairness of the games, accepting that some students may be describing their experiences of playing the game rather than considering probability more generally.

Some of the activities in this unit can be adapted to use contexts and materials that are familiar and engaging for students. In particular, when students create their own games in the final session encourage them to consider their friends and classmates when planning, and to create a game that will appeal to them and be fun to play. This could be achieved by incorporating favourite elements from other games, or items of current interest.

Required Resource Materials
• A large die (you can make one by cutting a large cube of foam rubber)
• Dice (one per student labelled 1 - 6)
• Copymasters of probability games
• Centimetre cubes
• Coins
Activity

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

1. Begin the session by showing the students the large die and asking them which number they think will come up if you roll it.
What number do you think I will roll?
Why do you think that?

Roll the die and see whether students' predictions were correct. Repeat a couple of times.
2. What are the possible numbers that I can roll?
List these on the board and tell the students that this list of all the possible outcomes is called the sample space.
3. What if I rolled the die twenty times. What do you think will happen? Why?
List these predictions on the board or on chart paper.
4. With the class roll the die twenty times keeping track with a tally chart. Summarise onto a class chart.
 1 2 3 4 5 6 lll llll l lll llll l lll
5. Give pairs of students a die and ask them to work together to roll it 20 times. As they finish ask them to record their results on the class chart.
 Pairs 1 2 3 4 5 6 Ms Smith 3 4 1 3 6 3 Ben & Tane 2 5 3 2 4 4
6. Discuss the results with the class. Look back at their earlier predictions.
Why are all our results different?
If you rolled the die another twenty times what do you think would happen? Why?
7. Now lets add our results together.
What do you think that we will find?
Use a calculator to sum down each of the columns

Number rolled

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

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

#### Exploring

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

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

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

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

Bunny hop (Copymaster 1)
As there is an equal chance of getting a head or a tail you would expect the bunny to be close to 0 after a number of turns.

Doubles (Copymaster 2)
Sample space

 + 1 2 3 4 5 6 1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 6 6, 1 6, 2 6, 3 6, 4 6, 5 6,

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

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

Duck racing (Copymaster 3)

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

 + 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12

Odds or evens (Copymaster 4)

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

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

Up or down (Copymaster 6)
This is very similar to the bunny hop game. You expect the climber to be close to 0 after a number of turns.

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

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

#### Reflecting

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

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

## Fold and cut

Purpose

In this unit students explore line or reflective symmetry and the names and attributes of two-dimensional mathematical shapes. They fold and cut out shapes to make shapes that have line symmetry.

Achievement Objectives
GM2-3: Sort objects by their spatial features, with justification.
GM2-7: Predict and communicate the results of translations, reflections, and rotations on plane shapes.
Specific Learning Outcomes
• Explain in their own language what line symmetry is.
• Describe the process of making shapes with line symmetry.
• Name common two-dimensional mathematical shapes.
• Describe the differences between common two-dimensional mathematical shapes in relation to number of sides.
Description of Mathematics

A shape that can be folded down a line to produce two matching halves is said to have line symmetry or reflective symmetry. The fold-line is called a line of symmetry. A line of symmetry can also be described as a mirror line or line of reflection because the part of the object that is on one side of the line is reflected onto the other side of the line.

The goal at this level is to have the students being able to describe reflective symmetry in their own language and understand this concept. This creates a foundation on which to build a more complex understanding of symmetry at higher levels of the curriculum, e.g. the order of reflective symmetry and rotational symmetry.

Learning the names and attributes of common two-dimensional mathematical shapes is important and necessary as students develop a geometry vocabulary. The following are common two-dimensional mathematical shapes and their attributes that could be introduced in this unit.  Not all these shapes need to be presented to all students in the class.  Teachers need to select the ones appropriate, based on the readiness of the students.

• Polygon - a shape with straight line sides
• Triangle - a shape with 3 straight sides
• Equilateral triangle - all sides the same length and all angles 60°
• Right angle triangle - one inside angle is a right angle, 90°
• Isosceles triangle - two sides are the same length and two angles are the same
• Scalene triangle - all sides are different lengths, all angles are different
• Quadrilateral - a shape with 4 straight sides
• Square - a shape with 4 sides all the same length, all angles 90°
• Rectangle - a shape with 2 pairs of parallel sides, all angles 90°
• Trapezium - a shape with 4 sides, including 1 pair of parallel sides
• Rhombus - a shape with 4 sides all the same length, angles may or may not be right angles
• Parallelogram - a shape with 2 pairs of parallel sides, angles may or may not be right angles
• Pentagon - a shape with 5 straight sides
• Hexagon - a shape with 6 straight sides
• Octagon - a shape with 8 straight sides

Note that pentagons, hexagons and octagons are any shapes with 5, 6 or 8 straight sides.  The length of sides do not need to be the same nor do the angles need to be the same.

Pentagons, hexagons and octagons with sides the same length and angles the same are called regular pentagons, regular hexagons and regular octagons.  A square is a regular quadrilateral and an equilateral triangle is a regular triangle.

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

• Provide templates that students can use to create symmetrical shapes.
• Have students use mirrors to help them think through how particular shapes can be made by folding and cutting.
• Provide students with a reference list of the names and attributes of a range of shapes they could work with throughout the unit.

The context in this unit can be adapted to recognise diversity and student interests to encourage engagement. Students can identify familiar objects with line symmetry, such as skateboards, swimming goggles, and running shoes. They may also like to investigate line symmetry that can be found in kōwhaiwhai patterns or other cultural motifs.

Required Resource Materials
• Paper
• Scissors
• Rulers
• Pencils
Activity

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

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

#### Session 1

1. Take a square piece of paper and fold it in half in front of the class.
2. Using scissors cut out the shape as shown below.  Before opening the paper ask the class:
How do you know the other half will be exactly the same?
When I open this piece of paper, what shape will the hole in the middle be?
3. Open the paper and open up the piece that was cut out. Talk about the attributes of the shape.
4. Repeat this process cutting out the following shapes.
5. Discuss the shapes when the paper was folded in half and when it was unfolded.  The aim of this discussion is to find out what the students know and notice. Questions like the following could be used:
Why did it work like that?
How many sides and how many angles?
What do you notice about the length of the sides?
Are any angles the same?
Does anyone know the name of this shape?

6. Challenge the students:  What other shapes could be made by folding a square piece of paper in half and cutting? and  What shapes do you think are impossible to make?
7. Hand out square pieces of paper and get the class to experiment and try to make some new shapes.

#### Session 2 - Straight Line Shapes

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

For most of this unit the focus is on straight line shapes. Using a ruler to draw the straight lines onto the folded paper before cutting is suggested. Working in small groups, the students are to make as many of the following as they can.

Make . . .

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

Place these shapes into three piles.

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

Once as many different shapes as possible have been made assign a category of shapes to pairs of students, e.g. shapes with 3 straight sides.  The pairs sort their shapes according to the way they look.  The students then share with the rest of the class why they sorted their shapes as they did.

Pairs who need help, could be encouraged to look at the length of sides or the angles in each shape:
Are any of the sides the same length?
How many angles are larger than a right angle?

#### Session 3 – Make the Shapes

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

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

Model doing one in front of everyone emphasising that you are looking for a line in the shape that you could fold on so both halves would be the same.

Get the students to predict, before they start on this task, which shapes will be the easiest to make, the hardest to make and whether any will be impossible.  Ask why they think they will be easy, hard or impossible.

Make some more challenges like the ones above for others in your class.

#### Session 4 - Alphabet Shapes

Make as many letters of the alphabet as you can by folding and cutting.

#### Session 5 - Reflecting

Ask the students to think about the things they have learnt this week, the names of shapes and about their reflective or line symmetry.

Give pairs of students 4 of the shapes listed below.  Ask them to describe them using their own words and the words they have been learning this week.  Also ask them to identify which shapes have line or reflective symmetry.

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