## Late 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.

## Robots

Level Two
Geometry and Measurement
Units of Work
In this unit students explore movement and direction concepts in the context of programming a robot to move. They will be developing sets of instructions to accomplish tasks, focusing on the use of right, left, forward, backwards and quarter turns.
• Describe the difference between movement and direction.
• Order a set of movement and direction instructions.
• Create a set of instructions.

## Number families and relationships

Level Two
Number and Algebra
Units of Work
The purpose of this unit of three lessons is to develop an understanding of how the operations of addition and subtraction behave and how they relate to one another.
• Recognise three numbers that are related through the operations of addition and subtraction.
• Recognise that there are two related addition and two subtraction equations in a ‘family of facts’.
• Explain, in their own words...

## How long does it take?

Level Two
Geometry and Measurement
Units of Work
This unit involves students in looking at the lengths of time various activities take and calculating how long is spent on these activities in a week.
• Estimate the time taken for daily activities in hours and minutes.
• Use advanced counting or partitioning strategies to solve problems involving minutes and hours.
• Check the reasonableness of answers obtained using a calculator.

## Cube and spinner challenges

Level Two
Statistics
Units of Work
In this unit we play several games based on coloured cubes and coins. The purpose is to get some idea of the relative rate at which things happen and think about the concept of a fair game.
• Recognise that not all things occur with the same likelihood.
• Observe that some things are fairer than others.
• Explore adjusting the rules of games to make them fairer.
Source URL: https://nzmaths.co.nz/user/75803/planning-space/late-level-2-plan-term-3

## Robots

Purpose

In this unit students explore movement and direction concepts in the context of programming a robot to move. They will be developing sets of instructions to accomplish tasks, focusing on the use of right, left, forward, backwards and quarter turns.

Achievement Objectives
GM2-5: Create and use simple maps to show position and direction.
Specific Learning Outcomes
• Describe the difference between movement and direction.
• Order a set of movement and direction instructions.
• Create a set of instructions.
Description of Mathematics

At Level 2 the position and orientation element of Geometry builds on work started at Level 1. Students continue to develop the ability to describe position and the direction of movement, and interpret others’ descriptions of position.

The ability to give clear instructions that describe direction and movement clearly is an important skill, which is useful in a wide variety of situations. The context of programming a robot in this unit requires students to think in a logical and systematic way. Finding mistakes, identifying the cause and fixing them will be a feature of the thinking prominent during this unit. This unit also allows skills to be developed that will be useful as students work with computers.

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:

• Have students work with a smaller grid to limit the scope of the instructions that can be given. A 3 x 3 grid, or a 4 x 4 grid may be a useful starting point for some students.
• Limit the number of commands available for students to work with. For example, start by working with just two commands F (move forward one square) and B (move backward one square).
• Limit the number of commands in a set of instructions. For example, start with just two or three cards.
• Remove the expectation that students will predict the outcome of the set of instructions, focusing instead on following the instructions correctly.

Focus on contexts for giving and following instructions which will appeal to students’ interests and experiences and encourage engagement. Examples may include:

• The idea that there is buried treasure at a particular location, and the instructions describe how to find the treasure.
• Adding a map of a familiar area to the grid. For example, the school grounds, the local marae, or the skatepark.
• The idea that the grid represents a car park, and instructions guide cars to the best place to park.
Required Resource Materials
Activity

#### Getting Started

1. This unit of work starts with a class discussion about robots. Find out what the students know about robots and their uses. The main point of the discussion is that the students start to understand that robots do not think for themselves. They move because someone has given the robot a set of instructions. Without these instructions the robot can not do anything. The following questions could be used:
What is a robot?
What are they used for?
Why are they used?
Can robots think?
Do they have a brain like a human brain?
If not, how do they know what to do?
What sort of instructions?
What is the name of a person that writes instructions for robots?  (programmer)

2. Explain to the students that they are going to be writing some instructions to see if they can get a robot to do things for them. Show the students the instruction cards and explain what each card instructs the robot to do.

3. Have the class sit around the outside of an 8 x 8 grid. This could be marked in chalk on concrete outside, in marker on a large piece of fabric, or with masking tape on the carpet. This is called the walk-on grid later in the unit. This grid will be used throughout the unit.
F means move forward one square.
B means move backwards one square.
R means turn to the right 1/4 turn (90°).
L means turn to the left 1/4 turn (90°).
End means the set of instructions is finished.
If the robot had the cards, F  F  F  F end,  what would it do?
What would the instructions,
R  L  R  L  end, do?

4. Explain to the students that we will be getting the robot to move and do things for us on an 8 x 8 grid. The squares in the grid need to be large enough for a student to stand in them, e.g. 50cm x 50cm. Also explain that they will sometimes be the programmer, writing the instructions, and sometimes the robot, following the instructions.

5. Place the following set of instruction cards, so the students can see them. Pegging the instruction cards onto a line would be a good way to display them. This way the teacher can change the order easily and the students can clearly see the instructions are a set of individual instructions.
F  F  F  F  R  F  F  F  F  end

6. Get one student to be the robot while the teacher calls out the instructions. Before starting have the students predict where the robot will end up.

7. Work through the instruction cards, one at a time, to see where the robot ends up. Reinforce the idea of “One card, one action” with the robot only doing what the instruction cards say, nothing more, nothing less. L means 1/4 turn to the left staying within the square the robot is in, it does not mean turn left then move into the next square.

8. Continue this discussion exploring how the robot works and the meaning of each instruction card. The impact of changing the cards, the order of the cards and the starting point of the robot need to be considered. The following questions could be used to facilitate this discussion.
What would happen if we changed the L to an R in the set of instructions above?
Where would the robot end up?
Would we end up at the same place if we used the same set of instruction cards but the robot started in a different place?
Where would the robot end up if we started here, but kept the same set of instruction cards?
Starting here,
where do you think the robot would end up with this set of instruction cards?
F  F  F  F  F  B  B  B  B  B  end
Starting here,
where do you think the robot would end up with this set of instruction cards?
Which direction will the robot be facing at the end?
B  B  B  L  B  B  B  L  B  B  B  L   B  B  B  end

#### Exploring

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

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

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

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

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

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

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

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

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

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

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

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

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

#### Reflecting

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

Attachments
robots-1.pdf151.92 KB
robots-2.pdf152.35 KB

## Number families and relationships

Purpose

The purpose of this unit of three lessons is to develop an understanding of how the operations of addition and subtraction behave and how they relate to one another.

Achievement Objectives
NA2-6: Communicate and interpret simple additive strategies, using words, diagrams (pictures), and symbols.
Specific Learning Outcomes
• Recognise three numbers that are related through the operations of addition and subtraction.
• Recognise that there are two related addition and two subtraction equations in a ‘family of facts’.
• Explain, in their own words, the inverse relationship between addition and subtraction.
• Recognise that addition is commutative but that subtraction is not.
• Solve number problems that involve application of the additive inverse.
Description of Mathematics

Algebra is the area of mathematics that uses letters and symbols to represent numbers, points and other objects, as well as the relationships between them. We use these symbols,=, ≠, <, >, and later ≤ and ≥ , to express the relationships between amounts themselves, such as 14 = 14, or 14 > 11, and between expressions of amounts that include a number operation, such as 11 + 3 = 14 or 11 + 3 = 16 – 2.

These relationships between the quantities are evident and clearly stated, by the very nature of an equation or expression, the purpose of which is to express a relationship.

The relationships between the number operations, and the way in which they behave, can be less obvious. Often these operations are part of arithmetic only, as computation is carried out and facts are memorised. However, recognizing and understanding the behaviours of and relationships between the operations, is foundational to success in algebra as well as arithmetic.

One way in which the relationship between the operations of addition and subtraction is often explored early on, and is made more obvious to students, is by connecting the members of a ‘family of facts.’ Whilst these ‘fact families’ have often been used to facilitate learning of basic facts, they have not always been used well to develop a deeper understanding of the relationship between these operations.

Because of this, students often encounter conceptual difficulties in understanding the nature of the operational relationships that exist between the three numbers. It is not unusual for students, when asked to write related facts, to write, for example, 3 + 4 = 7, 4 + 3 = 7, 10 – 3 = 7, 10 – 7 = 3, in which the equations are correct, but the relationship between 3, 4 and 7 is not understood.

As students encounter more difficult problems, and are required to develop a range of approaches for their solution, having an understanding of the inverse relationship that exists between operations is critical. To ‘just know and understand’ that subtraction ‘undoes’ addition, and that addition ‘undoes’ subtraction becomes very important, if responses to problems are to come from a position of understanding, rather than simply from having been taught a procedure. For example, it is important to know how and why problems such as 61 – 19 = ☐ can be solved by addition, saying 19 + ☐ = 61, or that the value of n can be found in 12n + 4 = 28, by subtracting 4 from 12n + 4 and from 28. By having a sound understanding of the inverse relationship between addition and subtraction (for example) students are better placed to solve equations by formal means, rather than by simply guessing or following a memorised procedure.

It is important to consider this larger purpose, as we explore ‘families of addition and subtraction facts’. Furthermore, in studying addition and subtraction together, an equal emphasis is placed on both operations. Unfortunately, it is not always the case that these operations are given equal emphasis in classrooms. Subtraction has been the ‘poor relation’ that receives less attention. In fact sometimes students are not helped to see that there is any relationship at all between addition and subtraction.

The activities suggested in this series of three lessons can form the basis of independent practice tasks.

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:

• have students continue to use materials (such as the tens frame and coloured counters) to establish, explore, and show relationships between three numbers, rather than progressing to working with numbers only without the support of materials.
• remove the expectation that students will record equations and focus on exploring the relationships between the three numbers.

Situating the families of facts in familiar additive contexts will appeal to students’ interests and experiences and encourage engagement. Examples may include:

• Native birds in a tree, with birds arriving and flying away. 3 birds are in the tree and 2 more arrive, how many birds now? 3 birds fly away, how many are left in the tree?
• People travelling in a van. How many in the front set? How many in the back seat? How many all together? How many left if the people in the front/back seat get out of the van?
• Collecting kai moana. How many pipi in the bucket? How many kina? How many altogether? If we take the pipi out, how many left in the bucket?
Required Resource Materials
• Digit cards
• Cards on which are written, separately +, -, = symbols
• Blank tens frames (Material Master 4-6)
• Plastic counters
• Hundreds board (Material Master 4-4)
Activity

#### Session 1

SLOs:

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

Activity 1

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

Activity 2

1. Return to the list created in Activity 1, Step 1. Identify the sets of numbers that have been found to be related by addition and subtraction. Record four equations for some of these. For example:
1, 2, 3
3, 4, 5
2, 4, 6  (2 + 4 = 6, 4 + 2 = 6, 6 – 4 = 2, 6 – 2 = 4)
4, 7, 10
1, 5, 9
3, 5, 8  (3 + 5 = 8, 5 + 3 = 8, 8 – 5 = 3, 8 – 3 = 5)
Ask, “Can you see what we could do to the other groups of numbers to make each of them into a family?” (Change one of the numbers) Accept suggestions and explore ideas.
Take one of these groups, for example 3, 4, 5. Have a student model with counters on a tens frame, 3 + 4. This will show that the third number is 7.

Conclude that 5 can be changed to 7. Explore the other options of changing one of the numbers: 4 to 2, or 3 to 1.
Explore one more example, for example 4, 7, 10. Discuss that instead of 10 this number should be 11. Alternatively, model seven

and highlight the fact that the 4 could be changed to 3. Ask, is there a third thing we could do? (Change 7 to 6).
2. Continue to make available to the students, digit cards, cards with addition, subtraction and equals symbols, tens frames and counters, paper and pencils.
Have students choose to work in pairs or on their own, to explore the other sets of ‘unrelated numbers’ on the list. They should make a change and write the four equations (+-) which result.
3. Have students (pair) share their recordings. Discuss what is the same or different about them (this depends on which number they change). They should draw a box around sets of four equations that they have written that are the same as a partner has recorded.

Activity 3

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

#### Session 2

SLOs:

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

Activity 1

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

Activity 2

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

Activity 3

Conclude by sharing and discussing student work.

#### Session 3

SLOs:

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

Activity 1

1. Begin by sharing the student work from Session 2, Activity 2, Step 6.
List student suggestions. In particular highlight the commutative property of addition. For example: If you know 17 + 5 = 22, you will also know 5 + 17 = 22. Also highlight the related subtraction facts.
2. Distribute the addition and subtraction grids (Copymaster 2) to each student. Use the larger class copies to model how to complete each grid. In particular show how to complete the subtraction grid, subtracting the numbers down the side from those along the top row, and putting a dot for those that ‘cannot be subtracted’, rather than discussing negative numbers at this point.
Highlight the importance of the students writing their observations about each grid once they are completed. These observations should include number patterns and the fact that the subtraction grid cannot be fully completed.
3. Once completed, have students share what they notice and record their observations.
4. On the class addition grid look for the same sums for both addends.
For example, 2 + 1 = 3 and 1 + 2 = 3, 2 + 3 = 5 and 3 + 2 = 5.

(Discuss the pattern and also notice the pattern of doubles)
Write this statement on the class chart and read it with the students:
We can carry out addition of two numbers in any order and this does not affect the result. Introduce the word commutative.
5. Ask why they can’t complete the subtraction grid in the same way they have the addition grid. Record a student statement that states, in their words, that addition is commutative, subtraction is not.

Activity 2

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

Activity 3

1. Introduce the game Families on Board. Model how it is played.
(Purpose: to identify three fact family members and to record four related equations.)
The game is played in pairs. The players have one Hundreds Board, 25 counters of one colour each, pencil and paper.
Tens frames showing ten, blank tens frames, and extra counters should be available to the students to model or work out equations if appropriate.

Round one: The Hundreds Board is screened so that numbers 1 – 20 only are visible. Students take turns to place 3 counters on three related numbers. Counters cannot all be placed in the same row.
For example: students cannot cover 3, 4 and 7 because they are all in the same row.
As they place their counters they say and write the four related equations. Students can use tens frames, if needed, to work out or demonstrate the equations and their relationship.
For example:
Player 1

Player 2

Turns continue.
The challenge is to complete the task between them, leaving only two numbers uncovered. If they have more than two uncovered on the first try, they try again with different combinations.

Round two:
The Hundreds Board is screened so that numbers 1 – 30 only are visible. Students take turns to place 3 counters on three related numbers. Counters cannot all be placed in the same row. As they place their counters they say and write the four related equations, using tens frames if needed.
The challenge is to complete the task between them, leaving no more than three numbers uncovered by counters.

Round three:
Numbers 1-50 are made visible. The task is completed. The challenge is to leave just two numbers uncovered by counters.
2. Conclude this lesson and series of lessons, by recording students’ reflections on their learning about addition and subtraction, and the relationship between them.
Attachments

## How long does it take?

Purpose

This unit involves students in looking at the lengths of time various activities take and calculating how long is spent on these activities in a week.

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
• Estimate the time taken for daily activities in hours and minutes.
• Use advanced counting or partitioning strategies to solve problems involving minutes and hours.
• Check the reasonableness of answers obtained using a calculator.
Description of Mathematics

Two aspects of mathematics are explored in this unit on time:

• Investigations of the length of time taken for various activities, working in hours and minutes.
• Calculations involving hours and minutes. It is anticipated students would be at the advanced counting or early additive strategy stages of the number framework and could use repeated addition, skip counting, multiplication or division to solve these problems.

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

• Students can use materials or draw diagrams to support their thinking as they solve problems involving numbers of minutes and hours.
• Provide simplified tasks for students to investigate, working with small numbers of minutes that add to less than an hour. For example, 2 minutes to brush your teeth (28 minutes total in a week), 5 minutes to make a sandwich for school lunch (25 minutes total in a week), or 3 minutes to pack your school bag (15 minutes total in a week).

The contexts in this unit can be adapted to recognise diversity and encourage engagement. For example, ask students to share activities that happen regularly in their families, or activities that they regularly enjoy. Adapt the investigations to include these familiar and enjoyable activities.

Required Resource Materials

Copymasters One, Two, Three, Four, Five, Six, and Seven

Activity

#### Getting Started

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

#### Exploring

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

Copymaster 5 can help in these comparisons.

#### Reflecting

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

## Cube and spinner challenges

Purpose

In this unit we play several games based on coloured cubes and coins. The purpose is to get some idea of the relative rate at which things happen and think about the concept of a fair game.

Achievement Objectives
S2-3: Investigate simple situations that involve elements of chance, recognising equal and different likelihoods and acknowledging uncertainty.
Specific Learning Outcomes
• Recognise that not all things occur with the same likelihood.
• Observe that some things are fairer than others.
• Explore adjusting the rules of games to make them fairer.
Description of Mathematics

A fair game is a game in which there is an equal chance of winning or losing. We say that a game is fair when the probability of winning is equal to the probability of losing. Changing the rules of a game can affect the likelihood of winning or losing, and therefore whether the game is fair.

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

• Simplify the challenges by using smaller numbers of cubes or segments than suggested. Cube challenges III and IV could be parallel tasks to cube challenges I and II, using different coloured cubes.
• Expect students to share their thinking about the fairness of the challenges, accepting that some students may be describing their experience of playing the challenge rather than comparing relative likelihoods.

The challenges in this unit can be adapted to recognise diversity and student interests to encourage engagement. For example:

• spinners could be created with school colours or the colours of a favourite team. Spinners can be made and tested online using the Spinner learning object.
• students could select items to use instead of cubes in the cube challenges. This could be anything that appeals to their interests and experiences, such as All Blacks cards, although they need to be things that are equally likely to be selected.
Required Resource Materials
Activity

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

For each challenge:

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

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

#### Session one challenges

Cube Challenge I:

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

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

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

Cube Challenge II:

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

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

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

#### Session two

Cube Challenge III:

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

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

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

Cube Challenge IV:

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

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

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

#### Session three

Spinner Challenge I:

Spinner:

Spin the spinner once

To win the challenge: spinner lands on green

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

Spinner Challenge II:

Spinner:

Spin the spinner once

To win the challenge: spinner lands on green

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

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

#### Session four

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

Spinner Challenge III:

Spinner:

Spin the spinner once.

To win the challenge: spinner lands on green.

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

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

Spinner Challenge IV:

Spinner:

Spin the spinner once.

To win the challenge: spinner lands on green.

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

#### Session five

1. Review a few of the challenges from the week. Ask students to think about the probability of all possible events in a challenge, ordering them from most likely to least likely and identifying events that have the same likelihood of occurring. Students need not know the theoretical probabilities involved but should be able to explain their reasoning.
2. Ask students to work in pairs to make a new challenge using cubes, spinners, or something else that they select. In each case they should have some idea of whether their challenge is fair or not.
3. Have students swap challenges and play them.
4. Talk about students’ challenges, and have students explain whether they think particular challenges are fair, and why.