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:

- Introduce the challenge and discuss students’ ideas about whether the challenge is fair, and why.
- Have the students play the challenge in pairs, recording how many games they play, and how many of these they win.
- Discuss how students’ ideas about whether the challenge is fair have changed now that they have tried it.
- 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.
- 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**

- 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.
- 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.
- Have students swap challenges and play them.
- Talk about students’ challenges, and have students explain whether they think particular challenges are fair, and why.

## Robots

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.

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:

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

## Getting Started

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)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.

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?

R L R L endWhat would the instructions,,do?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.

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

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.

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.

Continue this discussion exploring how the robot works and the meaning of each instruction card. The impact of changing the cards, the order of the cards and the starting point of the robot need to be considered. The following questions could be used to facilitate this discussion.

What would happen if we changed the L to an R in the set of instructions above?

Where would the robot end up?

Would we end up at the same place if we used the same set of instruction cards but the robot started in a different place?

Where would the robot end up if we started here, but kept the same set of instruction cards?

Starting here,where do you think the robot would end up with this set of instruction cards?

F F F F F B B B B B end

Starting here,where do you think the robot would end up with this set of instruction cards?

Which direction will the robot be facing at the end?

B B B L B B B L B B B L B B B end## Exploring

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

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

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

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

## Task 1

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

## Task 2

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

## Task 3

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

## Task 4

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

## Task 5

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

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

## Task 6

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

## Task 7

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

## Task 8

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

## Reflecting

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

Dear family and whānau,

At home this week the homework task is to talk to your family about robots. Find as many situations as you can where robots are used. Think into the future and predict the jobs robots may do in the future or jobs you wish you could programme a robot for.

## Number families and relationships

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.

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

isto 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

algebraas 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

deeperunderstanding 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

andunderstand’ 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 andwhyproblems 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 ‘

familiesof 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 ‘poorrelation’ that receives less attention. In fact sometimes students are not helped to see that there is anyrelationshipat all between addition and subtraction.The activities suggested in this series of three lessons can form the basis of independent practice tasks.

Links to the Number FrameworkAdvanced counting (Stage 4)

Early Additive (Stage 5)

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:

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

Session 1SLOs:

Activity 1Ask 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.)

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).”

Family of Factson the class chart. Ask whether this is a good name and why. Record student ideas, highlighting the fact that these numbers arerelatedthrough the operations ofaddition and subtraction. Families are related.Activity 21, 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).

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.Activity 3Conclude 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 2SLOs:

Activity 1cards (Copymaster 1) to students.Family Shuffle(Purpose: To recognise related addition and subtraction equations)

Explain how to play.

Each student has

one setof 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 beslid across or up and down within the array space,, 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.but not liftedTo increase the challengeof 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 214 + 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.”

“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

and5 + 1 = 6, 6 – 1 = 5).Discuss what is happening in these equations. Elicit the observation that

subtraction is ‘undoing’ addition.Record several of these in words and in equations on the class chart.

“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

and6 - 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.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 3Conclude by sharing and discussing student work.

Session 3SLOs:

Activity 1Ask “Why is knowing about families of related facts useful?”

List student suggestions. In particular highlight the commutative property of addition. For example: If you know 17 + 5 = 22, you will also know 5 + 17 = 22. Also highlight the related subtraction facts.

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.

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 wordcommutative.their words, that addition is commutative, subtraction is not.Activity 2. In giving instructions highlight the importance of the students recording equations and on explaining what is happening with the numbers in the problems.Nana’s partyActivity 3. Model how it is played.Families on Board(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 countersStudents can use tens frames,they say and write the four related equations., to work out or demonstrate the equations and their relationship.if neededFor example:

Player 1Player 2Turns 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 uncoveredby counters.Round three:

Numbers 1-50 are made visible. The task is completed. The challenge is to

leave just two numbers uncoveredby counters.Dear parents and whānau,

In maths this week, the students have been learning about the relationship between addition and subtraction. They have been exploring fact “families” with three numbers, such as 8 + 6 = 14, 6 + 8 = 14, 14 - 6 = 8 and 14 – 8 = 6. They have found out that subtraction “undoes” addition. (Subtraction is known as the inverse operation of addition, and vice versa).

They have played a game called,

. Your child will show you how they have played this. You might like to have a turn, then make up your own fact family together and try the game again.Family ShuffleWe hope you enjoy the game, and your discussions, as you make up and talk about new families.

Thank you.

You Don't Need the Number from

Book 8: Teaching Number Sense and Algebraic Thinking## How long does it take?

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.

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

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:

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.

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

## Getting Started

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?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?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?## Exploring

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?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

Did you find the statement that was unreasonable? How?What made you think it was the unreasonable one? How did you check?Dear parents and whānau,

In maths this week we are focusing on time and on the duration of activities.

This week your child will be timing how long various personal routines take to do. Your assistance and encouragement with the following task is appreciated. Thank you.

Time yourself one day and write down how long it takes to:

## Cube and spinner challenges

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.

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

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

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:

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 challengesCube 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 twoCube 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 threeSpinner 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 fourWork 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 fiveDear parents and whānau,

This week, in pairs, we have been using coloured cubes and spinners to play probability games that we have called 'challenges'. We have been deciding whether or not the challenges are fair. Here is an example.

Cube Challenge:One red and two blue cubes are in a bag. One person chooses one cube. To win the challenge the person must take out a red cube.This is not a fair 'challenge'. The most likely event is that a blue cube will be taken. Choosing red is less likely. In fact players here are twice as likely to lose the 'challenge' as to win it.

Ask your child to tell you more about the games. It would be good if you and your child could invent and play a 'challenge' of your own. Is it a fair 'challenge'? In other words is it equally likely that you would win or lose this 'challenge'?

Enjoy the challenge!