The Probability Scale: a week of ready-to-teach maths

Five days of lessons for Year 6 Probability. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus twenty event cards for a class washing line and Chance war, spinner templates, a mini-check and every answer.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Cut out the event cards once for the washing line, and once more for each pair for Chance war. Print the spinner templates for Day 5.
4. Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and answers are all on the one page.

No maths background needed

This pack is written for the busy generalist teacher. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week: a probability is a number from 0 (impossible) to 1 (certain), and the same chance can be written as a fraction, a decimal or a percentage, so 1/4, 0.25 and 25% are one chance. When outcomes are equally likely, the probability is the favourable outcomes over all the outcomes; when they are not, we estimate from what has happened before.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.

Words to watch

An outcome is one way an experiment can turn out; an event is the outcome or outcomes you are interested in. Equally likely means every outcome has the same chance, as on a fair die. A fair game gives every player the same chance of winning. An estimate of a probability is a best guess from data, not a certainty, and gets better with more data.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 puts chances on a scale; Day 2 writes them three ways; Day 3 works them out from equally likely outcomes; Day 4 estimates them from data; and Day 5 designs spinners, bags and fair games to match. It builds on the Year 5 work of listing outcomes and comparing likelihoods (AC9M5P01, AC9M5P02). It leads to the next unit, experiments and simulations (AC9M6P02), and to the Year 7 work on probabilities of events in sample spaces.

Materials for the week (one trip)

• From the classroom: a length of string and pegs for a washing line, dice, a few coins, scissors and coloured pencils.
• For Day 3: a bag with some coloured counters or cubes for each group, if you have them.
• For Day 5: a pencil and a paperclip for each pair to spin the spinner templates.

Words for the word wall

probability, chance, impossible, unlikely, even chance, likely, certain, outcome, event, equally likely, fraction, decimal, percentage, estimate, fair

Dear families

This week in maths, Year 6 puts chances on a scale from 0 to 1, writes them as fractions, decimals and percentages, works them out from dice, spinners and bags of counters, and estimates them from data.

Try this at home

• Look at a weather forecast together. What does a 70% chance of rain mean? What is the chance of no rain?
• Roll a die and ask: what is the chance of rolling a number bigger than 4? Write it three ways.
• Put some coloured pegs or pens in a bag. What is the chance of pulling out each colour?
• Talk about events that are impossible, unlikely, even, likely and certain in your week.

Chances this week

Fill one row a day. Write an event, its chance as a fraction or a percentage, and a word for it.

Printed from the free seegongsik Probability Scale teaching pack · seegongsik.com/au/y6/probability/AC9M6P01/pack

Impossible to certain

Every probability sits on a scale from 0 to 1: 0 is impossible, 1 is certain and 0.5 is an even chance. Children place events on a class washing line and on their own scales, match the five chance words to numbers, and see that events with the same chance sit at the same place.

We are learning to

• place events on a probability scale from 0 to 1,
• match the words impossible, unlikely, even chance, likely and certain to numbers,
• compare two events by their places on the scale.

Success criteria

• I can explain why rolling a 7 on a die sits at 0 and a coin landing tails sits at 0.5.
• I can place an event that is likely but not certain.

You need

The worksheet, one per child, string and pegs for a washing line, and the event cards (cut-outs), one set for the class.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the washing line; start Session B with Parts A and B.

Watch for these ideas

• Thinking a chance can be more than 1 or less than 0.
• Calling any possible event likely: rolling a 6 can happen but is unlikely.
• Thinking a 6 is harder to roll than a 3.

Answers

• Part A: impossible, 0; unlikely, between 0 and 0.5; even chance, 0.5; likely, between 0.5 and 1; certain, 1.
• Part B: A 1 (certain), B 0.5 (even chance), C 0 (impossible), D 0.9 (likely), E 0.25 (unlikely), F 0.1 (unlikely), G 0.5 (even chance), H about 0.67 (likely).
• Part C: B and G have the same chance (0.5); D and F (0.9 and 0.1) are the same distance either side of even.

Impossible to certain

A. Words and numbers

B. Place the events

Write each letter above the scale where its chance belongs.

C. Same and opposite

1. Which two events have exactly the same chance? How do you know? ________________________________

2. Which two events are the same distance from even chance, one each side? ____________________________

Three ways to write it

The same chance can be written three ways: as a fraction, a decimal or a percentage. 1/4, 0.25 and 25% all mark the same place on the scale, because a percentage counts hundredths and the decimal is the fraction divided out. Children convert between the three, place chances on a scale marked 0 to 1 above and 0% to 100% below, and compare chances written in different ways.

We are learning to

• write a chance as a fraction, a decimal and a percentage,
• place chances on a scale of 0 to 1 and of 0% to 100%,
• compare chances written in different ways.

Success criteria

• I can write 3/4 as 0.75 and 75%.
• I can say whether 0.4 or 3/8 is more likely, and how I know.

You need

The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the double scale in Part B.

Watch for these ideas

• Writing 0.3 as 3% instead of 30%.
• Thinking a bigger number means a bigger chance, so 45% looks bigger than 0.9.
• Thinking 1/8 is bigger than 1/4 because 8 is bigger than 4.

Answers

• Part A: 1/2 = 0.5 = 50%; 1/4 = 0.25 = 25%; 3/4 = 0.75 = 75%; 1/10 = 0.1 = 10%; 1/5 = 0.2 = 20%; 3/5 = 0.6 = 60%; 1/8 = 0.125 = 12.5%; 1 = 1 = 100%.
• Part B: P at 0.3 (30%), Q at 0.75 (75%), R at 0.45 (45%), S at 0.2 (20%), T at 0.9 (90%); R (45%) is closest to even chance.
• Part C, the more likely: 0.4 (0.4 beats 0.375); 2/3 (0.67 beats 0.65); the same (0.5 = 1/2); 30% (0.3 beats 0.25).

Three ways to write it

A. Fill the table

B. Place them on the double scale

Write each letter above the scale: P 0.3Q 3/4R 45%S 1/5T 0.9

C. Which is more likely?

Write both the same way first, then circle the more likely chance. If they are equal, write “same”.

Which of these is closest to even chance: 0.3, 3/4, 45%, 1/5, 0.9? ______ Why? ______________

Count the outcomes

When every outcome is equally likely, the probability of an event is the number of outcomes that give it over the number of outcomes altogether. A die has 6 equally likely outcomes, so rolling a number less than 5 is 4/6, which simplifies to 2/3. Children find probabilities on a die, on spinners with equal parts and in a bag of counters, write each in simplest form and as a percentage, and play Chance war.

We are learning to

• list the equally likely outcomes of an experiment,
• write a probability as the outcomes that count over all the outcomes, in simplest form,
• write the same probability as a decimal and a percentage.

Success criteria

• I can find the chance of rolling a number less than 5 and simplify it.
• I can find the chance of picking a yellow counter from a bag.

You need

The worksheet, one per child, a die for each pair, a bag of counters if you have one, and the event cards for Chance war.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part B; start Session B with the bag in Part C.

Watch for these ideas

• Counting the outcomes that give the event but not all the outcomes, or the other way round.
• Thinking a spinner with unequal parts gives every colour the same chance.
• Leaving 2/4 unsimplified when comparing, so 2/4 and 1/2 look different.

Answers

• Part A: a 3: 1/6 = about 17%; an even number: 3/6 = 1/2 = 50%; a number less than 5: 4/6 = 2/3 = about 67%; a 7: 0/6 = 0 = 0%.
• Part B: (1) blue: 2/4 = 1/2 = 50%, green: 1/4 = 25%, not yellow: 3/4 = 75%; (2) green: 2/5 = 40%, blue: 3/5 = 60%; (3) an even number: 4/8 = 1/2 = 50%, more than 6: 2/8 = 1/4 = 25%, a multiple of 3: 2/8 = 1/4 = 25%.
• Part C: blue: 1/2 (0.5, 50%); green: 1/6 (about 0.17, about 17%); yellow: 1/3 (about 0.33, about 33%); not blue: 1/2 (0.5, 50%).

Count the outcomes

A. A fair die

B. Spinners with equal parts

P(blue) is short for the probability of blue.

C. A bag of counters

Estimate the chance

Many events do not have equally likely outcomes: rain, a free throw, a late bus. We estimate their chance from what has happened before: if it rained on 6 of the last 20 days, the chance of rain is about 6/20, which is 30%. The chance that something does not happen is 1 take away the chance that it does. Children estimate chances from data, find the chance of not happening, and discuss how far to trust an estimate.

We are learning to

• estimate a probability from data as a fraction, a decimal and a percentage,
• find the chance that an event does not happen,
• explain why more data gives a better estimate.

Success criteria

• I can estimate the chance of rain from 6 rainy days out of 20.
• I can find the chance of no rain when the forecast says 65%.

You need

The worksheet, one per child, and a weather forecast with chances of rain if you have one.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the chances of not happening in Part B.

Watch for these ideas

• Thinking any event with two outcomes is an even chance, such as rain or no rain.
• Making the two chances add to 10 or to 50% instead of to 1 or 100%.
• Trusting an estimate from a few days as much as one from many days.

Answers

• Part A: 6/20 = 3/10 = 0.3 = 30%; 15/20 = 3/4 = 0.75 = 75%; 2/10 = 1/5 = 0.2 = 20%; 35/50 = 7/10 = 0.7 = 70%.
• Part B: no rain 70%; a miss 1/4; the bus is on time 0.8; no rain 35%.
• Part C: (1) the estimate from more days, because a few days can easily be unusual; (2) no, because summer and winter weather differ, so the data should come from the same season.

Estimate the chance

A. From data to a chance

Use what happened to estimate the chance. Write it as a fraction, a decimal and a percentage.

B. The chance it does not happen

Write each answer the same way as the chance you are given.

1. The chance of rain tomorrow is 30%. What is the chance of no rain? ________

2. The chance a free throw goes in is 3/4. What is the chance of a miss? ________

3. The chance the bus is late is 0.2. What is the chance of the bus is on time? ________

4. The forecast gives a chance of rain of 65%. What is the chance of no rain? ________

C. How sure?

1. Ali estimated the chance of rain from 3 days. Kim used 30 days. Whose estimate would you trust more? Why?

______________________________________________________________________________

2. Would rainy days counted in summer give a good estimate of rain in winter? Why or why not?

______________________________________________________________________________

Design it

Designing runs probability backwards: a spinner of 10 equal parts with a chance of 0.3 for blue needs 3 blue parts. A game is fair when every player has the same chance of winning. Children colour spinners and fill bags to match given chances, then judge games and make an unfair one fair.

We are learning to

• design a spinner with equal parts to match given probabilities,
• choose the counters for a bag to match a probability,
• decide whether a game is fair, and make it fair.

Success criteria

• I can colour a spinner of 10 parts so that blue has a chance of 0.3.
• I can explain why a game is not fair, and change it so that it is.

You need

The worksheet, one per child, coloured pencils, and the spinner templates (cut-outs) with a pencil and a paperclip for each pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the bags in Part B.

Watch for these ideas

• Colouring 3 parts for a chance of 0.3 on a spinner that has 8 parts, not 10.
• Thinking a game is fair because both players can win.
• Leaving parts uncoloured: the chances of all the colours must make 1.

Answers

• Part A: (1) 2 blue, 1 green, 1 yellow; (2) 2 blue, 3 green, 1 yellow; (3) 3 blue, 6 green, 1 yellow; (4) 2 blue, 0 green, 6 yellow.
• Part B: (1) 3 blue; (2) 2 green; (3) 7 yellow; (4) 6 counters: 3 blue, 2 green, 1 other.
• Part C: (1) not fair, 1/3 and 2/3; (2) fair, 1/2 and 1/2; (3) fair, 1/2 and 1/2; (4) not fair, 2/3 and 1/3. Game 1 is fair if each player wins on three numbers.

Design it

A. Colour the spinner

Colour the equal parts so each colour has the chance shown.

B. Fill the bag

1. A bag holds 5 counters. The chance of picking blue must be 3/5. How many blue counters? ______

2. A bag holds 8 counters. The chance of picking green must be 0.25. How many green counters? ______

3. A bag holds 10 counters. The chance of picking yellow must be 70%. How many yellow counters? ______

4. The chance of blue must be 1/2 and the chance of green 1/3. What is the smallest bag that works? ______________________

C. Fair or not?

1. Roll a die. You win on 1 or 2; your partner wins on 3, 4, 5 or 6.  Fair? ______

2. Toss a coin. Heads, you win; tails, your partner wins.  Fair? ______

3. Spin a spinner with parts blue, blue, green and yellow. Blue, you win; green or yellow, your partner wins.  Fair? ______

4. Pick from a bag of 3 blue, 2 green and 1 yellow counter. Blue or yellow, you win; green, your partner wins.  Fair? ______

Change game 1 so that it is fair: ______________________________________________________

Event cards

Cut out the twenty cards: one set for the class washing line, and one set for each pair for Chance war.

Chance war (Day 3)

1. Shuffle the cards and deal them out face down. Both players turn over their top card.
2. Each player says the probability of their event as a fraction, a decimal or a percentage.
3. The more likely event wins both cards. If the chances are equal, each player keeps their own card. Most cards wins.

Teacher note, each event’s probability: (1) 1/2, 0.5, 50%; (2) 1/6, about 0.17, about 17%; (3) 6/6 = 1, 100%; (4) 0/6 = 0, 0%; (5) 2/6 = 1/3, about 0.33, about 33%; (6) 4/6 = 2/3, about 0.67, about 67%; (7) 3/6 = 1/2, 0.5, 50%; (8) 1/4, 0.25, 25%; (9) 3/4, 0.75, 75%; (10) 3/10, 0.3, 30%; (11) 1/10, 0.1, 10%; (12) 2/5, 0.4, 40%; (13) 3/5, 0.6, 60%; (14) 1/8, 0.125, 12.5%; (15) 8/8 = 1, 100%; (16) 2/7, about 0.29, about 29%; (17) 7/12, about 0.58, about 58%; (18) 3/9 = 1/3, about 0.33, about 33%; (19) 2/11, about 0.18, about 18%; (20) 4/20 = 1/5, 0.2, 20%.

Spinner templates

Print one sheet for each pair. Colour a spinner, then spin it with a pencil and a paperclip.

How to spin

1. Hold a pencil point upright on the centre dot, through one end of a paperclip.
2. Flick the paperclip so it spins round the pencil, and read the part it stops on.
3. If it stops on a line, spin again.

What we know: The Probability Scale

Work on your own. Show your thinking if you can.

1. Mark I for impossible, E for even chance and C for certain on the scale. Then mark L where a likely event could go.
2. Write 3/4 as a decimal and as a percentage. ______ ______
3. A die is rolled. What is the chance of a number less than 3? Give it in simplest form. ______
4. A spinner has 5 equal parts and 2 of them are blue. What is the chance of blue, as a percentage? ______
5. Which is more likely, 0.7 or 2/3? ______ How do you know? ______________________
6. The forecast says a 35% chance of rain. What is the chance of no rain? ______
7. A team won 12 of its last 20 games. Estimate the chance it wins the next one, as a decimal. ______
8. A bag holds 10 counters. The chance of green must be 0.4. How many green counters? ______

Answers and marking guide

Answers

1. I at 0, E at 0.5, C at 1; L anywhere between 0.5 and 1, but not at either end.
2. 0.75 and 75%.
3. 2/6 = 1/3.
4. 40%.
5. 0.7: 2/3 is about 0.67, and 0.7 is more.
6. 65%.
7. 12/20 = 0.6.
8. 4 green counters.

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and moves a chance between a fraction, a decimal and a percentage without losing its place on the scale.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: the scale, the three ways to write a chance, counting outcomes, estimating from data, and designing.