Experiments and Simulations: 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 Model it cards for matching devices to situations, a set of six cards to collect, a spreadsheet card, 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 Model it cards and the Collect the set cards for Day 4, one set for each pair.
4. Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
5. For Day 5, have a spreadsheet open on the board. The spreadsheet card lists every step.

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: probability tells us what to expect, not what will happen. The expected result is the chance times the number of trials, and what we observe scatters around it. Short experiments scatter a lot; as the number of trials grows, the percentage we observe settles close to the chance, and repeated runs vary less, even though the count can drift further from the expected number.

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

A trial is one go at an experiment, such as one roll of a die. The expected result is what the chance predicts; the observed result is what actually happened. Variation is how much results change from one run of an experiment to the next. A simulation uses a device, such as a die or a spreadsheet, with the same chance as a real situation, so the situation can be tried many times quickly.

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 predicts results and tests them; Day 2 adds trials together and watches the percentage settle; Day 3 repeats runs and compares their spread; Day 4 builds simulations from dice and cards; and Day 5 hands the rolling to a computer to run thousands of trials. It builds on the Year 5 experiments that recorded frequencies (AC9M5P02) and on the probability scale from earlier this year (AC9M6P01). It leads to the Year 7 work of running simulations with a large number of trials and explaining the differences from predictions (AC9M7P02).

Materials for the week (one trip)

• From the classroom: a die and a coin for each pair, and coloured pencils.
• For Day 3: sticky notes or a board for a class dot plot.
• For Day 4: the cut-out cards, one set for each pair, and a few counters and paperclips if you want to build the devices.
• For Day 5: a computer or tablet with a spreadsheet for the board, and for groups if you have them.

Words for the word wall

trial, experiment, simulation, outcome, chance, expected result, observed result, frequency, percentage, running total, variation, range, dot plot, spreadsheet

Dear families

This week in maths, Year 6 runs chance experiments and simulations. We predict what should happen, toss coins and roll dice to see what does happen, and find out why more trials give results closer to the chance.

Try this at home

• Toss a coin 10 times and count the heads. Do it again. Why are the two counts different?
• Before rolling a die 30 times, predict how many 6s you will get. Then roll and compare.
• Keep a family tally of coin tosses all week. Does the percentage of heads get closer to 50%?
• Ask your child how a die could stand in for something that happens 1 day in 6.

Our family coin tally

Toss a coin 20 times a day. Add each day’s heads to the total and work out the percentage so far.

Printed from the free seegongsik Experiments and Simulations teaching pack · seegongsik.com/au/y6/probability/AC9M6P02/pack

What should happen?

Probability tells us what to expect, not what will happen. The expected result is the chance times the number of trials: each face of a die has a chance of 1/6, so in 60 rolls we expect each face about 10 times. Children work out expected results, roll a die 60 times in pairs, and compare what they observe with what they expected.

We are learning to

• work out the expected result of a chance experiment,
• record the results of a repeated experiment in a table,
• compare observed results with expected results.

Success criteria

• I can work out how many 5s to expect in 60 rolls of a die.
• I can explain why our results are not exactly what we expected.

You need

The worksheet, one per child, and a die for each pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the 60 rolls in Part B.

Watch for these ideas

• Expecting the observed results to match the expected results exactly.
• Thinking a face that has not come up for a while is due to come up next.
• Finding 1/6 of the rolls for an event such as a number less than 3, which has 2 of the 6 faces.

Answers

• Part A: (1) 1/2, 20; (2) 1/6, 10; (3) 2/6 = 1/3, 10; (4) 1/4, 20; (5) 3/10, 15; (6) 3/6 = 1/2, 60.
• Part B: answers vary; each face is expected 10 times, and gaps of 2 or 3 either way are common.
• Part C: (1) the 6, 13 times; (2) the 5, 5 fewer than expected; (3) the 4; (4) no: gaps this size are common in 60 rolls, and many more rolls would be needed to test the die.

What should happen?

A. Expected results

Write the chance of the event, then how many times you would expect it.

B. Our 60 rolls

C. Ari and Jo’s 60 rolls

1. Which face came up most often? ______

2. Which face is furthest from what was expected? By how much? ______________________

3. Which face came up exactly as often as expected? ______

4. Ari says the die is unfair because a 5 came up only 5 times in 60 rolls. Do you agree? Why?

______________________________________________________________________________

More trials

One run of 20 tosses can be well away from half heads. Added together, runs settle close to 50% heads, even though the count can drift further from exactly half: 6 over in 200 tosses is a smaller part than 3 over in 20. Children toss a coin 20 times, add the pairs’ results into running totals, and graph the percentage of heads.

We are learning to

• combine the results of repeated trials into running totals,
• write a result as a percentage of the trials,
• describe how the percentage changes as the number of trials grows.

Success criteria

• I can find the percentage of heads after 60 tosses, to the nearest whole percent.
• I can explain why the percentage gets closer to 50% even when the gap in heads does not shrink.

You need

The worksheet, one per child, a coin for each pair, and a running-total table on the board.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the class running total; start Session B with Part B.

Watch for these ideas

• Expecting the number of heads to get closer to exactly half as the tosses grow.
• Comparing counts from different numbers of tosses instead of percentages.
• Thinking a coin evens things up by giving more tails after a run of heads.

Answers

• Part A: answers vary.
• Part B: heads so far 13, 24, 32, 45, 56, 65, 77, 87, 96, 106; expected 10 to 100 in steps of 10; gaps +3, +4, +2, +5, +6, +5, +7, +7, +6, +6; percentages 65%, 60%, 53%, 56%, 56%, 54%, 55%, 54%, 53%, 53%.
• Part C: the line starts at 65% and settles near 50%, ending at 53%; (1) 6 more; (2) closer; (3) the gap is a much smaller part of 200 tosses than of 20.

More trials

A. Our 20 tosses

Heads: _______________________   Tails: _______________________   Heads as a percentage: ______

B. Class 6R’s running totals

Ten pairs each tossed a coin 20 times. Fill in the totals. Round each percentage to the nearest whole percent.

C. Graph the percentage of heads

Plot the percentage of heads after each 20 tosses and join the points. The scale runs from 40% to 70%, with a line every 2%.

1. After 20 tosses, heads were 3 more than expected. After 200 tosses? ______

2. Did the percentage of heads get closer to 50% or further away? ______________________

3. How can the gap in heads grow while the percentage gets closer to 50%? __________________________

Runs vary

Repeat an experiment and the result changes from run to run. This variation is a normal part of chance, not a mistake. Runs of 10 tosses nearly always give between 2 and 8 heads (20% to 80%); runs of 100 tosses vary much less as a percentage, nearly always between 40% and 60%. Children plot runs of 10 on a dot plot and compare its spread with a computer’s runs of 100.

We are learning to

• repeat an experiment and record the result of each run,
• show runs on a dot plot and describe their spread,
• compare the variation of short runs and long runs.

Success criteria

• I can find the lowest and highest results on a dot plot.
• I can explain why runs of 100 tosses vary less, as a percentage, than runs of 10.

You need

The worksheet, one per child, a coin for each child, and sticky notes for a class dot plot from 0% to 100%.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the class dot plot; start Session B with Part B.

Watch for these ideas

• Thinking a run that is far from half means the coin is unfair.
• Comparing spreads in heads instead of in percentages: runs of 100 spread over more heads, but less of the scale.
• Thinking more trials make the variation disappear altogether.

Answers

• Part A: answers vary; most runs of 10 give between 3 and 7 heads.
• Part B: (1) 20% and 80%; (2) 44% and 56%; (3) the runs of 10; (4) in heads, the runs of 100 (12 against 6); as a percentage, the runs of 10 (60% against 12%); (5) a run of 100, because long runs stay closer to the chance.
• Part C: most results in a narrow band around 50%, roughly 47% to 53%, and none near 20% or 80%.

Runs vary

A. Our runs of 10 tosses

Run 1: ______ heads    Run 2: ______ heads    The class’s lowest: ______    highest: ______

B. Short runs and long runs

Class 6R did 20 runs of 10 tosses. Plot each run as a percentage (3 heads of 10 is 30%) on the top scale. The bottom scale shows a computer’s 20 runs of 100 tosses.

Heads in each run of 10: 5, 7, 4, 6, 3, 5, 6, 8, 5, 4, 6, 5, 7, 4, 5, 2, 6, 5, 4, 6

1. Runs of 10: lowest ______ highest ______    2. Runs of 100: lowest ______ highest ______

3. Which runs vary more as a percentage? ______________________

4. Which runs have the bigger range in heads? __________   As a percentage? __________

5. To estimate the chance of heads from one run, which kind would you choose? Why? ______________

C. Predict

A computer did 20 runs of 1000 tosses. Shade where you think most of its results would land.

Most results between about ______ and ______, because ______________________________________

Simulate it

A simulation uses something with the same chance to stand in for a real situation: if it rains on 1 school day in 6, a die can stand in for the weather, with a 6 for rain. Children choose devices for situations, simulate collecting six cards from cereal boxes with a die, and use the class’s runs to estimate a chance that is hard to count.

We are learning to

• choose a device with the same chance as a real situation,
• run a simulation many times and record the results,
• use the results of a simulation to estimate a chance.

Success criteria

• I can choose a device, and the outcomes that mean yes, for a situation with a chance of 1/4.
• I can estimate the chance of needing more than 20 boxes from the class’s runs.

You need

The worksheet, one per child, a die for each pair, and the Model it and Collect the set cards (cut-outs), one set for each pair.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Choosing a device without deciding which outcomes mean yes: a die for 1/3 needs two faces, such as 1 and 2.
• Thinking one run of a simulation gives the answer.
• Forgetting to put the counter back in the bag, which changes the chance for the next pick.

Answers

• Part A: (1) coin, heads; (2) spinner of 4, 1 part coloured; (3) die, a 6; (4) die, 1 to 4; (5) bag of 10, 9 red counters. Other devices work too if their chance matches.
• Part B: answers vary; every run needs at least 6 boxes, and most need between about 9 and 20.
• Part C: (1) 7 and 35; (2) 14 boxes, 4 times; (3) 6 runs, 6/30 = 1/5 = 20%; (4) yes, more runs give an estimate closer to the true chance. Worked out exactly, the chance is about 15%, so 20% from 30 runs is a normal difference.

Simulate it

A. Choose a device

Choose a coin, a die, a spinner or a bag of counters with the same chance. Say which outcomes mean yes.

B. Collect the set

A cereal brand hides one of 6 animal cards in every box, each equally likely. Roll a die for each box. Tick a face the first time it comes up, and stop when all 6 are ticked.

C. What Class 6R found

Boxes needed in 30 runs: 12, 18, 14, 8, 8, 9, 7, 25, 12, 17, 10, 22, 14, 35, 15, 14, 30, 15, 9, 10, 13, 31, 14, 16, 11, 18, 11, 13, 22, 16

1. The fewest boxes any run needed: ______ The most: ______

2. Which number of boxes came up most often? ______

3. How many runs needed more than 20 boxes? ______ Estimate the chance of needing more than 20 boxes, as a fraction and a percentage. ______________

4. Would 300 runs give a better estimate? Why? __________________________________________

Let the computer roll

A spreadsheet rolls a die thousands of times in a moment with =RANDBETWEEN(1,6) and counts the sixes with =COUNTIF. From 60 to 600 to 6000 rolls, the sixes drift further from the expected number, yet their percentage closes in on 1/6, about 17%. Children read a computer’s results, judge whether a die is fair, and plan a simulation.

We are learning to

• use a digital tool to run a simulation with many trials,
• compare observed and expected results as counts and as percentages,
• use what we know about variation to judge a result.

Success criteria

• I can explain why 6000 rolls give a better estimate of the chance of a six than 60 rolls.
• I can decide whether 163 sixes in 600 rolls is surprising, and say why.

You need

The worksheet, one per child, a spreadsheet on the board (and for groups, if you can), and the spreadsheet card (cut-outs).

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Expecting a computer’s results to match the expected results exactly.
• Judging a result by its gap in count without thinking about the number of trials.
• Thinking more trials always land closer than fewer: usually, but not every time.

Answers

• Part A (sixes, expected, gap, percentage): 60 rolls: 12, 10, +2, 20%; 600 rolls: 91, 100, −9, about 15%; 6000 rolls: 1019, 1000, +19, about 17%; (1) bigger; (2) yes, closer to about 17%; (3) the 5, 29 fewer than expected.
• Part B: Die B, 63 more sixes than the expected 100, when a fair die’s sixes in 600 rolls nearly always land between about 80 and 120; the other die is ordinary.
• Part C: (1) =COUNTIF(A1:A600,3); (2) =RANDBETWEEN(1,2) in A1, filled down to A1000, with 1 for heads; (3) plans vary.

Let the computer roll

A. A computer rolls a die

1. As the rolls increase, does the gap in sixes get bigger or smaller? ______________

2. Does the percentage of sixes get closer to the chance of 1/6, about 17%? ______________

3. In the 6000 rolls, which face is furthest from what was expected? ______

B. Is the die fair?

Class 6R rolled two dice 600 times each. Die A gave 104 sixes. Die B gave 163 sixes. Which die would you test with more rolls? Why?

______________________________________________________________________________

C. Plan a simulation

1. A spreadsheet has rolls in A1 to A600. Write the formula that counts the 3s. ____________________

2. To toss a coin 1000 times, what goes in A1, and how far down do you fill it? ____________________

3. My question: ________________________________ Device or formula: ________________

Number of trials: ________ What I will record: __________________________________________

Model it cards

Cut out one set for each pair: twelve situation cards and six device cards.

How to play (Day 4)

1. Shuffle the situation cards face down and lay the device cards face up between you.
2. Turn over a situation. Each player writes down a device that could stand in for it, and which outcomes mean yes.
3. Check each other: a device with the right chance scores 1 point. If both are right with different devices, the one with fewer outcomes scores 1 more. After twelve situations, the most points wins.

Teacher note, the devices that fit each situation (yes outcomes for the simplest): (1) coin, heads; also spinner of 4, die, spinner of 8, bag of 10. (2) die, a 6. (3) spinner of 4, 1 part coloured; also spinner of 8. (4) bag of 10, 3 red counters. (5) die, 1 or 2. (6) die, 1 to 5. (7) spinner of 4, 3 parts coloured; also spinner of 8. (8) spinner of 8, 1 part coloured. (9) bag of 5, 2 red counters; also bag of 10. (10) bag of 10, 7 red counters. (11) spinner of 8, 3 parts coloured. (12) bag of 5, 4 red counters; also bag of 10.

Collect the set cards

Cut out one set of six for each pair. Turn a card face up the first time its number is rolled.

The spreadsheet card (Day 5)

What we know: Experiments and Simulations

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

1. A die is rolled 90 times. How many 4s would you expect? ______
2. A spinner has 5 equal parts and 2 of them are red. It is spun 50 times. How many reds would you expect? ______
3. Lee tossed a coin 10 times and got 7 heads. Lee says the coin must be unfair. Do you agree? Why? ______________________
4. Class A tossed a coin 20 times and got 60% heads. Class B tossed a coin 500 times and got 51% heads. Whose estimate of the chance of heads would you trust more? Why? ______________________
5. A coin gave 46 heads in 100 tosses, then 489 heads in 1000 tosses. Which result is further from half in heads? ______ Which is closer to 50% as a percentage? ______
6. Name a device that could simulate “a train is late 1 day in 4”, and say which outcomes mean late. ______________________
7. Five runs of 20 tosses gave 8, 13, 10, 11, 7 heads. What is the range? ______ Would runs of 200 tosses vary more or less, as a percentage? ______
8. A spreadsheet rolls a die 600 times and counts 97 sixes. How many sixes were expected? ______ Is 97 surprising? Why? ______________________

Answers and marking guide

Answers

1. 15.
2. 20.
3. No: 10 tosses vary a lot, and 7 heads in 10 is a common result for a fair coin.
4. Class B: 500 tosses give a more reliable estimate than 20.
5. The 1000 tosses are further from half in heads (11 against 4), but closer as a percentage (48.9% against 46%).
6. For example, a spinner of 4 with 1 part coloured for late; any device with a chance of 1/4 works.
7. 6 heads; less.
8. 100; no, 3 fewer is a small gap for 600 rolls.

A quick three-level guide

Eight questions, four ideas. A child at standard predicts expected results, expects runs to vary, and explains that more trials bring the percentage closer to the chance.

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: expected results, adding trials, variation between runs, simulating with a device, and simulating with a digital tool.