Repeated Chance Experiments: a week of ready-to-teach maths

Five days of lessons for Year 4 Probability. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus two cut-out spinners, a class results chart, 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. Print the spinner sheet once for each pair, and the class results chart once for the board.
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. Children run real experiments, so their own results will differ; each worksheet also has results from another class with fixed answers, and the plan says what to look for in your class’s results.

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.

Variation and the long run

Repeating a chance experiment shows two things at once. The results vary: two runs of the same experiment rarely match, and that is normal, not a mistake. And as more results are combined, they tend to settle close to what we predict, such as about half heads. Keep the class results chart from Day 1 to Day 4 so children can see both.

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

Days 1 and 2 repeat an experiment and compare the results with a prediction. Day 3 runs the same experiment several times and describes the variation. Day 4 combines everyone’s results to see them settle, and Day 5 uses repeated results to see how two outcomes relate. It builds on describing outcomes and ordering them by likelihood (AC9M4P01).

Materials for the week (one trip)

• From the classroom: a coin and a die for each pair, paper clips, pencils, scissors, and this pack printed.
• Print once for each pair: the spinner sheet. Print once for the board: the class results chart.
• Optional: sticky dots or counters to build the class dot plot on the board.

Words for the word wall

experiment, trial, repeat, outcome, result, tally, predict, prediction, about, variation, vary, run, combine, total, half, fair

Dear families

This week in maths, Year 4 repeats chance experiments with coins, dice and spinners, and looks at how the results vary and what they show when there are lots of them.

Try this at home

• Flip a coin ten times and count the heads. Do it again. Did you get the same number?
• Before a board game, predict how many times a 6 will come up in ten rolls. Keep a tally.
• Add up everyone’s results in the family. Is the total closer to half heads?
• Ask: if a result surprises us, does that mean the coin or the die is unfair?

My coin flips this week

Fill one row a day: flip a coin ten times and write the number of heads.

Printed from the free seegongsik Repeated Chance Experiments teaching pack · seegongsik.com/au/y4/probability/AC9M4P02/pack

One trial, many trials

One trial of a chance experiment, one flip or one spin, gives one outcome and tells us very little: one head does not show that a coin always lands heads. Repeating the experiment builds a tally, and a pattern starts to show. Children flip a coin twenty times and total the heads after 1, 5, 10 and 20 flips, then compare with Sam’s record and with each other.

We are learning to

• explain why one trial tells us very little,
• repeat a chance experiment and record every result,
• compare the results of the same experiment.

Success criteria

• I can explain why one flip is not enough to know about a coin.
• I can record twenty flips and total them as I go.

You need

A coin for each pair. The class results chart (cut-out sheet 2) on the board. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the twenty flips; start Session B with Part B.

Watch for these ideas

• Judging a coin or a spinner from one result.
• Expecting the flips to take turns, heads then tails.
• Thinking twenty flips must give exactly ten heads.

Answers

• Part A: answers vary; after 20 flips most children have between 6 and 14 heads.
• Part B: after 1 1 heads and 0 tails; after 5 2 heads and 3 tails; after 10 5 heads and 5 tails; after 20 11 heads and 9 tails.
• Part C: answers vary; the results differ because each flip is chance, even with the same kind of coin.
• Part D: one flip gives only one outcome. A fair coin can land heads first; only many flips show a pattern.

One trial, many trials

A. Flip a coin twenty times. Write H or T in each box, then fill in the totals.

B. Sam flipped a coin twenty times. Fill in Sam’s totals.

C. Did you and Sam get the same results? Why might they be different?

D. Mia flipped a coin once and it landed heads. She says: “This coin always lands on heads.” Why is one flip not enough to know?

Predict, then test

A prediction says what we expect before an experiment. A die has 6 faces that are equally likely, so in 30 rolls we expect about 5 of each number. Then we test: roll, tally and compare. The results will not match the prediction exactly; they vary around it, some a little above and some a little below. Children predict, roll a die 30 times, and compare their results with the prediction and with another class’s.

We are learning to

• predict the results of a repeated chance experiment,
• test a prediction by repeating the experiment,
• describe how close the results came to the prediction.

Success criteria

• I can predict about how many times each number will come up, and say why.
• I can say how far my results were from my prediction.

You need

A die for each pair. The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Expecting results to match the prediction exactly.
• Deciding a die is unfair from one set of rolls.
• Thinking a 6 is harder to roll than the other numbers.

Answers

• Part A: about 5 of each number, because the 30 rolls share out over 6 equally likely faces.
• Part B: answers vary; most numbers come up within 2 or 3 of 5.
• Part C: 1: 6, 2: 4, 3: 5, 4: 7, 5: 3, 6: 5 (total 30); most often 4, least often 5; the biggest difference from 5 is 2.
• Part D: no; results vary around the prediction, and 30 rolls is not many. A difference like this happens with a fair die.

Predict, then test

A. Predict. You will roll a die 30 times.

B. Test. Roll the die 30 times and tally each number.

C. Another class rolled a die 30 times.

D. A child in that class says the die must be unfair. Do you agree?

Results vary

Run the same experiment again and the results come out differently: that difference from run to run is variation. It is what chance does, not a mistake and not a broken spinner. Spinner 1 has 3 equal parts, so in 12 spins we expect about 4 of each colour. Children run 12 spins three times, then describe the variation in their own runs and in another pair’s, using the lowest and highest counts.

We are learning to

• repeat the same experiment several times,
• identify the variation between runs,
• describe variation using the lowest and highest results.

Success criteria

• I can say that runs of the same experiment give different results, and why.
• I can describe the variation, such as “blue came up between 2 and 6 times”.

You need

Spinner 1 from the spinner sheet (cut-out sheet 1), a paper clip and a pencil for each pair. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the three runs; start Session B with Part B.

Watch for these ideas

• Thinking a run that differs from the prediction was done wrongly.
• Expecting the next run to make up for the last one.
• Describing variation with one result instead of the lowest and highest.

Answers

• Part A: answers vary; most counts fall a little above or below 4.
• Part B: blue lowest 2, highest 6; green lowest 2, highest 6; orange lowest 3, highest 4; no run gave exactly 4 of each colour.
• Part C: answers vary, for example “In runs of 12 spins, blue came up between 2 and 6 times.”

Results vary

A. Spin spinner 1 12 times. Do it three times, and record each run.

B. Another pair did four runs of 12 spins.

C. Write a sentence that describes the variation in one colour.

More trials, steadier results

One run of an experiment can land far from the prediction; put many runs together and the total tends to come closer to it. Pair D below got 13 heads in 20 flips, well away from half, but the whole class together got 82 heads in 160 flips, very close to half (80). Children plot the pairs’ results, combine them, and then combine their own class’s results from Day 1.

We are learning to

• combine the results of many runs of an experiment,
• observe that combined results come closer to the prediction,
• explain why more trials tell us more than a few.

Success criteria

• I can total the results of many runs and compare the total with half.
• I can explain why all the class’s flips tell us more than one pair’s.

You need

The class results chart from Day 1 (cut-out sheet 2). The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Thinking a big run must give exactly half.
• Thinking the pair furthest from half did something wrong.
• Comparing the gaps from half without thinking about how many flips there were.

Answers

• Part A: dots at 12, 9, 10, 13, 8, 11, 10, 9; lowest 8, highest 13; pair D is furthest from 10.
• Part B: 82 heads out of 160 flips; half is 80; the total is 2 away from half.
• Part C: the whole class is closer to half. Pair D is 3 away from half in only 20 flips; the class is 2 away in 160 flips.
• Part D: answers vary; most classes land close to half. About 500 heads in 1000 flips.

More trials, steadier results

A. Draw a dot for each pair. Then find the lowest and highest.

B. Combine the results of the whole class.

C. Which is closer to half: one pair or the whole class? Explain.

D. Our class. Total: ______ heads out of ______ flips. Is it close to half? About how many heads would 1000 flips give?

Relationships between outcomes

Repeated results also show how outcomes relate to each other. Spinner 2 has 3 blue parts and 1 orange part, so blue should come up about 3 times as often as orange: in 40 spins, about 30 blue and 10 orange. On a die, a number from 1 to 4 (four faces) should come up about twice as often as a 5 or a 6 (two faces). Children predict, spin 40 times, and check whether their results show the relationship, allowing for variation.

We are learning to

• predict how outcomes relate, such as one coming up about three times as often,
• observe the relationship in the results of a repeated experiment,
• allow for variation when we check a prediction.

Success criteria

• I can use the parts of a spinner to predict how many times as often one colour will come up.
• I can check whether results show a relationship, even when they do not match exactly.

You need

Spinner 2 from the spinner sheet, a paper clip and a pencil for each pair, and a die. The worksheet, one per child.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Thinking blue and orange are equally likely because there are two colours.
• Expecting exactly 30 and 10, and deciding the spinner is wrong when they do not come.
• Thinking a relationship seen in a few spins must hold in every run.

Answers

• Part A: about 30 blue and 10 orange; blue about 3 times as often, because it has 3 of the 4 parts.
• Part B: answers vary; most pairs get between about 25 and 35 blues.
• Part C: yes; 3 × 9 = 27, close to 31, so blue came up about 3 times as often.
• Part D: a number from 1 to 4, about twice as often (four faces against two); in 30 rolls predict about 20 and 10; 21 and 9 fit, since 2 × 9 = 18 is close to 21.

Relationships between outcomes

A. Predict. Spinner 2 has 4 equal parts.

B. Test. Spin spinner 2 40 times and tally.

C. Another pair spun spinner 2 40 times: blue 31, orange 9.

D. Roll a die. Which should come up more often: a number from 1 to 4, or a 5 or a 6? About how many times as often?

Spinner sheet

One per pair. Cut out each spinner. Put a pencil point through a paper clip at the centre dot, hold the pencil still and flick the clip. The colour under the end of the clip is the outcome; if the clip stops on a line, spin again.

Keep both spinners in an envelope between lessons.

Our class results chart

For the board. On Day 1 each pair writes its heads in 20 flips and adds a dot to the dot plot. On Day 4 the class totals the chart.

Our dot plot

What we know: Repeated chance experiments

1. Why is one spin not enough to tell whether a spinner is fair? ______________________________
2. A coin is flipped 40 times. About how many heads would you predict? ________
3. Four groups each flipped a coin 10 times. Heads: 6, 3, 5, 7. What were the lowest and highest results? ____ and ____
4. Put the four groups together. How many heads out of how many flips? ____ out of ____ Is that close to half? ________
5. Which is likely to be closer to half heads: 10 flips or 100 flips? ________
6. This spinner is spun 80 times. About how many yellows would you predict? ________

   G green, Y yellow
7. Ella rolled a die 12 times and never got a 6. She says her die has no 6. Is she right? Explain. ________________
8. A die is rolled 60 times. Ring the most likely number of 6s:   0  ·  10  ·  60

Answers and marking guide

Answers

1. One spin gives only one outcome; any colour can come up once. Many spins are needed to see a pattern.
2. About 20.
3. 3 and 7.
4. 21 out of 40; yes, half is 20.
5. 100 flips.
6. About 20: yellow is 1 of the 4 equal parts.
7. No: 12 rolls is a short run, and a fair die can miss a number in a short run. One of its six faces still shows a 6.
8. 10: a 6 is 1 of the 6 faces.

A quick three-level guide

Eight questions, three ideas. A child at standard repeats an experiment and predicts its results, describes how runs vary, and sees combined results settle close to the prediction.

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: one trial and many, predicting and testing, variation, combining results, and relationships between outcomes.