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

Five days of lessons for Year 5 Probability. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus two spinners to make, bag cards for a guessing game, another class’s results to read, a mini-check and every answer. Every day, children run a chance experiment of their own.

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 and the bag cards once per pair. Gather coins, paper clips, paper cups, counters and opaque bags.
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: one trial of a chance experiment tells us very little, but many trials show a pattern. Record every result, count how many times each outcome came up (its frequency), and compare the frequencies. An outcome that comes up more often in many trials is estimated to be more likely, and more trials give a better estimate. This works even when we cannot tell the chances by looking.

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, and Day 3 uses the coin results from Day 1. Every plan marks a split point if your timetable runs two shorter sessions.

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

It builds on listing outcomes and deciding whether they are equally likely (AC9M5P01), and on Year 4, where children saw results vary when an experiment is repeated (AC9M4P02). In Year 6, chances get numbers from 0 to 1 (AC9M6P01), and experiments grow to many more trials with digital tools and are compared with what was expected (AC9M6P02).

Materials for the week (one trip)

• Per pair: a coin, a paper clip and a pencil, a paper cup, an opaque bag, and 10 counters in each of three colours (blue, orange and green, or any three).
• For the teacher: a mystery bag of 10 counters set up before Day 5 (see the Day 5 plan), and scissors and glue for the spinners.
• Print once per pair: the spinner sheet (Day 2) and the bag cards (Day 5).

Words for the word wall

trial, repeated experiment, result, tally, frequency, frequency table, total, relative frequency, out of, estimate, likelihood

Dear families

This week in maths, Year 5 runs chance experiments again and again: tossing coins, spinning spinners, tossing paper cups and drawing counters from a bag. Children record every result, count how often each outcome came up, and use those counts to say which outcomes are more likely. One toss tells us little; fifty tosses tell us a lot.

Try this at home

• Toss a coin 20 times together and tally heads and tails. Was it exactly 10 and 10? Is that a surprise?
• Drop a bottle top 20 times. Does it land top up or top down more often? Could you have told by looking?
• Ask your child why ten tosses are not enough to decide whether a coin is fair.

Our experiments, day by day

Fill one row a day. Write what we tested and what our results showed.

Printed from the free seegongsik Repeated Experiments teaching pack · seegongsik.com/au/y5/probability/AC9M5P02/pack

Toss it again

A trial is one go at a chance experiment, such as one toss of a coin. One trial tells us very little, because its result is down to chance. Repeating the same trial many times, in the same way, shows a pattern: over many tosses, heads and tails each come up about half the time, though hardly ever exactly half, and a short run can look lopsided. So we record every result as it happens, with a tally, and judge from many trials, not a few.

We are learning to

• carry out the same trial many times, in the same way,
• record every result with a tally,
• see how results build up and vary.

Success criteria

• I can tally the results of a repeated experiment without missing any.
• I can explain why a few trials are not enough to judge a coin.

You need

A coin per pair. A class tally chart on the board for heads and tails. The worksheet.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Deciding that a coin is unfair after only a few tosses.
• Thinking a tail is ‘due’ after several heads in a row.
• Losing count, so the heads and tails do not add up to the number of tosses.

Answers

• Parts A and C: the pair’s and the class’s own results; each round’s heads and tails add to 10.
• Part B: rounds 1: heads 7, tails 3; 2: heads 3, tails 7; 3: heads 4, tails 6. So far 7 and 3, then 10 and 10, then 14 and 16. Zoe: no; after 30 tosses there are 14 heads and 16 tails, and ten tosses are too few to judge a coin.

Toss it again

A. Our 30 tosses: tally each round of ten

B. Zoe and Raf’s 30 tosses (H = heads, T = tails)

After round 1, Zoe said, ‘This coin favours heads.’ After all 30 tosses, do you agree? Why?

C. Our class

Our class tossed ______ times: heads ______, tails ______. Is that exactly half heads? Is it close to half?

Spin and record

A frequency table turns a long run of results into a summary: each outcome, its tally, and its frequency, the number of times it came up. The frequencies add up to the number of trials, which checks that nothing was missed or counted twice. Spinner A has four equal parts, so its outcomes are equally likely; Spinner B’s sections are different sizes, so its outcomes are not. Recording both lets the data show the difference.

We are learning to

• record results in a frequency table,
• check that the frequencies add up to the number of trials,
• compare the results of a spinner with equally likely outcomes and one without.

Success criteria

• I can turn a tally into frequencies and check the total.
• I can say what our two frequency tables show about the two spinners.

You need

The spinner sheet (cut-out sheet 1), one per pair, glued to card. A paper clip and a pencil per pair. The class coin tally from Day 1.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Spinner A; start Session B with Spinner B.

Watch for these ideas

• Frequencies that do not add up to the number of spins.
• Reading a group of five tally marks as four, or counting the crossing mark twice.
• Expecting Spinner B to give exactly half blue, or Spinner A exactly equal frequencies.

Answers

• Part A: the pair’s own spins; the totals are 20 and 40.
• Part B: blue 18, orange 12, green 4, yellow 6; total 40, the same as the number of spins. Most often blue, least often green.
• Part C: Spinner B’s outcomes are not equally likely: blue, half of the spinner, comes up far more often than green or yellow. Spinner A’s frequencies are usually closer to even, though rarely exactly even.

Spin and record

A. Our spins

B. Group 3 spun Spinner B 40 times. Write the frequencies.

Total: ______   Does it match the number of spins? ______   Most often: ______   Least often: ______

C. Compare your tables for Spinner A and Spinner B. What do they show? Why?

Out of how many?

A frequency means more beside the number of trials: 12 heads is a lot in 20 tosses but not many in 50. The relative frequency is the frequency out of the number of trials, written as a fraction: 7 heads in 10 tosses is 7/10, read ‘7 out of 10’. To compare results with different numbers of trials, make the bottom numbers the same first: 7/10 = 14/20. A relative frequency close to 1/2 means about half the trials.

We are learning to

• write a result as a fraction of the trials,
• compare results from experiments with different numbers of trials,
• say whether a share is close to half.

Success criteria

• I can write a relative frequency as a fraction, the right way up.
• I can compare two results that have different numbers of trials.

You need

Each pair’s Day 1 worksheet and the class coin totals from Day 1. The worksheet.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Writing the fraction upside down, with the number of trials on top.
• Putting the other results at the bottom, such as heads out of tails.
• Comparing two frequencies as if the experiments were the same size.

Answers

• Part A: 17/30, 19/40, 4/30, 13/20.
• Part B: Mia 9/20 = 18/40; Kai 21/40. Kai had heads in the bigger share. Half is 20/40, so Kai is closer to half (1/40 away; the other is 2/40 away).
• Part C: the pair’s and the class’s own tosses; the class’s share is usually closer to half.

Out of how many?

A. Write each result as a fraction of the trials.

B. Two pairs tossed coins.

1. Who got heads in a bigger share of their tosses?   ______________

2. Half of 40 is ______. Whose share of heads is closer to half? How do you know?

C. Our class (use Day 1)

Which share is closer to half? Why might that be?

Compare the outcomes

Frequencies let us compare outcomes even when we cannot tell their chances by looking: the outcome with the higher frequency came up more often. When outcomes are equally likely, as with a coin, the frequencies come out close but rarely equal, and a small gap is just chance. When one frequency is far bigger over many trials, as when a drawing pin lands one way far more often, the outcomes are probably not equally likely.

We are learning to

• compare outcomes by their frequencies,
• show results in a column graph,
• tell a small gap from chance from a big gap that shows outcomes are not equally likely.

Success criteria

• I can say which outcome came up more often, and by how many.
• I can use frequencies to judge whether outcomes look equally likely.

You need

A paper cup per pair. A class tally chart on the board for the three ways a cup can land. The worksheet.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Calling a coin unfair because heads and tails did not come up exactly equally.
• Comparing the heights of columns without reading the scale.
• Expecting three outcomes to come up equally often just because there are three of them.

Answers

• Part A: the pair’s own tosses; the frequencies add up to 30.
• Part B: columns 37, 9, 4 on the scale. On its side came up most: 37 − 9 = 28 more than upside down. Not equally likely: on its side came up far more often in 50 tosses.
• Part C: no; 26 heads and 24 tails differ by only 2, a gap chance gives all the time, so the results fit a coin with equally likely outcomes.

Compare the outcomes

A. Our 30 cup tosses

B. Class 5C tossed a paper cup 50 times.

2. Which way did the cup land most often? ______________   3. How many more times than upside down? ______

4. Are the cup’s three outcomes equally likely? How do the results show it?

C. Class 5C also tossed a coin 50 times.

Heads came up 26 times and tails 24 times. Ella says, ‘The coin must be unfair.’ Do you agree? Why?

Which bag?

Frequencies from many trials let us estimate how likely each outcome is, even when we cannot see how the experiment is set up. If counters are drawn from a hidden bag and put back each time, the colour drawn most often is probably the most common colour in the bag, and the share of the draws is a rough guide to the share of the bag: if about half the draws are blue, about half the counters are probably blue. An estimate is not a certainty, and more draws give a better one.

We are learning to

• estimate how likely outcomes are from their frequencies,
• use many draws to estimate what is in a hidden bag,
• explain why more trials give a better estimate.

Success criteria

• I can order outcomes from most to least likely using their frequencies.
• I can say why my estimate might be wrong, and how to make it better.

You need

A mystery bag set up before the lesson, for example 7 blue, 2 orange and 1 green (keep it hidden). Per pair: the bag cards (cut-out sheet 2), an opaque bag and 10 counters of each colour. Mini whiteboards for the quiz.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the mystery bag; start Session B with Part B and the game.

Watch for these ideas

• Treating an estimate as certain.
• Deciding a colour is not in the bag because it has not been drawn yet.
• Forgetting to put the counter back, which changes the bag for the next draw.

Answers

• Part A: the class’s own draws; the frequencies add up to 30.
• Part B: 1. blue, orange, green. 2. b (6 blue, 3 orange, 1 green): in 40 draws it would give about 24 blue, 12 orange, 4 green, close to 23, 12, 5. 3. no; draws are down to chance, so another bag could give these results, though less often. More draws would make us surer.
• Part C: the pair’s own games.

Which bag?

A. Our class’s 30 draws from the mystery bag

B. Class 5D drew from a bag of 10 counters 40 times, putting the counter back each time.

1. Order the colours from most likely to least likely.   ________   ________   ________

2. Which bag do you think it was? Circle one.

3. Can you be sure? Why or why not?

C. Which bag? game

Two spinners to make

One sheet per pair (Day 2). Cut out both spinners and glue them onto thin card. Spinner A has four equal parts, one for each colour; Spinner B has sections of different sizes.

How to spin

Teacher note: Spinner B is blue for half of it, orange for a quarter, and green and yellow for an eighth each.

Which bag? Bag cards

One set per pair (Day 5). Each card shows a bag of ten counters. Cut the cards out and keep them face down.

How to play

1. Player 1 takes a bag card without showing it and puts those ten counters in the bag.
2. Player 2 draws one counter without looking, says its colour, and puts it back. Player 1 tallies it. Shake the bag.
3. After 20 draws, Player 2 looks at the tally and guesses the bag card. Then swap.

Teacher note: twenty draws is often, but not always, enough to tell these bags apart.

What we know: Repeated Experiments

Work on your own.

1. A spinner was spun 30 times. Write each frequency and the total.

   Colour	Tally	Frequency
   green
   blue
   yellow

   Total: ______
2. Why is it better to toss a coin 50 times than 5 times to decide whether it is fair?
3. Heads came up 13 times in 25 tosses. Write the relative frequency as a fraction.   ______
4. A bottle top landed top up 32 times and top down 18 times in 50 drops. Which came up more, and by how many? Are the two outcomes equally likely? Explain.
5. In 50 spins a spinner gave green 7, blue 28, yellow 15. Order the colours from most likely to least likely.   ________   ________   ________
6. Ava got 6 heads in 10 tosses. Raj got 11 heads in 20 tosses. Who got heads in a bigger share of their tosses? Show how you know.
7. Sam tossed a coin 4 times and got 4 heads. He says the coin is unfair. Do you agree? Why?
8. A bag holds 10 counters. In 30 draws, putting the counter back each time, white came up 21 times and orange came up 9 times. Which bag is it most likely to be? Circle one.
   a.b.c.

Answers and marking guide

Answers

1. green 13, blue 9, yellow 8; total 30
2. a few tosses can easily come out lopsided by chance; many tosses show a steadier pattern
3. 13/25
4. top up, 32 − 18 = 14 more; probably not equally likely: the gap is big over 50 drops
5. blue, yellow, green
6. Ava 6/10 = 12/20; Raj 11/20, so Ava
7. no: 4 tosses are far too few to judge; a fair coin gives 4 heads in a row from time to time
8. a: 21 white and 9 orange in 30 draws is the same share as 7 white and 3 orange in 10

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can say why, by judging from many trials and reading each result out of the number of trials.

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: repeating, recording frequencies, out of how many, comparing, and estimating.