Statistical investigations: a week of ready-to-teach maths

Five lessons for Year 8 Statistics. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus twelve question cards, twelve planning cards, a year group of 180 students to sample from with lines of random numbers, a report frame, a mini-check and every answer with its working.

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 two cut-out sheets, one copy per pair: the question cards (Day 1) and the planning cards (Day 2).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

A statistical investigation is a cycle: pose a question that data can answer, plan and collect the data, display and summarise it, analyse it, and report what it shows. Fair methods choose the sample so that everyone has an equal chance; ethical methods ask permission, let people say no and keep answers anonymous. A sample can only estimate the population, so a good report says what was found, how, and how sure we can be, and a finding often starts a new question.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

The data in this pack

The 180 students and their sleep are made up for this pack from fixed random draws, so every group can sample the same year and the class can check its estimates against the whole year on Day 5. A class that wants its own data can survey itself on Day 3, with consent and anonymous answers; the made-up year stays as the worked example.

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

The week runs once around the cycle: Day 1 poses the question, Day 2 plans a fair and ethical collection, Day 3 collects the data, Day 4 displays and analyses it, and Day 5 reports it with its uncertainty. It brings together the earlier Year 8 statistics units (collecting data, sampling and distributions, sample size and variation) and the median, mean and range from Year 7.

Materials for the week (one trip)

• From the classroom: pencils, calculators, scissors, a paper bag or envelope per pair, this pack printed.
• Cut-out sheet 1, one per pair: twelve question cards (Day 1).
• Cut-out sheet 2, one per pair: twelve planning cards (Day 2).

Words for the word wall

statistical investigation, statistical question, population, sample, random sample, sample size, bias, fair, ethical, consent, anonymous, data, dot plot, median, mean, range, inference, estimate, uncertainty, limitation, report

Dear families

This week in maths, Year 8 is running a statistical investigation from start to finish: asking a clear question, collecting data fairly and ethically, and reporting what the data show and how sure we can be.

Try this at home

• Think of a question about your family that data could answer, such as “How long do we each spend getting ready?”
• Find a survey result in the news. Who was asked, how many, and how were they chosen?
• Talk about why a survey should ask permission and keep names off the answers.
• Spot a news claim that sounds too sure. What would make it more honest?

Questions and data I met this week

Fill one row a day. Write a question that data could answer, who you would ask, and how you would choose them.

Printed from the free seegongsik Statistical investigations teaching pack · seegongsik.com/au/y8/statistics/AC9M8ST04/pack

Posing a question

An investigation is a cycle: question, plan and collect, display and summarise, analyse, report. It starts with a question data can answer. Pairs sort twelve question cards, mend the weak ones, and the class settles the question it will investigate.

We are learning to

• name the five stages of a statistical investigation,
• tell a statistical question from one that is not,
• rewrite a weak question so it names a group, a measurement and a time.

Success criteria

• I can say why a question is, or is not, a statistical question.
• I can turn a vague question into one that says who, what and when.

You need

Cut-out sheet 1 (the question cards) and scissors per pair; the worksheet each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the rewrites.

Watch for these ideas

• A question about one person, or with only one answer.
• A yes-or-no question that measures nothing.
• A question that does not say who is asked or when.

Answers

• Part A: a good statistical question: B, D, F, I, J; about one person or one value: C, K, L; too vague, or yes or no: A, E, G, H.
• Part B, for example: H “How many hours do Year 8 students at our school sleep on a school night?”; E “How many hours of sport did students in our year play last week?”; G “How do students at our school rate canteen lunches, from 1 to 5?”
• Part C: this pack follows “How many hours of sleep do Year 8 students at our school get on a school night?” (population: the 180 Year 8s).
• Part D: the answer to “How old am I?” is one value; the school’s ages vary from person to person, so they need data.

Posing a question

A statistical question is about a group, asks for something you can measure or count, and expects different answers from different people.

A. Sort the question cards

B. Mend three weak questions

Choose three cards from the other two piles. Rewrite each so it says who is asked, what is measured and when.

C. Our class question

Write the question the class will investigate this week.

D. Stretch: one value or many?

Why is “How old am I?” not a statistical question, while “How old are the students at our school?” is?

A fair and ethical plan

A plan says who the population is, how big the sample is, how it is chosen and exactly what is asked; an ethical plan also asks permission, lets people say no and keeps answers anonymous. Pairs sort twelve planning cards, then write the class plan.

We are learning to

• choose a population and a sample size for a question,
• choose a random method that gives everyone an equal chance,
• collect data ethically: permission, choice, no names, totals only.

Success criteria

• I can spot a step that is unfair or unethical, and fix it.
• I can write a plan another group could follow.

You need

Cut-out sheet 2 (the planning cards) and scissors per pair; the worksheet each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the plan.

Watch for these ideas

• Choosing friends or volunteers because they are easy to ask.
• Keeping names “just to check”.
• Asking the question in different words to different people.

Answers

• Part A: fair and ethical: B, C, I, J, L; unfair: D, G, H, K; unethical: A, E, F.
• Part B, for example: card G → number all 180 students and choose with random numbers; card F → record the answers with no names.
• Part C, for example: population the 180 Year 8s; 20 per group; random numbers from 1 to 180, skipping repeats; “On a school night, about how many hours do you sleep, to the nearest half hour?”; permission, anyone may skip, no names, totals only.
• Part D: only that class can be chosen, and its students share a timetable and a teacher, so they may differ from the rest of the year.

A fair and ethical plan

Fair: the way the sample is chosen and the way the question is asked do not favour any answer. Ethical: people agree to take part, may say no, and cannot be picked out from the results.

A. Sort the planning cards

B. Fix two cards

Choose one unfair card and one unethical card.

C. Our plan

D. Stretch: just our class?

Ana says, “Let’s just survey our class. It is part of Year 8.” Why is that not a random sample of the year?

Collecting the data

The plan meets the data. Each group uses its line of random numbers to choose 20 of the 180 students, records their sleep and whether their phone stays in the bedroom, and checks the sample was chosen exactly as planned.

We are learning to

• select a random sample from a numbered population,
• record data accurately on a recording sheet,
• check the collection against the plan.

Success criteria

• I can choose my 20 students and record their data without slips.
• I can explain why we skip a number we already have.

You need

The worksheet each; a different line of random numbers for each group (two groups can share a line by reading it from the right).

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the tally.

Watch for these ideas

• Dropping an answer because it looks too big or too small.
• Reading the wrong row or column of the grid.
• Changing the sample after seeing the answers.

Answers

• Part A (line 1, skipping the second 176): students 026 049 020 060 119 176 108 154 122 064 029 018 114 091 153 123 102 083 141 096; hours 7, 7.5, 6.5, 8, 7.5, 9.5, 5, 6.5, 6.5, 8, 6, 6.5, 7.5, 6.5, 7.5, 8, 9, 6, 10, 7; 10 of 20 keep their phone in the bedroom.
• Part B (line 1): 5: 1, 6: 2, 6.5: 5, 7: 2, 7.5: 4, 8: 3, 9: 1, 9.5: 1, 10: 1.
• Part C: every student came from the line, in order; nothing was swapped, dropped or added.
• Part D: check it was copied correctly; if it was, keep it. It is real data, and leaving it out would bend the result.

Collecting the data

The grid is the 180 Year 8 students at a made-up school: hours of sleep on a school night, and P if their phone stays in the bedroom overnight. Student 117 is in row 11, column 7.

Your line of random numbers: take the first 20 different students, and skip a number you already have.

A. Our sample of 20

B. Tally the hours

C. Check against the plan

Did you follow every step of the plan? Write any problem, and how you fixed it.

D. Stretch: an odd answer

One answer looks far bigger than the rest. What should you do with it, and why?

Display and analyse

A display shows the pattern, and a summary carries it into the report. Groups draw a dot plot of their 20 values, find the median, mean and range, then compare the eight groups’ medians to see how sure the class can be about the whole year.

We are learning to

• display a sample in a dot plot,
• summarise it by shape, centre (median and mean) and spread (range),
• use several samples to judge how sure an estimate is.

Success criteria

• I can draw our dot plot and find the median, mean and range.
• I can give a likely range for the whole year’s median, and say why we cannot be exact.

You need

The Day 3 data, the worksheet and a calculator each; the eight group medians on the board as groups finish.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the group medians.

Watch for these ideas

• Finding the median without putting the values in order.
• Treating one sample’s median as the whole year’s exact value.
• Claiming a cause from a difference between two groups.

Answers

• Part A (line 1): one peak around 6.5 to 7.5 hours, from 5 to 10, with a few long sleepers on the right.
• Part B: median 7.25 (the mean of the 10th and 11th values), mean 146 ÷ 20 = 7.3, range 5 hours.
• Part C: the eight medians run from 6.75 to 7.5, so the whole year’s median is probably between those; each sample is only 20 of the year.
• Part D (line 1): phone in the bedroom 10 students, median 6.75; not 10, median 7.5. A difference in one sample is not proof, and even across the year it would not show the phone is the cause.

Display and analyse

A. Our dot plot

One dot for each of your 20 students, stacked over their hours.

B. Summary

C. Eight groups, eight medians

Each group took its own random sample of 20 from the same 180 students. Their medians, in hours:

Line 1: 7.25Line 2: 6.75Line 3: 7.25Line 4: 7Line 5: 7Line 6: 7Line 7: 7Line 8: 7.5

Between which values is the median of the whole year likely to be? Why can you not be sure of it exactly?

D. Stretch: phones and sleep

Does your sample show that phones cause less sleep? Explain.

Reporting with uncertainty

A report answers the question in context, says how the data were collected, and states how sure we can be. Groups write their report, check it against the whole year, and play the quiz as a game.

We are learning to

• report an answer in the context of the question,
• describe the method, including its fairness and ethics, in a sentence or two,
• state the uncertainty and the limits of a sample.

Success criteria

• I can write a report another person could check.
• I can say how sure I am, and why.

You need

The Day 3 and Day 4 sheets and the worksheet each; the whole year’s median and mean (below) to reveal; the self-check quiz at the bottom of the unit page.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the reveal and the quiz.

Watch for these ideas

• Reporting a sample’s figure as the exact truth.
• Leaving out how the sample was chosen.
• Adding a cause the data cannot show.

Answers

• Part A (line 1), for example: “We chose 20 of the 180 Year 8s with random numbers and asked each the same written question, with permission and no names. Median 7.25 hours, mean 7.3, range 5. A typical Year 8 student sleeps about 7 hours on a school night; the eight group medians ran from 6.75 to 7.5, so the year’s median is probably between those.”
• Part B: the whole year: median 7 hours, mean 7.3; every group median was within 0.5 of it.
• Part C, for example: do students who keep their phone in the bedroom sleep less on a school night?
• Part D: “exactly” claims more than a sample can show; say “about 7.4 hours” and give the sample size and a likely range.

Reporting with uncertainty

A. Our report

One or two sentences in each row. Someone who was not in the class should be able to check every claim.

B. Check against the whole year

Your teacher will tell you the whole year’s median and mean.

Was the year’s median inside the likely range you gave on Day 4? How close did your estimate come?

C. A new question

What new question does your finding raise? It starts the cycle again.

D. Stretch: exactly?

A newspaper says, “Year 8 students sleep exactly 7.4 hours a night.” What is wrong with “exactly”? Rewrite the sentence.

Twelve question cards

For Day 1, one set per pair

Cut out the cards. Sort them: a good statistical question, a question about one person or one value, or one that is too vague or only yes or no.

Twelve planning cards

For Day 2, one set per pair

Each card is one step someone might put in a survey plan. Cut out the cards and sort them: fair and ethical, unfair, or unethical.

Unfair steps bias the sample or the answers; unethical steps do not respect the people who take part.

What we know: Statistical investigations

Work on your own. Show your working.

1. Put the stages of an investigation in order: report, display and summarise, pose a question, plan and collect, analyse and interpret. ______________________
2. Which is a statistical question, and why? (a) How tall am I? (b) How tall are the Year 8 students at our school? (c) Is height important? ______________________
3. Rewrite “Is homework too hard?” as a statistical question. ______________________
4. A plan says, “Survey the first 30 students to arrive at school.” What is the problem, and how would you fix it? ______________________
5. Name two things that keep a survey ethical. ______________________
6. Hours of homework last week for a random sample of 12 students: 3, 5, 4, 6, 2, 5, 9, 4, 5, 3, 6, 4. Find the median, mean and range. ______________________
7. Four groups’ random samples gave medians of 6, 6.5, 7, 6.5 hours of screen time on a weekend day. Give a likely range for the whole year’s median, and say why you cannot be exact. ______________________
8. A report says, “Our sample of 20 proves that every Year 8 student sleeps 7.5 hours.” Write two corrections. ______________________

Answers and marking guide

Answers with working

1. pose a question, plan and collect, display and summarise, analyse and interpret, report; a report can lead to a new question.
2. (b): it is about a group, asks for a measurement, and the answers vary from person to person.
3. For example: “How many minutes of homework do students in our year do on a school night?”
4. Early arrivals may differ (bus timetables, morning training): a convenience sample. Number every student and choose 30 with random numbers.
5. Any two: ask permission and explain what the answers are for; let people skip or stop; record no names; report only totals.
6. In order: 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 9. Median (4 + 5) ÷ 2 = 4.5; mean 56 ÷ 12 ≈ 4.7; range 7.
7. Probably between 6 and 7 hours: each sample is only part of the year, and samples vary.
8. A sample cannot prove anything about every student, and students vary: “the median in our sample of 20 was 7.5 hours, so a typical student sleeps about 7.5 hours”.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions and reports an estimate with its uncertainty.

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: posing a question, planning, collecting, display and analysis, and reporting.