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

Five lessons for Year 7 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 question cards, sample cards, 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 at 100%, one set per pair: the question cards (Day 1) and the sample cards (Day 2).
4. Find a 30 cm ruler for each pair for the ruler-catch test on Day 2.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

A statistical investigation answers a question with data. It runs as a cycle: ask a question that data can answer, plan and collect the data, display and summarise it, then interpret it and report what it shows, which often raises the next question. The question decides the variable and how to measure it, a fair sample makes the answer worth trusting, and the report gives the shape, centre and spread with numbers and says how far the finding reaches.

The class investigation: the ruler catch

On Day 2 the class collects its own data to answer one question: do we catch a falling ruler sooner with our writing hand? Partners drop a ruler for each other, five catches with each hand, and each student keeps the median of the five. The distance is a continuous variable measured to the nearest centimetre, and a shorter distance is a faster catch. The worked answers use a sample class of 24 students printed on the sheets, so every answer can be checked; a class with its own results takes the same steps with them.

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 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 starts with the question; Day 2 plans whom to ask and how to measure, then collects the class’s data; Day 3 displays and summarises it; Day 4 compares the two hands and tests claims against the data; and Day 5 writes the report. The week builds on the Year 6 investigation cycle (AC9M6ST03) and uses the displays and measures from AC9M7ST01 and AC9M7ST02. In Year 8, census, sampling, experiment and observation are compared as ways of collecting data (AC9M8ST01).

Materials for the week (one trip)

• From the classroom: pencils, rulers, calculators, this pack printed at 100%.
• A 30 cm ruler for each pair, marked in centimetres from one end (Day 2).
• Cut-out sheet 1, one set per pair: 12 question cards (Day 1).
• Cut-out sheet 2, one set per pair: 12 sample cards (Day 2).

Words for the word wall

statistical question, variable, discrete, continuous, investigation, population, sample, fair, biased, census, method, data, display, mean, median, mode, range, shape, symmetric, skewed, outlier, conclusion, limitation

Dear families

This week in maths, Year 7 is running a statistical investigation from start to finish: asking a question, collecting data fairly, summarising it, and reporting what it shows and how far the finding can be trusted.

Try this at home

• Try the ruler catch together: one person drops a ruler without warning and the other catches it. Who catches it sooner?
• Think of a question about your household that data could answer, such as how long everyone takes to get ready in the morning.
• When a news story quotes a survey, ask: who was asked, and how many people?
• Ask: would asking only your friends give a fair answer about everyone in Year 7? Why not?

Data this week

Fill one row a day with a question you could investigate, and the data you would collect to answer it.

Printed from the free seegongsik Statistical investigations teaching pack · seegongsik.com/au/y7/statistics/AC9M7ST03/pack

Questions worth asking

An investigation is only as good as its question. Students sort questions that data can answer from questions with one fixed answer, sharpen vague questions, and end with the class’s question for the week.

We are learning to

• tell a statistical question, answered by data that varies, from a question with one fixed answer,
• rewrite a vague question so it names who, what is counted or measured, and the unit,
• name a variable as discrete (counted) or continuous (measured).

Success criteria

• I can explain why a question is or is not a statistical question.
• I can rewrite a vague question so that someone could collect data to answer it.

You need

The worksheet and the question cards (one set per pair).

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Thinking any question with a number in it is statistical: “How many students are in our class today?” has one answer.
• Leaving the variable vague: “fast” or “popular” cannot be measured until the class says how.
• Calling a measurement discrete because it is rounded: a time to the nearest second is still continuous.

Answers

• Part A: not statistical B, E, I, K; discrete A, F, H, L; continuous C, D, G, J.
• Part B: answers vary, for example (1) how many minutes students in our class spend on a phone on a school day (continuous); (2) how many Year 7 students buy lunch at the canteen each day for a week (discrete); (3) how long students in our class take to run 50 m (continuous).
• Part C: the catch distance, in centimetres, for each hand; continuous, as it is measured; writing hand and other hand.
• Part D: answers vary, for example the number of training sessions each week (discrete) and the time spent training (continuous).

Questions worth asking

A. Sort the question cards

A statistical question is answered by collecting data that varies. Write each card’s letter in its pile.

B. Sharpen the question

Each question is too vague to investigate. Rewrite it so it says who, what is counted or measured, and the unit.

C. Our question for the week

Do we catch a falling ruler sooner with our writing hand than with our other hand?

D. Stretch

Write two statistical questions about sport: one about a discrete variable and one about a continuous variable.

Planning and collecting

A fair answer needs a fair sample and a method everyone follows. Students sort ways of choosing whom to ask, write a method for the ruler catch that anyone could repeat, and collect the class’s data.

We are learning to

• tell a fair sample, where everyone in the group could be chosen, from a biased one,
• write a method precise enough for someone else to repeat, and record data to the nearest centimetre.

Success criteria

• I can say whom a biased sample leaves out.
• I can follow our method and record my catches in the table.

You need

The worksheet, the sample cards (one set per pair) and a 30 cm ruler for each pair.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Thinking a bigger sample fixes a biased one: a hundred friends are still only friends.
• Changing the method partway, such as counting down before a drop, so the catches no longer compare.
• Reading the ruler at a different place each time, or rounding some catches and not others.

Answers

• Part A: fair B, E, G, J, L; biased A, C, D, F, H, I, K. For example, A: friends tend to be alike, and everyone else is left out; B: every Year 7 student has the same chance.
• Part B: for example, the catcher rests a forearm on the desk with thumb and finger open level with the 0 cm mark; the dropper holds the top of the ruler and lets go without warning; the catcher reads the mark at the top of the thumb to the nearest centimetre; five catches with each hand, and a ruler that falls through is dropped again.
• Part C: results vary. The median of five catches is the third once they are in order.
• Part D: catches can improve with practice, or slip as a hand tires, so taking turns stops one hand always getting that effect.

Planning and collecting

A. Fair or biased?

A fair sample gives everyone in the group a chance of being chosen. Write each sample card’s letter in its pile.

B. Our method

Write the steps so that anyone could repeat our ruler catch and get data that compares with ours.

C. Collect

Record each catch to the nearest centimetre. Put the five in order; the median is the third.

D. Stretch

Half the pairs start with the writing hand and half with the other hand. Why does that make the comparison fairer?

Display and summarise

A display and four summary statistics turn a list of results into something a question can be answered from. Students draw a dot plot of the sample class’s catches, find the four measures, do the same for a discrete data set, then summarise their own class.

We are learning to

• display discrete and continuous data in a dot plot with an even scale,
• find the mean, median, mode and range, including from a frequency table.

Success criteria

• I can draw a dot plot that shows every result.
• I can find all four measures and say what each one tells us.

You need

The worksheet, calculators, and the class list of medians from Day 2.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Finding the median from the list as recorded instead of in order.
• Dividing the total number of books by the number of columns in the table instead of the number of students.
• Leaving off the axis the values nobody scored, which squashes the scale and hides gaps.

Answers

• Part A: mean 361 ÷ 24 ≈ 15.0 cm, median (14 + 15) ÷ 2 = 14.5 cm, modes 14 and 15 cm, range 16 cm; skewed right, with one slow catch.
• Part B: 25 students, 62 books; mean 62 ÷ 25 = 2.48, median 2, mode 2, range 8; skewed right; discrete, as books are counted in whole numbers.
• Part C: results vary.
• Part D: without the 8, mean 54 ÷ 24 = 2.25 and median 2; the mean falls and the median stays.

Display and summarise

A. The sample class

Writing-hand catches (cm) of the 24 students in a sample class, one median each: 14, 11, 16, 13, 18, 12, 15, 14, 21, 13, 17, 12, 15, 16, 10, 14, 19, 13, 15, 26, 12, 14, 16, 15

B. A discrete variable

Books read last term by the students in another class.

C. Our class

Use the class list of medians from Day 2.

D. Stretch

In Part B, one student read 8 books. Find the mean and the median without that student. Which measure changes, and why?

Analyse and compare

Comparing two groups means comparing whole distributions, not single results. Students compare the two hands by shape, centre and spread, check the comparison student by student, and judge which claims the data supports.

We are learning to

• compare two distributions by their shape, a measure of centre and the range,
• decide whether a claim is supported, not supported, or cannot be judged from the data.

Success criteria

• I can write a comparison with a number for the centre and a number for the spread.
• I can explain when a claim goes further than the data.

You need

The worksheet, calculators, and the Day 3 worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Comparing only the fastest catch with each hand instead of the whole distribution.
• Reading a shorter catch as a worse result: here the smaller number is the faster catch.
• Stretching a finding about one class to every Year 7 student.

Answers

• Part A: writing hand skewed right, median 14.5 cm, mean 361 ÷ 24 ≈ 15.0 cm, range 16 cm; other hand skewed right, median 16.5 cm, mean 412 ÷ 24 ≈ 17.2 cm, range 15 cm. In this class the writing hand usually caught about 2 cm sooner, with a similar spread.
• Part B: 20 sooner with the writing hand, 1 the same, 3 sooner with the other hand.
• Part C: (1) not supported: 3 were sooner with the other hand; (2) supported: the median is 14.5 cm; (3) supported: medians 14.5 and 16.5 cm; (4) can’t tell: one class only; (5) not supported: ranges 16 and 15 cm; (6) can’t tell: handedness was not recorded.
• Part D: medians 14 cm and 16 cm, still 2 cm apart, so the conclusion stands.

Analyse and compare

A. Two hands

The sample class’s catches, on the same scale. A shorter catch is a faster one.

B. Student by student

C. Supported, not supported, or can’t tell?

D. Stretch

Student 20 was distracted by a noise. Leave that student out and find both medians again. Does the conclusion change?

Report the findings

A report gives the question, method, display, summary statistics and shape, a conclusion and its limits. Students order a report, write their own conclusion and critique a weak one. Then the quiz.

We are learning to

• report findings in terms of shape and summary statistics, with the limits of the data,
• critique the sample, method and claims of a report.

Success criteria

• I can write a conclusion that answers the question with numbers.
• I can name a limitation of an investigation and say how to fix it.

You need

The worksheet, the class’s Day 3 results, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• A conclusion with no numbers, or numbers with no conclusion.
• Leaving out the limits: one class does not speak for every Year 7 student.
• Reporting the mean alone for skewed data.

Answers with working

1. B, D, E, A, F, C.
2. Answers vary: the shape, a median and a range for each hand, and which hand caught sooner, as in sentence F.
3. For example: one class of 24; catches read to the nearest centimetre; fix: test more classes, chosen at random.
4. Any two: a tiny sample; all basketballers, so biased; the claim covers all of Year 7; no spread is given.
5. Skewed right: most have 0 to 2 pets; median 1 (mean 32 ÷ 20 = 1.6); range 7, from one student.
6. Answers vary, for example: do our catches get faster after practice?
7. Plans vary: check for the question, a fair sample, the variable and its unit, the method, and a display.
8. The mean, 361 ÷ 24 ≈ 15.0 cm; the median, 14.5 cm, is more typical: the 26 cm catch pulls the mean up.

Report the findings

Problems (7 and 8 are the stretch)

1. Put the six sentences of this report in order. Write the letters.
   A. The writing hand had a median of 14.5 cm and a range of 16 cm; the other hand had a median of 16.5 cm and a range of 15 cm. Both were skewed right, with one slow student.

   B. We asked: do Year 7 students catch a falling ruler sooner with their writing hand than with their other hand?

   C. We tested only one class, so next we could test other classes and see whether practice changes the result.

   D. Each of the 24 students in one Year 7 class caught a dropped ruler five times with each hand and kept the median of the five, to the nearest centimetre.

   E. We drew a dot plot for each hand on the same scale.

   F. In this class the writing hand usually caught the ruler about 2 cm sooner, so it reacted faster.

   1.

   2.

   3.

   4.

   5.

   6.

2. Write the conclusion for your own class’s results, with the shape, a median and a range for each hand.
3. Give two limitations of the investigation in problem 1, and one way to fix one of them.
4. “We asked the five players in our basketball team how many hours they train each week. The mean was 6 hours, so Year 7 students train for 6 hours a week.” Give two problems with this report.
5. The number of pets of the 20 students in another class. Describe the distribution in one sentence: shape, centre and spread.
6. Write a statistical question the class could investigate next, following on from the ruler catch.   ______________________
7. Stretch. Plan an investigation of a continuous variable: the question, whom you would ask and how you would choose them, what you would measure and in what unit, and the display you would draw.
8. Stretch. A report on the class in problem 1 says “the average writing-hand catch was 15.0 cm”. Which average is that? Why might the median describe a typical catch better here?

Question cards

Day 1, one set per pair. Cut out the 12 cards and sort them into three piles: not a statistical question, a statistical question about a discrete variable, and a statistical question about a continuous variable.

Sample cards

Day 2, one set per pair. The question is: how long do Year 7 students at our school sleep on a school night? Each card is a way of choosing whom to ask. Cut out the 12 cards and sort them into fair samples and biased samples.

What we know: Statistical investigations

Work on your own. Show your working.

1. Which is a statistical question? Name its variable and say whether it is discrete or continuous.
   (a) How old is our principal?
   (b) How long do Year 7 students spend on homework each night?
   (c) How many days are in March?   ______________________________
2. Discrete or continuous?
   (a) the number of push-ups a student does in a minute   ________
   (b) the length of a standing long jump   ________
3. To find out how much pocket money Year 7 students get, a student asks the Year 7 students in the canteen queue. Why is this sample biased? Describe a fairer one.
4. Long jumps (m) of 7 students: 2.4, 3.1, 2.8, 2.4, 3.6, 2.9, 2.5. Find the mean, median, mode and range.   ______________________

5. Times to run 1 km for 18 students, to the nearest minute. Describe the shape. Without calculating, which is larger, the mean or the median? Explain.   ______________
6. Class 7C’s ball throws have a median of 18 m and a range of 12 m; Class 7D’s have a median of 22 m and a range of 25 m. Write two sentences comparing them.   ______________
7. A survey of 40 students at one school found a median of 8.5 hours of sleep on a school night, with a range of 3 hours. Supported, not supported, or can’t tell?
   (a) Half the students surveyed slept 8.5 hours or more.   ________
   (b) Every student surveyed slept 8.5 hours.   ________
   (c) Students at every school in Australia sleep 8.5 hours.   ________
8. An investigation asked 10 students from one class how long they spend on screens each day. Give one limitation and a way to fix it.   ______________

Answers and marking guide

Answers with working

1. (b): its answer comes from data that varies; the variable is the time spent on homework, which is continuous.
2. (a) discrete: push-ups are counted; (b) continuous: a jump is measured.
3. Biased: it asks only students who buy at the canteen, who may get more money; fairer: draw names at random from the Year 7 roll.
4. Mean 19.7 ÷ 7 ≈ 2.81 m, median 2.8 m (the 4th of 7), mode 2.4 m, range 3.6 − 2.4 = 1.2 m.
5. Skewed right; the mean (126 ÷ 18 = 7) is larger than the median (6.5), pulled up by the slow runners in the tail.
6. For example: 7D threw further in the middle, by 4 m; 7C was more consistent, with a range of 12 m against 25 m.
7. (a) supported: that is what the median means; (b) not supported: the range is 3 hours; (c) can’t tell: one school only.
8. For example: 10 students from one class is a small sample and may not be typical; ask more students, chosen at random from every class.

A quick three-level guide

Eight questions, four ideas. A student at standard poses a statistical question, plans a fair sample, summarises data with the four measures and its shape, and reports a comparison with numbers and limits.

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: questions, planning and collecting, summarising, comparing, and reporting.