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

Five lessons for Year 10 Statistics in which the class runs one investigation from question to report: is sleep related to reaction time? 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 to sort, twelve limitation cards to match, a backup data set, 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. Cut the 12 question cards before Day 1 and the 12 limitation cards before Day 5, and keep each set in an envelope.
4. For Day 3, find one 30 cm ruler for each pair. Ask students on Day 2 to note how long they sleep that night.
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 moves through four stages: pose a question that data can answer, collect the data, analyse it with displays and summaries, and interpret what it shows. With bivariate data, two variables are recorded for each person, and the type of each variable decides the display: a scatterplot for two numerical variables, a two-way table for two categorical ones, and parallel boxplots for a numerical variable compared across groups. A good report answers the question and says honestly what the data cannot show: who it is about, how sure we can be, and whether one thing caused another.

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. If the class’s own data is thin or lost, the Day 4 sheet carries a backup data set of 24 students.

How we write these

In the ruler-drop test, one student holds a 30 cm ruler with its zero mark between the other student’s open thumb and finger, and lets go without warning; the catch distance is read at the top of the thumb. A shorter distance means a quicker reaction. Sleep is last night’s, to the nearest half hour. Quartiles are the medians of the lower and upper halves.

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 poses the question; Day 2 plans how to answer it; Day 3 collects and displays the data; Day 4 analyses it; and Day 5 interprets it and reports with its limits. It brings together the scatterplots and two-way tables of this year with the sampling and bias of Year 9, and it prepares the investigations of senior mathematics.

Materials for the week (one trip)

• From the classroom: one 30 cm ruler per pair, rulers for drawing, calculators, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 question cards (Day 1).
• Cut-out sheet 2, one per pair: 12 limitation cards (Day 5).
• If you have them: computers with a spreadsheet to enter the class data on Day 3. No other equipment to buy.

Words for the word wall

investigation, statistical question, bivariate data, numerical variable, categorical variable, population, sample, random sample, bias, measurement error, scatterplot, two-way table, parallel boxplots, association, line of best fit, gradient, conclusion, limitation, extrapolation, cause

Dear families

This week in maths, Year 10 is running a statistical investigation from start to finish: asking whether sleep is related to reaction time, planning how to measure both, collecting class data with a ruler-drop test, analysing it, and writing a report that is honest about what the data can and cannot show.

Try this at home

• Try the ruler-drop test with someone at home: hold a ruler above their open thumb and finger, drop it without warning, and read where they catch it.
• Ask: a survey of 100 people online says most people love a product. Who answered, and who did not?
• Ask: two things rise together. Does that mean one causes the other? What else could link them?

My sleep week

Each morning, write how long you slept to the nearest half hour, and how alert you felt from 1 (not at all) to 5 (very).

Asking a statistical question

An investigation starts with a question that data can answer: it names the two variables, how each is measured, and the group it is about. Students sharpen vague questions, sort variables into numerical and categorical, and sort twelve questions by the display each one needs.

We are learning to

• turn a vague question into a statistical question about two variables and a group,
• tell numerical variables from categorical ones,
• choose the display a question needs from the types of its two variables.

Success criteria

• I can rewrite “Is sleep good for you?” as a question data could answer.
• I can sort the twelve question cards and say why each needs its display.

You need

The question cards (cut-out sheet 1), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Questions with no variable that can be measured, such as “good” or “bad”.
• Calling a variable numerical because its categories are numbers.
• A scatterplot for every question, whatever the variables.

Answers

• Part A: for example: 2 Is there a relationship between height (cm) and standing long-jump distance (cm) for Year 10 students at our school? 3 Is there a relationship between minutes on a phone after 9 pm and the mark on the next day’s quiz for students in our year?
• Part B: numerical: hours of sleep last night, ruler-drop catch distance (cm), minutes of homework a night; categorical: year level (9 or 10), dominant hand (left or right), plays a team sport (yes or no).
• Part C: the card key is with the mini-check answers.
• Part D: for example: scatterplot, is hand span related to the number of piano keys a student can reach? two-way table, is owning a pet related to walking to school? parallel boxplots, do students who play sport sleep longer?

Asking a statistical question

A. Sharpen the question

Rewrite each question so data could answer it: name two variables, how each is measured, and who the question is about. The first is done for you.

B. Numerical or categorical?

C. Sort the cards by display

Two numerical variables: a scatterplot. Two categorical variables: a two-way table. A numerical variable compared across groups: parallel boxplots.

D. Stretch: your own questions

Planning the collection

A conclusion is only as good as the data behind it. Students judge three plans for the same investigation, list what else could change a catch distance and how to hold it steady, and write the plan the class will follow on Day 3.

We are learning to

• choose a sample that represents the population the question is about,
• plan a measurement that is repeated and done the same way every time,
• spot the things besides the explanatory variable that could change the results.

Success criteria

• I can say which of the three plans is sound and what is wrong with the other two.
• I can list three things that could change a catch distance and how we will control each.

You need

The worksheet; a 30 cm ruler to show the test.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Thinking a bigger sample fixes a biased one: an online poll of thousands still hears only those who choose to answer.
• Measuring once and trusting it.
• Forgetting consent, or recording names when numbers would do.

Answers

• Part A: plan 1 is not: six friends are few and not like the whole year; one catch is unreliable; “usually” is a guess; plan 2 is sound: a clear question, a random sample of a fair size, and repeated measurements; plan 3 is not: a vague question, a self-selected sample, and a question that leads the answer.
• Part B: for example: which hand (test the same hand, or both and record each); practice (give one practice drop first); warning (drop at a random moment); where the hand starts (zero mark level with the top of the thumb); tiredness at that time of day (test everyone in the same lesson).
• Part C: the class plan, which should read like plan 2.
• Part D: for example: number the 180 students 1 to 180 and use a random number generator to choose 30 different numbers.

Planning the collection

A. Judge the plans

Three plans for an investigation into sleep and reaction time. Which is sound? What is wrong with the others?

B. What else could change a catch?

C. Our class plan

D. Stretch: choose at random

Collecting and displaying

Today the class collects its data. Each student takes three catches with each hand and records the medians, the class pools its results with last night’s sleep, and every student draws the scatterplot and a comparison of the two hands.

We are learning to

• carry out a measurement the same way every time and record it,
• pool class data in one table,
• draw the display each question needs: a scatterplot, and boxplots on one scale.

Success criteria

• I can record three catches with each hand and find my two medians.
• I can draw the class scatterplot and the two boxplots on the same scale.

You need

One 30 cm ruler per pair; the worksheet; a class table on the board or in a spreadsheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Plotting all three catches instead of the median.
• Reading the ruler at the bottom of the hand instead of the top of the thumb.
• Drawing the two boxplots on different scales.

Answers

• Part A: each student’s own results; the median is the middle of the three catches.
• Part B: classes vary; most see a weak to moderate negative association: students who slept more tend to catch sooner. If the class data is thin, use the backup data on the Day 4 sheet.
• Part C: the other hand usually catches a little further down the ruler; in the backup data the medians are 13 and 16 cm and the boxes overlap.
• Part D: for example 13 cm = 0.13 m, so t = √(2 × 0.13 ÷ 9.8) ≈ 0.16 s; 16 cm gives about 0.18 s.

Collecting and displaying

A. My catches

Catch distance in cm, read at the top of the thumb. Hours of sleep last night, to the nearest half hour: ________

B. The class scatterplot

Plot one point for each student in the class table: hours of sleep across, dominant-hand median up the side.

C. The two hands

Find the five-number summary of the class’s dominant-hand medians and of its other-hand medians. Draw both boxplots on this scale.

D. Stretch: distance to time

Analysing

Analysis turns a display into words and numbers. On the backup data of 24 students, and then on their own class data, students describe the association, fit a line by eye and read its gradient, and compare the two hands with five-number summaries.

We are learning to

• describe an association by its direction, strength and form, in context,
• fit a line by eye and say what its gradient means,
• compare two groups by their medians, spreads and overlap.

Success criteria

• I can describe the sleep data in full and predict a catch distance for 8 hours of sleep.
• I can say how much further the other hand catches, and how sure I am that the hands differ.

You need

Rulers; calculators; the worksheet; the class data from Day 3.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Reading a negative gradient as “worse”: here a smaller distance is a quicker catch.
• Calling the association strong because the line is steep.
• Deciding the hands differ from the medians alone, ignoring the overlap.

Answers

• Part A: a line close to distance = 24.5 − 1.4 × sleep; the gradient, about −1.4 cm per hour, says each extra hour of sleep goes with a catch about 1.4 cm sooner.
• Part B: a moderate negative linear association, no clear outliers: students who slept more tended to catch sooner. At 8 hours the line gives about 13.3 cm (accept 12 to 14).
• Part C: medians 13 and 16 cm, IQRs 4 and 4.5 cm; the other hand catches about 3 cm further in the middle, but the boxes overlap a lot, so the difference is modest and another sample of 24 could show less.
• Part D: classes differ in size, in who was tested and in how carefully the test was run; a different sample gives a different pattern.

Analysing

The backup data: 24 Year 10 students. Sleep last night (hours) and the median catch distance (cm) for each hand.

A. A line by eye

B. Describe it

C. The two hands

Five-number summaries (cm): dominant 8, 12, 13, 16, 18; other 8, 13.5, 16, 18, 21. Draw both boxplots, then compare.

D. Stretch: your class

Interpreting and reporting

A report answers the question and is honest about its limits: who the data is about, how it was measured, and what it cannot show. Students find the flaws in a report, match flawed claims to their limitations, write their own conclusion, and play the self-check quiz.

We are learning to

• write a conclusion that answers the question for the group the sample represents,
• name the limitations of a study: sample, measurement, cause and extrapolation,
• tell a claim the data supports from one it does not.

Success criteria

• I can find three flaws in the lunch-table report and say how to fix each.
• I can write a conclusion for our investigation with at least two limitations.

You need

The limitation cards (cut-out sheet 2), one set per pair; the worksheet; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; start the next with Part C.

Watch for these ideas

• Conclusions about everyone from a sample of a few friends.
• “Proves” and “makes” where the data shows only an association.
• Limitations so general they say nothing, such as “we could have done better”.

Answers

• Part A: seven friends are too few and not chosen at random, so the result cannot be about all teenagers; “usually” sleep and one catch are unreliable measurements; “proves … makes” claims cause from an association; 12 hours is outside the data, so the 8 cm prediction cannot be trusted.
• Part B: the card key is with the mini-check answers.
• Part C: for example, for the backup data: “In this sample of 24 Year 10 students there was a moderate negative association between last night’s sleep and catch distance. The sample came from one school, and the association does not show that sleep causes quicker reactions.”
• Part D: an experiment would choose at random who sleeps longer or shorter, but cutting teenagers’ sleep could harm them, so it needs adult volunteers, small changes and consent, or a study that measures other variables instead.

Interpreting and reporting

A. Find the flaws

Another class wrote this report. Underline three flaws, then explain each one and how to fix it.

B. Match the cards

Match each flawed sentence to the limitation it shows. Write the letters.

C. Our conclusion

D. Stretch: testing for cause

Which display?

For Day 1, one set per pair. Sort the questions into three piles by the display each needs: a scatterplot, a two-way table or parallel boxplots.

Teacher note: the key is with the mini-check answers.

Flaws and limitations

For Day 5, one set per pair. Six cards are sentences from reports; six name a limitation. Match each sentence to the limitation it shows.

Teacher note: the key is with the mini-check answers.

What we know: Bivariate investigations

Work on your own. Answer in full sentences.

1. Rewrite as a statistical question: “Is exercise good for sleep?” ______________________
2. Name the two variables and say whether each is numerical or categorical: “Is there a relationship between minutes of exercise a day and hours of sleep?” ______________________
3. The question “Do students who walk to school sleep longer than students who are driven?” needs, circle one:   parallel boxplots  ·  a scatterplot  ·  a two-way table
4. A question is about all Year 10 students at a school, but the data comes from one maths class. Give one problem and a better sample. ______________________
5. In a study of 30 students, sleep and catch distance had a moderate negative association. Write the conclusion a careful investigator would write. ______________________
6. Why should the line of best fit from that study not be used to predict the catch distance for 14 hours of sleep? ______________________
7. Name a third variable that could link sleep and reaction time without one causing the other. ______________________
8. One catch distance was recorded as 140 cm on a 30 cm ruler. What should the investigator do? ______________________

Answers and marking guide

Answers

1. For example: is there a relationship between minutes of exercise a day and hours of sleep for Year 10 students at our school?
2. Minutes of exercise a day and hours of sleep: both numerical.
3. The first: parallel boxplots (sleep is numerical, how students get to school is categorical).
4. One class may not be like the whole year (one teacher, one timetable, one ability group); choose students at random from the whole year list.
5. For example: in this sample of 30 students, those who slept more tended to catch sooner, a moderate association; it may not hold for other students, and it does not show that sleep causes quicker reactions.
6. 14 hours is far outside the data: we do not know that the pattern carries on there, and very long sleep need not mean ever quicker catches.
7. For example: general health, stress, screen time before bed, or caffeine.
8. Check it: 140 cm cannot happen on a 30 cm ruler, so it is likely a recording error (perhaps 14.0 cm). Correct it from the record or leave it out, and say so in the report.

A quick three-level guide

Eight questions, four ideas. A student at standard plans a fair study and reports what the data shows for its sample, and no more.

Card keys

Question cards (Day 1): scatterplot: B, D, F, I, L; two-way table: A, E, H, K; parallel boxplots: C, G, J. Limitation cards (Day 5): S with M; P with V; Q with X; N with T; W with R; U with O.

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: the question, the plan, collecting, analysing and reporting.