Scatterplots and correlation: a week of ready-to-teach maths

Five lessons for Year 10 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 scatterplot cards to sort by direction, twelve cards that match plots to descriptions, 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 direction cards before Day 2 and the 12 description cards before Day 4, and keep each set in an envelope.
4. Rulers are used every day and calculators on Days 3 and 5. A clear ruler helps on Day 3, where students see points on both sides of a line.
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

When two numerical variables are measured on the same individuals, such as the arm span and the height of each student, a scatterplot shows each individual as one point. The pattern of points shows the association between the variables: its direction (do the points rise or fall?), its strength (how closely do they follow the trend?) and its form (is the trend straight or curved?). A straight trend can be summarised by a line of best fit, which predicts well inside the data and poorly far outside it. And however strong an association is, it does not by itself show that one variable causes the other.

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.

How we write these

The variable that may explain the other goes on the horizontal axis and is called the explanatory variable; the other is the response. Strength is judged by eye as strong, moderate or weak. The correlation coefficient r comes in senior mathematics, but every card in this pack was checked with it (strong 0.75 or more either way, moderate 0.5 to 0.75, weak 0.25 to 0.5). Lines of best fit are drawn by eye; the answers give the least-squares line, which a careful line by eye comes close to, and accept a range.

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 builds the scatterplot; Day 2 reads its direction; Day 3 summarises a straight trend with a line; Day 4 judges strength and form; and Day 5 separates association from cause. It builds on plotting points on the Cartesian plane and on the gradients of linear graphs from earlier years, and on Year 9, where students compared data sets by centre, spread and shape. It prepares the bivariate data analysis of senior mathematics, where the correlation coefficient and the least-squares line are calculated.

Materials for the week (one trip)

• From the classroom: rulers (a clear one helps on Day 3), sharp pencils, calculators, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 direction cards (Day 2).
• Cut-out sheet 2, one per pair: 12 description cards (Day 4).
• If you have one: a tape measure, so the class can measure its own arm spans and heights on Day 1. No other equipment to buy.

Words for the word wall

bivariate data, explanatory variable, response variable, scatterplot, association, positive, negative, no association, strong, moderate, weak, linear, non-linear, outlier, line of best fit, gradient, interpolation, extrapolation, correlation, causation, third variable

Dear families

This week in maths, Year 10 is learning to plot two measurements of the same things against each other, such as arm span against height, and to describe the pattern: which way it goes, how tight it is and whether it is straight or curved. We also learn why two things that rise together do not prove that one causes the other.

Try this at home

• Measure arm spans (fingertip to fingertip, arms out wide) and heights at home. Are they close for each person?
• Ask: on hot days more ice creams are sold and more people swim. Does eating ice cream make people swim? (No: the hot weather drives both.)
• Ask: older cars tend to be cheaper. Is every older car cheaper than every newer one? (No: “tends to” describes a trend, not a rule.)
• Ask: can you think of two things that rise together without one causing the other?

My two measurements week

Each day, write the day’s highest temperature and how many glasses of water you drank. On Friday, plot your five points: is there a pattern?

Building a scatterplot

A scatterplot shows two numerical variables measured on the same individuals, one point for each individual. Students decide which variable goes on each axis, plot twelve students’ arm spans and heights on scaled axes, and describe the pattern.

We are learning to

• decide which variable may explain the other (horizontal axis) and which responds (vertical axis),
• choose scales and plot paired data accurately, with a zigzag where an axis does not start at zero,
• describe a pattern in a sentence that names both variables.

Success criteria

• I can plot all twelve points and check three of them with a partner.
• I can say which variable goes on the horizontal axis, and why.

You need

Rulers; sharp pencils; the worksheet; a tape measure if the class wants to measure its own arm spans.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Swapping the two values of a pair, or plotting one variable on both axes.
• Uneven scales, or starting at 0 and squeezing every point into a corner.
• Joining the dots: a scatterplot has no line from point to point.

Answers

• Part A: twelve points rising from Isla at the bottom left to Jai at the top right, close to a straight line.
• Part B: as arm span increases, height tends to increase. Not for every pair: Ben has a longer arm span than Hamish but is shorter; Kiara has a longer arm span than Aroha but is shorter; Liam has a longer arm span than Dev but is shorter.
• Part C (horizontal, then vertical): (1) hours of sunshine in a day, ice creams sold that day; (2) age of a car, value of a car; (3) people at a barbecue, sausages eaten; (4) minutes spent practising free throws, free throws made out of 20; (5) daily maximum temperature, electricity used for cooling; (6) kilometres driven since filling up, fuel left in the tank. The variable that may explain the other goes across.
• Part D: 7 points sit above the line (taller than their arm span: Aroha, Chloe, Dev, Grace, Hamish, Isla, Jai) and 5 below; none is more than 3 cm from it, so Leonardo’s rule holds closely for this class.

Building a scatterplot

A. Plot the pairs

Arm span (fingertip to fingertip, arms out wide) and height, in cm, of twelve Year 10 students, in the order they were measured. Aroha’s point (164, 167) is plotted for you.

B. Describe the pattern

C. Which axis?

D. Stretch: Leonardo’s rule

Direction

The direction of an association says whether one variable tends to rise or fall as the other rises. Students describe directions in context, predict directions before seeing any data, and sort twelve scatterplots into positive, negative and no association.

We are learning to

• describe an association as positive, negative or none,
• write the direction in context: as one variable increases, the other tends to increase or decrease,
• see that direction does not depend on how tight or how straight the pattern is.

Success criteria

• I can sort the twelve cards and justify each pile.
• I can predict a direction from the variables alone and give a reason.

You need

The direction 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

• Reading “negative” as “bad” or as “no association”.
• Calling a loose or curved pattern “no association”.
• Describing direction without naming the two variables.

Answers

• Part A: (a) negative: as a car’s age increases, its price tends to decrease. (b) positive: as the maximum temperature increases, the number of swimmers tends to increase. (c) none: height tells us nothing about the spelling mark.
• Part B: 1 negative (colder days need more heating); 2 positive (pay goes up with each hour); 3 none (foot size has nothing to do with maths); 4 negative (air cools as you climb); 5 negative (more training, faster runs, smaller times); 6 negative (driving uses fuel).
• Part C: the card key is with the mini-check answers.
• Part D: for example a falling curve that levels off, such as the temperature of a cooling drink against time, or the value of a phone against its age: negative, because one falls as the other rises, though not in a straight line.

Direction

A. Read the direction

B. Predict before the data

C. Sort the cards

D. Stretch: negative but curved

The line of best fit

A line of best fit sums up a straight trend in one line. Students draw one by eye through travel data, read its gradient in context, predict inside and outside the data, and see why a straight line fails on a curve.

We are learning to

• draw a line of best fit by eye that follows the trend, with about as many points above it as below,
• find the gradient of the line and say what it means in context,
• tell a prediction inside the data (interpolation) from one outside it (extrapolation).

Success criteria

• I can draw a line through the trip data and use it to predict the time for 8 km.
• I can explain why the line’s prediction for 40 km should not be trusted.

You need

Rulers (a clear one shows the points on both sides); calculators; the worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Forcing the line through the origin, or through the first and last points.
• A gradient with no units: here it is minutes per kilometre.
• Trusting a prediction far outside the data.

Answers

• Part A: a line close to time = 5.9 + 2.4 × distance; at 8 km about 25 minutes (accept 23 to 28: lines by eye differ).
• Part B: about 2.4 minutes per km: each extra km adds about 2.4 minutes; the 6 minutes at 0 km is walking and waiting. At 40 km it gives about 103 minutes, far beyond the data (up to 14 km): do not trust it.
• Part C: the tea cools fast at first and then slowly, so the points curve: above the line at both ends and below it in the middle. The line predicts −9 °C after 30 minutes, below freezing, but the tea cannot cool below the 22 °C room.
• Part D: 30 = 5.9 + 2.4 × distance gives a distance of about 9.9 km (accept answers from their own line, near 9 to 11 km).

The line of best fit

A. Draw a line by eye

Distance from home to school and the usual bus trip, for twelve students.

B. Read the line

C. When a straight line fails

A cup of tea cooling in a room at 22 °C, measured every 2 minutes. The gold line is its straight line of best fit.

D. Stretch: work backwards

Strength and form

Strength says how closely the points follow the trend; form says whether the trend is straight (linear) or curved (non-linear). Students rank plots by strength, see one unusual point weaken an association, and match six plots to three-word descriptions.

We are learning to

• describe strength as strong, moderate or weak by how tightly the points follow the trend,
• describe form as linear or non-linear,
• say how an outlier changes an association, and look into it before removing it.

Success criteria

• I can describe a scatterplot in three words: strength, direction and form.
• I can describe the sleep data with and without its unusual point.

You need

The description cards (cut-out sheet 2), 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

• Taking a steep trend for a strong one: strength is closeness to the trend, not slope.
• Calling a tight curve weak: it is strong and non-linear.
• Deleting an outlier because it spoils the pattern.

Answers

• Part A: strongest to weakest R, P, Q (strong, moderate, weak); all three are positive and linear.
• Part B: with every point, a weak negative association; without the point (9 hours, 0.42 s), a strong negative linear one: more sleep goes with quicker reactions. The student may have been unwell or distracted, or the test may have been mistimed: check, and report the data with and without the point rather than delete it quietly.
• Part C: the card key is with the mini-check answers.
• Part D: strong and non-linear: points hugging a curve; weak and linear: a loose band of points around a straight trend.

Strength and form

A. Rank by strength

B. One unusual point

Hours of sleep the night before and reaction time in an online test, for 12 students.

C. Match the cards

Match each scatterplot card to its description card. Write the letters.

D. Stretch: sketch two

Association is not cause

An association does not show that one variable causes the other: a third variable can drive both, or the cause can run the other way. Students find third variables, describe a scatterplot in full, judge four claims of cause and play the self-check quiz.

We are learning to

• explain why an association alone cannot show cause,
• name a third variable that could create an association, or a cause that runs the other way,
• write a full description: direction, strength, form, unusual points and context.

Success criteria

• I can name the third variable behind ice-cream sales and beach rescues.
• I can describe the phone and sleep data and say what it does not show.

You need

Calculators; 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

• “They are correlated, so one causes the other”, or the opposite slip: “no cause, so no link”.
• Forgetting that the cause could run the other way.
• Descriptions with no context: name the variables and their units.

Answers

• Part A (the first is done on the sheet): 2 age; 3 the size of the fire; 4 the number of people in the town.
• Part B: for example: a moderate negative linear association: students with more phone time after 9 pm tended to sleep less, about 0.7 hours less per extra hour, with no point far from the pattern. It cannot show cause: late sleepers simply have more phone time, and homework or worry could affect both.
• Part C: 1 no (breakfast may go with more sleep or help at home); 2 yes (a description, with no cause claimed); 3 no (bigger suburbs have more of both); 4 yes (chance chose the seedlings, so it can show cause).
• Part D: for example, phones away at 9 pm on nights chosen at random and used as usual on the others; compare the sleep. Chance evens out other differences.

Association is not cause

A. Find the third variable

B. Describe it in full

Minutes on a phone after 9 pm and hours of sleep that night, for 12 students.

C. Does the claim hold?

D. Stretch: test for cause

Which direction?

For Day 2, one set per pair. Sort the scatterplots into three piles: positive, negative and no association. Some patterns are tight and some are loose; some are straight and some curve.

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

Plots and descriptions

For Day 4, one set per pair. Six cards show a scatterplot; six give a description. Match each plot to its description.

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

What we know: Scatterplots and correlation

Work on your own. A calculator is allowed. Show your working.

1. A coach records each sprinter’s hours of training a week and time for 100 m. Which variable belongs on the horizontal axis, and why? ______________________
2. Describe the association in the scatterplot above: direction, strength and form, in context. ______________________
3. A line of best fit drawn by eye passes through (2, 15.3) and (10, 13.3). Find its gradient and say what it means. ______________________
4. Use the line to predict the time for 6 hours of training a week. Would you use it for 30 hours? Why? ______________________
5. Points that lie close to a curve that rises and then levels off show an association that is, circle one:   weak and linear  ·  no association  ·  strong and non-linear
6. People who own more books tend to have higher reading scores. Does owning books cause higher scores? Give another explanation. ______________________
7. Explain the difference between a negative association and a weak association. ______________________
8. An outlier sits far below a strong positive pattern. What does it do to the strength, and what should you do about it? ______________________

Answers and marking guide

Answers with working

1. Hours of training: it may explain the time, so it is the explanatory variable; the time responds.
2. A strong negative linear association: sprinters who train more tend to run faster (smaller times), and the points lie close to a straight line.
3. (13.3 − 15.3) ÷ (10 − 2) = −0.25: each extra hour of training a week goes with a time about 0.25 s faster.
4. 15.3 − 0.25 × (6 − 2) = 14.3 s. No: 30 hours is far outside the data, and the line gives 8.3 s, faster than the world record of 9.58 s.
5. The third: strong and non-linear.
6. Not shown: people who like reading may both buy more books and read more (cause the other way round), or family background could affect both.
7. Negative is the direction (one variable falls as the other rises); weak is the strength (the points are loosely scattered). A strong negative association exists.
8. It weakens the association. Find out why it happened (an error, or a real unusual case) before deciding, and report the data with and without it.

A quick three-level guide

Eight questions, four ideas. A student at standard names the direction, the strength and the form, in context, and never reads cause into an association.

Card keys

Direction cards (Day 2): positive: B, D, E, I, K; negative: A, F, G, J; no association: C, H, L. Description cards (Day 4): 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: plotting, direction, the line of best fit, strength and form, and cause.