D. Data · Two-Variable Data, Scatter Plots and ProbabilityDescribing Relationships: Positive, Negative or None, Strong or WeakD1.5 D1.6About 45 minutesLearning goalsI can describe the relationship in a scatter plot as positive, negative or none.I can describe a relationship as strong or weak by how closely the points follow a line.I can describe a relationship in words that use the two variables.Words to knowpositive relationshipAs one variable increases, the other tends to increase. The points rise from left to right.negative relationshipAs one variable increases, the other tends to decrease. The points fall from left to right.no relationshipThe points show no rising or falling pattern. Knowing one variable doesn't help you predict the other.strong relationshipThe points lie close to a straight line.weak relationshipThe points follow a rising or falling direction but are spread widely around it.trendThe general direction of the points.RememberIn the last few lessons you collected two-variable data and made scatter plots. Now comes the payoff: reading the pattern of the points to describe how the two variables are related.Which way do the points go?Look at the whole cloud of points and read it from left to right, the way you read a sentence.Positive relationship: the points go up to the right. As the first variable increases, the second tends to increase. Taller people tend to have longer arm spans.Negative relationship: the points go down to the right. As the first variable increases, the second tends to decrease. On warmer days, fewer cups of hot chocolate are sold.No relationship: the points are scattered with no upward or downward direction. Shoe size and the number of minutes a student reads each day.Positiveboth go upNegativeone up, one downNoneno patternThree scatter plots without numbers. Left: the points rise (positive). Middle: the points fall (negative). Right: no rising or falling pattern (none).How strong is it?Imagine drawing one straight line through the middle of the points, so about as many points are above it as below it.If the points are close to that line, the relationship is strong. You could predict one variable from the other quite well.If the points are spread far from the line but still have a direction, the relationship is weak. The trend is there, but predictions would be rough.Another way to picture it: draw an oval around all the points. A thin oval means strong; a fat oval means weak; a roughly round blob means no relationship.Strong positiveclose to a lineWeak positivespread out, still risingBoth plots show a positive relationship, with a dashed line through the middle of the points. On the left the points hug the line (strong). On the right they are spread far above and below it (weak).Worked exampleExample 1: Height and arm spanDescribe the relationship in the Okafor family's height and arm-span scatter plot (from the Making Scatter Plots lesson).Direction: the points rise from left to right, so the relationship is positive.Strength: the points lie very close to a straight line. No point is far away from the others. So it is strong.In context: say what it means using both variables.There is a strong positive relationship: people who are taller tend to have longer arm spans. In fact, each person's arm span is within about 3 cm of their height.Temperature and Hot Chocolate Sales−16−12−8−404010203040506070Daily high temperature (°C)Cups soldSource: community rink canteen, 10 winter days10 winter days at a community rink. Points from (−15, 62) on the coldest day down to (4, 22) on the warmest day. Values left of 0 are below zero.Worked exampleExample 2: Hot chocolate at the rinkDescribe the relationship between the daily high temperature and the cups of hot chocolate sold.Direction: as you move right (warmer), the points go down (fewer cups). That is negative.Strength: the points are close to a straight line from the top left to the bottom right. Strong.Check with numbers: at −15°C, 62 cups were sold; at 4°C, only 22 cups.A strong negative relationship: on colder days the canteen tends to sell more hot chocolate, and on warmer days it sells less.Worked exampleExample 3: Describe from a tableA class records each student's shoe size and minutes read yesterday: (5, 20), (6, 45), (6.5, 10), (7, 30), (7.5, 60), (8, 15), (8.5, 40), (9, 25), (10, 50), (10.5, 5). Describe the relationship.Imagine (or sketch) the points with shoe size across and minutes up.As shoe size goes up, the minutes jump up and down: 20, 45, 10, 30, 60, 15, 40, 25, 50, 5. There is no steady rise or fall.The largest shoe size (10.5) has one of the smallest reading times, and the middle sizes have both high and low times.No relationship. Knowing a student's shoe size doesn't help you predict how long they read.Say it in wordsA complete description has three parts:Strength: strong or weak (skip this if there is no relationship).Direction: positive, negative or none.Context: "As ___ increases, ___ tends to ___."Use the word tends. A relationship describes the overall pattern, not every single point. In a positive relationship, a taller person can still have a shorter arm span than a slightly shorter person (compare Persons G and H).Just for funA scatter plot tells its friend, "I think my relationship is positive... but honestly, some days it feels a little weak and spread out."Hands-on activityFamily measurementsYou need: a measuring tape, paper, pencil, and as many people as you can (family, or friends on a video call measuring themselves).Measure each person's height and foot length (heel to longest toe, in cm), with their permission.Also record the number of letters in their first name.Make two scatter plots: height vs foot length, and height vs letters in name.Describe each relationship: strong or weak, positive, negative or none.What to look for: height and foot length usually show a positive relationship (especially if you include younger children and adults). Letters in a name should show no relationship. If you have only a few people, ask more relatives or friends to measure themselves and send you their numbers: more points make a pattern easier to judge.Your turn1The points on a scatter plot go down from left to right and are close to a line. Describe the relationship.Show answerStrong negative.2Describe the likely relationship between the age of a used car and its price.Show answerNegative: older cars tend to cost less. (It may not be very strong: some older cars in great condition still cost a lot.)3Describe the likely relationship between the number of hours a phone screen is on and its battery level.Show answerNegative: the more hours the screen is on, the lower the battery tends to be.4Describe the likely relationship between a student's birth month and their math mark.Show answerNone. Birth month doesn't help predict math marks.5In the hot chocolate graph, how many fewer cups were sold on the 4°C day than on the −15°C day?Show answer40 fewer. 62 − 22 = 40.6Persons G (170, 172) and H (172, 169) break the pattern a little: H is taller but has a shorter arm span. Does that mean there is no positive relationship? Explain.Show answerNo. A relationship describes the overall trend. Most of the points still rise to the right, so it is still a strong positive relationship. One small exception is normal.More practice: warm-up1Points rise from left to right. Positive or negative?Show answerPositive.2Points form a round blob with no direction. What relationship?Show answerNone.3Points fall from left to right but are spread widely. Describe it.Show answerWeak negative.4What does a thin oval around the points tell you?Show answerThe relationship is strong.5Describe the likely relationship between distance driven and gas used.Show answerPositive (and likely strong): the farther you drive, the more gas you use.More practice: on your own1Describe the relationship in the soccer team's water data: (14, 9), (17, 10), (19, 12), (21, 12), (23, 14), (25, 15), (26, 17), (28, 18), (30, 19), (31, 21).Show answerStrong positive. As the temperature rises, the water the team drinks rises too, and the points stay close to a line (every step right is a small step up, or level once).2Describe the likely relationship between outdoor temperature and a household's heating bill over the months of a year in Ontario. Explain.Show answerNegative, likely fairly strong: in colder months the furnace runs more, so the bill is higher; in warmer months it is lower.3Data: (1, 40), (2, 38), (3, 35), (4, 36), (5, 30), (6, 28). Positive, negative or none? Strong or weak?Show answerStrong negative. The second numbers go down steadily (only one small rise, from 35 to 36), and the points stay close to a falling line.4Write a sentence in context for a weak positive relationship between hours of sleep and quiz scores.Show answerSample: "Students who slept more tended to score a bit higher, but there were many exceptions, so sleep alone is not a good predictor of quiz scores."5Kiran says, "The points rise, so it's strong." What is he mixing up?Show answerHe is mixing up direction and strength. Rising means positive. Strength depends on how close the points are to a line, not which way they go.Challenge1Give an example (not from this lesson) of two variables you would expect to have a strong negative relationship. Explain why.Show answerSample: the number of kilometres a bus has driven since filling up and the amount of fuel left in its tank. Every kilometre uses fuel, so as distance goes up, fuel left goes down in a steady way.2The hot chocolate data goes from −15°C to 4°C. Do you think the negative pattern would continue on a 25°C summer day? Explain.Show answerProbably not exactly. The line would predict a negative number of cups, which is impossible. Also, an outdoor rink isn't open in summer. A pattern found for one range of data might not continue outside that range.3Can two variables have a strong relationship that is not a straight line? Use the pendulum data (20, 9.1), (40, 12.6), (60, 15.6), (80, 17.8), (100, 20.2) to explain.Show answerYes. The pendulum times rise by 3.5, 3.0, 2.2 and 2.4 s for each 20 cm. The rises mostly get smaller, so the points curve gently. The relationship is still strong and positive, because the points follow a clear smooth pattern. In Grade 8 we describe strength by closeness to a line, so it's good to notice when a curve fits better.Check your understandingQuiz1. As the temperature drops, more people go skating at an outdoor rink. On a scatter plot of temperature (x) and skaters (y), the relationship is...negativepositivenone2. The points lie very close to a rising line. The relationship is...strong positiveweak positivestrong negative3. Which pair of variables most likely has no relationship?Hours studied and test scoreLast digit of a phone number and heightHeight and shoe size4. What makes a relationship weak?The points are spread far from a lineThe points fall to the rightThere are only a few points5. Which sentence best describes a positive relationship between practice time and free throws made?Everyone who practises more makes more free throws.Players who practise more tend to make more free throws.Practising makes you worse.6. In the hot chocolate graph, what was sold on the 0°C day?0 cups38 cups31 cupsMark lesson completeFor parents and teachersBig idea: The Ontario curriculum (D1.5) asks Grade 8 students to use mathematical language, including strong, weak, none, positive and negative, to describe the relationship between two variables. The next lesson adds outliers.Common difficulties:Confusing direction with strength ("it goes up, so it's strong").Thinking one exception ruins the relationship. Relationships describe tendencies.Forgetting context. "Strong positive" is not finished until your child says what it means for the two variables.How to help: Play "positive, negative or none?" with everyday pairs: hours of daylight and month, price of a pizza and its size, age and reaction time. Ask for a reason each time. The lesson also builds critical thinking (A1) by asking students to explain patterns in context.PreviousChoosing and Justifying the Best GraphNextOutliers in Two-Variable Data