Unit #4: Understanding Relationships

Date Title Assignment Completed

4.1 Interpreting Scatter Plots

4.2 Lines of Best Fit and Making Predictions

4.3 Correlation of Data

Quiz 1: Lessons 4.1 to 4.3

4.4 Distance Time Graphs

4.5 Interpreting Graphs

4.6 Relationship Consolidation

4.7 Review of Relationships

Relationship Unit

4.1 Interpreting Scatter Plots

Describe the following scatter plots:

Warm-Up:

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Scatter Plot Review

A scatter plot is a graph that shows the _________________________ between ________ variables.The points in a scatter plot often show a pattern, or _____________________________.

Example:Julie gathered information about her age and height from the markings on the wall in her house.

Age (years) 1 2 3 4 5 6 7 8

Height (cm) 70 82 93 98 106 118 127 135

a) Label the vertical axis.

b) Describe the relationship between the variables.

b) Describe the trend of the data points.

Independent and Dependent Variables

The independent variable is located on the ___________ axis. The independent variable is located ___________ in the table of values. This variable does not depend on the other variable

Independent variable: ______________________________

The dependent variable is located on the ____________ axis. The dependent variable is located ___________ in the table of values. This variable depends on the other variable.

Dependent variable: ___________________________________

Practice Problem: Determine the independent variable and the dependent variable and then create a scatter plot for each situation.

1. Outdoor air temperature and amount of ice cream sold at a grocery store. Independent Variable: _________________________

Dependent Variable: __________________________

2. Number of student crossing Eramosa Rd and the time of day.

3. The day of the month you were born and the last digit of your phone number

4. A person’s age and their average yearly income

Practice Problem: Compare the following two scatter plots, the first one is correct, the second one is not.List the requirement when creating a scatter plot.

Justin’s Walk

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Practice Problem: The following table compares the number of hours that a student spent studying, and their test score.

a) What does one square represent on the x-axis?

b) What does one square represent on the y-axis?

c) Create a scatter plot of the data:

Study Hours Test Score3 805 902 756 807 901 506 152 657 851 407 100

d) What is the dependent variable? How do you know? What is the independent variable? How do you know?

e) Circle the point on the graph that does not seem to follow the trend. This is called an outlier. Give a possible explanation for this point.

f) There are three points with an x-value of 7. How is this possible?

g) Is it easier to see the relationship in the table of values or in the graph? Explain.

h) Describes the relationship shown by the data.

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Assignment 4.1 Interpreting Scatter Plots

1. Suppose you were to survey your classmates to see if there is a relationship between math marks and the number of hours spent watching TV.

a. What column heading would you use in a table of values designed to organize the data from your survey?

b. Which variable would you choose as the independent variable? Explainc. How would you interpret the ordered pair (2, 65) if it were to appear on a scatter plot of your

data?

2. The scatter plot shows the sales of bottled water at a refreshment booth at the farmer’s market for different days during the summer.

a. What information does point A represent?b. What information does point B represent?c. What does the scatter plot show about the

relationship between the variables?

3. A survey of 200 students finds out that students need less sleep as they get older.

a. Identify the independent variable and the dependent variable.

b. Sketch a scatter plot of this situation.

4. Consider the graph shown on right. Which relationship is most likely to be represented by this graph? Explain your choice

i. height vs. weightii. pay vs. number of hours worked

iii. gas remaining vs. distance travellediv. volume of water in a bucket vs. its mass

5. Classify the variables in each pair as independent or dependent.a) distance travelled and speedb) quality of study and exam markc) time of year and cost of vacationd) amount of wool needed and number of sweaters

6. This scatter plot shows the capacity of gas tanks and the distance travelled on one tank of gas.a) Which is the independent variable? Which is the dependent

variable?b) Describe the relationship between the capacity of the gas tank

and the distance travelled.

7. The table shows the heights and arm spans for a group of students:

Height (cm) Arm Span (cm)162 160174 175162 163171 168157 171175 173154 153179 177

a) Draw a scatter plot of the data.b) Describe the relationship between a person’s height and arm span.c) Identify any outliers and explain how they are different from the rest of

the data.

8. The table shows the values of some used cars.

Value ($1000s) Age (years)36 322 625 429 431 337 221 7

a) Identify the independent variable and the dependent variable.b) Draw a scatter plot of the data in the table.c) Describe the relationship between the age of a car and its value.d) Jane bought a 7-year-old car for

$28 000. Did she pay too much? Explain.

 5. a) independent: speed; dependent: distance travelledb) independent: quality of study; dependent: exam markc) independent: time of year; dependent: cost of vacationd) independent: number of sweaters; dependent: amount of wool needed

 6. a) independent: capacity of gas tank; dependent: distance travelledb) as the capacity of the gas tank increases, so does the distance travelled

 7. a)

b) A taller person tends to have a longer arm span and a shorter person tends to have a shorter arm span.c) There is one outlier. The height 157 cm and arm span 171 does not fit the pattern of the other data. In most cases, height

and arm span are almost equal. 8. a) independent: age; dependent: value

b)

c) As a car gets older, its value tends to decrease.d) Jane paid too much. When I plot the point (7, 28) on the scatter plot, I can see it is an outlier.

The value of another 7-year-old car is only $21 000 and a 4-year-old car has a value of $29 000.

4.2 The Line of Best Fit and Making Predictions

Warm Up:The graph below represents the relationship between earnings and time worked.

a) Which point represents the person who worked the least amount of hours?

b) Which point represents the person who earned the most amount of money?

c) Which point represents the person who has the highest rate of pay?

a) How many hours where driven when 40 passengers were aboard the bus?

b) How many passengers were aboard the bus after 10 hours where driven?

c) How many passengers were aboard the bus after 6 hours where driven? Is this answer to realistic? Why or why not?

d) Can you determine how many passengers were aboard the bus after 26 hours where driven? Why or why not?

Making Predictions from DataA skateboarder starts from various points along a steep ramp and practices coasting to the bottom. This table lists the skateboarder’s initial height above the bottom of the ramp and his speed at the bottom of the ramp.

a) Identify the independent variable and the dependent variable. Explain your reasoning.

b) Make a scatter plot of the data.

c) Describe the relationship between the variables.

d) Identify any outliers. What might cause an outlier in the data?

e) Draw a straight line that passes close to the data points.

How many data points lie on your line?

Above your line?

Below your line?

f) Compare your line with two other students’ lines. Are all three lines the same? Why or why not?

g) Using the line that you drew; predict the speed of the skateboarder at an initial height of 1m.

h) Will the rest of the students get the same answer to question g) as you do? Explain.

i) Create some rules that everyone can follow so that they all create similar line.

Line of Best FitThe line of best fit of a scatter plot is the line that is drawn on the graph so that the distance from the line to all of the points is minimized. To be able to make predictions, we need to model the data with a line of best fit.

Practice Problems: Draw a Line of Best fit for the scatter plot below:

Think about it: Do you include an outlier point in you consideration when creating a line of best fit?

Recall: We always identify any outliers by circling the point on the scatter plot.

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InterpolateWhen you interpolate, you are making a prediction __________ the data.You are interpolating when the value you are finding is somewhere between the first point and the last point.

Rules for drawing a line of best fit:

1. The line must follow the _____________________.2. The line should __________ through as many points as possible.3. There should be __________________________________ of points above and below the line.4. The line should pass through points all along the line, not just at the ends.5. The line must be drawn with a _________________________.

These predictions are usually fairly _________.

ExtrapolateYou are extrapolating when the value you are finding is before the first point or after the last point. This means you may need to extend the line.

When you extrapolate, you are making a prediction _____________ the data.

It often requires you to ____________the line.

These predictions are less ______________.

Making PredictionsUse your line of best fit to estimate the following:

Question Answer Method of PredictionA student’s mark if they studied for 4 hours.A student’s mark if they did not study.How long a student studied for if they received 60 out of 100.How long a student studied for if they received 100 out of 100.

Lines of “BEST” fitIs each line of best fit a good model for the data? Why or why not?

a) b)

c) b)

Lines of best fit are useful when we want to predict what we expect to happen.

Practice Problems:

1) Malia records the number of ice cream cones she sells each day and the maximum daily temperature, as shown on the graph.

According to this graph, approximately how many ice cream cones will Malia sell on a day when the maximum temperature is 36°? Justify your answer.

Declan has a candy-making business. The graph shows the total number of candies his business has produced by the end of each day for the first four days.

Plot each of the following points on the grid.a) (5, 500)b) (9, 850)c) (10, 1300) d) (14, 1400)

If this trend continues, which of the following points represents a day with more candies produced than expected?

State a point that would represent a day with less candies produced than expected?

Practice: Creating Scatter Plots and Lines of Best Fit

Plot the data on the scatter plot below, choosing appropriate scales and labels.

Age Earnings ($)25 2200030 2650035 2950037 2900038 3000040 3200041 3500045 3600055 4100060 4100062 4250065 4300070 3700075 37500

Note: The symbol _______ is used to signal a “break” in the axis when the scale does not start at zero to avoid a large empty space in one corner of the graph.

1) Draw a line of best fit. Describe the trend in the data.

2) Predict the earnings of a 29 year old. Which method of prediction did you use?

3) Predict the earnings of an 80 year old. Which method of prediction did you use? Is this accurate?

4) Explain why the data for ages over 65 do not follow the trend in the data.

Assignment 4.2: The Line of Best Fit and Making Predictions

1. Study the scatter plot below, then answer the following questions.

a) If you were asked, What was the herring population when 4 fishermen fished?” What would be your answer?

b) But what if you were asked, “What was the herring population when 12 fishermen fished?” What would be your answer?

c) Could you give an answer if asked, “What was the herring population when 7 fishermen fished?” If so, what would be your answer?

d) If you were to draw a Line of Best Fit to represent the correlation of the data, how would you draw it? List the five requirements should your line of best fit should follow. Draw the Line of Best Fit to represent the correlation of the data in the graph above.

e) Now use your Line of Best Fit to the herring population with...i) 4 fishermen ii) 12 fishermen iii) 7 fishermen iv) 18 fishermen v) No fishermen

f) State which type of prediction you used in question e).

g) How do the herring population predictions with 4 fishermen, 7 fishermen and 12 fishermen differ from the predictions with 18 and 0 fishermen?

2. Define interpolation.

3. Define extrapolation.

4. Graph the data and draw the Line of Best Fit. Describe the relationships.

For graph a)i) Determine the height at 6 years. Is it interpolation or extrapolation?ii) Determine the age for a person 75cm. Is it interpolation or extrapolation?

For graph b)i) Determine the speed at 45min. Is it interpolation or extrapolation?ii) Determine the time at 80km/h. Is it interpolation or extrapolation?

For graph c)i) Determine the mark if a person studied for 6 hours. Is it interpolation or extrapolation?ii) Determine number of hours required to obtain 95%. Is it interpolation or extrapolation?

5. The graph below represents the relationship between the number of calories and the volume for various drinks.

a) Draw a line of best fit.b) Describe the relationship between the variables.c) Which of the following points represents a drink

with more calories than expected for its volume? Explain how you know.

4.3: Describing Trends and Relationships in Data

Warm Up:Study the graph below a) What is the independent variable?

b) What does the symbol on the x-axis represent?

c) Describe what the point W represents?

d) Which point represents a girl whose shoe size is smaller than expected for a girl of her height?

d) Describe what the Y represents.

Explain why each line of best fit is not a good model for the data.a) b)

What is correlation?

Type of Correlation:A scatter plot shows a ____________ correlation when the pattern rises up to the right.

This means that the two quantities increase together.

A scatter plot shows a ____________ correlation when the pattern falls down to the right.

This means that as one quantity increases the other decreases.

A scatter plot shows _____________ correlation when no pattern appears.

Hint:If the points are roughly enclosed by a circle, then there is no correlation.

Hint:To visualize this, enclose the plotted points in an oval.If the oval is thin, then the correlation is strong.If the oval is fat, then the correlation is weak.

Strong or Weak? If the points nearly form a line, then the correlation is ___________________.

If the points are dispersed more widely, but still form a rough line, then the correlation is

_______________________.

Practice Problem: Write two key words to describe each relation, then draw a line of best fit.

Linear or Non Linear? If the points form a relatively straight line on a scatter plot, we say that there is a _______________

relationship.

A _________________________ of best fit can be used to model a linear relationship.

If the points do not form a relatively straight line on a scatter plot, but a trend is still evident, we say that

there is a _______________ relationship.

A _________________________ of best fit can be used to model a linear relationship.

Practice Problems:

1. Write two key words to describe each relation, then draw a line or curve of best fit.

2. Would you expect a positive correlation, a negative correlation, or no association between the two data sets? Explain why to a partner. Where possible, identify the independent and dependent variable, then sketch a scatter plot that describes the relationship.

a) The amount of free time you have and the number of classes you take

b) The sales of snow shovels and the amount of snowfall

c) Length of a baby at birth and the month in which the baby was born

d) Amount of gas remaining in a car and distance travelled by that car.

e) Mark on a test and hair length

3. Identify a variable that has….

a) a positive correlation with school marks

b) a negative correlation with school marks

c) no correlation with school marks.

4. Give an example of two variables that would have a strong non-linear correlation. Sketch a scatter plot for

these variables and explain your choice.

5. Create a mind map of the type of correlation possible. Include scatter plots in your mind map.

Assignment 4.3: Describing Trends and Relationships in Data

1. The manager of a Petro Canada service station in Collingwood and Brampton, Ontario changes the price of unleaded gasoline and records the amount of gas sold per hour at each price. The results are shown in the table below:

CollingwoodPrice (c/L) 100.2 98.4 96.3 94.5 95 95.9 99.2 102.3Amount sold (L/h) 293 242 264 272 292 289 271 241

BramptonPrice (c/L) 100.2 98.4 96.3 94.5 95 95.9 99.2 102.3Amount sold (L/h) 202 227 250 286 278 272 212 183

a) For each town,i) Make a scatter plot of the data. Draw a line of best fit if possible.ii) Describe the trend and correlation in the data.iii) Does the price of gas affect how much is purchased?

b) How does the Brampton data compare to the Collingwood data? Give reasons for the differences.

2. Would you expect a positive correlation, a negative correlation, or no association between the two data sets? Where possible, identify the independent and dependent variable, then sketch a scatter plot that describes the relationship.

a. Kilometers driven on the highway versus the amount of gas left in the tank. b. Temperature outside versus the number of people at the beach. c. Time spent reading versus the number of pages left in the book. d. Time spent exercising versus weight. e. Body temperature versus the length of hair. f. Age versus height. g. Age versus the number of teeth in your mouth.

4.4: Distance Time Graphs

Warm Up1. Student starts 1m away from the CBR and walks away at a constant speed (slowly).

2. Student starts 3m away from the CBR and walks towards at a constant speed (slowly).

Reflect:

a) How are the graphs different? How are they alike?

b) How does the starting position show up on the graph?

3. Student starts 5m away from the CBR and then walks very quickly towards it.

Reflect:

a) How is the speed of the walker shown on the graph?

b) How do we calculate a person’s speed?

4. Student increases speed while walking towards the CBR™.

5. Student decreases speed while walking away from the CBR™.

6. Student walks away from ranger, at 2 metres stops for 5 seconds, then returns at the same pace.

Reflect:

a) What happens on the graph if a person is not moving?

b) Does it matter which direction the person is facing?

c) What letter of the alphabet could you create on a distance time graph? How do you know?

Distance vs. Time Graphs Summary

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Practice Problems: Sketch the graph that is described in each story.

a) Begin 5 metres from the sensor.

Walk towards the sensor for 6 seconds at a steady rate of 1 metre in 2 seconds.

Stop for 5 seconds.

Run back to your starting position at a steady rate of 1 metre per second.

Stop.

b) Begin at the sensor. Walk very slowly at a steady rate away from the sensor

for 3 seconds.

Increase your speed and walk at this new speed for 3 seconds.

Stop for 3 seconds.

Walk very slowly at a steady rate towards the sensor for 3 seconds.

Gradually increase your speed to a run and go back to the sensor.

c) The graph shows Brian’s journey. Use the graph to answer the questions below.

i) How many times did he stop?

ii) At what times did he stop?

iii) For how long did he stop each time?

iv) When did he travel the fastest?

vi) What was the total length of his journey?

vii) How many miles had he travelled by 12.30pm?

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Assignment 4.4: Distance Time Graphs

1. Describe the motion represented by each graph. State the direction, time and distance of each section.

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c) d)

2. Mark walks to his friend’s house. Partway there, he realizes he forgot a CD at home. He runs back home to pick up the CD, then walks back to his friend’s house. Construct a distance-time graph to represent this situation.

3. A helicopter left the landing pad at the top of a skyscraper and then quickly flew downwards towards the ground and maintained a 5 foot distance above the ground for a while before it had to fly up above a small hill and land at the bottom of the far side of the hill.

4. Jen left her house and drove to school in the morning, as shown in the accompanying graph. Explain what is happening during each part of the graph below

5. Stewart Little ventures out from the safety of the Little residence and takes a very long trip. His trip is graphed in the graph to the right. Answer the questions below based on the graph of Stewart’s trip.

a) Which section of the graph was Stewart travelling the fastest?

b) Which section of the graph was Stewart travelling the slowest?

c) How many stops did Stewart make? d) How long into his trip did Stewart stop for the

first time? How long was this stop?e) Create a story that outline what may have

happened to Stewart.

6. Kerri and Ashley start from Kerri’s house and go for a little trip to run some errands. After they are done with the errands, they want to watch a movie at Ashley’s house. Their distance from Kerri’s house for this trip is plotted as a function of time below. Use this Distance v. Time graph in order to answer the questions below.

a) How far from the Kerri’s house are they after 4.0 hours?

b) How many times did they pass Kerri’s house before they ended their trip? Explain your answer and give the approximate times when they passed the house.

c) At 11 hours into their trip, were they going towards or away from Kerri’s house? What about at 14 hours?

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7. A racecar is travelling across a straight track. Its position from the start line is plotted on the graph with respect to time. Use the graph to fill in the table relating time to the position of the car.

Time (s) Position (m)0.01.02.03.04.05.06.07.0

a) Where did the racecar start?b) What do you notice about the distance being covered as each second passes?c) How would you describe the motion of the racecar?

8. Another racecar is travelling along the same track. Its position from the starting line is plotted on the graph with respect to time. Use the graph to fill in the table relating time to the position of the car.

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4.5: Interpreting Graphs

Warm Up

1) What is happening between points A and B?

2) What is happening between points B and C?

3) What is happening between points C and D?

4) What is happening between points D and E?

5) What is happening between points E and F?

6) Tell a story that describes what is happening in the graph.

Interpreting GraphsExample 1) Suppose tap water, flowing from a faucet at a constant rate is used to fill each of these containers. Match each of the following graphs with the appropriate container. Justify your choice.

a) Over what distance was the race run? Who won?

b) Who got a head start? How many seconds was the head start?

c) Who ran faster during the race?

d) For how many seconds had Adrian been running when he was overtaken by Joseph?

e) How far from the start did Joseph overtake Adrian?

f)

Example 2) Chris runs each day as part of his daily exercise. The graph shows his distance from home as he runs his route.

a) Which segment is Chris travelling the fastest/slowest?

b) The formula to calculate speed is

c) Calculate his rate of change (speed) for each segment of the graph.

Example 3) The graph represent the distance between a squirrel and a tree, d, in metres, as a function of time, t, in seconds. Calculate the speed of each section of the graph. Show your work

Assignment 4.5: Interpreting Graphs

1. Ian and his friends were sitting on a deck and eating sunflower seeds. Each person had a bowl with the same amount of seeds. The graphs below all show the amount of sunflower seeds remaining in the person’s bowl over a period of time. Write sentences that describe what may have happened for each person.

a) b) c) d)

2. Indicate which graph matches the statement. Give reasons for your answer.

a) A bicycle valve’s distance from the ground as a boy rides at a constant speed.

a) b) c) d)

b) A child swings on a swing, as a parent watches from the front of the swing.

4. Liquid is poured at a steady rate into the bottle shown in the diagram. As the bottle is filled, the height, h, of the liquid in the bottle changes. Which of the five graphs below shows this change? Give a reason for your choice.

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5. 11 o’clock, Micha’s mother sends him to the corner store for milk and tells him to be back in 30 minutes. Examine the graph.

a) Why are some line segments on the graph steeper than others?

b) Calculate the rate of change (speed) of each of the line segments:

c) Over what interval(s) of time is Micha travelling the fastest? Compare steepness, not direction.

d) Over what interval(s) of time is Micha travelling the slowest? Compare steepness, not direction.

e) How long did it take Micha to reach the store? How do you know?

f) How long did Micha stay at the store?g) How long did it take Micha to get home from the store?h) How can you use the graph to tell which direction Micha is

travelling?i) Did Micha make it home in 30 minutes? How do you

know?

6. The tortoise and the hare run a race against each other. The distance time graph for their race is shown below. The motion of the tortoise is shown by the line AB. The motion of the hare is shown by the line CDEF.

a) Over what distance was the race run?b) What is the speed of the tortoise?c) How many seconds of a head start did the tortoise have over the hare?d) Describe what the hare was doing between the points C and D, D and E, E and F.e) What is the speed of the hare between the points C and D, D and E, E and Ff) How long has the tortoise been moving when it was overtaken by the fare?g) Who won the race and why?

7. The information shown in a graph does not have to represent distance from a CBR. For each of the following, choose the description that most closely matches the graph.

(a) (i) The elevator starts on the second floor, goes to the fifth floor where it takes on passengers and then goes to the seventh floor.

(b) (ii) Sarah takes half an hour to get dressed in the morning, hurries to her friend’s house so they can walk to the bus stop to catch the bus to work.

(c) (iii) As the weather cools in the autumn, sales at Matt’s ice cream shop have been decreasing. During one rainy week only the regulars who work in the same building purchased their usual snacks. When Matt introduces a new product, sales begin to improve.

(d) (iv) Daniel’s company assets decrease during the months when he is preparing his new store but improve dramatically when the store opens. When a competitor opens a nearby store, Daniel’s fortunes take a down turn.

(e) (v) Sandy has been saving a regular amount each week. When she joins her family on a vacation she spends part of her savings but resumes her saving habit when she returns.

(f) (vi) Jeff’s strength training program calls for him to add time to his workout gradually. After a few weeks Jeff decides to suddenly increase the time he is spending but finds he can maintain but not increase the new level for several weeks. Then he is able to gradually increase the workout time again.

4.6 Consolidation: Relationships

Joe works at a fruit factory. All fruit are counted and placed in wooden crates for delivery to various stores. He is given the job of recording the number of watermelon in each crate and their corresponding weight. Joe creates a scatter plot to show his boss.

1. List the rules for drawing a line of best fit, and then draw a line of best fit on the scatter above in pencil.

2. Describe the relationship between the variables and type of correlation of the scatter plot above.

3. Predict the weight when there are 24 watermelons in the crate. Explain how you made your prediction and state the type of prediction you made.

4. Predict the weight of an empty crate. Explain how you made your prediction and state the type of prediction you made.

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5. Joe is transferred to the lemons department. He notices that they use the same crates as the watermelon department. Predict what a line of best would look like if Joe measured the number of lemons in the crate and their corresponding weight. Draw the line of best fit on the scatter plot in a different colour and label your line. Explain how you determined the placement of this line of best fit.

6. Joe gets transferred back to the watermelon department and gets told that the watermelons will now be shipped in plastic crates (which are much lighter than the wooden crates). Joe wonders what his original scatter plot would look like if the watermelons were in plastic crates rather than wooden crates. Help him visual the scatter plot of the watermelons in plastics crates and their corresponding weight by drawing a new line of best fit in a different colour and labeling the line. Explain how you determined the placement of this line of best fit.

7. Estimate the weight of one watermelon. Show your calculations.

2. Devin went for a bicycle ride. The graph below shows his trip.Note: Distance is the number of kilometres from home.

a) Calculate his speed during the first hour (AB) and the second hour (BC). Show your work.

b) How does the speed between A and B compare with the speed between B and C?

c) Explain what segment CD tells you about Devin’s motion.

d) Which section of the graph shows that Devin was changing speeds? Explain

e) What information can you determine from segment EF?

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4.7 Review of Relationships

Scatter plots can be classified using the following terms:

Type of correlation: _______________ or __________________ or _____________________

Strength of correlation: ____________ - points are close to the pattern, or

____________ - points are spread further apart

Direction: _______________ - moving up toward the right, or

_______________ - moving down toward the right

______________: Points that don’t seem to fit in with the rest of the data;

Describe the following scatter plots using appropriate mathematical terms.

________________ ________________ ______________

________________ ________________ ______________

________________ ________________ ______________

________________ ________________ ______________

________________ ________________ ______________

________________ ________________ ______________

________________ ________________ ______________

________________ ________________ ______________

Linear Relationships

A ____________________________ is made by laying down a ruler (clear is best) so that it touches as many points as possible. There should be approximately the same number of points ______________________the line. The line of best fit ___________________ have to go through the origin.

Non-Linear Relationships

A __________ of best fit is made by drawing a smooth, free-hand curve through the points.

Making Predictions

If you can draw a line of best fit, you can use it for further analysis, such as _______________ or _________________.

Interpolation

Making an estimation within the range of recorded data.

Extrapolation

Making a prediction outside the range of recorded data.

Assignment 4.7 Review of Relationships

1. The table below shows the winning distances for the men’s discus event at the Olympic Summer games.Year Distance in Metres1948 53.01956 54.01964 61.01972 64.51980 671988 69.01996 73

a. Draw a scatter plot of the data.b. Draw the line of best fit. c. Use the line of best fit to find the distance in 1968. d. What kind of prediction is being used in part c?

2. A basketball coach recorded the length of time each player was in a game and the number of points the player scored as an ordered pair (time in minutes, points scored). The data was collected over ten games.(151, 51), (18, 19), (164, 38), (86, 28), (136, 39), (110, 40), (55, 8), (163, 62), (192, 50), (98, 32), (71, 25).

a) What are the variables?b) What is the independent variable?c) Create a table of values.d) What is the dependent variable?e) Draw a scatter plot.f) Describe the relationship.

3. Tom drew the line of best fit on the graph for the Women’s Olympic Discus Results.

a. Does the line of best fit drawn accurately depict the flow of data. Explain.

b. Explain 4 rules of drawing a line of best fit, then draw a new line of best fit

c. Based on the new line of best fit, what would you expect the distance to be in the year 2000 for the women’s discus event?

4. The following graph represents the growth of a certain bacteria.

a. Describe the relationship and correlation. b. Draw the line or curve of best fit. c. Explain why the number of bacteria is the dependent variable.

5. If a person collected data about how tired students feel during the day, and how much food they have eaten, which variable is independent and which is dependent? Explain.

6. What type of correlation would you expect to find if you compared:a. The relationship between the volume of gasoline remaining in your gas tank with the distance

driven since the tank was filled?b. The relationship between the average outdoor temperature with the number of slushies sold

by a store?c. The relationship between a players distance from the basket ball net board and the number of

baskets they made?d. The relationship between the length of a student’s hair and their mark on this test?e. The relationship between the number of months since a student’s birthday and the number of

months until that student’s birthday?f. The relationship between the number of letters in a student’s first name and the number of

letters in the student’s last name?

7. Weight Watchers boasts that their program will help people loose 30 pounds every 3 months as they work towards their ideal weight. Suppose a 300 pound person is working towards an ideal weight of 140 lbs.

According to Weight Watchers, the following results should occur.Months 0 3 6 9 12 15 18

Weight 300 270 240 210 180 150 120However, when a 300 lbs person tracked their weight loss and found the following set of results.

Months 0 3 6 9 12 15

Weight 300 270 243 219 197 177

Graph both sets of Data on the same graph.

a) Describe the graph of the predicted weight loss and compare it to the graph of what actually happened. What is the shape of each graph?

b) According to the predicted weight loss graph, how many months will it take the person to reach their ideal weight of 140 lbs? (Show using your graph)

c) In reality, how long would it take to reach 140 lbs based on the graph of the person’s actual weight loss? (Show using your graph)

8. Sketch a graph that shows water height vs time for the following situation Start with an empty bathtub with the plug in

a. Turn on the tapb. Turn off the tapc. Get in the bathtubd. Wash yourselfe. Get out of the bathtubf. Remove the plug

9. The graph below depicts Jack’s walk home from school.

a) Calculate Jack’s speed for each section of the graph.

b) Write a story to describe the trip. Be sure to use appropriate times, speeds, and distances in your description.

10. For each type of correlation listed below

a) create a scatter plot showing the type of correlation b) label the axes with an example of the correlation

i) weak positive correlation ii) strong negative correlationiii) no correlation

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Jack's Walk Home