Lesson 4.1 Interpreting Scatter Plots Learning Goal: By the end of this lesson I will be able to create and interpret scatter plots. WARM UP

Describe the following scatter plots:

Tibia Length (cm)

Leg Length (cm)

Distance from the Basket

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Age of House

Hou

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Price (

$)

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 1: 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

Height (cm) 70 82 93 98 106 118 135

a) Label the vertical axis. b) Describe the relationship between the

variables. c) 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: ___________________________________

Example 2: Determine the independent variable and the dependent variable and then create a scatter plot for each situation. a) Outdoor air temperature and amount of ice cream sold at a grocery store. Independent Variable: _________________________ Dependent Variable: __________________________

b) Number of students crossing Adler Street and the time of day.

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

d) A person’s age and their average yearly income.

Example 3: 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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Example 4: 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:

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.

Study Hours Test Score

3 80

5 90

2 75

6 80

7 90

1 50

6 15

2 65

7 85

1 40

7 100

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Lesson 4.2 Lines of Best Fit and Making Predictions Learning goal: By the end of this lesson I will demonstrate how to draw a line of best fit, interpolate, and extrapolate 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 DATA A 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. LINE OF BEST FIT

The 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.

Success Criteria 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 _________________________.

Example 1: Draw a line of best fit for the scatter plot below.

Interpolate When you interpolate, you are making a prediction __________ the data. These predictions are usually fairly _________. Extrapolate When you extrapolate, you are making a prediction _____________ the data. It often requires you to ____________the line. These predictions are less ______________. MAKING PREDICTIONS Use your line of best fit to estimate the following:

Question Answer Method of Prediction

A 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 fit are useful when we want to predict what we expect to happen.

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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.

You are interpolating when the value you are finding is somewhere between the first point and the last point.

You 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.

Example 2: 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. Example 3: Dechen 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. If this trend continues, which of the following points represents a day with more candies produced than expected?

a) (5, 500) b) (9, 850) c) (10, 1300) d) (14, 1400)

Example 4: Plot the data on the scatter plot below, choosing appropriate scales and labels.

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. a) Draw a line of best fit. Describe the trend in the data. b) Predict the earnings of a 29 year old. Which method of prediction did you use? c) Predict the earnings of an 80 year old. Which method of prediction did you use? Is this accurate? d) Explain why the data for ages over 65 do not follow the trend in the data.

Age Earnings ($)

25 22000

30 26500

35 29500

37 29000

38 30000

40 32000

41 35000

45 36000

55 41000

60 41000

62 42500

65 43000

70 37000

75 37500

Lesson 4.3 Correlation of 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.

Is each line of best fit a good model for the data? Why or why not? a) b) c)

What is correlation? TYPES 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.

STRONG vs. 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 _______________________.

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

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.

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.

Example 2: Write two key words to describe each relation, then draw a line or curve of best fit.

Example 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) 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

Example 3: The local ice cream shop keeps track of how much ice cream they sell versus the temperature on that day, here are their figures for 12 days.

a) Identify the: Independent Variable - Dependent Variable -

b) Create a scatter plot of the data.

c) Is this relationship Linear or Non – linear? Explain! d) Describe the correlation. e) If the temperature is 20°C, how much sales from ice cream did the shop get? Is this an example of interpolation or extrapolation? Explain how you know. f) If the ice cream shop made $350 in sales, what is the temperature? Is this an example of interpolation or extrapolation? Explain how you know.

Ice Cream Sales vs

Temperature

Temperature

°C

Ice Cream

Sales

14.2° $215

16.4° $325

11.9° $185

15.2° $332

18.5° $406

22.1° $522

19.4° $412

25.1° $614

23.4° $544

18.1° $421

22.6° $445

17.2° $408

Lesson 4.4 Distance Time Graphs Learning Goals: By the end of this lesson I will interpret and discuss information from a distance time graph. WARM UP

Is each line of best fit a good model for the data? Why or why not?

a)

b)

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Activity: We will be collecting data from our classmates walking, jogging, and running down the hall. Before we start, draw your predictions of the shape of the graph for each movement type.

Walking Jogging Running

1. In the hallway measure out 25 m in a straight line. Have a student with a stopwatch stand at 0 m, 5 m,

10 m, 15 m, 20 m and 25 m. (The student at 0 m doesn’t need a watch). 2. Another student begins walking along the line (They should start about 2.0 m from the first timer. As

the walker reaches the first student (at 0 m ) the first student calls out "Start" and all the timers start their watches. As the walker passes each timer they stop their watches. Record the times in the table below.

time (s) 0

position (m) [forward] from start 0 5.0 10.0 15.0 20.0 25.0

3. Repeat step 2 using a student that is jogging and record your results below.

4. Repeat step 2 for a student that is running and record your results below

5. On the same axis graph the distance vs time data for each of the three movement types. Use different symbols for each data set and draw a line of best fit for each set. Use a legend for your graph.

REFLECT

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

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

time (s) 0

position (m) [forward] from start 0 5.0 10.0 15.0 20.0 25.0

time (s) 0

position (m) [forward] from start 0 5.0 10.0 15.0 20.0 25.0

c) How is the speed of the student shown on the graph?

d) How do we calculate the student’s speed?

e) How might the graph change if the student was not at a position of 0 m when we started timing? Draw an example of a graph for a student starting 5 m away from a motion sensor (CBR attached to a graphing calculator) and walking very quickly towards it.

f) Predict the shape of the graph if the student continuously increased their speed as they move away from the sensor.

g) Predict the shape of the graph if the student continuously decreased their speed as they moved away from the sensor.

h) Predict the shape of the graph if at first a student walks away from the sensor at a constant speed for 2 meters, they then stop for 5 seconds, then move towards the sensor at the same speed they were initially walking.

Distance vs. Time Graphs Summary

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Example: Describe each of the following graphs: Be sure to mention direction, distance, time, and speed. Graph 1st section 2nd section 3rd section

Lesson 4.5 Interpreting Graphs Learning goal: By the end of this class I will discuss and interpret information from various graphs. Example 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.

Example 2: Adrian and Joseph decided to race one another. The graph shows their motion during the race.

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?

Example 3: 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 4: 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