Course: Grade 9 Applied Mathematics (MFM1P) Unit 3 ... · Course: Grade 9 Applied Mathematics (MFM1P) Unit 3: Looking for Relationships, Lines and Curves of Best Fit

Course: Grade 9 Applied Mathematics (MFM1P)

Unit 3: Looking for Relationships, Lines and Curves of Best Fit

MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008) 3-2

Unit 3

Exploring Relationships: Lines and Curves of Best Fit

Section Activity Page

3.1.1 Graphing Review 3 3.1.2 Relationships 4 3.1.3 Data Collection – Is there a Relationship

Here? 5

3.1.4 Class Data Sheet 6 3.1.5 Graphing the Data 7 3.2.1 Plotted Points 8 3.2.2 Scatter Plots – Types of Correlation 9 3.2.3 Line of Best Fit 10 3.2.P Practice 11 3.2.4 Relationships Summary 12 3.3.1 Could I Be a Forensic Scientist? 14

3.3.2a Introduction to FATHOM 15 3.3.2b Introduction to TI-83 TI-84 16 3.3.2c Introduction to TI-Nspire CAS 17 3.3.3 Choosing the Best Model 20 3.3.P Practice 22 3.4.1 Creating Scatter Plots and Lines of Best Fit 23 3.4.3 Forensic Analysis 24 3.4.J Journal Activity 25 3.5.1 Investigations 26 3.7.1 First Differences 36 3.7.2 Using What You Have Discovered 41 3.8 Unit 3 Review (Practice) 42 3.W Definition Page 47 3.S Unit Summary Page 49 3.R Reflecting on My Learning 50

3.RLS Reflecting on Learning Skills 51


3.1.1: Graphing Review Plot the wingspan data for these.18 birds on the grid below.

Source: Faculty of Mathematics, University of Waterloo, “Linear Relations: Graphing and Analyzing”

(Wingspread of Birds)

1. Describe the pattern of the dots on the scatter plot.

2. Describe the relationship between bird length and wingspan.


3.1.2: Relationships Complete the following statements by yourself, then share your answers with your partner. Explain the reasons for your choice. Indicate if you and your partner agree or disagree.

Is There a Relationship? My Partner and I:

__ agree As a person gets taller their armspan ______________________. (gets wider, gets smaller, stays the same)

__ disagree

__ agree The longer a person's legs are ______________________ they run. (the faster, the slower, will make no difference to how fast)

__ disagree

__ agree As a person's foot size increases, their walking stride _____________________. (gets longer, gets shorter, stays the same) __ disagree

__ agree As a person's forearm gets longer, their armspan _______________. (gets longer, gets shorter, stays the same length)

__ disagree

__ agree The longer a person's thumb is ______________________ their index finger. (the longer, the shorter, will make no difference to the length of) __ disagree

__ agree As a person gets taller, their foot size ______________________. (gets longer, gets shorter, is not affected)

__ disagree


3.1.3: Data Collection – Is There a Relationship Here? With a partner, measure and record each measurement to the nearest centimetre. Enter your data into the class data collection chart. a) total height ____________ cm b) forearm ____________ cm c) arm span from fingertips to fingertips ____________ cm d) foot length _______________ cm e) walking stride length _______________ cm f) hand span ____________________cm


3.1.4: Class Data Sheet

Name Height (cm)

Forearm (cm)

Arm span (cm)

Foot length (cm)

Walking Stride

length (cm)

Hand span (cm)

Maggie 134 22 130 20 120 13

Homer 162 26 160 24 140 18

Stefan 169 27 170 26 145 19

Teniesha 150 24 149 23 132 16

Debbie 143 23 143 22 122 15

Kevin 167 26 161 23 145 17

Dwight 178 28 178 27 155 19

Shahad 165 25 163 24 142 18


3.1.5: Graphing the Data Using class data from Day 1, choose two measurements that you would like to investigate. Create a scatter plot of your chosen relationship on grid paper.

Using your graph answer the following questions: 1. Which phrase describes the direction of the plotted points in the graph?

a) The plotted points rise upward to the right. b) The plotted points fall downward to the right. c) The plotted points are scattered across the graph.

2. Describe the relationship between the two quantities. 3. How could you use this graph to predict additional measurements? Explain.


Age of House

Hou

se P

rice

($)

Distance from the Basket

Num

ber o

f Bas

kets

3.2.1: Plotted Points

1. The graph shows the plotted points rising upwards to the right. • Agree • Disagree • Pass

1. The graph shows the plotted points falling to the right. • Agree • Disagree • Pass

2. As the length of the tibia increases the length of the leg increases. • Agree • Disagree • Pass

2. As the distance from the net increases the number of baskets made decreases. • Agree • Disagree • Pass

3. The graph can be used to determine the length of a person's leg if you know the length of the tibia bone. • Agree • Disagree • Pass

3. The graph can be used to determine the number of baskets you will make if you know the distance from the basket. • Agree • Disagree • Pass

1. The graph shows the plotted points

scattered. • Agree • Disagree • Pass

2. As the age of the house increases the price of the house is either large or small. • Agree • Disagree • Pass

3. The graph can't be used to determine the price of the house if you know how old it is. • Agree • Disagree • Pass

Tibia Length (cm)

Leg Length (cm)


3.2.2: Scatter Plots - Types of Correlation Correlation helps to describe the relationship between 2 quantities in a graph. Correlation can be described as positive or negative, strong or weak or none.

Positive or Negative 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.

Strong or Weak Correlation

If the points nearly form a line, then the correlation is __________________. To visualize this, enclose the plotted points in an oval. If the oval is narrow, then the correlation is strong.

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

If the points are dispersed even more widely, but still form a rough pattern of a line, then the correlation is ___________________. If the oval is wide, then the correlation is weak.

No Correlation

A scatter plot shows _______________ correlation when no pattern appears. Hint: If the points are roughly enclosed by a circle, then there is no correlation.


3.2.3: Line of Best Fit Line of Best Fit To be able to make predictions, we need to model the data with a line or a curve of best fit.

Use the information below to draw a scatter plot. Describe the correlation and draw the line of best fit. The teachers at Holy Mary high school took a survey in their classes to determine if there is a relationship between the student’s mark on a test and the number of hours watching T.V. the night before. Mark % 75 70 68 73 59 57 80 65 63 55 85 70 55 Number of Hours

1 2 3 2 4 4.5 1 3 3.5 4 1 2.5 4

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.


3.2.P: Practice

For each of the graphs below: 1) Draw a line of best fit if possible. If a straight line can be drawn, label the graph as linear.

If a straight line cannot be drawn, label the graph as non-linear. 2) Label each graph as showing a relationship or no relationship. 3) The following instructions are for the linear graphs only.

a) Describe the correlation of each scatter plot as positive or negative. b) Describe the correlation as weak or strong.

Linear / Non-linear Relationship / No Relationship

Positive / Negative Strong / Moderate / Weak

a Linear / Non-linear

Relationship / No Relationship Positive / Negative

Strong / Moderate / Weak

b Linear / Non-linear



c



d Linear / Non-linear



e Linear / Non-linear



f



g Linear / Non-linear



h Linear / Non-linear



i


3.2.4: Relationships Summary A scatter plot is a graph that shows the ____________ between two variables. The points in a scatter plot often show a pattern, or ____________.

From the pattern or trend you can describe the ________________. 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 trend in the data. c) Describe the relationship. Variables The independent variable is located on the ___________ axis.

This variable does not depend on the other variable.

The dependent variable is located on the ____________ axis.

This variable depends on the other variable.

Independent variable: _______________ Dependent variable: _____________

Note: The independent variable comes first in the table of values.


3.2.4: Relationships Summary (continued) Making Predictions Use your line of best fit to estimate the following:

Question Answer Method of Prediction How tall was Julie when she was 5 years old?

How tall will Julie be when she is 9 years old?

How old was Julie at 100 cm tall?

How tall was Julie when she was born? Interpolate When you interpolate, you are making a prediction __________ the data.

These predictions are usually _________. Extrapolate When you extrapolate, you are making a prediction _____________ the data.

It often requires you to ____________the line. These predictions are less reliable.

Hint: 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.


3.3.1: Could I Be a Forensic Scientist? Name: _________________________________________ Date: __________________ Exploring the Problem Remnants of a human skeleton were found at an archaeological dig that is thought to be the ruins of an ancient civilization. From the bones discovered, the scientists have determined the following: • length of the forearm is 23 cm • armspan is 185 cm • handspan is 23 cm • foot length is 24 cm The scientists call you in as an expert in anthropology who is currently researching relationships between body measurements to help them determine an estimated height of the skeleton in question. As the expert, your job will be to: • estimate the height of the skeleton; • explain the procedure you used to determine the height of the skeleton; • include evidence (tables, graphs, and other models) to support your conclusion; • explain the limitations of your method or discuss a different way to conduct your

investigation. Clarifying the Problem Review the problem and highlight any important information. Formulating an Hypothesis • Decide which pairs of variables you think could show a relationship that would aid the

scientists in their predictions. • Explain your reasoning.

• What are the variables? • What exactly are you being asked to find? • Are there certain variables that would be more

useful than others?


3.3.2a: Introduction to FATHOM Creating Scatter Plots Method Step 1) Pull down a case table. Enter the heading YEAR in the <new> column and enter

Height in the next column. Enter the data as shown.

Step 2) Pull down an empty scatter plot. Grab and drag the YEAR heading to the horizontal axis, and the HEIGHT heading to the vertical axis. Your scatter plot is done!

Step 3) Pull down the GRAPH menu and choose MOVABLE LINE. This will be your line of best fit when you move it to its best position.

Step 4) Pull down the GRAPH menu and choose SHOW SQUARES. Try to position the line such that the sum of the squares is a MINIMUM. Watch the SUM change as you reposition the line.

Step 5) Change the horizontal and vertical scales by grabbing and dragging them towards zero. This will change the scale of the scatter plot and allow you to make predictions beyond the data collected.

Step 6) Predict the Height for Year 10 Predict the Height for Year 20 Predict the Year when the Height will be 30

Step 7) Add a text box to record your description of the scatter plot and the predictions by pulling down the INSERT menu and choosing TEXT. Type "Scatter Plot” by "your name." Describe your scatter plot with three sentences. * One sentence will describe the correlation. * The next sentence will describe the relationship and how strong it is. * The third sentence will use examples to support your conclusion. Use the line of best fit to make your predictions.

Collection 1 Year Height (cm)

1 1 2 2 2 4 3 3 5 4 4 7 5 5 10


3.3.2b: Introduction to Calculator TI-83 or TI-84 Creating a Line of Best Fit Method Step 1) Press STAT. Press 1:Edit… Enter the data for Year in L1 and Height in L2.. Step 2) Press STATPLOT. Press 1:Plot1…Off. Press ENTER to turn on the statplot. Set

the following options as shown. Step 3) Press ZOOM. Press 9:ZoomStat. Step 4) Press APPS. Choose TRANSFRM. Press ENTER. Press Y=. Type the equation

AX + B for Y1. Step 5) Press Window then press up arrow then to get to SETTINGS. Scroll down and

enter values for A, B and STEP from your teacher. Step 6) Press GRAPH. Use up and down arrows to move between A or B. Use and

to decrease or increase the value. Continue until the line becomes a line of best fit. Record your equation:

_______________________________ Step 7) Press APPS. Choose TRANSFRM. Press 1:UNINSTALL. Step 8) Press STAT. Arrow over to CALC. Press 4:LinReg (ax+b). Press 2ND then 1.

Press 2nd then 2. Press ENTER. Record the Linear Regression values below (Round the values of A and B to one decimal place):

_______________________________ Step 8) Press Y=. Type in the equation given above for Y2. Press Graph. Step 9) Press TRACE and use and to move the cursor until you get the X value you

want. Predict the Height for Year 10 : _________ Predict the Height for Year 20: _________ Predict the Year when the Height will be 30: _________ Step 10) Make a sketch your graph. Describe your scatter plot with three sentences. * One sentence will describe the correlation. * The next sentence will describe the relationship and how strong it is. * The third sentence will use examples to support your conclusion.


3.3.2c: Introduction to Calculator TI-Nspire CAS Creating a Line of Best Fit Method Getting to know your calculator Green Letters - Alphabet Grey Buttons - Math Functions White buttons - Numbers Dark Grey Buttons - Math Operations - home – add a new page Menu - menu options available – similar to a computer menu

- menu items – use the arrows and then press enter OR press the number only (this is how the instructions are given)

Mouse (NavPad) up show all pages left right CTRL + mouse page left page right down show one page Esc – go back a step Ctrl z – undo the last step Entering Data

1. Press HOME 2. Choose 6:New Document 3. Press to Select No to Save Changes and press Enter. 4. Choose 3:Add Lists and Spreadsheets

5. Move the cursor up to the spot beside A 6. Enter the column heading eg: Height (minimum 4 characters, no spaces) 7. Press Enter 8. To widen the column press Menu

a. 1:Actions b. 2:Resize c. 1:Resize Column Width d. move the mouse left or right to widen the column e. press Enter

9. Move to cell A1 10. Enter your data – Type in each value then press Enter 11. Move the cursor up to the top beside B 12. Repeat the above steps for all your data


3.3.2c: Introduction to Calculator TI-Nspire CAS Creating a Line of Best Fit Method (Continued) Graphing the Data 1. Press Home 2. Choose 5:Data and Statistics 3. A blank graph will be displayed. You will need to specify the independent and dependent

variables. 4. Move the cursor to the bottom of the screen (Click to add variable) and click the Mouse

Button and choose the independent variable (Armspan) from the list. Press Enter.

5. Move the cursor to the left side of the screen (Click to add variable) and click the Mouse Button and choose the dependent variable (Height) from the list. Press Enter.

6. Repeat this section for all other Height vs _______________ graphs. To move between the different graphs – Press Ctrl + the mouse arrow Drawing a line of Best Fit – Manually 1. Press Menu 2. Choose 3:Actions 3: Choose Add Movable Line 4. A line will now appear pointing to this edge, the cursor changes to this symbol pointing to the centre, the cursor changes to this symbol pointing to this edge, the cursor changes to this symbol To move your line

– point to the part of the line you wish to move - press the mouse button down until the cursor changes to the closed hand and

stays that way. - move your line using the cursor keys - press Enter when your line becomes the line of best fit

(Hint: You may need to move all 3 parts of the line)

NOTE: An equation appeared with your line. Point at the equation and move the equation to

the bottom to see it.


3.3.2c: Introduction to Calculator TI-Nspire CAS Creating a Line of Best Fit Method (Continued) Draw a line of Best Fit – Regression Linear 1. Press Menu 2. Choose 3: Actions 3: Choose Regression 4. Choose 1: Show Linear (mx + b) Compare your line with the Linear Regression Place lines on each graph.


3.3.3: Choosing the Best Model - Could I Be a Forensic Scientist? Use the graph to examine each pair of variables from your data set. Sketch or print your scatter plots and place your graphs in the boxes below and describe the correlation.

Height vs. Length of Forearm Height vs. Armspan

Height vs. Handspan Height vs. Foot Length


3.3.3: Choosing the Best Model - Could I Be a Forensic Scientist? (continued) Inferring and Concluding 1. Describe the relationships in the graphs. 2. Which relationship shows the strongest correlation? 3. Do any of the graphs show no relationship?

4. Which model is the best predictor of the height? Give reasons for your answers. 5. What is the height of the skeleton? Give evidence to support your answer.


3.3.P - Practice 1. List two variables that will show a positive correlation. Eg. As the population of a city increases, garbage increases i. _________________________ ii. _________________________ 2. List two variables that will show a negative correlation. Eg. As the temperature increases, the amount of snow decreases i. _________________________ ii. _________________________ 3. List two variables that will show no correlation. i. _________________________ ii. _________________________


3.4.1: Creating Scatter Plots and Lines of Best Fit Test the hypothesis: The older you are, the more money you earn. 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. 1) Draw a line of best fit. Describe the trend in the data. 2) Does the data support the hypothesis? Give reasons to support your answer.

(Refer to the scatter plot.) 3) Explain why the data for ages over 65 do not correspond with the hypothesis. 4) Explain what the point (41, 35000) represents.

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


3.4.3: Forensic Analysis Anthropologists and forensic scientists use data to determine information about people. Scientists can make predictions about the height, age, and sex of the person they are examining by looking for relationships in large amounts of data. 1. Construct a graph of the length of the humerus bone vs. the length of the radius. 2. Circle the point on the graph that represents the data for a radius that is 21.9 cm long.

How long is the humerus? _____________.

3. Put a box around the point on the graph that represents the data for a humerus that

is 27.1 cm long. How long is the radius? ______________.

4. Describe the trend. 5. Describe the relationship: As the length of the radius gets longer, the humerus

____________________________.

6. a) Draw a line of best fit. b) Use the line of best fit to predict the length of the humerus, if the radius is 24.5 cm long.

Did you interpolate or extrapolate?

c) Use the line of best fit to predict the length of the radius, if the humerus is 25 cm long. Did you interpolate or extrapolate?

Length of Radius

(cm)

Length of Humerus

(cm) 25.0 29.7 22.0 26.5 23.5 27.1 22.5 26.0 23.0 28.0 22.6 25.2 21.4 24.0 21.9 23.8 23.5 26.7 24.3 29.0 24.0 27.0


3.4.J: Journal Activity Write a response to: What have you learned about how scientists use data in their jobs?


3.5.1: Investigations Investigation 1 – Ball Bounce Purpose To determine if there is a relationship between the drop height of a ball and its rebound height. Hypothesis I think that as the drop height increases, the rebound height ________________ because… ____________________________________________________________________________ Procedure • Attach measuring tape to the wall so you can measure the heights. • Drop the ball from various heights and record the rebound height. • Always drop the ball so that the bottom of the ball is just over the drop height. Models Conclusion • Describe the relationship in your own words • Was your hypothesis correct? • Describe any factors that may have affected your results.


3.5.1: Investigations (Continued) Going Further: Investigation 1 – Ball Bounce Answer the questions for your investigation. 1. Use your graph to determine the rebound height if the ball was dropped from a height of

120 cm.

2. How high was the ball dropped from if it rebounded 40 cm? 3. Sketch the line of best fit for your ball on the graph below. Sketch and label a new line

representing a: a) super ball (more bouncy)

b) beach ball (less bouncy) Give reasons for your answers.


3.5.1: Investigations (continued) Investigation 2 – Pendulum Swing Purpose To determine if there is a relationship between the mass of the swinging object and the time it takes for it to make five complete swings. Hypothesis I think that as the mass increases, the time to complete five swings will _____________ because… ____________________________________________________________________________ Procedure • Attach one weight (paper clip) to end of the pendulum string. • Release the pendulum from a 35° angle and start the timer. • Measure and record the length of time for five complete swings. • Repeat, after increasing the mass at the end of the pendulum by one paper clip each time. Models Conclusion • Describe the relationship in your own words • Was your hypothesis correct? • Describe any factors that may have affected your results.


3.5.1: Investigations (Continued) Going Further: Investigation 2 – Pendulum Swing Answer the questions for your investigation. 1. What do you notice that is different about this graph than the others? 2. Use your graph to find the length of time it takes if 8 paper clips are used.


3.5.1: Investigations (continued) Investigation 3 – Cylinder Size Purpose To determine if there is a relationship between the height of various cylindrical containers and their diameter. Hypothesis I think that as the height increases, the diameter ___________________ because… ____________________________________________________________________________ Procedure Measure and record the height and the diameter of the cylinders. Models

Conclusion • Describe the relationship in your own words • Was your hypothesis correct? • Describe any factors that may have affected your results. • Going Further: Do you think there would be a relationship between the diameter and the

circumference of the cylinder? Explain.



3.5.1: Investigations (continued) Investigation 4 – Bag Stretch Purpose To determine if there is a relationship between the height of a bag suspended by elastics over the floor and the number of books in the bag. Hypothesis I think that as the number of books increases, the distance of the bag from the floor

___________________because…________________________________________________

Procedure • Hang the shopping bag from elastics so that the bottom is about 1 m above the floor. • Measure and record the distance from the bottom of the bag to the floor. • Add one book to the bag. Measure and record the distance from the bottom of the bag

to the floor. • Repeat, adding one book at a time until all the books are in the bag. Models

Conclusion • Describe the relationship in your own words • Was your hypothesis correct? • Describe any factors that may have affected your results.


3.5.1: Investigations (Continued) Going Further: Investigation 4 – Bag Stretch Answer the questions for your investigation. 1. How many books will it take for the bag to touch the floor? 2. If a stretchier rubber band was used, how would this affect the graph? Sketch the line of

best fit for your investigation on the graph below. Sketch and label a new line representing a stretchier rubber band.

Give reasons for your answer.


3.5.1: Investigations (continued) Investigation 5 – Water Drains Purpose To determine if there is a relationship between the height of water in a container and the time it takes to drain the container. Hypothesis I think that as time increases, the height of the water in the container ___________________ because…_______________________________________________ Procedure • Fill the bottle with water up to the point where the container slopes towards the top. • Record the height of the water in the bottle. • When your timer is ready, poke a hole in the bottom and top of the bottle. • Record the height of water every 20 seconds until the water is past the cylindrical part of the

bottle. Models

Conclusion • Describe the relationship in your own words • Was your hypothesis correct? • Describe any factors that may have affected your results.


3.5.1: Investigations (Continued) Going Further: Investigation 5 – Water Drains Answer the questions for your investigation. 1. How long would it take for the water to drain completely from the bottle? 2. Estimate the height of the water after 30 seconds. 3. If the container had a larger diameter, but was still the same height, how would this

affect the graph if the hole in the bottom stayed the same? Sketch the line or curve of best fit for your investigation on the graph below. Sketch and label a new line or curve representing a container with a larger diameter.

Give reasons for your answer.


3.7.1: First Differences Problem 1 A. Jody works at a factory that produces square tiles for bathrooms and kitchens. She helps

determine shipping costs by calculating the perimeter of each tile. i) Calculate the perimeter and record your answers in the Perimeter column of the table.

Side Length (cm)

Perimeter (cm)

First Differences 1

2

3

4

5

ii) Describe what happens to the perimeter of each tile when the side length increases by one

centimetre. _______________________________________________________________

iii) Construct a graph of the perimeter vs. the side length. Include labels and titles. a) Which variable is the independent variable? b) Which variable is the dependent variable? c) Use the graph to describe the relationship

between the perimeter and side length of a tile. d) Describe the shape of the graph. iv) Calculate the first differences in the First

Differences column of the table. What do you notice about the first differences?

v) Summarize your observations. a) When the side length increases by one

centimetre, the perimeter increases by ________. b) The plotted points suggest a… c) The first differences are…


3.7.1: First Differences (continued) B. Jody is paid $8.50/hour to calculate perimeters.

i) Calculate her pay and record your answers in the Pay column of the table.

Number of Hours Pay ($)

First Differences 1

2

3

4

5

ii) Describe what happens to her pay when the number of hours she works increases by one

hour. ___________________________________________________________________

iii) Construct a graph of her pay vs. the number of hours she works. Include labels and titles. a) Which variable is the independent variable? b) Which variable is the dependent variable? c) Use the graph to describe the relationship

between her pay and the number of hours she works.

d) Describe the shape of the graph. iv) Calculate the first differences in the First


v) Summarize your observations. a) When the number of hours worked increases

by one, the pay increases by _________.

b) The plotted points suggest a… c) The first differences are…


3.7.1: First Differences (continued) C. Raj, another employee at the factory, also works with the tiles. He helps to determine the

shipping costs by calculating the area of each tile. i) Calculate the area and record your answers in the Area column of the table.

Length of sides (cm) Area (cm2)

First Differences 1

2

3

4

5

ii) Describe what happens to the area of each tile when the side length of a tile increases by

one centimetre. ____________________________________________________________

iii) Construct a graph of the area vs. the length of the

sides of the tiles. Include labels and titles. a) Which variable is the independent variable?

b) Which variable is the dependent variable? c) Use the graph to describe the relationship

between the area and the side length of the tile. d) Describe the shape of the graph. iv) Calculate the first differences in the First


v) Summarize your observations. a) When the side length increases by one centimetre,

the area increases by __________________.

b) The plotted points suggest a… c) The first differences are…


3.7.1: First Differences (continued) Problem 2 Chuck works on commission for sales. He earns $12.00 for each of the first 3 boxes he sells. He earns $24.00 each for boxes 4, 5, and 6, and $36.00 each for selling boxes 7, 8, 9, and 10. i) Calculate Chuck’s earnings for the following numbers of boxes of files and record your

answers in the Earnings column of the table.

Number of Boxes Earnings ($)

First Differences 1

2

3

4

5

6

7

8

9

10

ii) Describe what happens to his earnings when the number of boxes he sells increases

by one box.


3.7.1: First Differences (continued) iii) Construct a graph of his earnings vs. the number

of boxes he sells. Include labels and titles. a) Which variable is the independent variable? b) Which variable is the dependent variable? c) Use the graph to describe the relationship

between his earnings and the number of boxes he sells.

d) Describe the shape of the graph. iv) Calculate the first differences in the third column of the table. What do you notice about the

first differences? v) Summarize your observations. a) When the number of boxes he sells increases by one box, his earnings increase by… b) The plotted points suggest a… c) The first differences are…


3.7.2: Using What You Have Discovered Deep Sea Divers The table below shows data collected as divers descend below sea level. Calculate the first differences. Use the first differences to determine if the relationship is linear or non-linear. Check your solution by graphing. Include labels and titles.

The relationship is: Hot Air Ballooning The table shows data collected as a hot air balloon leaves the ground. Calculate the first differences. Use the first differences to determine if the relationship is linear or non-linear. Check your solution by graphing. Include labels and titles.

The relationship is:

Time (min)

Depth (m) First

Differences 0 -2

1 -4

2 -6

3 -8

4 -10

Time (sec)

Height (m) First

Differences 0 2

1 4

2 6

3 8

4 10


3.8: Unit 3 Review Scatter Plots 1. David went for a bike ride. The table shows his distance from home at different times.

2. The relationship between the Fahrenheit and Celsius scales can be seen in the table below. a) Graph the data. Plot Temperature (°C) along the horizontal axis and Temperature (F) along the y- axis. b) Determine the temperature in °C for each of the following: i) 100°F ______ ii) 150°F _____ c) Determine the temperature in °F for each of the following: i) 100°C ______ ii) 150°C _____

Temperature (°C) 0 20 40 60 80 100 Temperature (F) 32 68 104 140 176 212

a)Graph the data. Plot Time (min) along the horizontal axis and Distance (km) along the vertical axis. b)How far from home was David after each time? i) 15 min _____ ii) 55 min _____ c)After how many minutes was David each distance from home? i) 4 km _____ ii) 13 km ______


3.8: Unit 3 Review (Continued) 3. Below is a comparison of the number of oil changes in a year and the cost of auto repairs.

Oil Changes (per year)

Annual Repair Cost ($)

0 850 5 310 5 270 7 125 5 400 8 110 6 150 8 95 5 300 0 1290 1 560

10 0 3 400 2 650 1 750 3 450

a) Create a scatter plot. Plot Oil Changes on the x-axis and Annual Repair Cost on the y-axis. b) Draw a line of best fit. c) Use your line of best fit to:

i. Estimate the annual repair cost if someone had 2 oil changes. _________ ii. Estimate the number of oil changes for an annual repair cost of $200. _________

4. The graph shows the number of people that visited the Long Island Aquatic Club outdoor

pool each day from July 14th to July 27th. a) How many people were at the pool each day? July 17th _________ July 23rd _________ b) On what day(s) were each number of people at the pool? 20 _________ 45 _________ c) Which day had the highest attendance? _________ d) Which day had the lowest attendance? _________


3.8: Unit 3 Review (Continued) 5. The table shows, for 10 students, the number of hours their spent studying for their final

exam and the mark they received on the exam.

Study Time (hours)

10

0

1

3

5

4

7

9

8

5

Exam Mark (%)

100

50

60

70

65

80

95

80

90

55

a) Create a scatter plot for this data. Plot Study Time on the horizontal axis and Exam Mark on the vertical axis.

b) Draw a line of best fit.

c) Use your line of best fit to answer the questions:

i. How many hours would someone have to study to get a mark of 70%?

ii. What mark would someone get if they studied for 6 hours?

6. The following scatter plot shows the number of bacteria living in a culture at various

temperatures.

a) Draw a line of best fit. b) How many bacteria will live in a temperature of:

40 degrees Celsius? _________ 80 degrees Celsuis? _________

c) Predict the temperature if: 8 bacteria are in the culture ________ 14 bacteria are in the culture ________


3.8: Unit 3 Review (Continued) 7. The following graph represents the long term trends in smoking by teenagers aged 15 – 19.

a) Draw the line of best fit. b) What is the average number of cigarettes

smoked per day by 15-19 year old in 1989? c) In what year were 15-19 year olds smoking

an average of 14 cigarettes/day? 8. The table of values below shows life expectancy and year of birth.

Year of Birth 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 Life Expectancy

(years) 47.3

50.0

54.1

59.7

62.9

68.2

69.7

70.8

73.7

75.4

a) Create a scatter plot. Plot year of birth on the x-axis and life expectancy on the y-axis. b) Draw a line of best fit. c) What is the average life expectancy for someone born in: 1945? _________ 1965? _________ d) Predict the year of birth of someone with a life

expectancy of: 55 years. _________ 65 years. _________


3.8: Unit 3 Review (Continued) 9. The table below shows the height of a certain type of tree over several years.

Age (years) 4 10 6 12 11 8 13 13 10 8 3 6 Height (metres) 2 6 4 5 6 3 5 8 4 5 3 3

a) Create a scatter plot. Plot Age on the x-axis and Height on the y-axis. b) Draw a line of best fit. c) What is the height of this tree after: 5 years? _________ 9 years? _________ d) Predict the age of a tree that is: 1 m tall. _________ 7 m tall. _________ 10. The table shows the number of successful shots Alex made a various distances from the

basket. Distance from the basket (m) 3 5 7 8 9 10

Number of shots made 22 17 16 10 8 3

a) Create a scatter plot.

Plot Distance on the x-axis and Shots Made on the y-axis.

b) Draw a line of best fit. c) Predict the number of shots Alex will make

from 4 m?

d) Predict Alex’s distance if she makes 15 shots.


3.W: Definition Page Term Picture / Sketch / Examples Definition

Scatter Plot

Dependent Variable

Independent Variable

Continuous Data

Discrete Data

Variable

Linear

Correlation

Non-Linear

Finite Differences

First Differences


3.W: Definition Page (continued) Term Picture / Sketch / Examples Definition

Trend

Interpolate

Extrapolate

Algebraic Model

Graphical Model

Numerical Model

Hypothesis


3.S Unit Summary Page Create a Mind Map for the following graphing calculator process of:

• entering data in lists • making a scatter plot • setting the window • making a line of best fit


3.R Reflecting on My Learning (3, 2, 1)

3 Things I know well from this unit

2 Things I need explained more

1 Question I still have


3.RLS: Reflecting on Learning Skills

Students should be aware of the importance that these skills have on your performance. After receiving your marked assessment, answer the following questions. Be honest with yourself. Good Learning Skills will help you now, in other courses and in the future.

• E – Always • G – Sometimes • S – Need Improvement • N – Never

Organization • E G S N I came prepared for class with all materials • E G S N My work is submitted on time • E G S N I keep my notebook organized.

Work Habits • E G S N I attempt all of my homework • E G S N I use my class time efficiently • E G S N I limit my talking to the math topic on hand • E G S N I am on time • E G S N If I am away, I ask someone what I missed, • E G S N I complete the work from the day that I missed.

Team Work • E G S N I am an active participant in pairs/group work • E G S N I co-operate with others within my group • E G S N I respect the opinions of others

Initiative • E G S N I participate in class discussion/lessons • E G S N When I have difficulty I seek extra help • E G S N After I resolve my difficulties, I reattempt the problem • E G S N I review the daily lesson/ideas/concepts

Works Independently • E G S N I attempt the work on my own • E G S N I try before seeking help • E G S N If I have difficulties I ask others but I stay on task • E G S N I am committed to tasks at hand

Yes No I know all the different ways available in my school, where I can seek extra help. Yes No I tried my best. What will I do differently in the next unit to improve? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________


Course: Grade 9 Applied Mathematics (MFM1P) Unit 3 ... · Course: Grade 9 Applied Mathematics (MFM1P) Unit 3: Looking for Relationships, Lines and Curves of Best Fit

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