Course: Grade 9 Applied Mathematics (MFM1P)
Unit 3:
Looking for Relationships, Lines
and Curves of Best Fit
Unit 3
Exploring Relationships: Lines and Curves of Best Fit
Section
Activity
3.1.1 Graphing Review
3.1.2 Relationships
3.1.3 Data Collection – Is there a Relationship
Here?
3.1.4 Class Data Sheet
3.1.5 Graphing the Data
3.2.1 Plotted Points
3.2.2 Scatter Plots – Types of Correlation
3.2.3 Line of Best Fit
3.2.P Practice
3.2.4 Relationships Summary
3.3.1 Could I Be a Forensic Scientist?
3.3.2a Introduction to FATHOM
3.3.2b Introduction to TI-83 TI-84
3.3.2c Introduction to TI-Nspire CAS
3.3.3 Choosing the Best Model
3.3.P Practice
3.4.1 Creating Scatter Plots and Lines of Best Fit
3.4.3 Forensic Analysis
3.4.J Journal Activity
3.5.1 Investigations
3.7.1 First Differences
3.7.2 Using What You Have Discovered
3.8
Unit 3 Review (Practice)
3.W
Definition Page
3.S
Unit Summary Page
3.R
Reflecting on My Learning
3.RLS Reflecting on Learning Skills
Page
3
4
5
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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?
As a person gets taller their armspan ______________________.
(gets wider, gets smaller, stays the same)
The longer a person's legs are ______________________ they run.
(the faster, the slower, will make no difference to how fast)
My Partner and I:
__ agree
__ disagree
__ agree
__ disagree
As a person's foot size increases, their walking stride
_____________________.
__ agree
(gets longer, gets shorter, stays the same)
__ disagree
As a person's forearm gets longer, their armspan _______________.
__ agree
(gets longer, gets shorter, stays the same length)
__ disagree
The longer a person's thumb is ______________________ their
index finger.
__ agree
(the longer, the shorter, will make no difference to the length of)
__ disagree
As a person gets taller, their foot size ______________________.
__ agree
(gets longer, gets shorter, is not affected)
__ disagree
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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3.1.4: Class Data Sheet
Maggie
134
22
130
20
120
Hand
span
(cm)
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
Name
Height
(cm)
Forearm
(cm)
Arm
span
(cm)
Foot
length
(cm)
Walking
Stride
length (cm)
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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Leg
Length
(cm)
Tibia Length (cm)
Number of Baskets
3.2.1: Plotted Points
Distance from the Basket
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
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
House Price ($)
1. The graph shows the plotted points
scattered.
• Agree
• Disagree
• Pass
Age of House
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
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.
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 %
Number
of Hours
75
1
70
2
68
3
73
2
59
4
57
4.5
80
1
65
3
63
3.5
55
4
85
1
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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2.5
3-10
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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.
a
d
g
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
b
e
h
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
c
f
i
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
Linear / Non-linear
Relationship / No Relationship
Positive / Negative
Strong / Moderate / Weak
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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: _______________
Note:
The independent
variable comes first in
the table of values.
Dependent variable: _____________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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 _________.
Hint:
You are interpolating when
the value you are finding is
somewhere between the
first point and the last point.
Extrapolate
When you extrapolate, you are making a prediction _____________ the data.
It often requires you to ____________the line.
These predictions are less reliable.
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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
•
•
•
What are the variables?
What exactly are you being asked to find?
Are there certain variables that would be more
useful than others?
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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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3.3.2a: Introduction to FATHOM Æ 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 Æ
Predict the Height for Year 20 Æ
Predict the Year when the Height will be 30 Æ
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
2
1
2
4
2
3
5
3
4
7
4
5
10
5
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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3.3.2b: Introduction to Calculator TI-83 or TI-84 Æ 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 Æ 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 Å and Æ
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 Æ and Å 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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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3.3.2c: Introduction to Calculator TI-Nspire CASÆ Creating a
Line of Best Fit Method
Getting to know your calculator
Green Letters
White buttons
- Alphabet
- Numbers
Grey Buttons
- Math Functions
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
left
right
show all pages
CTRL + mouse
page left
down
page right
show one page
Esc – go back a step
Ctrl z – undo the last step
Entering Data
1.
2.
3.
4.
Press HOME
Choose 6:New Document
Press Æ to Select No to Save Changes and press Enter.
Choose 3:Add Lists and Spreadsheets
5.
6.
7.
8.
Move the cursor up to the spot beside A
Enter the column heading eg: Height (minimum 4 characters, no spaces)
Press Enter
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
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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3.3.2c: Introduction to Calculator TI-Nspire CASÆ 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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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3.3.2c: Introduction to Calculator TI-Nspire CASÆ Creating a
Line of Best Fit Method (Continued)
Draw a line of Best Fit – Regression Linear
1.
2.
3:
4.
Press Menu
Choose 3: Actions
Choose Regression
Choose 1: Show Linear (mx + b)
Compare your line with the Linear Regression
Place lines on each graph.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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3.3.P - Practice
1.
2.
3.
List two variables that will show a positive correlation.
Eg.
As the population of a city increases, garbage increases
i.
_________________________
ii.
_________________________
List two variables that will show a negative correlation.
Eg.
As the temperature increases, the amount of snow decreases
i.
_________________________
ii.
_________________________
List two variables that will show no correlation.
i.
_________________________
ii.
_________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
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
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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
Length of
Radius
(cm)
25.0
22.0
23.5
22.5
23.0
22.6
21.4
21.9
23.5
24.3
24.0
Length of
Humerus
(cm)
29.7
26.5
27.1
26.0
28.0
25.2
24.0
23.8
26.7
29.0
27.0
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?
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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3.4.J: Journal Activity
Write a response to:
What have you learned about how scientists use data in their jobs?
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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…
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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
Differences column of the table. What do you
notice about the first differences?
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…
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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
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 area increases by __________________.
b) The plotted points suggest a…
c) The first differences are…
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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…
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
Time
(min)
Depth
(m)
0
-2
1
-4
2
-6
3
-8
4
-10
First
Differences
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.
Time
(sec)
Height
(m)
0
2
1
4
2
6
3
8
4
10
First
Differences
The relationship is:
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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3.8: Unit 3 Review
Scatter Plots
1. David went for a bike ride. The table shows his distance from home at different times.
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?
c)After
how many minutes was David each distance from home? i) 4 km
i)
15 min _____
_____
ii)
55 min _____
ii)
13 km ______
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)
Temperature (F)
0
32
20
68
40
104
60
140
80
176
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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212
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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
Annual Repair Cost
(per year)
($)
0
5
5
7
5
8
6
8
5
0
1
10
3
2
1
3
850
310
270
125
400
110
150
95
300
1290
560
0
400
650
750
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? _________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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.
6.
What mark would someone get if they studied
for 6 hours?
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 ________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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. _________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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3.8: Unit 3 Review (Continued)
9. The table below shows the height of a certain type of tree over several years.
4
10
6
12 11
8
13 13 10
8
3
Age (years)
6
4
5
6
3
5
8
4
5
3
Height (metres) 2
6
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
3
5
7
8
9
10
the basket (m)
Number of shots
22 17 16 10 8
3
made
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.
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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3.W: Definition Page (continued)
Term
Picture / Sketch / Examples
Definition
Trend
Interpolate
Extrapolate
Algebraic Model
Graphical Model
Numerical Model
Hypothesis
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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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
• E G S N
• E G S N
I came prepared for class with all materials
My work is submitted on time
I keep my notebook organized.
Work Habits
• E G S N
• E G S N
• E G S N
• E G S N
• E G S N
• E G S N
I attempt all of my homework
I use my class time efficiently
I limit my talking to the math topic on hand
I am on time
If I am away, I ask someone what I missed,
I complete the work from the day that I missed.
Team Work
• E G S N
• E G S N
• E G S N
I am an active participant in pairs/group work
I co-operate with others within my group
I respect the opinions of others
Initiative
• E G S
• E G S
• E G S
• E G S
I participate in class discussion/lessons
When I have difficulty I seek extra help
After I resolve my difficulties, I reattempt the problem
I review the daily lesson/ideas/concepts
N
N
N
N
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?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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MFM 1P - Grade 9 Applied Mathematics – Unit 3: Lines and Curves of Best Fit (DPCDSB Dec 2008)
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