Graphing and the Coordinate Plane

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Graphing and the Coordinate Plane
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• This is a chameleon
• His name is Sam. Sam likes to eat bugs and flies.
  He always has a lot to eat, because he is very
  good at finding the right place.

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Graphing Points on a Line

• Here is a line
• The arrows at each end show that the line really
  goes on forever.
• Each place on the line is called a point. A few
  of the points on this line are marked with red
  dots
• We can number some of the points to make them
  easier to find. The numbers get bigger from left
  to right
• Right now, Sam is sitting on point 4

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• On this line, only the even numbers are labeled.
  The other numbers are marked like this I. This
  mark is called a tick mark.
• If Sam wants to find point 5, what should he do?
• Sam starts at 0,
• and crawls forward. Sam knows that 5 is 1 more
  than 4, so he counts one tick mark after 4.
• Now Sam is at point 5.
• Sam makes a big green dot to show where he has
  been. He labels the dot too, so you can tell what
  it is.
• Sam just graphed point 5.

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Negative Numbers on a Line

• So far, when Sam wanted to graph a point, he
  started at zero and went forward. What would
  happen if he wanted to go the other way? After
  all, there are lots of points before the one we
  labeled zero.
• Let's label some more points, going backwards
  from zero. We'll use the "-" symbol to show that
  these numbers are less than zero.
• The numbers before zero on the number line are
  called negative numbers. We read a number like -4
  as "negative four."

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• Let's ask Sam to graph negative two.
• Sam always starts at zero.
• Sam knows he has to graph a negative number, so
  he turns around.
• He moves two units away from zero, because
  negative two is two less than zero.
• Finally, Sam marks the point he found with a big
  green dot.
• Sam found -2

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The Coordinate Plane

• The Coordinate Plane is made up of two number
  lines.
• Each of these lines is an axis. (Together they
  are called axes.) The axes are like landmarks
  that we can use to find different places in the
  plane.

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• We can label the axes to make them easier to tell
  apart.
• The axis that goes from side to side is the
  x-axis. It is sometimes called the horizontal
  axis because it runs horizontally.
• The axis that goes
• straight up and down
• is the y-axis. It is
• sometimes called the
• vertical axis because
• it runs vertically.

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The 4 Quadrants

• The x and y axes divide the plane into four
  sections. These sections are called quadrants.

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Let's zoom in on one corner of the plane. (This
corner is called the first quadrant.)

• We have marked some of the points on each axis to
  make them easier to find. The point where the two
  axes cross has a special name it is called the
  origin.
• The gray lines will help us find points. When you
  make your own graphs, you can use the lines on
  your graph paper to help you.

(0,0)
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We'll begin by graphing point (0, 0).

• Sam starts at the origin and moves 0 units along
  the x-axis, then 0 units up. He has found (0,0)
  without going anywhere!
• Sam marks the point
• with a green dot, and
• labels it with its coordinates.

Sam has finished graphing point (0, 0).
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Finding Points in the Plane

• We can find every point in the plane using two
  numbers. These numbers are called coordinates. We
  write a point's coordinates inside parentheses,
  separated by a comma, like this
• (5, 6). Sometimes coordinates written this way
  are called an ordered pair.
• The first number in an ordered pair is called the
  x-coordinate. The x-coordinate tells us how far
  the point is along the x-axis.
• The second number is called the y-coordinate. The
  y-coordinate tells us how far the point is along
  the y-axis.

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Let's try an example.

• Fly is sitting in the plane.
• Sam knows that the fly is at point (4, 3). What
  should he do?

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• Sam starts at the origin. So far, he has not
  moved along the x-axis or the y-axis, so he is at
  point (0, 0).

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• Because he wants to find (4, 3), Sam moves four
  units along the x-axis.

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• Next, Sam turns around and shoots his tongue
  three units. Sam's tongue goes straight up, in
  the same direction that the y-axis travels.

Sam has found point (4, 3). He eats the fly
happily.
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Next, let's graph point (0, 3).

• Notice that point (0, 3) is
• on the y-axis and its
• x-coordinate is 0. Every point
• on the y-axis has an
• x-coordinate of 0, because
• you don't need to move
• sideways to reach these points. Similarly, every
• point on the x-axis has a y-coordinate of 0.

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Let's graph the point (2, -2).

• Sam begins at point (0, 0).
• He moves 2 units along the x-axis.
• The y-coordinate of the point Sam wants to graph
  is -2. Because the number is negative, Sam sticks
  his tongue down two units. This makes sense,
  because negative numbers are the opposite of
  positive numbers, and down is the opposite of up.
• Before he leaves, Sam labels the point he
  graphed.

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Estimating Points

• Sometimes, the point you want to graph is in
  between points that are marked on the axes. When
  this happens, you must estimate where to put your
  point.
• For example, let's help Sam graph (5, 13) using
  these axes

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• Sam always starts graphing at the origin.
• The x-coordinate of the point is 5, so Sam needs
  to find 5 on the x-axis. 5 is exactly halfway
  between 0 and 10, so Sam moves between 0 and 10.

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• Next, Sam must find the y-coordinate, 13. He
  knows that 15 is halfway between 10 and 20. 13 is
  a little bit less than 15, so Sam tries to put
  his point a little below the halfway point.
• Sam labels the point so we can tell exactly where
  it is.

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Some Rules for All Graphs

• Unless you are just plotting a point, like we did
  with Sam, you will be graphing points that relate
  to a situation or thing. All of your graphs
  should have
• A title
• At the top of the graph and underlined
• It should represent what you are graphing (use
  your variables)

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Some Rules for All Graphs cont

• Labeled Axis
• Use a straight-edge to draw all lines
• Use the blue lines that are provided for you on
  the graph paper.
• Axes should be drawn a few lines in and up from
  the edge of the paper
• You must state what is represented on the x-axis
  and what is represented on the y-axis include
  units when necessary

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Some Rules for All Graphs cont

• The appropriate scale
• We need the graph to fill up the most paper. To
  find the right scale, we divide the range of the
  values by the number of tick marks on that axis.
  (Range is the highest value the lowest value).
  
• Then we round to a number that is easy to count
  by.

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Independent (Manipulated) Variable vs. Dependent
(Responding) Variable

• The independent variable causes a change in the
  dependent variable.
• The independent variable is always plotted on the
  x-axis and is usually listed first in a table
• The dependent variable is always plotted on the
  y-axis and is usually listed second in a table.

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How to Graph
Time vs. Distance

• Hold the graph paper the tall way.
• Title it using the variables.
• Label the axes dont forget to include units.
• Draw axes a couple of lines up and over
• Count the number of lines going across the x-axis
  starting at the zero mark
• 20 lines

Distance (m)
Time (min)
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Scale the x-axis
Time (min) Distance (m)
0
1
2
3
4
5
6
7
8
9
10

• Find your range for the x-axis (in science its
  the highest data point because we always start
  from zero)
• Time 10-010 so range is 10
• Divide the range of the x-axis by the of lines
  on the x-axis 10/200.5
• 0.5 is an easy-to-count by number so count EVERY
  blue
• line as 0.5

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Nice Counting Numbers

• Whole Numbers
• 1
• 2
• 5
• 10
• 15
• 20
• 25
• 50
• 100
• Etc.

• Decimals
• 0.1
• 0.2
• 0.25
• 0.5

Once in a while you might have to count by a
different no so nice number!
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Scale the x-axis
Time vs. Distance
Distance (m)
0 1 2 3 4 5 6 7 8 9 10
Time (min)
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Scale the y-axis
Time (min) Distance (m)
0 0
1 10
2 40
3 35
4 50
5 65
6 70
7 90
8 85
9 100
10 110

• Repeat for the y-axis tic marks 30 lines
• Range 110/303.6667 so round to 5 Count the
  y-axis by 5s

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Make Ordered Pairs
Time vs. Distance

• (0,0)
• (1,10)
• (2,40)
• (3,35)
• (4,50)
• (5,65)
• (6,70)
• (7,90)
• (8,85)
• (9,100)
• (10,110)

Plot data
120115110 100 90 80 70 60 50 40 30 20 10 0
Distance (m)
Relationship The average distanced traveled is
fairly constant for each time period.
0 1 2 3 4 5 6 7 8 9 10
Time (min)
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ReviewAll Graphs need

• A title
• At the top and underlined
• Labeled Axes
• Axes scaled appropriately (every tick mark
  increases by the same amount each axes can be
  scaled differently)

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Some Graphs need

• A Key (when necessary) If you are putting more
  than one line on a graph, it must have a key to
  distinguish the difference

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Lets try making our own graph from some given
information

• Example
• Karen drove her scooter at a constant speed of 5
  miles per minute. That means, for every 1 minute
  that she drove her scooter, she went 5 miles
  further from where she was. Draw a graph to
  represent Karens scooter trip for the first 5
  minutes that she drove.

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1.) Make a table to represent her time and
distance

• We know that for every minute that she drives she
  goes five miles, so lets match up the number of
  minutes with the amount of miles that Karen is
  away from her start point.

Time (min) 0 1 2 3 4 5
Distance (miles) 0 5 10 15 20 25
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2.) Write as ordered pairs

• (0,0)
• (1,5)
• (2,10)
• (3,15)
• (4,20)
• (5,25)
• These are the points that we will plot on our
  graph.

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• 3.) Draw the x and y axis on graph paper using
  the blue lines and a straight edge. (Be sure to
  leave enough room to fit the numbers for the tick
  marks and the words for your labels.)
• 4.) Title the graph. (Be sure to underline the
  title using a straight edge.)
• 5.) Label the axis. (Time (min) goes on the x
  and Distance (miles) goes on the y). Put all
  tick marks an numbers on your graph. You may
  only write the even numbers.
• 6.) Plot the points that you have in your table.

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Different Types of Graphs
Tables, charts and graphs are convenient ways to
clearly show your data.
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There are three basic graph forms.
Line Graph
Bar Graph
Circle (or Pie) Graph

• Notice on the next few slides how each of the
  following examples are used to illustrate the
  data.
• Choose the best graph form to express your
  results.

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Bar Graph

• A bar graph is used to show relationships between
  groups.
• The items being
• compared do not need
• to affect each other.
• It's a fast way to
• show big differences.
• Notice how easy it is
• to read a bar graph.

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Circle Graph or Pie Graph

• A circle graph is used to show how a part of
  something relates to the whole.
• This kind of graph is needed to show percentages
  effectively.

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Line Graph

• A line graph is used to show continuing data how
  one thing is affected by another.
• It's clear to see how things are going by the
  rises and falls a line graph shows.

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The same data displayed in 3 different types of
graphs.
Bar Graph
Line Graph
Circle (Pie) Graph
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Choosing the Right Graph

• Use a bar graph if you are not looking for trends
  (or patterns) over time and the items (or
  categories) are not parts of a whole.
• Use a pie chart if you need to compare different
  parts of a whole, there is no time involved and
  there are not too many items (or categories).
• Use a line graph if you need to see how a
  quantity has changed over time.  Line graphs
  enable us to find trends (or patterns) over time.

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More Examples of Different Graphs
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Circle Graph

• Used to show how the parts relate to the whole

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Bar Graph

• A bar graph contains horizontal or vertical bars.
  
• A good way to compare data that can be grouped
  into a category.
• The bars do not touch.

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Histograms

• Special type of bar graph
• Compares different intervals of data rather than
  categories
• The ranges used for the intervals must be the
  same size
• Bars should touch

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Line Graphs

• Drawn dot-to-dot
• Shows trends
• To compare trends between two or more things, you
  plot different lines for each and include a key

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Scatter Plot

• A scatter plot is a graph made by plotting
  ordered pairs in a coordinate plane to show the
  correlation between two sets of data.

y-variable
x-variable
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Scatter Plots

• Used to display data showing how the responding
  or dependent variable (y-axis) changes in
  response to the manipulated or independent
  variable (x-axis)
• Used when the manipulated variable is continuous
  (when there are measurements possible between the
  measurements you recorded interpolate)
• Used to go beyond the data
• by looking at trends
• extrapolate.

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Line of Best Fit

• Lines not drawn point to point
• Lines are continuous
• Used to show trends in data

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How do you determine the best-fit line through
data points?
y-variable
Try to get an even number of data points on
each side of the line
x-variable
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Positive Correlation

• A scatter plot describes a positive trend if, as
  one set of values increases, the other set tends
  to increase. 

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Negative Correlation

• A scatter plot describes a negative trend if, as
  one set of values increases, the other set tends
  to decrease.

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No Trend

• A scatter plot shows no trend if the ordered
  pairs show no correlation

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Example of scatter plot data
Time (minutes) Depth (cm)
2 7
4 8
6 13
8 19
10 20
12 24
14 32
16 37
18 38
20 41

• Emily measured the depth of water in a bathtub at
  two-minute intervals after the tap was turned on.
  The table shows her data. Make a scatter plot for
  the data.

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The graph shows a positive correlation, as time
increases So does depth.
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Another Scatter Plot Example
100 75 50 25 0

• Again, lines are not drawn point to point.

D i s t a n c e (km)
This graph represents distance slowing over
time or average deceleration.
0 10 20 30 40 50
Time (min)