1. Graphing and the Coordinate Plane

2. 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.

3. 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:

4. 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.

5. 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."

6. 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

7. 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.

8. 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.

9. The 4 Quadrants The x and y axes divide the plane into four sections. These sections are called quadrants.

10. 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)

11. 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).

12. 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.

13. 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?

14. 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).

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

16. 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.

17. 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.

18. 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.

19. 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:

20. 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.

21. 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.

22. 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)

23. Some Rules for All Graphs con’t 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

24. Some Rules for All Graphs con’t 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.

25. 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.

26. How to Graph Hold the graph paper the tall way. Title it using the variables. Label the axes; don’t 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 Time vs. Distance Distance (m) Time (min)

27. Scale the x-axis Find your range for the x-axis (in science it’s the highest data point because we always start from zero) Time: 10-0=10 so range is 10 Divide the range of the x-axis by the # of lines on the x-axis: 10/20=0.5 0.5 is an easy-to-count by number so count EVERY blue line as 0.5 Time (min) Distance (m) 0 1 2 3 4 5 6 7 8 9 10

28. Nice Counting Numbers Decimals: 0.1 0.2 0.25 0.5 Whole Numbers: 1 2 5 10 15 20 25 50 100 Etc. Once in a while you might have to count by a different no so nice number!

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

30. Scale the y-axis Repeat for the y-axis: tic marks = 30 lines Range = 110/30=3.6667 so round to 5; Count the y-axis by 5s Time (min)Distance (m) 00 110 240 335 450 565 670 790 885 9100 10110

31. Make Ordered Pairs (0,0) (1,10) (2,40) (3,35) (4,50) (5,65) (6,70) (7,90) (8,85) (9,100) (10,110) 0 1 2 3 4 5 6 7 8 9 10 120 115 110 100 90 80 70 60 50 40 30 20 10 0 Time (min) Distance (m) Time vs. Distance Plot data Relationship: The average distanced traveled is fairly constant for each time period.

32. Review: All 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)

33. 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

34. Let’s 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 Karen’s scooter trip for the first 5 minutes that she drove.

35. 1.) Make a table to represent her time and distance We know that for every minute that she drives she goes five miles, so let’s match up the number of minutes with the amount of miles that Karen is away from her start point. Time (min) 012345 Distance (miles) 0510152025

36. 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.

37. 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.

38. Different Types of Graphs Tables, charts and graphs are convenient ways to clearly show your data.

39. Circle (or Pie) Graph There are three basic graph forms. 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. Bar Graph Line Graph

40. 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.

41. 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.

42. 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.

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

44. 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.

45. More Examples of Different Graphs

46. Circle Graph Used to show how the parts relate to the whole

47. 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.

48. 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

49. 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

50. 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. x-variable y-variable

51. 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) interpolate 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 extrapolate.

52. Line of Best Fit Lines not drawn point to point Lines are continuous Used to show trends in data

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

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

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

56. No Trend A scatter plot shows no trend if the ordered pairs show no correlation

57. Example of scatter plot data 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. Time (minutes) Depth (cm) 27 48 613 819 1020 1224 1432 1637 1838 2041

58. The graph shows a positive correlation, as time increases So does depth.

59. Another Scatter Plot Example Again, lines are not drawn point to point. 0 10 20 30 40 50 100 75 50 25 0 Time (min) D i s t a n c e (km) This graph represents distance slowing over time or average deceleration.