1 Drawing Graphs Introduction In recent years, the CSEC Integrated Science Examination, Paper 02, deals with graphical analysis. That is, data is presented in a tabular format and the student is asked to draw a graph and interpret information from it. Graphs are also commonly used when representing the results of a laboratory experiment and is an integral part of the School Based Assessment (SBA). What is a graph? 1. A diagram that exhibits a relationship, often functional, between two sets of numbers as a set of points having coordinates determined by the relationship. 2. A pictorial device, such as a pie chart or bar graph, used to illustrate quantitative relationships. (http:www.thefreedictionary.com/graphs) Basically, a graph is a representation of the relationship between two or more variables. Types of Graphs The following are the most common graphs that may be encountered at CSEC: Pie chart Bar graph Line graph (The focus will be on this type)
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3 Pie chart video: http://study.com/academy/lesson/what-is-a-pie-chart-definition-examples-quiz.html Bar graph video: http://study.com/academy/lesson/bar-graph-definition-types-examples.html Line chart video: http://study.com/academy/lesson/what-is-a-line-graph-definition-examples.html Interpreting graphs: http://study.com/academy/lesson/reading-and-interpreting-line-graphs.html Or visit http://intscience.weebly.com/cool-videos.html to download and watch these videos. Parts of a graph
4 Getting started 1. Look at the data to be plotted. Example: Table 1 showing the distance covered by the Stones family in a seven hour road trip. Time (h) 1 2 3 7 Distance (km) 10 20 30 70 In the example above, Time and Distance are the two variables being compared. 2. Insert relevant information on the graph page, especially if it is being done in the SBA notebook. Information such as Date, Lab # and Page. The Date and Lab # should be the SAME as the Date and Lab # of the lab report. 3. The height (vertical distance) of the graph page is greater than the width (horizontal distance). That is, the number of grids vertically is greater than the number of grids horizontally. 4. A 1cm grid consists of 5 subdivisions vertically and horizontally while a 2cm grid consists of double that amount, that is, 10 subdivisions vertically and horizontally.
5 Date: Lab: Page: Vertical distance 2 cm grid Horizontal distance 1 cm grid
6 Axes A graph consists of two axes. The horizontal line drawn at the bottom of the graph is called the x-axis. The vertical line drawn at the leftmost part of the graph is called the y-axis. It should be noted that the x and y axes must touch one another. In the case of CSEC Integrated Science, this is the only orientation that the axes will take since negative values are never used. (Refer to Parts of a graph ) Which variable goes on the x-axis and which on the y-axis? Variables Variables can be classified as either Independent or Dependent. The independent variable is put on the x-axis. It is the variable that does not change or depend on another. The dependent variable is placed on the y-axis. It is the variable that is being measured or is dependent on the other. In the case of our example, Time is the independent variable and Distance is the dependent variable since Distance travelled cannot determine the Time, and likewise, Time keeps ticking regardless of how far the family travelled. x-axis y-axis Time (h) 1 2 3 7 Distance (km) 10 20 30 70 In most cases, the table of data lists the independent variable (x-axis) first! Before drawing in the axes lines, determine which variable contains the smaller range. Remember that a graph page contains different number of grids vertically and horizontally.
7 Looking once again at the example, both orientations may be used since both have the same ranges, that is, Time ranges from 1 to 7 whereas Distance ranges from 10 to 70, which is essentially 7 units each. Only if the range on the x axis is bigger than the y axis, Orientation 2 should be used. Orientation 1 x-axis contains the smaller range Orientation 2 x-axis contains the larger range
Distance (km) 8 Labels After the axes have been drawn in (pencil and ruler), they should both be labelled. This is done by simply stating the variable and its corresponding unit in brackets. Time (h) The axis label should be placed leaving room to insert the values for each variable. Title At this point where the axes have both been labelled, a title of the graph can be inserted at the top of the graph page. The title is stated as follows: Graph 1 showing Dependent variable against (or versus) the Independent variable. Example: Graph 1 showing Distance (km) against Time (h)
9 Remember the title of the table in the example? Table 1 showing the distance covered by the Stones family in a seven hour road trip. Simply replace Table 1 with Graph 1 to get the title of your graph! Example 2: Graph 1 showing distance covered by the Stones family during a seven hour road trip. Graph 1 showing Distance covered by the Stones family during a seven hour road trip.
10 Scale Values corresponding to each variable must now be inserted on each axis. This must be done at equal intervals away from the starting point (usually zero) trying to occupy as much of the graph page as possible. Looking at the example: Time (h) 1 2 3 7 Distance (km) 10 20 30 70 The x-axis ranges from 1 to 7 but some graph pages have up to 18 1cm intervals. Therefore, it would be better to space markers at 2cm intervals to maximize space. Time (h) Markers at 2cm intervals Next, insert values at each marker using a simple common multiple. Note: A common mistake of students is to insert the exact values found in the table.
11 1 2 3 7 1h 1h 1h 3h? Time (h) 1 2 3 4 5 6 7 Time (h) Similarly, insert the markers and values on the y-axis. The y-axis goes from 10 to 70 therefore a multiple of 10 is used. Since the y-axis can have up to 23 1cm intervals, it would be better to space out values every 2cm as well.
Distance (km) 12 70 60 50 40 30 20 10
13 It is not necessary to stop at the last value. Additional values may be inserted to make the graph look complete or in case any extrapolation of data is required. The Scales used on each axis must be inserted on the graph where it does not interfere with the graph itself. This should, therefore, be inserted AFTER the graph has been completed. (See Plotting Coordinates ) In General: Scale: On the x axis cm represents unit(s) On the y axis cm represents unit(s) Example: Scale: On the x axis 2cm represents 1 unit On the y axis 2cm represents 10 units OR Scale: On the x axis 2cm represents 1 hour On the y axis 2cm represents 10 km
14 Plotting Coordinates A coordinate indicates the point where two variables meet or intersect. The x axis value is stated first in the form of (x, y) Time (h) 1 2 3 7 Distance (km) 10 20 30 70 From the table, the first coordinate is 1, 10. (Re: Time is on the x axis and Distance on the y axis) Step 1 Look for the x value (1 on the x axis). This value runs vertically.
15 Step 2 Look for the y value (10 on the y axis). This value runs horizontally. Point of Intersection (1, 10) The point where both lines intersect is the (x, y) coordinate. In this case, (1, 10). The point is marked by either an x or an. Note: the Red and Green dotted lines serve as a guide and should not be drawn in the graph.
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17 Step 3 Continue plotting the points from the table. 2 nd coordinate (2, 20) 3 rd coordinate (3, 30) and so on. Step 4 Draw in the graph line. This can be done by joining the coordinates using a ruler and pencil. In most cases, the graph line expected is stated. For this example, the best straight line is drawn. (See Graph Lines )
18 Scale: On the x axis 2cm represents 1 h On the y axis 2cm represents 10km The Scale can now be inserted in a clear part of the graph! Two or more graph lines can be inserted using the same axes. In this case, a Legend should be inserted to differentiate between them. This can be done using different colour ink for the graph line (preferably black or blue), different plot symbols or labelling the end of the graph lines.
19 Example: Legend BOYS GIRLS Interpretation of Scale Subunits Plotting of the coordinates in the example given is straight-forward, but, supposed there were values that did not run along the labelled lines? Time (h) 1 2 3 4.5 7 Distance (km) 10 20 30 45 70 Let s look for the x value of 4.5. 1cm 4.5 There are 10 subdivisions between 4 and 5. Since 4.5 (or 4 ½) is half-way between 4 and 5, this value will fall on the 1cm marker.
20 Another way of finding the value is to determine the value of each sub-division. 1/10 = 0.1 Difference between the upper value (5) and the lower value (4) No. of subdivisions Value of each subdivision 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 Time (h) So, 4.5 is found on the 5 th subdivision between 4 and 5. Let s look for the y axis value of 45. 45 Once again, 45 is mid-way between 40 and 50. That is, it lies on the 1cm mark between the upper value (50) and the lower value (40).
Distance (km) 21 The value of each subdivision could also be determined as before. 10/10 = 1 Difference between the upper value (50) and the lower value (40) No. of subdivisions Value of each subdivision 50 49 48 47 46 45 44 43 42 41 40
22 The new coordinate on the graph appears as follows: Let s assume that the scale was altered so that 1cm represented 1 unit instead of 2cm representing 1 unit. This would mean that the number of subdivisions between 4 and 5 on the x axis would now be 5 instead of 10. Likewise, the number of subdivisions between 40 and 50 on the y axis would also be 5 instead of 10. This will change the value of each subdivision.
Distance (km) 23 On the x axis: 1/5 = 0.2 Difference between the upper value (5) and the lower value (4) No. of subdivisions Value of each subdivision 4 4.2 4.4 4.6 4.8 5 Time (h) On the y axis: 10/5 = 2 Difference between the upper value (50) and the lower value (40) No. of subdivisions Value of each subdivision 50 48 46 44 42 40 Note: There is NO 4.5 line on the x axis and NO 45 line on the y axis. So where do these values go?
Distance (km) 24 (4.5, 45) Time (h) Think about it!!! What if 1cm represented 5 units. What would be the value of each subdivision? What if 4cm represented 10 units. What would be the value of each subdivision? What if 2cm represented 25 units. What would be the value of each subdivision? Difference bet. upper & lower No. of subdivisions Value of each subdivision 5 5 1 10 20 0.5 25 10 2.5
25 Reading the graph Extrapolation of data The graph line represents the relationship between two variables. It can be used to find unknown values if the corresponding value is known. Meaning, if the x value is known, the y value can be found using the graph line and vice versa. Note: The graph line does not have to be straight to do this. Example: What distance would the Stones family travelled in 3.7 hours? (3.7, y)
26 Find the value 3.7 on the x axis. Draw a line along it until it touches the graph line. (red arrow) At this point, draw a line horizontally until it touches the y axis. (green arrow) From the example, the corresponding y value is 37km. In other words, the Stones family travelled 37km in 3.7 hours. (3.7, 37) Example 2: How long would it take the Stones family to travel 62km? (x, 62)
27 Find the value 62 on the y axis. Draw a line along it until it touches the graph line. (red arrow) At this point, draw a line vertically until it touches the x axis. (green arrow) From the example, the corresponding x value is 6.2 hours. In other words, it took the Stones family 6.2 hours to travel a distance of 62km. (6.2, 62) There are times where values may NOT fall on the graph line. In these cases, the graph line could be extended to fall within the range of the unknown value. Unlike before when plotting coordinates, the lines drawn on the graph to find unknown values should be left there. This shows the examiner where the values came from in your answer. Reading the lines Straight lines The shape of the graph gives an overview of the relationship between two variables. Use the variables on both axes to describe the shape of the line, read from left to right. Let s look at a few relationships commonly found. Keep in mind that there are different ways to say the same thing. Proportional relationship As the x value increases, the y value increases by an equal number of units. Example: If x increases from 1 to 2 (doubles) by 1 unit, the y value increases from 10 to 20 (doubles) by 1 unit.
28 Inversely Proportional relationship As the x value increases, the y value decreases by an equal number of units. Example: If x increases from 1 to 2 by 1 unit the y value decreases from 30 to 20 by 1 unit. Weak positive relationship As the x value increases, the y value increases by a smaller amount. Example: If x increases from 1 to 2 by 1 unit the y value increases from 22 to 25 by less than 1 unit. Strong positive relationship As the x value increases, the y value increases by a greater amount. Example: If x increases from 1 to 2 by 1 unit the y value increases from 10 to 30 by greater than 1 unit.
29 Weak negative relationship As the x value decreases, the y value increases by a smaller amount. Example: If x decreases from 1 to 2 by 1 unit the y value decreases from 11 to 9 by less than 1 unit. Strong negative relationship As the x value decreases, the y value decreases by a greater amount. Example: If x decreases from 1 to 2 by 1 unit the y value decreases from 40 to 15 by greater than 1 unit. Constant / No change As x increases, y remains the same
Temp ( 0 C) 30 Constant / No change As y increases, x remains the same Curves Gradual then Drastic then Drastic then Gradual then drastic increase gradual increase gradual decrease drastic decrease Different relationships on the same curve / lines Use a combination of straight and curve line graphs to describe the graphs below. Remember to use the variables to describe what happens to the graph line as one moves from left to right. Example 1: Melting Curve Time (mins)
Height (cm) 31 General description: At zero mins, there is a decrease in temperature from 40 to 30 0 C until 1.5mins where there is no change until 2.5 mins. The temperature then falls from 25 to 10 0 C until 4 mins. Example 2: Growth Curve General description (no values used): There is no change in height initially then a drastic increase which eventually slows down as age increases. Age (years)
32 Other common graphs What about graphs that only have ONE set of numerical values? Example: Bar graph Table 1 showing carbon dioxide emissions per capita Country Australia US Canada Russia CO 2 emissions 18.6 18.0 16.3 12.0 (metric tonnes) Note: The alphabetical values are placed on the x axis at equal intervals apart. Sometimes names are replaced by years. The same procedure applies by just spacing the data given at equal intervals apart regardless of value.
33 Example 2: Pie Chart Table 1 showing CO 2 emissions in the United States of America Agriculture Commercial and Residential Electricity Industry Transportation 8% 11% 33% 20% 28% https://geog397.wiki.otago.ac.nz/images/9/9f/sources-overview.png In CSEC Integrated Science Paper 02, there is the tendency to list TWO y axis values. However, the graph usually involves plotting just one set of coordinates. Example 3: CSEC June 2011 The x axis values are the years (again this is usually placed first on a table). The two y axis values are China and USA.
34 If asked to plot a graph to represent the data for China, the information taken from the table would be as follows: Similarly, if asked to plot a graph to represent the data for USA, the x axis values remain the same but the y axis shifts to the column with USA values:
35 Examples from CSEC Integrated Science Try the following questions from CSEC on new graph pages. Ensure that all the elements are present. Labelled Axes Title Scale Legend You can download these past papers in its entirety at The Science Exchange http://intscience.weebly.com/downloads.html June 2011 *Plot a graph to represent the data for China using the same axes and scale. USA has already been plotted. (Remember to use different plotting symbols or ink. Similarly, graph lines can be labelled.)
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37 June 2012
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39 June 2013 *Plot a graph to represent the data for the tobacco smokers
40 June 2014
41 June 2015
42 June 2016
43 Conclusion It is the hope of the author that the preceding information will not only assist students, but also teachers when encountering graphs at all levels of Integrated Science. Recently, the Ministry of Education of Trinidad and Tobago has reformulated the Science Curriculum to stress on the various aspects of Integrated Science, that is, Biology, Chemistry and Physics. However, the fact that CSEC Integrated Science still exists, the guidelines remain 100% relevant to the syllabus. Having dealt with students who have not understood the concept of graphs for some time, I am confident that these guidelines will cut down on the time spent drawing graphs in the classroom, increasing students self-efficacy and teacher motivation. Feel free to visit my website at http://intscience.weebly.com/ and Facebook page The Science Experiment for your comments / questions / suggestions. Ron Mahabalsingh July 19 th 2016