2 Introduction to Graphing We start with an activity dealing with graphing.We then turn to analytical and graphing skills.
3 Introduction to Graphing Analytical and Graphing SkillsA very basic question in many science investigations is, “How does one thing depend on another?”Each of these ‘things’ is a variable since we can change or vary its value. An important skill in both math and science is the ability to analyze the relationship between variables.We start by looking at some simple relationships.
4 VariablesFinding a relationship between variables means first identifying the variables, then performing a ‘controlled’ experiment to see how one variable affects the other.A controlled experiment involves being able to ‘neutralize’ all except the two variables you are investigating.Eventually, you wish to come up with a mathematical relationship (an equation) between the two variables. This often involves making a graph showing the relationship.
5 Constructing Knowledge from Data Eventually, you wish to come up with a mathematical relationship (an equation) between the two variables. This often involves making a graph showing the relationship.What you collect during the investigation are the data obtained from observations and measurements.You construct knowledge from these data using the graphing process. This is a recurrent theme in the science standards.
6 Constructing Knowledge from Data Height and Bounce Activity.Drop a ball from various heights, collect data of how high the ball bounces back.Make a data table to see if you can find a mathematical relationship between the height dropped and the height of bounce.
7 Constructing Knowledge from Data Height and Bounce Activity.The two variables in this activityare height of drop and height ofbounce.The variable that you have direct control over is called the independent variable. In this investigation it is the height of drop.The height of bounce is dependent on this variable and is called the dependent variable.We will now study two data tables.
9 Constructing Knowledge from Data How do you find a mathematical relationship between the height dropped and the height of bounce?If you study the data in Table A carefully, you will probably notice that the height of drop is 3 times the height of bounce.Table A
10 Constructing Knowledge from Data The data in Table B are more likely to be the kind that your students will find.Here the relationship is not so obvious so we need to do graphical analysis of the data to find the relationship.Table B
11 Constructing Knowledge from Data To construct our graph, we need to remember which is the independent variable, and which is the dependent variable.Recall that we control the Height Dropped so it is the independent variable. It is marked on the x-axis.We remember that the Height of Bounce depends on the height dropped, so it is the dependent variable and it is marked on the y-axis.
12 Constructing Knowledge from Data Then we plot the points. The data table gives us the (x, y) pairs to use.
13 Constructing Knowledge from Data Then we plot the points. The data table gives us the (x, y) pairs to use.
14 Constructing Knowledge from Data We can find the slopeof the line that indicatesthe relationship.We remember that theformula for slope isThis means that for every increase in x of 10 cm, the increase in y will be 4 cm (because 0.4 x 10 = 4).
15 Constructing Knowledge from Data If you find yourself unsure in any situation, you can also plot the graph using Microsoft Excel, add the trendline, and see what the more exact answer is. In this example, we got it exactly.
16 Constructing Knowledge from Data How steep are these roofs?Obviously the three houses are different. Find a way to measure them so that we can compare them.
17 Constructing Knowledge from Data Steepness implies slope which is rise/run.House House House 310/5 = /2= /10 = 0.5
18 Constructing Knowledge from Data Steepness implies slope which is rise/run.House 1 is ___ times steeper than house 3?House 1 is ___ times steeper than house 2?House 2 is ___ times steeper than house 3?
19 Constructing Knowledge from Data Steepness implies slope which is rise/run.House 1 is _4_ times steeper than house 3.House 1 is 0.4 times steeper than house 2.House 2 is _10_ times steeper than house 3.
20 Analyzing Data with Graphs The ability to make and interpret graphs and tables is very important in mathematics and science. Children should begin learning these skills as early in their education as possible.Graphs permit a visual description of how two variables are related and enable students to make predictions about the data and see the relationships between variables.There are five steps to be followed when preparing any graph.
21 Analyzing Data with Graphs Standard procedures for making graphsPrepare a data table showing numbers to be plotted. Note: it really helps if the independent variable is in the left column and the dependent variable in the right column. Remember our tables.Table A Table B
22 Analyzing Data with Graphs Standard procedures for making graphsLabel the horizontal (x) and vertical (y) axes with the same headings used in the data table.Choose an appropriate scale unit and number our axes so that all the numbers of the data will fit in. The scale for the horizontal and vertical axes need not be the same.
23 Analyzing Data with Graphs Standard procedures for making graphsOur graph with scales and labels
24 Analyzing Data with Graphs Standard procedures for making graphsGraph or plot the numbers
25 Analyzing Data with Graphs Standard procedures for making graphs5. Draw a straight line, smooth curve or bar graph using your points.Straight line—note that it DOES NOT pass through the origin.
26 Analyzing Data with Graphs Standard procedures for making graphs5. Draw a straight line, smooth curve or bar graph using your points.Smooth curve—exponential data
27 Analyzing Data with Graphs Standard procedures for making graphs5. Draw a straight line, smooth curve or bar graph using your points.Bar Graph--(Excel calls this a vertical column graph).
28 Learning from GraphsFrom Excel, we know that the equation of the line is y = 0.4x – 2.8.
29 Learning from Graphs A value read straight from the data table Using the equation y = 0.4x – 2.8, we can predict what how high the ball will bounce without actually bouncing it.First we try a data point we know. Our Data Table says that a height of 30 gave a bounce of 9.Using the formula we gety = 0.4 (30) – 2.8 = 9.2This is pretty close considering that the line we get from Excel is the line of best fit.
30 Learning from GraphsA height of 30 should give a bounce of 9. We measured this from the actual investigation.
31 Learning from Graphs A value within the data range Our highest height is 60 and our lowest height is 20. We can select a value between these two boundaries.Selecting within the boundaries is known as interpolation (inter = between the boundaries). We select 35.Using the formula we gety = 0.4 (35) – 2.8 = 11.2We have learned that a height of 35 should give a bounce of approximately 11 without doing it.
32 Learning from GraphsA height of 35 should give a bounce of about 11.
33 Learning from Graphs A value outside the data range Our highest height is 60 and our lowest height is 20. We can also select values outside the boundaries.Selecting outside the boundaries is known as extrapolation (extra = outside the boundaries). We select 65.Using the formula we gety = 0.4 (65) – 2.8 = 23.2We have learned that a height of 65 should give a bounce of approximately 23 without doing it.
34 Learning from GraphsA height of 65 should give a bounce of about 23.
35 Learning from GraphsBe very careful with extrapolation, especially if you are far away from the range of the data.Every time you do a calculation and get an answer, you need to stop and think about if the answer makes sense or not.The equation will always give you a mathematical answer but it may not be a meaningful answer.