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Skill 1.B
Plotting Data

Science Practice 1 is all about creating representations and 1.B indicates students will be asked to create quantitative graphs with appropriate scales and units, including plotting data.

This skill will be 0% of the multiple-choice section of the AP Physics 1 exam and will be only part of 20-35% of the free response dedicated to all of science practice 1. The individual skills do not have a % breakdown provided.

How will you recognize when to use this skill?

On the AP exam in May, Question 3 will be a lab-based question that should assess if you can properly graph data. There will be a table with data that you are asked to plot. This is different from when you are sketching a graph because this skill starts with a set of data points. We will also do this throughout the year on our lab investigations.

Example Prompt:

"On the grid shown in Figure, plot the data or variations of the data that can be used to determine the acceleration of the object from a straight-line graph with the position on the y axis. Scale the graph, as appropriate. Clearly label the horizontal axis, including units." (underlining added for reference below)

Skills Assessed when Plotting Data

Note the underlined parts of the prompt above remind you of the requirements.

Deciding what to plot on each axis

Sometimes the question will tell you what to graph (for example, graphing the data in a table), but more often you will be asked what quantities to plot in order to create a "straight-line graph" that can be used to solve for a value. To decide what to graph you should use your other skills, like 2.A Derivation to identify an equation that will link the data provided in the lab to the desired quantity. Arranging that equation to compare with the equation for a straight line will help you know what to put on each axis.

In our example above we are given position and time data and asked to graph a straight line with the position on the y axis. The work at right comparing the physics equation linking displacement and time to the slope intercept form of a linear equation shows that in order to create a straight line we need to plot t^2 on the x axis, Δx on the y, and then 1/2 a will be the slope of the line. So, we need to fill in the empty column with [time (s)]^2 and square all the values.

This process is called linearization, and it often takes many tries to feel any confidence in. Be patient but keep coming back to practice and learn it. You can find more detail and some practice on my Linearizing Data page.

The equation that links position, time, and acceleration is Δx = (v_0)t + 1/2a t^2. If it can be assumed from the problem that the initial velocity is zero, then this can be simplified to Δx = 1/2a t^2. In the equation y = mx +b there are 2 variables, so we want to make sure that our equation with position and time has it solved for the variable we need on the y axis, which it is Δx and so that is already set up right.

2. Scaling the graph

Requirement: Your graph must have a uniform linear scale that will fit your data points to use at least 1/2 of the graph space on each axis.
Having a uniform linear scale means each horizontal box is worth the same amount; the same applies to the vertical scale, although it can be a different scale on each axis. PLEASE do not just label your X values from your data set along the X axis. This will always make a linear graph, but it will always be meaningless.

How to scale a graph by hand:

Find the spread you need to show on your graph.
- I often use zero and the maximum value.
- In the example shown the vertical axis span is 3.1 m and the horizontal axis is 25 s^2
Count the number of vertical and horizontal spaces on your graph.
- On the grid used to graph in the example I counted 8 major spaces horizontally and 7 vertically.
Divide the spread by the number of spaces on each axis and then round up to a convenient number to work with.
- This convenient number has to be less than 2x the result you got by dividing. It should also be easily divided into the number of minor tick mark segments on the graph.
- For the Y axis 3.1/7 = 0.44, I rounded up to 0.5 for each major space and 0.1 for each minor. That scale was already set.
- For the X axis .25/8 = .03125, I rounded up to .04 for each major space and .0125 for each minor. If could have chosen something like .035 for the major axis with a minor spacing of 0.07, but I find 0.0125 easier to work with. I could also have gone up to .05 or even .06 for the major units (since they are less than 2x the result of .03125), but the bigger I go the more cramped the graph will be.

3. Labeling the axes

This is the easiest part, and yet is the most frequent error made! Label each axis with a word or symbol to represent the measurement AND the units. These can have a scale included for really large numbers or really small numbers (like x10^-2 as shown in the example), but do not need that.

4. Plotting the data

- Each data point should be plotted as a dot. Be careful in making sure you do not flip the numbers between X and Y, read your scale correctly, and estimate the position correctly between lines.

5. Prepping to Analyze

- Draw a best fit line if there is a trend demonstrated by the data or if you will be finding.
  - Best fit lines can be linear or curved, but should never be jagged connect-the-dot style lines. If linear, use a ruler.
  - Best fit lines show a general, smoothed out trend shown by the data.
  - The best fit line should pass through/close to as many of the points as possible while keeping a balance of deviations above and below the line.
  - Once the best fit line is drawn, we ignore the data points except to discuss error.
- Consider what each graph feature means and decide if any of these can help you answer a follow up question.
  - Y intercepts are often the starting value
  - X axis will represent when 0 of the y axis value is reached, which may have a special meaning, like hitting the ground, stopping, having no potential energy, etc.
  - Slope will involve something found by dividing the Y and X axis quantities.
  - Area will involve something found by multiplying the Y and X axis quantities.

Practice

Graph 1: What if anything is wrong with the graph?

There are 2 errors in this graph.

1) There is no vertical axis labeling.

2) The vertical axis is not scaled to take up half of the vertical space.

Graph 2: What if anything is wrong with the graph?

This graph has 2 errors:

1) The horizontal axis might look like it is evenly spaced, but it is not. The first major line has a value of 0.5 seconds and the next major line is at 0.333 s. These are not of equal value and are not actually increasing either. Never use fractions on your axis labels!

2) The axis labels on this only have one part: the units. They need a measurement label AND a unit label. The vertical axis should be something like height (meters) or distance (meters) and the horizontal axis should say time (seconds). Common symbols are fine to use as labels as well, like t (s) for the time axis.

Graph 3: What if anything is wrong with the graph?

This data is linear with random error. The lines connecting the dots are not a smooth approximation of that the data shows, instead emphasizing the error. The student should have used a ruler to draw a best fit line.

Graph 4: What if anything is wrong with the graph?

Notice that the first 3 data points and all the rest are below the line? This student forced the best fit line to go through 0,0, but there should be a + intercept and a lower slope that can be closer to mmore of the points and represent the general trend better (by having a balance of above and below deviations both at the beginning and the end)

Graph 5: What if anything is wrong with the graph?

2 issues on this one:

1) the vertical axis is not scaled properly because the last point is outside the graph area.

2) The horizontal axis label for the unit is does not match the quantity. m/s is the correct unit for velocity and if you square velocity its units will be squared also, so it should be (m/s)^2 or m^2/s^2 for the units.

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Skill 1.BPlotting Data