Author: Jon Valasek
Source: This lesson was first conducted at the Mississippi School for Math and Science in 1991 by the author and Claudia Carter in a team taught science and mathematics problem solving course.
Students will determine the density of one drop of water using experimental laboratory methods and graphing techniques, and then independently find the density of another liquid such as cooking oil. This lesson can be teacher directed or conducted as an open investigation.
What should students know as a result of this lesson?
What should the students be able to do as a result of this lesson?
Introduce this investigation by posing the following problem to the class: Throughout the year we will be using micropipets filled with various liquids to carry out experiments. We will be asked to add 1 milliliter of this or 5 milliliters of that so we need to know how much to add from the micropipet. But these micropipets are not marked. So how can we use these micropipets to measure volumes of liquids? Accept all answers and choose, "by counting the number of drops."
Assessment: Note the participation of the students and ask "Do you think we can use these micropipets to accurately measure a quantity of liquid?"
Then ask, "So if we know the number of drops how can we determine the volume measured in milliliters of a quantity of drops?" This question should cause a number of responses to be generated. Divide the class into small working groups of 2 or 3 and have them brainstorm the answer to the question.
Have each group report their solutions. A possible solution that you can direct the groups to consider would be to use a graduated cylinder and count the number of drops required to fill the cylinder to the 1, 2, 3 etc. milliliter marks.
Have groups perform the method they adopted and record results.
Assessment: The results of this exercise are a valuable teaching moment. Some of the items the teacher should consider are: how close did the students follow the procedures? How was the volume measured? To the nearest whole, tenth, or hundredth of a milliliter? How was the volume recorded? What number of drops was recorded? Was it at regular intervals? What was the lowest and highest number of drops for which volume was recorded? Could the students measure the volume of one drop of water? Why not? A thorough discussion of these questions should allow the students to reach a consensus on volume measurement considerations and data recording procedures. You could reinforce the results of the discussion by having selected individuals restate the major premises of measurement and data recording. To wrap up this portion of the activity, ask, "Does the way we measure and what type of data depend on what problem we are trying to solve?" The answer is that it should not influence the way we measure data but does affect the data we collect. We are interested in volume and number of drops to answer the question, "How many milliliters are in one drop of water?" We should not bias our results by reading the graduated cylinder improperly.
Then ask, "Will the procedure you devised be accurate and work for all volumes?" Demonstrate to the class by taking a pipet filled with water and putting one drop from the pipet into an empty 10 mL graduated cylinder. Note that the orientation of the pipet determines the amount of water that is dispensed. A good practice is to hold the pipet in a vertical position at all times when dispensing the liquid to ensure uniformity. Pass the graduated cylinder around the classroom and ask if anyone can tell you the volume of that drop of water.
You could use this opportunity to discuss the concept of accuracy as relates to the measuring instrument you are using. This discussion would include the scale of the measuring instrument and interpretation of values between two lines of measurement.
Assessment: Have the students read and record the volume of water in a beaker, graduated cylinder, and buret.
The groups might want to redo their number of drops, volume exercise but before they do, have the groups brainstorm a way to accurately determine the volume of any number of drops of water. Accept reports from the groups by listing possible procedures on the board. The groups' discussions should focus on measuring the mass, volume, and number of drops of water; developing a data table to record the measurements; and after discussion reach a consensus on the procedure to follow. One possible procedure would be to place an empty graduated cylinder on a balance and zero the balance. Next drop the same increment of drops into the graduated cylinder and mass and record the volume of each increment. As an example, add 5 drops and mass and read volume, add 5 more drops and mass and read volume and so forth. Create a table to show the volume in milliliters, mass in grams, and number of drops.
Assessment: Check that each group has recorded the mass, volume, and number of drops of water for a number of different volumes of water.
After each group has preformed the measurements, discuss what to do with the data in order to answer the question originally posed.
Through this discussion pose that there must be a better way to find the volume of any number of drops of liquid.
While the focus of this lesson is not on 'density' at this point, the teacher could also introduce the concept that all matter has volume and mass and that substances' mass to volume ratio (mass divided by volume) called density is known. So if we know water's mass we can find its volume. Density will be discussed in the elaboration portion of the lesson.
Guide the discussion to relate that scientists sometimes show data in graphs to convey relationships and derive mathematical formulae.
Explain that graphs have a vertical and a horizontal dimension called an axis. Each axis has a quantity scale that includes the range of the data. The scale is in equal increments normally from zero to a number beyond the range of the data. Ask students to determine the scale of the volume and number of drops. Assess that they correctly develop a scale for each variable.
Explain that the independent variable (what they are changing) normally goes on the horizontal axis (x-axis) and the dependent variable (what they are observing or measuring) goes on the vertical axis (y-axis). Ask, "So where does the volume and number of drops go on the graph?" The correct answer is that the volume should go on the y-axis and the number of drops should go on the x-axis. Assess: Make sure that the student put the appropriate label on each axis.
Now that the students have graphs of volume and number of drops and mass and number of drops, ask, "Using your graphs find the volume of 20 drops of water." Allow each group to report and notice the differences, if any.
At this point introduce the concept of significant digits. How many digits should the answer be reported in? Normally a 10 mL graduated cylinder has markings to the nearest tenth of a milliliter, so the volume could be read to the nearest hundredth of a milliliter. So the volume of 20 drops of water could be 1.02 mL and should be reported as same.
Assessment: Now that the students are able to read their graphs, have them determine the volume of 37 drops of water. This will be hard to do since their data is for multiples of 5 drops.
Assess to see that everyone has made the lines properly. Now instruct the students to read where the parallel line meets the y-axis and indicate that the intersection is the volume for 37 drops of water.
Assessment: Have students use their graphs to determine the volume of 1 drop of water. Check for accuracy.
Next ask, "Is there a more accurate method to determine the volume of 1 drop of water?" Entertain all responses. Suggest that one could use the mathematical formula for a straight line to make the determination. Have students recall that the equation for a straight line is y = mx + b, where m is the slope (rise over run) and b the y-intercept. Since b is equal to 0, the equation becomes y/x = slope. Have the students find the slope by taking two data points and substitute the values into the equation (y2 - y1) divided by ( x2 - x1 ).
Assessment: Check for proper substituting and solving for the slope, then have students find the volume of one drop of water using the formula y (volume) divided by number of drops equals slope. Check to see that the volume of one drop of water is close to .040 mL (the value for a straight stem Beral(tm) pipet).
Now that the students have demonstrated proficiency in graphing volume versus number of drops have them collect data on mass and number of drops using procedures previously developed.
Assessment: Check to see that the data tables and resulting graphs are properly accomplished by visiting each group. The data and resulting graph should be similar to the volume versus number of drops graph. Also have the students establish a line of best fit, write the straight-line equation, find the slope of the line, and solve for the mass of one drop of water.
All matter has mass and occupies space. The ratio of mass divided by volume is called density. Have the students graph from data previously generated the mass versus volume, find the line of best fit, and determine for any point of the line the slope (density).
Assessment: Check to see that each student has properly graphed the mass and volume and found the density for any point. The slope of the line should be close to one. Thus, the density of water is 1.0
After the students have successfully found the density of water (a fact to remember) have them find the density of cooking oil independently.
Knowledge of elementary algebra would be useful.
Common Core Standards:
Graphs are use to present data and show relationships. From these relationships, values not easily obtained in an experiment can be accurately derived from mathematical equations based on the graph.
Students should be able to properly graph data and use the graph or mathematical equation to find a variable.
Grouping Suggestions Students should be placed in groups of two or three. A novel idea on grouping is found in, "Cooperative Learning in the Science Classroom," by Emily Lin, the Science Teacher Vol 73 No 5 Summer 2006, pp34-39.
Pacing/Suggested Time: The lesson can be completed in two 45-minute periods. If on a block schedule the lesson should be divided so that half is accomplished on day one (engagement and exploration) and the remainder (explanation and evaluation) on day two.