Counting methods - Pairs of students worked out methods for determining the number of colonies on the plates. Many students elected to count individual colonies, marking each colony once they were in the count to ensure that each colony would be counted only once. As the number of colonies on the plates increased with time, students developed other methods for counting. These included:

• division of labor- each partner took every other time point and counted their share of the plate, or each pair divided single plates and counted half and combined their numbers for the total.

• sectoring- students used pie segments to sector the plates, counted one sector, and multiplied by the total number of sectors.

• estimation- students compared the increase in colonies between plates, estimated the increase and multiplied their previous count by the increase (however, in most cases this turned into an estimation of doubling for each time point).

   Tables

Here are some sample data from a class that decided to consider both average and median as possible candidates for the actual count. Their class also decided to eliminate from consideration the faulty counts of some groups.

   Graphs

Students graphed the number of bacteria/yeast as a function of time and then discussed differences/similarities to the simple doubling model. Students compared the lines from the doubling time graph with their actual data and observed the similarities/differences between the real data and the doubling model. In addition, the doubling model over time graphs can also be used to compare the rates of growth of the real cultures.

   Equations

Students used the slopes of line segments to compare the real data to the doubling model. Students determined the decrease in the growth of the culture, compared to the doubling model by comparing rates of growth of line segments of the two graphs.

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   Understanding population growth

Growth of microbe populations slows down-

By comparing the graphs of the real data to the doubling model, students determined that the growth of the population of microbes slowed down (bacteria) and plateaued (fungi). Students reached these conclusions by comparing the slope of the line at the time segment before and after the slow-down occurred. It was apparent that the later line segment was less steep, the slope of the lines indicated how much slower the rate of growth was at the end. The students interpreted the slope as a rate of growth, defined as cells/min.

The decrease in the rate of growth suprised many students, who expected the data to follow the doubling model. As they thought about why the growth would plateau, students hypothesized that the conditions had changed. Perhaps there were too many cells, and possibly not enough food to maintain the population. (VIDEO). The students also thought about ways to test these ideas, although time did not permit the students to carry out their tests. These included: seeing if there were dead cells in the population, giving the cells more food and seeing if they would grow faster; finding out how much food they eat and how long it takes to eat that amount of food. Students felt that if we added more food to the system at the point when the population plateaued, the population would grow more. Eventually, however, the cells would still run out of food and the levels would plateau again, and possibly even drop.

   Comparing growth rates

Bacteria grow faster than fungi

Students noticed that in this case the bacteria achieved higher numbers of cells than the yeast, and in a faster time. Students supported this impression by using the tables to compare the growth at a similar time point. Their comparisons showed that there were more cells of bacteria at that time than yeast (i.e. 310 min. for bacteria and 360 min. for yeast). The students also compared the slope of lines for the steepest segments of growth for both the yeast and bacteria. The slope showed that the bacteria were doubling faster than the yeast. Using the graphs from the doubling period activity , students compared the lines on the same axes as the real data and determined that the bacteria were more similar to the line for the 30 min. doubling period. The yeast were similar to the line for the 3 hour doubling period. The students also noticed that for the bacteria, the number of cells decreased in the earliest time points, which seemed counter-intuitive. They attributed this decrease to counting or sampling error .

As they compared their class data students noticed a lot of variation in the numbers, especially on the later time point plates that had more colonies on them. Students attributed this variation to a number of sources of counting error:

Two merged dots- In a number of cases, colonies began to grow into one another. Students thought that it was possible that some students could have been counting these as one colony instead of two. This might lead to different counts.

Sampling error (volume vs. number)- When the bacteria plates were counted the numbers decreased across the first few time points (see table X). Students proposed that sampling error might be producing this result. Perhaps when the samples were taken, the sampler failed to take the same amount of culture each time; alternatively, perhaps the bacteria aggregated in a certain area of the flask, so one sample might not have provided a good representation of the population.

Counting different quadrants- Many students eased their counting burden by dividing the plate into equal sectors, counting one sector and multiplying by the total number of sectors to get an estimate for the entire plate. Students proposed that differences in the number of colonies within the chosen sector may have led to the range of numbers observed at each time point. They felt this was especially likely if the colonies were not equally distributed on the plate.

Estimation of doubling between time points- At some of the later time points there were large numbers of bacteria colonies, consequently a few students estimated by visually guessing at how many more colonies there appeared to be, and then multiplying the previous plate count by that multiple. In most cases, the students who used this method thought that there was a doubling event between each time point, and multiplied accordingly. In the end, most students agreed that this was less accurate than the other methods that had been employed.

By comparing the graphs of the real data to the doubling model, students determined that the growth of the population of microbes slowed down (bacteria) and plateaued (fungi). Students reached these conclusions by comparing the slope of the line at the time segment before and after the slow-down occurred. It was apparent that the later line segment was less steep, the slope of the lines indicated how much slower the rate of growth was at the end. The students interpreted the slope as a rate of growth, defined as cells/min.

The decrease in the rate of growth suprised many students, who expected the data to follow the doubling model. As they thought about why the growth would plateau, students hypothesized that the conditions had changed. Perhaps there were too many cells, and possibly not enough food to maintain the population. (VIDEO). The students also thought about ways to test these ideas, although time did not permit the students to carry out their tests. These included: seeing if there were dead cells in the population, giving the cells more food and seeing if they would grow faster; finding out how much food they eat and how long it takes to eat that amount of food. Students felt that if we added more food to the system at the point when the population plateaued, the population would grow more. Eventually, however, the cells would still run out of food and the levels would plateau again, and possibly even drop.

Bacteria grow faster than fungi

Students noticed that in this case the bacteria achieved higher numbers of cells than the yeast, and in a faster time. Students supported this impression by using the tables to compare the growth at a similar time point. Their comparisons showed that there were more cells of bacteria at that time than yeast (i.e. 310 min. for bacteria and 360 min. for yeast). The students also compared the slope of lines for the steepest segments of growth for both the yeast and bacteria. The slope showed that the bacteria were doubling faster than the yeast. Using the graphs from the doubling period activity , students compared the lines on the same axes as the real data and determined that the bacteria were more similar to the line for the 30 min. doubling period. The yeast were similar to the line for the 3 hour doubling period. The students also noticed that for the bacteria, the number of cells decreased in the earliest time points, which seemed counter-intuitive. They attributed this decrease to counting or sampling error .