| Representing doubling with drawings |
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Often students initially represented doubling pictorially, drawing circles to represent the cells. After a few doublings, students decided that this method was too laborious and switched to a table format.
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| Representing doubling with tables |
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Most of the time students changed to adopt a list format when the circles became too numerous. At this point they decided that multiplying by 2 at each step was a more efficient way to find the number of cells at each new doubling.
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Doublings |
1 |
2 |
3 |
4 |
5 |
# Cells |
2 |
4 |
8 |
16 |
32 |
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With both the drawing and the lists, students puzzled over what constituted 10 doublings. Some students thought 0 -> 1 cell was the first doubling. Therefore, after 10 doublings they had half as many cells as the students who considered the first doubling to be 1 -> 2.
We introduced a more structured table format as a way of organizing the doubling model data. This format helped resolve students' thinking about the first doubling event and also made the biological implications more evident.
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#Doublings |
#Cells |
0 |
1 |
1 |
2 |
2 |
4 |
3 |
8 |
4 |
16 |
5 |
32 |
6 |
64 |
7 |
128 |
8 |
256 |
9 |
512 |
10 |
1024 |
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| Representing doubling with graphs |
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We asked students to represent the data graphically to show the growth of the population of cells. Students used both line graphs and bar graphs to represent the doubling model. [student examples] After examining the two types of graphs, students determined that both looked similar (both started gradually but then shot up), and that both provided the same information (by draw ing a line across the top of the bars, one could make a line graph). While graphing the doubling data, students worked with the range of scale needed for the data. For example, students discussed the intervals needed to fit 1024 cells on one axis. Students were bothered that this scale did not allow one to the see changes that occur in the early doublings. Some students remedied this by creating two graphs, one with the entire data set and a second on a smaller scale that highlighted the earl ier doubling events.
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| Representing doubling with equations |
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To understand the changes in population size, students wrote equations to describe segments of line graph. Using a piece-wise linear model (y = mx + b) to determine the slope of the line between two points, students concluded that the doubling model represents piece-wise line segments with changing slopes. [student examples, video]
Although this class did not pursue, classes that have explored more advanced mathematics can represent doubling with equations of exponential function.
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