

10. Display a transparency made from Master 3.3, Collecting the Data, and explain that you are going to use this table to tally the number of people in the class who would have developed cancer at each age if the onehit or twohit hypotheses for the development of cancer were true.
Explain that to discover the number of people who would have developed cancer, the students need to examine their random number tables according to the following instructions:
. Tell students that first the class is going to approximate what might happen if the onehit hypothesis were true (that is, if one mutation were sufficient to cause a normal cell to become cancerous). Ask students to imagine that if they found the first "unlucky" number in the column labeled "Gene 1," it meant a gene in one of their cells experienced a cancercausing mutation. Explain that if the onehit hypothesis were correct, the age at which the unlucky number first appears in the column labeled "Gene 1" would be the age at which they developed colon cancer.
Note that some students likely will not encounter the unlucky number and, therefore, will not develop cancer.
Only students who had the first unlucky number (in the example, 10) in the column labeled "Gene 1" should indicate that they developed cancer.
Tell students that it does not matter in what order the mutations occur. Emphasize that in order to have developed cancer under this model, they would need to have both the first unlucky number in the column labeled "Gene 1" and the second unlucky number in the column labeled "Gene 2" (in the example, both a 10 in the first column and a 4 in the second column). The age at which they developed cancer is the age by which they had experienced both mutations.
Note that many students likely will not encounter both unlucky numbers and, therefore, will not develop cancer.
These running totals represent the cumulative number of people who developed colon cancer at or before each age.
11. Ask students whether they see any pattern in the incidence of cancer in this population (the class).
Students may see that fewer people developed cancer under the second model (the twohit hypothesis) than under the first (the onehit hypothesis) and that those who did develop cancer under the twohit hypothesis tended to do so later in life. Encourage students to express this observation as a generalization (for example, the chance, or "risk,"of developing cancer early in life diminished as the number of events involved in its development increased). If students are able to recognize this, ask them what this observation suggests about the development of cancer. Do not give students this answer—that it suggests that more than one event may be involved in the development of cancer—but indicate that they should think about this question as they complete the activity.
12. Distribute one copy of Master 3.4, Graphing the Data, to each student and direct students to work in pairs to construct two graphs that illustrate the chances that a person in this class would have developed colon cancer by a certain age if the onehit or twohit hypothesis were true.
13. Ask students to examine the graphs they generated in Step 12 and decide which hypothesis best fits the actual data on the incidence of colon cancer.
Students likely already recognize that fewer cancer cases were encountered when two mutations (two hits) were required. However, they also may see from the graph that neither hypothesis fits the colon cancer data well.
14. Ask students to predict what the results of a simulation such as this might be if three, four, or as many as five mutations (five hits) must occur prior to the onset of cancer.
Students may be able to suggest that cancer would become an increasingly rare event and would tend to occur later and later in life as the number of mutations required for its development increases.
15. Distribute one copy of Master 3.5, Using the Hit Simulator, to each student. Direct students to organize into their teams and follow the instructions provided to use the Internetbased simulation to test their predictions and decide what type of model for the development of colon cancer best fits its observed incidence.
This simulation is a sophisticated tool that students can use to observe how mutation frequency and the number of mutations (hits) required for the development of cancer affect the incidence of cancer in a population. Although students can simply experiment with the simulation, their experience likely will be more meaningful if they follow the guidelines provided on Using the Hit Simulator.
Give students approximately 20 minutes to learn to use the simulation, test their predictions, and answer the questions on Using the Hit Simulator.
These questions focus students' attention on the activity's major concepts. Encourage students to express their understanding of cancer using the language of cells and genes. 
16. Convene a class discussion in which you invite students to share their answers to the questions on Using the Hit Simulator.
. Investigate the Effect of Changing the Number of Hits Required
Question 1 How does the incidence of cancer change as you require a greater number of hits for a cell to become cancerous?
Students should see that the greater the number of hits required, the fewer the number of people who develop cancer and the later in life they tend to develop it. Students may note that with the number of hits set at 1 and the mutation rate set at 0.5 (50 percent), nearly everyone in the population gets cancer by age 25. As they increase the number of hits required, the curve shifts to the right (people get cancer later in life), though most people still develop it eventually.
Question 2 Recall the graph of the incidence of colon cancer that you observed at the beginning of this activity. Did the incidence of cancer in any of the runs you just completed match the incidence of cancer recorded in that graph? Explain your answer.
Students should recognize that none of the runs matched the actual incidence of colon cancer. You may wish to remind students that they learned in Activity 1 that only about 1 in 3 people in the United States develops cancer sometime in his or her life.
Question 3 What can you conclude from this observation?
Students should see that because the curve shifted to the right (toward the development of cancer later in life) as more hits were required, the results suggest that more than 1 hit likely is involved in the development of cancer. Astute students also may say that because almost everyone eventually developed cancer in these simulations, the mutation rate of 0.5 (50 percent) likely is too high.
. Investigate the Effect of Changing the Mutation Rate
Question 4 How does the incidence of cancer change as the mutation rate increases?
The incidence of cancer increases as the mutation rate increases. Students should see that with the simulator set at a mutation rate of 0.1 (10 percent), a smaller proportion of the population develops cancer than when the simulator is set at 0.5 (50 percent). With the simulator set at a mutation rate of 1 (100 percent), everyone gets cancer between the ages of 0 and 5.
Question 5 Recall the graph of the incidence of colon cancer that you observed at the beginning of this activity. Did the incidence of cancer in any of the runs you just completed match the incidence of cancer recorded in that graph? Explain your answer.
Students should recognize that none of the runs matched the actual incidence of colon cancer. They should recognize, however, that the curve(s) generated with the mutation rate set at 0.1 or lower was/were more in line with the observed incidence than the curves generated with higher mutation rates.
Question 6 What can you conclude from this observation?
Students should recognize that these results suggest that the actual mutation rate is somewhat lower than 0.5 (50 percent), and maybe even lower than 0.1 (10 percent).
. Investigate the Effect of Changing Both the Number of Hits Required and the Mutation Rate
Question 7 What can you conclude from your observations?
Students should see that the curves generated by some of these runs begin to resemble the incidence of colon cancer observed on the graph they examined at the beginning of the activity. Encourage students to suggest combinations of numbers of hits and mutation rates that seem to give realistic results, but caution students not to use this simulator (which was designed for educational purposes*, not research) to try to make an absolute determination of number of hits and mutation rate.
You may wish to point out that a mutation rate of 0.04 is the same rate that was used in the random number table exercise. Challenge students to demonstrate this by comparing the graphs they made of the onehit and twohit hypotheses with the curves generated by the simulator. Note that using the Internetbased simulator allows them now to test the predictions they made in Step 14.
*Note that the graph of the incidence of colon cancer used in Step 2 is actually the number of people in a population of 100,000 who will be diagnosed with colon cancer at each age. The graphs students created in class and the graphs generated by the Hit Simulator are somewhat different because they plot cumulative numbers (the total number of people who will have developed colon cancer at or before that age).
. Summary
Question 8 What clue did the change in risk of colon cancer provide scientists about the cause of cancer?
Whereas Activity 2 illustrates the contribution that cell biologists and geneticists have made to understanding cancer, Activity 3 illustrates the contribution that epidemiologists have made. One of the most exciting aspects of cancer research in recent years has been the construction of an understanding of cancer that unifies the work of many types of scientists studying cancer for more than 100 years. 
Students should be able to explain that the increase with age in colon cancer incidence suggested to epidemiologists that more than one mutational event was involved in cancer's development. Similar graphs of agedependent cancer incidence, plotted for many other types of adult cancer, provided additional support for the hypothesis of multistep carcinogenesis. In fact, one of the goals of research today is to identify each of the steps and genes involved in the long and complex succession of events that occurs to create the malignant growth of cancer cells.
Note that this question returns students to the challenge they were given in Step 4.
17. Challenge your students to evaluate the models they used to test the different hypotheses for the development of cancer (that is, to think about the ways in which the random number table exercise and the Internetbased simulation do and do not match reality).
Remind students that all models are inaccurate in some respects. For example, the mutational events within cells may not be completely random, as the models assume. The models also assume that the probability of each individual mutational event is the same, and this may not be the case. There is some evidence, for example, that some mutations increase the probability that other mutations will occur. In addition, the models do not consider that some mutations may be detected and repaired. Nevertheless, the fact that the models the students used are not perfect does not mean they are not useful tools for understanding how disease processes work.
The questions on Testing an Explanation are challenging, but they represent an excellent opportunity for you to evaluate your students' understanding of the activity's major concepts and their ability to apply their understanding to novel situations. 
18. Close the activity by distributing one copy of Master 3.6, Testing an Explanation by Looking at Additional Data. Ask students to use their understanding of cancer as a multistep process to explain each of the observations listed.
Question 1 Cancer is a disease of aging.
Students should be able to explain it takes time for all of the mutations involved in the development of cancer to accumulate, and that this explains why the incidence of cancer increases with age (that is, why cancer is more likely to "strike" in the middle or later years than in childhood, youth, or young adulthood).
Question 2 You've come a long way, baby.
Students should be able to explain that as these women began to smoke, they began to accumulate cancercausing mutations in their lung cells. Because the accumulation of these mutations to the point where a cell becomes cancerous takes time, the results of the increase in the number of women smoking (in the form of an increase in lung cancer among women) did not begin to appear for 20 to 25 years.
Question 3 Genes and increased susceptibility.
Students should recognize that if a person is born with a cancercausing mutation already present in his or her cells, he or she has already experienced the first step toward the development of cancer and, thus, has a higher risk of accumulating all of the mutations required for the development of cancer than a person who does not carry such a mutation.
