I post the lottery problem on the projector ask each student to read it and write down an initial reaction. They should explain their thinking as to why they would take the $500 per week or the 1 cent, 2 cents, 4 cents, etc.
Extention/scaffolds: It is important to note in this lesson that students could take the exponential function in two different directions. (1) They could make a list of the amounts generated each week ex: 1 cent, 2 cents, 4 cents, 8 cents, etc. This will give them an understanding of how an exponential function starts out changing fairly slowly but its rate of change accelerates. Some more advanced students may notice that to answer this question fully (2) they need to keep track of the totals 1 cent, 3 cents, 7 cents, 15 cents. This involves the sum of a geometric series which, while outside the scope of the algebra course, could be introduced to some students at this time so that they could verify their work using the formula (s=a(1-r^n)/(1-r)).
Environment: For this investigation I organize students in groups of 2 or three with students of similar ability. This way all members of the group can contribute and understand the ideas being shared. This lesson really requires students to play around with the math and make noticings and discoveries through their work. If one student is cognitively above that of their peers the other members of the group will not have an opportunity to make their own meaning.
In groups, students investigate the problem more deeply. Some students will naturally gravitate towards making a table with weeks and payouts for each of the prize categories. Other students, who may not have such an organized way of thinking may take a different approach. What I am looking for is that groups can validate their choice with mathematics. I always get a kick out of hearing students changing their thinking when they realize how the exponential function begins to change much more quickly and "catch up" to the linear function (and eventually outpace it).
While I will challenge some of my stronger students to determine a formula for the exponential function, I want to ensure that all members of the class are able to build the function based on the information that is given to them.
NOTE: for the purposes of this lesson in the algebra 1 course we are only going to write the function for the input of the week and the output of the dollar amount on that week. This will need to be explained to some students who may have written their outputs in terms of the sum of the geometric series (mentioned above in the extension/scaffolds). This is where we will talk about what it means to double. Students are very good at giving recursive formulas (a formula for the nth term in terms of the (n-1)st term) but not as adept at coming up with explicit formulas. Usually we can see that each term is doubled but it takes some questioning to realize that the second term is being multiplied by 2, the third by 4, the fourth by 8, etc. But then once students see this it it not a difficult leap to come up with the powers of 2. Since the first amount is 1 cent we can multiply each term by 0.01. To wrap this portion of the lesson up, I ask students to graph the two functions on their calculator in an appropriate domain and range and to verify that their mathematical work from the investigation is supported by the graph.
Now that students have seen an exponential function and a linear function graphed together it is a good time to ask the following question. "An exponential function like we examined today always eventually have a greater y-value than a linear function" (1) Sometimes, (2) Always (3) Never. Explain why.
If students are having trouble getting started you can ask them a scaffolding question: "What if I had offered $1000 a month instead of $500? Would it still be better to take the 1 cent, 2 cent, 4 cent, etc.? What if I had offered $2000 per month?