and numeric What are the differences between the given situation and your new representations situation? Answers will vary depending on the response to question 2 The search for regularity or hope is that students will create a pattern that leads to an exponential trends model. What kind of function models the situation described at the top of Part I of For those students who are the handout? Why? Linear, constant rate of change. “Saving Pennies” struggling, you might What if we added 100 pennies the first and second days, but then added provide concrete objects or 500 pennies the third day? Is this still the same function? No, this is not a pictures to help constant rate of change. The visual display below may be helpful for conceptualize and solve a some students. problem this Continually ask, “Does make sense?” What if our pennies were collected in this way? This is increasing at a constant interval (each day), but not by a constant amount, so this is not a constant rate of change, and therefore not linear. What is different about your new situation? Multiplied by a constant amount each day. In other words, it increases at a constant ratio at a constant interval. The function that models this behavior is an exponential function. An exponential function grows or decays at a constant ratio over a constant interval. Show the example below (which hopefully was produced by one of the groups). What if we doubled the number of pennies the first and second days, but then added 500 pennies the third day? Or quadrupled the number of pennies the third day? Is this still the same function? No, this is not a
and numeric representations search for regularity or trends For those students who are struggling, you might provide concrete objects or pictures to help conceptualize and solve a problem Continually ask, “Does this make sense?” What are the differences between the given situation and your new situation? Answers will vary depending on the response to question 2 The hope is that students will create a pattern that leads to an exponential model. What kind of function models the situation described at the top of Part I of the “Saving Pennies” handout? Why? Linear, constant rate of change. What if we added 100 pennies the first and second days, but then added 500 pennies the third day? Is this still the same function? No, this is not a constant rate of change. The visual display below may be helpful for some students. What if our pennies were collected in this way? This is increasing at a constant interval (each day), but not by a constant amount, so this is not a constant rate of change, and therefore not linear. What is different about your new situation? Multiplied by a constant amount each day. In other words, it increases at a constant ratio at a constant interval. The function that models this behavior is an exponential function. An exponential function grows or decays at a constant ratio over a constant interval. Show the example below (which hopefully was produced by one of the groups). What if we doubled the number of pennies the first and second days, but then added 500 pennies the third day? Or quadrupled the number of pennies the third day? Is this still the same function? No, this is not a