As a warm-up to today’s lesson, have students review exponent rules by completing the questions on pages 21 and 22 of today's Flipchart - identifying change and writing exponentials (p. 20-28).
Once students have completed the warm-up, you may want to review Homework 1 - Exponential Functions if you have not done so already. It is essential that students are comfortable with question 4, particularly part a, before being successful on today's isotope problem. See reflection in this section for more details.
Overall, my students were not successful on question 2 part b of this isotope problem. I wish I had focused more on this homework question and done more modeling for students on how to relate two variables and rewrite an equation in terms of another variable. While taking questions on the homework, I would like to model for students how I would go about finding the equation that relates d decades to y years. Or in other words, how to write an equation for d in terms of y. To model this for students, I would make a table of values and find an equation that works. I will make it happen that I come up with y=10d. Then I will show students how I would solve for d and then substitute into the original equation.
Have students complete the Student handout Arcade Problem from yesterday. Remind students to use the steps provided to them in their notes to find the exponential equation.
After students have had about 15 minutes of work time and/or when they finish the problem, present pages 23-27 of the Flipchart - identifying change and writing exponetials (p. 20-28). Pages 23-24 are just for students to check their work. Then students are asked to start comparing the representations. The goal here is to help students make the connections between the graphs, equations, and tables and find patterns for quickly writing the functions from a table. At this point we should discuss with students initial values and growth factors (common ratios) of exponential functions and identifying these in a table.
Students should now apply the newly discovered patterns and shortcuts to writing exponential functions using the isotope decay problems as context. Students will probably sail through the first question, but if they seem to move quickly through the second problem it’s probably wrong. The second problem now has a half-life of 3 minutes so students will have to account for this not changing by 1 minute. I am curious to see how students will approach this. I see three approaches they may take:
1) Finding from two points (like page 19 of flipchart)
2) Writing the exponential function in terms of number of half-lives and then substituting in the half lives as a function of time.
3) Identifying the rate of change in the table and using the initial value to write the equation.
Of course all three are great approaches, I would like to take some time today at the end of the hour or tomorrow at the beginning of class to review all three approaches. My guess is that we will not have time in this class period, so I will definitely be coming back to this tomorrow.
Overall, my students were not very successful on the isotope problem. I had a few students do well, but a majority of my students struggled. Most of my students solved the first problem correctly using the patterns they identified, such as this student: Counting the Change, Isotope question 1, student 1.jpg. I had a few students attempt to use a system of equations. Some were successful with this method: Counting the change, isotope question 1, student 3.jpg. Some were not successful: Counting the change, isotope question 1, student 2.jpg.
Not one of my students got the correct answer for question 2, part b. As I thought may happen, I got the same incorrect answer repeatedly. This was definitely the most popular wrong answer: Counting the change, isotope question 2, student 4.jpg. Most of my students wrote the equation as a function of half-lives, not as a function of time (or didn’t even attempt part b at all!). I did have a few students try a system of equations approach, but were just unsuccessful: Counting the change, isotope question 2, student 5.jpg.
Since many of my students had the same misconception. I am going to do a Favorite No activity at the start of class tomorrow using this student’s answer:Counting the change, isotope question 2, student 4.jpg. I mainly want to demonstrate to students how this equation won’t work for t. It is in terms of half-lives. Then I want to open it up to the class to help me write an equation that relates t and d. Then I also want students to take time to solve this using a system. Approaching this problem using a system of equations brings up some opportunities for discussion of exponential function properties and the rules of exponents. Help students to see that the cube root of one half is the same as writing one half to the power of one third.
As a quick closure today, I am going to have students do a Thumbs Assessment. I just want students to stop and take a moment to communicate how well they think they understand question 2 of the isotope problem. A thumbs up will indicate they are confident in their answers. A sideways thumb will communicate they did something, but are unsure of their answer. A thumbs down would be given if a student was just so lost they didn’t even know where to start. I will use students’ responses to get a feel for the classes’ understanding as a whole. This will help me determine if I should have students present their methods at the start of class tomorrow or if I should lead the discussion of the three approaches.