Get a Half Life!: Student Worksheet

Name: ___________________________________

A decay function is one in which the values decrease by a constant factor, but it is not linear! These functions are often used to model the decay of a radioactive element. The half-life of a substance is the time it takes to reduce its initial effectiveness by one-half.

Students work in pairs.

Each pair needs:

• 1 cup of m&m's
  
• 1 paper plate or box
  
• 1 graphing calculator

1. Collect the Data
   1. Pour m&m's out and count your beginning sample size (N). This will be your N value when T = 0. Enter this value in the table below and put the m&m's back in the cup.
   2. Shake the cup gently and pour the m&m's out in the box or on the paper plate.
   3. Remove all the m&m's with an "m" showing. (You may eat these m&m's only!)
   4. Count the remaining m&m's and enter this value of N into your table next to T = 1.
   5. Put the remaining m&m's back in the cup.
   6. Repeat this process until there are no longer any m&m's with the letter "m" when you empty your cup.
   
   

   Trial Number (T)	0	1	2	3	4	5	6	7
   # of remaining m&m’s (N)

   
   
2. Use the Calculator
   1. Clear out the calculator:
      Press [2nd][Y=]. Enter [4]. Press [ENTER]. This turns off all plots.
      Press [Y=], [CLEAR], down arrow and [CLEAR] until all equations are cleared.
      Press [STAT], [ENTER] to view tables. To clear Lists, press up arrow to highlight L1, press [CLEAR], [ENTER]. Use arrow keys to highlight any other lists and clear.
   2. Enter the data from the table above:
      Press [STAT], choose EDIT by pressing [ENTER] and enter values of T in L1 using down arrow keys and [ENTER].
      Use right arrow key to move over to L2 and enter the values of N in L2.
   3. Graph the Data
      Press [STAT PLOT] ([2nd][Y=]), choose 1 by pressing [ENTER].
      Highlight: ON
      

      Broken line graph
      Xlist: L1
      Ylist: L2
      Mark: first one

      Press [ZOOM], then [9] to set up appropriate window and see graph.
   4. Have the calculator determine the best equation for the data.
      Linear:
      Press [STAT]. Choose CALC[5].
      Type in L1, L2 (press [2nd][1][,][2nd][2] then [ENTER]). You will see y = ax + b and values for a, b, and r.
      Press [Y=], set cursor next to Y1=, press [VARS][5], arrow over to EQ, press [7], the [GRAPH].
      
      Quadratic:
      Press [STAT], choose CALC [6], then type in L1, L2 (like above) and press [ENTER].
      Press [Y=]. Set your cursor next to Y2=.
      Press [VARS][5], arrow over to EQ, press [7], the [GRAPH]. (Adjust the window here to see more of this graph)
      
      Exponential:
      Press [STAT] and choose CALC . Choose A by using the down arrow key until you see "A ExpReg". Press [ENTER].
      Type in L1, L2 as before and then [ENTER]. You will see y = a*b^x (y = abx) and values for a, b, and r.
      Press [Y=] and set your cursor next to Y3=.
      Press [VARS], choose [5], arrow over to EQ, press [7], then [GRAPH].
   
   
3. Analyze the Data
   1. Which graph most clearly approximates a "curve of best fit" for your data?
   2. Why is this called a "half-life"?
   3. What is a good estimate for the length of a "half-life" of m&m's based on your research?

As a result of this activity, students learn to model a real-life situation by collecting data from a few trials, graphing the data, and then drawing a curve of best fit. The student must decide which of the equations best fits his/her data. Sometimes when a student gathers his/her own data, the graph does not make a perfect line or curve.