LEARN NC

5 Half-life

By Jennifer Elmo

Provided by Kenan Fellows Program.

This activity integrates Chemistry and Algebra II by using the concepts of half-life and exponential decay. Half-life is a way for students to see a real-life use of exponential decay functions.

Learning outcomes

Students will be able to:

  • Determine the half-life of “candium”
  • Determine the equation for half-life
  • Set up and solve for the resulting amount and the time for the half-life of “candium” given a set of conditions whereby they have to use the equation and not just their graph

Teacher planning

Time required

45–60 minutes

Materials needed

Each student or pair of students will need:

  • 100 pieces of candy with a letter on one side (such as M&M’s or Skittles)
  • a plastic bag
  • paper towels
  • graphing calculator

Student handouts

Half-life lab
Document by the author
Open as PDF (52 KB, 2 pages; also available as Microsoft Word document)
Half-life post-lab questions
Document by the author
Open as PDF (13 KB, 1 page; also available as Microsoft Word document)

Pre-activities

Before beginning this lesson, students should be able to:

  • Define half-life
  • Determine the equation of a line using their calculators
  • Solve half-life problems
  • Use logarithms to solve for the exponent in the half-life equation

Activities

In this lab exercise, students will collect decay data of their candy and write their results on the board and share their trials with the class.

  1. Distribute handouts and lab materials to students.
  2. Tell students that in this activity they will simulate radioactive decay with candy. The candy can be used to discover the relationship between the passage of time and the number of radioactive nuclei that decay. A candy piece with the plain side up will represent an atom that has not yet decayed. When this atom “decays,” it will land with the printed side up.
  3. Instruct students to count out 100 pieces of candy and place in the plastic bag.
  4. Have them seal the bag and gently shake for ten seconds.
  5. Students should then gently pour out all the candy on a paper towel.
  6. Next, they should remove all the pieces of candy that have “decayed” — landed with printed side up.
  7. Students can then count the pieces that remain or haven’t “decayed.” They will record this data in the first data table on their lab sheets under the individual trial column.
  8. Give students time to repeat steps 2–4 until all the pieces have decayed.
  9. Make sure they record their data on the white board at the front of the room, and then record the class data on the second data table on the lab sheet.
  10. Finally, they will average the class data for each half-life and record these averages in the appropriate data table.
  11. Once students have completed entering their data into the data tables, distribute the Half-life Post-lab Questions.
  12. Students will enter their individual data and the class averages into lists in their graphing calculators. They should graph both sets of data on the screen at the same time. They will choose different marks for their own data and the class data, and choose the first “type” of graph that doesn’t connect the lines. You should inspect each student’s screen at this point.
  13. Have students then perform a linear regression and an exponential regression (expreg) on the class data and list the values that the calculator gives for each regression.
  14. Allow students time to answer the next two items on their lab sheets:
    • Which function — linear or exponential — is more accurate for this data? Why? Explain your answer using information from the calculator.
    • Choose the more accurate function from the question above, and perform that
      regression on your individual data. Which is more accurate: your own data or class data? Explain your answer using information from the calculator.
  15. When students have completed the questions above, have them turn the plot with their individual data off. They will next perform the most accurate regression on the class data again and paste this regression into Y= and graph this best-fit regression on the screen. You should inspect each student’s screen again at this point.
  16. Ask students to graph their data and the class data on graph paper on the same set of axes. They can attach this paper graph to the post-lab questions.

Assessment

Evaluate students responses to the post-lab questions.

Critical vocabulary

half-life
the average time needed for half the nuclei in a sample of a radioactive substance to undergo radioactive decay (the half-life of a substance does not equal half of its full duration of radioactivity)
logarithm
the power to which a base, such as 10, must be raised to produce a given number
If nx = a, the logarithm of a, with n as the base, is x; symbolically, logn a = x.
decay
to disintegrate or diminish by radioactive decay; a spontaneous transformation of an elementary particle into two or more different particles
exponential function
a function in which an independent variable appears as an exponent

Comments

This activity works best when the Algebra II teacher has students do it after they learn about half-life in Chemistry and exponential functions and logarithms in Algebra II.

North Carolina curriculum alignment

Mathematics (2004)

Grade 9–12 — Algebra 2

  • Goal 2: Algebra - The learner will use relations and functions to solve problems.
    • Objective 2.03: Use exponential functions to model and solve problems; justify results.
      • Solve using tables, graphs, and algebraic properties.
      • Interpret the constants, coefficients, and bases in the context of the problem.
    • Objective 2.04: Create and use best-fit mathematical models of linear, exponential, and quadratic functions to solve problems involving sets of data.
      • Interpret the constants, coefficients, and bases in the context of the data.
      • Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.

Science (2005)

Grade 9–12 — Chemistry

  • Goal 1: The learner will develop abilities necessary to do and understand scientific inquiry.
    • Objective 1.01: Design, conduct and analyze investigations to answer questions related to chemistry.
      • Identify questions and suggest hypotheses.
      • Identify variables.
      • Use a control when appropriate.
      • Select and use appropriate measurement tools.
      • Collect and organize data in tables, charts and graphs.
      • Analyze and interpret data.
      • Explain observations.
      • Make inferences and predictions.
      • Explain the relationship between evidence and explanation.
      • Identify how scientists share findings.
  • Goal 4: The learner will build an understanding of energy changes in chemistry.
    • Objective 4.04: Analyze nuclear energy.
      • Radioactivity: characteristics of alpha, beta and gamma radiation.
      • Decay equations for alpha and beta emission.
      • Half-life.
      • Fission and fusion.

  • Common Core State Standards

    • Mathematics (2010)

      • High School: Functions

        • Linear, Quadratic, & Exponential Models
          • FUN.LQE.1Distinguish between situations that can be modeled with linear functions and with exponential functions. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals....
          • FUN.LQE.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

  • North Carolina Essential Standards

    • Science (2010)

      • Chemistry

        • Chm.1.1 Analyze the structure of atoms and ions. Chm.1.1.1 Analyze the structure of atoms, isotopes, and ions. Chm.1.1.2 Analyze an atom in terms of the location of electrons. Chm.1.1.3 Explain the emission of electromagnetic radiation in spectral form in...