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Curriculum Grab Bag

How Old Are These Fossils?
An Integrated Math/Science Activity
(Teacher notes; student materials follow)

This activity was developed in the Foundation Integrated Studies Program (FISP) at Homestead High School in Cupertino, California with my design team partner Debbi Hersh. Originally designed for grades 9-10, the following version is an adaptation for grades 3-6. The activity engages students in the Carbon 14 dating process used by scientists to date ancient artifacts. While the complete application of this process as used by scientists is beyond the level of elementary math and science students, the essential concepts of half life and geometric sequences are made accessible to elementary students in a simplified, hands on activity that allows students to discover and experience the basic principles. The activity is adaptable to various grade levels by adjusting the amount of structure supplied to students. The teacher notes below provide support on how to do this. Some of the problems will be very difficult for the majority of elementary level students. These are included for individual students who want a challenge, or in cases where you and the class are especially inspired and you would like to demonstrate, or have a student give a presentation. Depending on the support and time given to warm-up problems the activity can take from 1-3 days.

A warm-up problem establishes the fundamental mathematical idea of half life (geometric sequence with common ratio of 1/2). This idea is then applied to a classroom simulation where student groups (or individuals) are designated as either "scientists" or "fossils." Using popcorn as Carbon-14, the fossil groups discard half of their popcorn each minute, and the scientists must determine how many minutes the have passed for various fossil groups by counting the remaining popcorn. For teachers and students so inclined, the idea can be applied to real Carbon 14 dating problems.

Context for teaching

The interdisciplinary aspect of this activity is best taken advantage of by using it when you are studying a related theme in science. It does not have to be an exact fit to provide meaning. Most students are curious about how scientists determine the age of ancient artifacts, and the topic is often charged with controversy.

Objectives

  • To provide meaning and relevance for the study of mathematics
    By actively experiencing mathematics used as tool to answer mysterious questions about the origin of life, students gain an enhanced sense of meaning, relevance and interest in the study.
  • To expose students to complex mathematics topics in simple forms
    The concept of geometric sequences is traditionally studied in high school algebra classes. When experienced in initially at the elementary level in a simpler forms, students are better prepared to absorb and retain the concepts in later study.
  • To provide practice in problem solving non routine problems
    Readiness to engage non routine problems in mathematics is developmental in students, and is enhanced with experience and practice in applications where they ultimately are supported to reach success with the problem.
  • To provide meaningful computation practice
    The activities require students to perform repeated multiplication and division problems which, in addition to reinforcing the skill of applying the algorithm, reinforce an understanding of the operations as inverses of each other.

The Activity

Part 1: A Warm-up Special Problem--The Henderson's Gummybears

In general, allow students to work with the problem on their own or in groups before you give structure or hints. Provide hints to select groups or individuals as needed to allow each individual student to be challenged and/or supported to a degree appropriate for them. Hints: -make a chart that demonstrates how many bears she has left each day -use a model to act it out; to visualize. Refer to your problem solving resources (flow chart) As students pursue various strategies of their own design, gently lead each student or group to make a table as shown below. Students will need to be equipped with this table when they do the next activity, "How Old Are These Fossils?"

day of week start. Sun. Mon. Tues. Wed. Thurs. Fri. Sat. Sun.
day number
0
1
2
3
4
5
6
7
8
number of gummybears
128
64
32
16
8
4
2
1
0

Students may ask what to do when there is only 1 gummybear. You can agree that when that happens, she gives the whole bear away. You might consider asking students what would happen if bears could be cut in half. Would she ever run out? (Only if she did it forever; the limit as the number of days approaches infinity equals 0). Another version of this problem would be to have students stand a specified distance from the wall and to walk have the distance to the wall on your commands. This could be done on another day to provide added support of the essential concept that needs to be established: Halving a quantity in successive stages.

Part 2: Discussion: Carbon 14 and the dating of fossils

Before doing "How Old Are These Fossils?" the concept of Carbon 14 dating needs to be introduced to the students, either in a science class or as a brief discussion during math. In the gummybear problem, half the quantity was removed each day. This could have been arranged where half the quantity was removed each week, each year, or each decade. The term half life is referred to the amount of time it takes for a quantity to diminish by half. The half life of gummybears was 1 day. Many other substances in nature decay or evaporate naturally with the same halving process as the gummybears. One of these substances is Carbon 14, a compound present in all living things. The half life of Carbon 14 is 5730 years. Expressed simply, if a scientist knows how much Carbon 14 is in an animal when it is alive, and can measure how much is in its fossil, then based on the half life of Carbon 14 the amount of time that has passed since its death can be determined. This process of dating will be simulated with popcorn by students in class.

Part 3: Classroom Simulation: How Old Are These Fossils?

Divide the students into about 10-12 groups of 2 and/or 3. Designate half of the groups as "scientists" and the other half as "fossils." The fossils are equipped with a cup of 128 pieces of popcorn. For the first 5-10 minutes of the activity the scientists must be separated from the fossils; either by leaving the classroom or going behind a partition. On your cue, each fossil will remove half of the corn in the cup each minute. Start each fossil at a different time over a 6 minute period with hand signals so that the scientists don't catch on, then stop. Allow the scientists then to circulate among the fossils and gather data and determine how many minutes each one was "decaying." If time allows, the rolls can be reversed, and the process repeated. To determine the age of each fossil group students will need to refer to the table created in the gummybear problem. Don't volunteer this strategy. Give the students the opportunity to make this connection on their own.

Part 4: The Follow-up Problems

These problems increase in difficulty. Without substantial support from you, they would be too difficult for most students to figure out on their own. Even with support they are beyond the abilities of most students in the lower grades.
1) Ans: answers will vary A hint: Ask students:
a) How many times did each fossil group lose half of its corn?
b) For Carbon 14, how many years pass each time half of the amount of Carbon 14 is lost?
c) If the corn is Carbon 14, multiply 5730 by the number of times the group divided its corn, or by the number of minutes old.

Problems 2-4 require two steps:
a) Determine how many half lives there are in each given number of years. Each number needs to be divided by 5730, or how many 5730s are there in each number. Rather than dividing, suggest to students that they multiply 5730 by various integers, 1,2, etc. until they get the desired number.
b) Now the problem is identical to the gummybear problem. Divide 80 by 2, divide the quotient by 2, and so forth. Repeat this by the number half lives.

2) Ans: 40 gm
3) Ans: 20 gm
4) Ans: 5 gm
5) Ans: 17,190 years. This problem is an actual dating problem. First determine how many "halvings" (half lives) of 24 are required to obtain 3. Then multiply this number by the half life period of 5730 years.
6) Ans: 28,650 years. This is identical in process to problem 5. It takes 5 "halvings" (half lives) of 24 to get .375. So 5(5730)=28,650.

 

Student Handouts

Gummybear warm-up:

Problem of the Day
The Henderson's Gummybears
(C14 Warm-up Problem)

The Hendersons have a very big family. At the beginning of summer vacation, Ms. Henderson decided to give the children a package of 128 gummy bear candies. She didn't want to give them away all on one day, so on the first day she gave them half of the bears. On the next day she gave them half of what she had left that day. She continued giving them half of what she had each day until they were all gone. Answer the questions below and show all your work

1) If Ms. Henderson started giving the bears to the kids on Sunday, how many bears would she have on Thursday? Explain in words what helped you find the answer:

 

2) How many days will pass until all of the bears are gone?
Explain in words what helped you find the answer:

 

How Old Is That Fossil Data Sheet:

 

NAME:_______________

PARTNER:_____________

How Old Are These Fossils?
Data Sheet and Questions

Each group started with 128 kernels of corn. The corn represents Carbon 14. In today's experiment, each minute they throw away half of their corn, just like the way Ms. Henderson was giving away gummybears. Visit three fossil groups, count the amount of corn left in their cup and determine how many minutes they were throwing way corn.

I AM A (CIRCLE ONE)

Paleontologist
Fossil

DATA

Fossil Measured (Name of Student Group) Number of Pieces of Corn Present Calculations Minutes of Decay
1)
     
2)
     
3)
     

 

PROBLEMS:

1) The half-life of Carbon 14 is 5730 years. This means that it takes 5730 years for half of the Carbon 14 to go away. Pretend the corn in today's experiment was Carbon 14. Calculate how many years old each of the fossils in today's experiment would be.

 

A sample of wood contains 80 gm of Carbon 14.
2) How much of the Carbon 14 would be left after 5730 years?

 

3) How much would be left after 11,460 years?

 

4) How much would be left after 22,920 years?

 

5) An animal carcass had 24 grams of Carbon 14 when it was live and 3 grams present when its fossilized remains were discovered. How old is the fossil?

6) How old would the same fossil be if there was .375 gm of Carbon 14 present when it was discovered?

 

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