58
Lead in Children's Bodies
Student Reading Assignment
Exposure to lead and its toxic effect on children is a hazard we cannot ignore. Children
can be poisoned by drinking water in houses that have lead pipes or lead solder in their
plumbing. They can be poisoned by breathing in dust contaminated by peeling paint containing
lead, or eating chips of paint. Children are also at risk from lead-contaminated soil and dust in
their play areas. Tests of the soil near the Bovoni Dump on St. Thomas in July 2001 indicated
high levels of lead. Prior to the use of unleaded fuel, tons of lead were deposited in areas with
heavy traffic congestion. This lead is still in the soil, and children inhale the dust when they
breathe if they play in those areas. Lead from the soil and dust that children inhale is even
more dangerous than lead a child might swallow because children absorb into their bodies
about 40% of the lead in what they eat or drink, but they absorb about 90% of the lead particles
in the air they breathe. In children, high lead levels can cause mental deterioration and impair
the child's ability to learn. Lead also endangers the unborn child by harming development of the
fetus and by causing premature births.
Children should be treated for lead poisoning if tests show they have more than 45 µg
(micrograms) of lead for each tenth of a liter of blood. In a 60 lb. child, this would be a total of
about 0.9 mg of lead in the blood. Thus, we see that a very small amounts of lead taken into a
child's body can harm the child. What further complicates the problem is that only about 1.5%
of the lead is naturally removed from the blood each day. This means that without treatment, it
will take a long time for the lead in a child's blood to drop to an acceptable level .
1. Suppose a child has 1.0 mg of lead in her blood. Assume this child's body removes 1.5% of
the lead in the blood each day. Use a calculator for the following questions.
a. Assuming this child is removed from all sources of lead, how much lead is in her body
after 1, 2, 3, and 4 days? Use 4 decimal place accuracy.
t
1
2
3
4
Lead remaining
after t days (mg)
b. How many days will it take until the amount of lead in the child's blood drops below 0.9
mg?
c. Develop a function f (t ) that gives the amount of lead in this child's blood after t days,
assuming the child is not exposed to any more lead.
_______________________________________________________________________________________
© Copyright by Rosalie A. Dance & James T. Sandefur, 1998
This project was supported, in part, by the National Science Foundation.
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
59
Opinions expressed are those of the authors and not necessarily those of the Foundation.
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
60
Lead in Children's Bodies
In-Class materials
2. A child with 70-100 µg of lead per deciliter of blood needs urgent treatment. For a 60 lb. girl,
this would be about 1.4 to 2 mg of lead in the blood. Suppose this girl has 2 mg of lead in her
blood and eliminates 1.5% each day. An acceptable amount of lead in her blood would be
about 0.4 mg. (0.4 mg is "acceptable", but it is not "desirable.")
a. How much lead is in her blood after 1, 2, and 3 days, without treatment?
b. Develop a formula for the amount of lead in her blood after t days. Graph this function on
your calculator. Use the graph to determine how long it will take for the amount of lead in her
blood to be cut in half, from 2 mg to 1 mg.
c. Determine how long it will take to be cut in half again, from 1 to 0.5 mg.
d. What is the half-life of lead in the blood? What is half-life?
e. How long will it take until the level of lead drops to the acceptable level of 0.4 mg?
f. How long will it take until the level of lead drops to a more desirable level of 0.2 mg?
Medical treatment for lead poisoning is to give drugs that increase the rate of elimination of lead
from the blood through the urine. A problem with this treatment is that the drugs also cause the
body to eliminate essential metals such as zinc. Another problem with the treatment is that the
chemicals which cause us to eliminate a higher percentage of lead also cause us to absorb
more of the lead from our environment. For the treatment to be successful, the patient needs to
be removed from the environment which caused the problem.
Information in this activity was obtained from the Merck Manual, Merck & Co., Inc., Whitehouse Station, NJ; from an
interview with Howard Meilke in "Lead Poisoning Report," Acumen Technologies, Inc.; and from Goodman and
Gilman's The Pharmacological Basis of Therapeutics: A.G. Gilman, L. S. Goodman, T.W. Rall, and F. Murad,
Macmillan Pub. Co., NY, 1985; the 2001 Summer Science Enrichment Academy Project advised by Dr. Frank
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
61
Rinehart at the University of the Virgin Islands, St. Thomas campus; and from the Cooperative Extension Service,
University of the Virgin Islands.
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
62
Teaching Guide for
Lead in Children's Bodies
Introduction: The mathematics in this lesson strengthens and extends students’ understanding
of positive integer exponents and requires careful thinking about percents. Students will solve
an exponential equation either graphically or numerically; this is intended to strengthen their
understanding of the solution of an equation. The lesson provides a foundation for
understanding exponential functions to be built on in future courses.
It is important that students do the “reading assignment” for homework as preparation for the
class. The lesson should open with a whole-class discussion of the reading, leading into the
reading’s problems, question 1. The presence of lead in the soil and in plumbing is a significant
hazard to children in the Virgin Islands and elsewhere. It is valuable to impress upon students
the truth and significance of the information and the mathematics in this lesson, since this may
well influence their commitment to learning mathematics.
Answers and teaching suggestions:
1. Suppose a child has 1.0 mg of lead in her blood. Assume this child's body removes 1.5% of the lead
in the blood each day. Use a calculator for the following questions.
a. Assuming this child is removed from all sources of lead, how much lead is in her body after 1, 2, 3,
and 4 days? Use 4 decimal place accuracy.
t
1
2
3
4
Lead remaining
after t days (mg)
.985
.9702
.9557
.9413
It is important that students use 4 decimal places because the answers for the correct
approach, the lead remaining on day t+1, L(t+1) = .985L(t) or L(t+1) = L(t) - .015L(t) do not
differ from the answers from an incorrect approach, L(t+1) = L(t) - .015 in the first two
decimal places for t<4.
b. How many days will it take until the amount of lead in this child's blood drops below 0.9 mg?
It will take 7 days. This question is intended to force the students to want to automate their procedure.
On a graphing calculator, they can input
1
.985 ANS
ENTER
ENTER
ENTER
ENTER…..
until they see an output less than .9.
If they do this, it will facilitate their answering the next question.
c. Develop a function f (t ) that gives the amount of lead in this child's blood after t days, assuming
the child is not exposed to any more lead.
If students recognize that each day the lead remaining is .985 times the amount of lead that
was in the body at the end of the previous day, it is a short step to realizing that on day 5, for
example, the amount remaining is 1x.985x.985x.985x.985x.985=(.985)5. And in general, on
day t, the amount remaining is f(t) = (.985)5.
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
63
The result is a little less obvious if students took the approach L(t+1) = L(t) - .015L(t); in this
case, it may require questioning to get them to notice that L(t) - .015L(t) = 0.985L(t). If they
see this, they can reason as above.
For some students, the notation f(t) is an obstacle. In that case, encourage them to use L,
for lead, or y or any convenient variable. Seeing f(t) and similar notation in this introductory
manner helps familiarize them with it and minimizes the discomfort that some students
experience in subsequent courses.
Discussion of question 1 in the class following students’ reading and working question 1 should
be thorough. Students should be given time to discuss their work in groups of 3 or 4 first. Then
a summarizing discussion can be led by a student or by you. Be sure students are comfortable
with the basics, such as the decimal form of the percents and the important notion that if 1.5% is
removed then 100%-1.5% = 98.5% remains. Be sure students see the pattern and are
comfortable with the basic use of the exponent in saying, for example, .985x.985x.985=.9853.
Finally, help students to become comfortable with the generalization for day t, f (t ) = (.985) t .
Then students should work in their groups to continue their explorations in question 2.
2. A child with 70-100 µg of lead per deciliter of blood needs urgent treatment. For a 60 lb. girl, this
would be about 1.4 to 2 mg of lead in the blood. Suppose this girl has 2 mg of lead in her blood and
eliminates 1.5% each day. An acceptable amount of lead in her blood would be about 0.4 mg. (0.4 mg is
"acceptable", but it is not "desirable.")
a. How much lead is in her blood after 1, 2, and 3 days, without treatment?
Encourage students to enter this information into a table, as they did in question 1a.
t
Lead remaining
after t days (mg)
1
2
3
1.97
1.94045
1.91134
Students can get these answers quickly on a calculator by putting in
2
.985 ANS
ENTER
ENTER
ENTER
ENTER
b. Develop a formula for the amount of lead in her blood after t days. Graph this function on your
calculator. Use the graph to determine how long it will take for the amount of lead in her blood to be cut
in half, from 2 mg to 1 mg.
The formula we are after, y = 2(.985)t, differs from the formula in question 1 only in its initial
value, 2 instead of 1. Nevertheless, it is not always easily seen by students. You will probably
need to ask questions to help students focus on what they did at each step, until they recognize
the pattern: each day, the lead remaining was 2 x .985 x .985 x .985 x ….
Students should recognize that what they are being asked to do is to solve the equation
2(.985)t = 1 graphically. They can do this by graphing y = 2(.985)t and y = 1 on the same screen
and finding the points of intersection.
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
64
2
1.5
1
0.5
50
100
150
200
The graphs intersect at about (45.86, 1), so sometime during the 46th day, the amount of lead
reaches the 1 mg level.
c. Determine how long it will take to be cut in half again, from 1 to 0.5 mg.
Graphing y = 2(.985)t and y = 0.5, students will find the point of intersection at (91.72, 2). Thus
the time it takes for the lead to be reduced from 1 mg to 0.5 mg is 91.72 – 45.86 = 45.86, or
about 46 more days. Be sure they see that this is what the question asks.
d. What is the half-life of lead in the blood? What is half-life?
Students need to know what half-life is to answer the first question. They can look it up or you
can explain that half-life is the length of time it takes a substance to be reduced by half. The
concept only has meaning if this is a constant value. We see from parts b and c above that it
took 46 days for the lead to be reduced from 2 mg to 1, and 46 days for it to be reduced from 1
mg to ½ mg. Corroboration that 46 days is the half life will be given in part f.
e. How long will it take until the level of lead drops to the acceptable level of 0.4 mg?
About 106 days.
f. How long will it take until the level of lead drops to a more desirable level of 0.2 mg?
Ask students how much longer it should take to drop from .4 mg to .2 if the half life is really 46
days. It should take 46 more days. When they test to find the intersection of y = 2(.985) t and
y = 0.2, they will find that this occurs at about t = 152 days, just as they thought it would since
106+46=152.
When you summarize what students have learned from this lesson, it would be valuable to
distinguish for them between what they have learned that they will be expected to use in your
course and what they have learned that provides a foundation for mathematics that will be
learned in subsequent pre-calculus and calculus courses, as well as being very important
conceptually for understanding so much of our environment.
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.