Standard 15 - CONCEPTUAL BUILDING BLOCKS OF CALCULUS
K-12 Overview
All students will develop an understanding of the conceptual
building blocks of calculus and will use them to model and analyze
natural phenomena.
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Descriptive Statement
The conceptual building blocks of calculus are important for
everyone to understand. How quantities such as world population
change, how fast they change, and what will happen if they keep
changing at the same rate are questions that can be discussed by
elementary school students. Another important topic for all
mathematics students is the concept of infinity - what happens as
numbers get larger and larger and what happens as patterns are
continued indefinitely. Early explorations in these areas can broaden
students' interest and understanding of an important area of
applied mathematics.
Meaning and Importance
Calculus is the mathematics used to describe processes evolving in
space or time. How quantities change - the velocity of a
car as its position changes over time or the area of a square as its
sides lengthen - and what happens in the long run are
central themes of mathematics and its application to the real
world. Calculus is used to describe an exact result as the limit of
a sequence of approximations. Calculus is essential for
understanding the physical world and indispensable in economics and
industry. Engineers and physicists use calculus to calculate motion
in response to forces. Businessmen and economists use calculus to
find optimal solutions to pricing and production. An intuitive feel
for the mathematics of infinity, limit, and change are accessible and
necessary for all students.
Although some students will go on to study these concepts in a
formal calculus course, this standard does not advocate the
formal study of calculus in high school for all students or even for
all college-intending students. Rather, it calls for providing
opportunities for all students to informally investigate the central
ideas of calculus. Considering these concepts will contribute to a
deeper understanding of the function concept and its usefulness in
representing and answering questions about real-world situations.
K-12 Development and Emphases
Students at the elementary level should understand the concept of
linear growth (constant increments). For instance, a savings
account grows linearly if equal deposits are made at regular intervals
and the account bears no interest. This idea and its extensions can
be introduced and mastered without using the mathematical formalism of
functions, which is introduced later in the middle grades. Beginning
in elementaryschool, children can accumulate records of processes
which exhibit linear growth. Some examples are mileage accumulated by
traveling between home and school, cumulative expenses for school
lunches, and cumulative volume of cereal consumed if everyone in the
class eats a bowl every morning. Children should learn to recognize
linear growth and compare it to the more irregular pattern of
increases in their own height, the height of a bean plant, cumulative
rainfall, class consumption of paper, and real expenditures. Children
in elementary school can be introduced to exponential growth
(ever-increasing increments) through the discovery that if every pair
of rabbits produces two rabbits each month (one new rabbit for every
existing rabbit) then in less than two years there would be more than
a million rabbits!
Middle school students should be moving beyond the concrete and
pictorial representations used in the elementary grades to more
symbolic ones, involving functions and equations. They should use
graphing calculators and computers to develop and analyze graphical
representations of the changes represented in the tables, and to
produce linear and quadratic regression models of the data. They
should apply their knowledge of decimals to solving problems involving
compound interest, making use of a calculator to determine, for
example, the yield of a given investment or the length of time it
would take for an investment to double. In high school, students can
apply their knowledge of exponents, algebra, and functions to solve
these and other more difficult problems, with applications to growth
in economics and biology (e.g. population explosion), algebraically
and graphically.
Throughout their school years, students should be examining a
variety of situations where populations and other quantities change
over time, and use the mathematical tools at their disposal to
describe and analyze this change. As they progress, the situations
considered should become more complex; students who experiment with
constant motion in their early years will be able to understand the
motion of projectiles (a ball thrown into the air, for example) by the
time they complete high school.
Similarly, students should be aware how changing the linear
dimensions of an object - such as its height, length, or
diameter - affects its area and volume. In the early years
children should be involved in hands-on experiments which illustrate
this; for example, they might find that doubling the diameter of a
circular can (of fixed height) increases the volume four-fold by
filling the smaller can with water or rice and emptying it into the
larger one. By the time students are familiar with variables, this
intuition will provide them with the information they need to
understand formulas such as those involving volume.
In many settings, the kind of change that takes place over time is
repetitive and cumulative, and an important question that should be
discussed is what happens in the long run. The principal tool for
understanding and discussing such questions is the concept of infinite
sequences and the types of patterns that emerge from them. Thus a
second central theme is that of infinity.
Students are fascinated with the mysteries of large numbers and
"infinity," and that excitement should be nourished and be
used, as with other "teachable moments," to motivate the
learning of more mathematics. Primary students enjoy naming their
"largest" number or proudly declaring that there is no
largest! In the early years, large numbers and their significance
should be discussed, as should the idea that one can extend simple
processes forever (e.g., keep adding 2, keep multiplying by 3).
Once students have familiarity with fractions and decimals, these
notions can be extended. What happens when you keep
dividing by 2? By 10? Can you find a fraction between 0.999 and 1?
What decimal comes just before 1? Students should explore
and experiment with infinitely repeating decimals and other infinite
series, where they can make tables and look for patterns. They should
learn that by repeated iteration of simple processes you can get
better and better answers in both arithmetic (with increased
decimalaccuracy) and in geometry (with more accurate estimates of the
area and volume of irregular objects).
Although the concept of a limiting value (or a limit) may
appear inaccessible to K-8 students, this basic notion of calculus can
be explored through the process of measuring the area of a region.
Students can be provided with diagrams of a large circular (or
irregular) region, say a foot in diameter, and a large supply of tiles
of different square sizes. By covering the space inside the region
(with no protrusions!) with 4" tiles, then with 2" tiles,
then with 1" tiles, then with .5" tiles, students can gain
an appreciation that the smaller the unit used, the larger the area
obtained. They will recognize that the space cannot be filled
completely with small tiles, yet, at the same time, the sum of the
areas of the smaller tiles gets closer and closer to the actual area
of the region.
In summary, these kinds of experiences will provide a good
foundation for the notions of limits, infinity, and changes in
quantities over time. Such concepts find many applications in both
science and mathematics, and students will feel much more comfortable
with them if they begin to develop these concepts in the early
grades.
Note: Although each content standard is discussed
in a separate chapter, it is not the intention that each be
treated separately in the classroom. Indeed, as noted in the
Introduction to this Framework, an effective curriculum is one
that successfully integrates these areas to present students with rich
and meaningful cross-strand experiences.
Standard 15 - Conceptual Building Blocks of Calculus - Grades K-2
Overview
Students in the early primary grades bring to the classroom
intuitive notions of the meaning of such terms as biggest, largest,
change, and so forth. While they may not know the names of large
numbers, they certainly have a sense of "largeness." The
cumulative process indicators related to this standard for grades K-2
deal primarily with investigating patterns of growth and change over
time.
Students in grades K-2 should investigate many different types of
patterns. For some of these patterns, such as 2, 4, 6, 8, ... , the
same number is added (or subtracted) to each number to get the next
number in the sequence. When these patterns are represented with a
bar graph, the tops of the bars can be connected by a straight line,
so the pattern represents linear growth. Older students should
also see patterns that grow more rapidly, such as 2, 4, 8, ... .
These growing patterns involve exponential growth; each
number in the series is multiplied (or divided) by the same number to
get the next one. In this situation, when the tops of the bars on a
graph are connected, they do not form a straight line. These types of
patterns can be investigated very easily by using calculators to do
the computation; students enjoy making the numbers bigger and bigger
by using a constant addend (e.g., 2 + 2 = = = ) or a constant
multiplier (e.g., 2 x 2 = = = ). (Note that some
calculators require different keystrokes to achieve this effect.) By
relating these problems to concrete situations, such as the growth of
a plant, students begin to develop a sense of change over
time.
Students also begin to develop a sense of change with respect to
measurement. Students begin to measure the length of objects
by using informal units such as paperclips or Unifix cubes; they
should note that it takes more small objects to measure a given length
than large ones. By the end of second grade, they begin to describe
the area of objects by counting the number of squares that cover a
figure. Again, they should note that it takes more small squares to
cover an object than it does large ones. They should also begin to
investigate what happens to the area of a square when each side is
doubled. Students also need to develop volume concepts by filling
containers of different sizes. They might use two circular cans, one
of which is twice as high and twice as wide as the other, to find that
the large one holds eight times as much as the small one. Measurement
may also lead to the beginnings of the idea of a limiting value
for young children. For example, the size of a dinosaur footprint
might be measured by covering it with base ten blocks. If only the
100 blocks are used, then one estimate of the size of the footprint is
found; if unit blocks are used, a more precise estimate of the size of
the footprint can be found.
Students in grades K-2 should also begin to look at concepts
involving infinity. As they learn to count to higher numbers,
they begin to understand that, no matter how high they count, there is
always a bigger number. By using calculators, they can also begin to
see that they can continue to add two to a number forever and the
result will just keep getting bigger.
The conceptual underpinnings of calculus for students in grades K-2
are closely tied to their developing understanding of number sense,
measurement, and pattern. Additional activities relating to this
standard can be found in the chapters discussing these other
standards.
Standard 15 - Conceptual Building Blocks of Calculus - Grades K-2
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in kindergarten
and grades 1 and 2.
Experiences will be such that all students in grades K-2:
1. Investigate and describe patterns that
continue indefinitely.
- Students model repeating patterns with counters or
pennies. For example, they repeatedly add two pennies to their
collection and describe the results.
- Students create repeating patterns with the calculator.
They enter any number such as 10, and then add 1 for 10 + 1
=== . . . . The calculator will automatically repeat the
function and display 11, 12, 13, 14, etc. each time the = key is
pressed. (Some calculators may need to have the pattern entered
twice: 10 + 1 = + 1 === . . . etc. Others may use a key
sequence such as 1++10 ===. . . .) Students may repeatedly
add (or subtract) any number.
- Second graders create a pattern with color tiles. They
start with one square and then make a larger square that is two
squares long on each side; they note that they need four tiles to do
this. Then they make a square that is three squares long on each
side; they need nine tiles to do this. They make a table of their
results and describe the pattern they have found.
- Students investigate a doubling (growing) pattern with
Unifix cubes. They begin with one cube and then "win"
another cube. Then they have two cubes and "win" two more.
They continue this pattern, each time "winning" as many
cubes as they already have. Repeating this process, they begin to see
how quickly the number of cubes grows. They investigate this further
using a calculator.
- Students start with a rectangular sheet of paper that
represents a cake. They simulate eating half of the cake by cutting
the sheet in half and removing one of the halves. They eat half of
what is left and continue this process. They describe the pattern,
noting that after they repeat this about ten times, the cake is
essentially gone.
2. Investigate and describe how certain quantities
change over time.
- Students keep a daily record of the temperature both inside
and outside the classroom. They graph these temperatures and look at
the patterns.
- Students keep a monthly record of their height and record
the data collected on a bulletin board. At the end of the school
year, they describe what happened over time.
- Students play catch with a ball in the school playground.
One person counts out the number of times the ball is thrown, the
other counts out the distance that it travels, a third person adds
that distance to the total, and a fourth person records the totals.
Afterwards they discuss how the total distance changes over time; they
recognize that the sameamount is added repeatedly.
- Students study the changes in the direction and length of
the shadow of a paper groundhog at different times of the day. They
relate these observations to the position of the sun (e.g., as the sun
gets higher, the shadow gets shorter).
- Students discuss how ice changes to water as it gets
hotter. They talk about how it snows in January or February but rains
in April or May.
- Students plant seeds and watch them grow. They write
about what they see and measure the height of their plants at regular
time intervals. They discuss how changes in time result in changes in
the height of the plant. They also talk about how other factors might
affect the growth of the plant, such as light and water.
3. Experiment with approximating length, area, and
volume, using informal measurement instruments.
- Students measure the width of a bookcase using the 10-rods
from a base ten blocks set. They record this length (perhaps as 6 rods
or 60 units). Then they measure the bookcase using ones cubes; some
of the students decide that it is easier just to add some ones cubes
to the 10-rods that they have already used. They find that the
bookcase is actually closer to 66 units long. They decide that they
can get a better estimate of length when they use smaller units.
- Students use pattern blocks to cover a picture of a
turtle. They count how many of each type of block (green triangle,
yellow hexagon, etc.) they used. They make a bar graph that shows how
many blocks each student used. They discuss why some students used
more blocks than others and what they could do to increase or decrease
the number of blocks used.
- Students play with containers of various sizes,
transferring water from one container to another. They note that it
takes two cups of water to fill a small milk carton. A pitcher holds
three milk cartons of water, but four milk cartons overflow the
pitcher. Then they find that it takes seven cups to fill the pitcher
even though three milk cartons is only six cups. They decide that the
smaller container gives a better idea of how much the pitcher will
hold.
- Students find the area of huge dinosaur footprints that
they find taped to the classroom floor. They first try to fit as many
green 4" tiles as possible into a footprint without any
overlapping, and without any tiles sticking out of the footprint.
Before removing the green tiles, they cover them with blue 2"
tiles, and count the number of blue 2" tiles used. Then they
remove the green tiles and try to fit more blue 2" tiles into the
footprint without overlapping; they discover that they can fit more
and discuss why that is the case. They repeat this, using red 1"
tiles. They notice that with smaller tiles, less of the footprint is
uncovered, so that the smaller tiles provide a better estimate of the
footprint's size.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during Spring
1997. In time, we hope to post additional resources relating to this
standard, such as grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
Standard 15 - Conceptual Building Blocks of Calculus - Grades 3-4
Overview
Students in grades 3 and 4 continue to develop the conceptual
building blocks of calculus primarily through their work with patterns
and changes over time. Students investigate a variety of patterns,
using physical materials and calculators as well as pictures. In some
of the patterns investigated, a constant is added to, or subtracted
from, each number to get the next number in the sequence. These
patterns, involving repeated operations, show linear growth;
when these patterns are represented with a bar graph, the tops of the
bars can be connected by a straight line. Examples of such patterns
include skip counting, starting the week with $5 and paying 75 cents
each day for lunch, or enumerating the multiples of 9. Other patterns
should involve multiplying or dividing a number by a constant to get
the next number in the sequence. These growing patterns illustrate
exponential growth, as is the pattern which results when you
start with two guppies (one male and one female) and the number of
guppies doubles each week. Patterns should also include looking at
changes over time, since these types of patterns are extremely
important not only in mathematics but also in science and social
studies. Students might chart the height of plants over time, the
number of teeth lost each month throughout the school year, or the
temperature outside the classroom over the course of several
months.
Students continue to develop their understanding of
measurement, gaining a greater understanding of the approximate
nature of measurement. Students can guess at the length of a stick
that is between 3 and 4 inches long, saying it is about 3 1/2 inches
long and recognizing when this is a better approximation than either 3
or 4. They can use grids of different sizes to approximate the area
of a puddle, recognizing that the smaller the grid the more accurate
the measurement. They can begin to consider how one might measure the
amount of water in a puddle, coming up with alternative strategies and
comparing them to see which would be more accurate. As they develop a
better understanding of volume, they may use cubes to build a solid,
build a second solid whose sides are all twice as long as the first,
and then compare the number of cubes used to build each solid. The
students may be surprised to find that it takes eight times as many
cubes to build the larger solid!
Students continue to develop their understanding of infinity
in grades 3 and 4. Additional work with counting sequences, skip
counting, and calculators further reinforces the notion that there is
always a bigger number. Taking half of something (like a rectangular
cake or a sheet of paper) repeatedly suggests that there is always a
number that is still closer to zero.
As students develop the conceptual underpinnings of calculus in
third and fourth grades, they are also working to develop their
understanding of numbers, patterns, measurement, data analysis, and
mathematical connections. Additional ideas for activities relating to
this standard can be found in the chapters discussing these other
standards.
Standard 15 - Conceptual Building Blocks of Calculus - Grades 3-4
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 3 and
4.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 3-4 will be such that all students:
1. Investigate and describe patterns that
continue indefinitely.
- Using the constant multiplier feature of a calculator,
students see how many times 1 must be doubled before one million is
reached. They might first guess the number of steps to one million,
and to half a million.
- Students start with a long piece of string.
They fold it in half and cut it in two, setting aside one piece. Then
they take the remaining half, fold it in half, cut it apart, and set
aside half. They continue this process. They discuss how the length
of the string keeps getting smaller, half as much each time, so that
after about ten cuts, there is essentially no string left. Some
students may understand that the process could keep going for several
more steps, if we could only cut more carefully, and some may realize
that in theory the process could continue forever.
- Students count out 1, 2, 3, 4, 5, 6, ... , and recognize
that this pattern could continue forever. They also count out other
patterns, like the even numbers, or the square numbers, or
skip-counting by 3s starting with 2, and recognize that these patterns
also could be continued indefinitely.
- Students investigate the growth patterns of
sunflowers, pinecones, pineapples, or snails to study the natural
occurrence of spirals.
2. Investigate and describe how certain quantities
change over time.
- Students begin with a number like 5, add 3 to it, add 3 to it
again, and repeat this five times. They record the results in a table
and make a bar graph which represents the numbers that they have
generated. They draw a straight line connecting the tops of the bars.
They experiment with numbers other than 5 or 3 to see if the same
thing occurs.
- Students begin with a number less than 10, double it, and
repeat the doubling at least five times. They record the results of
each doubling in a table and make a bar graph which represents the
numbers they have generated. They discuss whether they can connect
the tops of the bars with a straight line.
- Students measure the temperature of a cup of water with ice
cubes in it every fifteen minutes over the course of a day. They
record their results (time passed and temperature) in a table and plot
this information on a coordinate grid to make a line graph. They
discuss how the temperature changes over time and why. Initially the
temperature will increase rapidly, but as the water warms up, its
temperature will increase more slowlyuntil it essentially reaches room
temperature.
- Students are given several examples of bar graphs with
straight lines connecting the tops of the bars. They are asked to
describe a motion scenario which reflects the data. For example, they
might indicate that a graph reflects their running to an after-school
activity, staying there for an hour, and then slowly walking home to
do their chores.
- Students keep a monthly record of their height and record
the data collected on a bulletin board. At the end of the school
year, they describe what happened over time. They also find each
month the average height of all the students in the class, and discuss
how the change in average height over the year is similar to, and how
it is different from, the change in height over the year of the
individual students in the class.
- Students plant some seeds in vermiculite and some in soil.
They observe the plants as they grow, measuring their height each week
and recording their data in tables. They examine not only how the
height of each plant changes as time passes but also whether the seeds
in vermiculite or soil grow faster.
3. Experiment with approximating length, area, and
volume, using informal measurement instruments.
- Students measure the length of their classroom using their
paces and compare their results. They discuss what would happen if the
teacher measured the room with her pace.
- Students use pattern blocks to cover a drawing of a
dinosaur with as few blocks as they can. They record the number of
blocks of each type used in a table and then discuss their results,
making a frequency chart or bar graph of the total number of blocks
used by each pair of students. Then they try to cover the same
drawing with as many blocks as they can. They again record and
discuss their results and make a graph. They look for connections
between the numbers and types of blocks used each time. Some students
simply trade blocks (e.g., a hexagon is traded for six triangles),
while other students try to use all tan parallelograms since that
seems to be the smallest block. (It actually has the same area as the
triangle, however.)
- Students compare the volumes of a half-gallon milk carton,
a quart milk carton, a pint milk carton, and a half-pint milk carton.
They also measure the length of the side of the square base of each
carton and its height. They make a table of their results and look
for patterns. The students notice that the difference between the
height measurements is not the same as the difference between the
volume. The differences in volume grow more quickly than the
differences in the heights. They see how many small cubes or marbles
fill up each of their containers, and they try to explain why more
than twice as many fit into a quart container than a pint
container.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during Spring
1997. In time, we hope to post additional resources relating to this
standard, such as grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
Standard 15 - Conceptual Building Blocks of Calculus - Grades 5-6
Overview
Students in grades 5 and 6 extend and clarify their understanding
of patterns, measurement, data analysis, number sense, and algebra as
they further develop the conceptual building blocks of calculus. Many
of the basic ideas of calculus can be examined in a very concrete and
intuitive way in the middle grades.
Students in grades 5 and 6 should begin to distinguish between
patterns involving linear growth (where a constant is added or
subtracted to each number to get the next one) and exponential
growth (where each term is multiplied or divided by the same
number each time to get the next number). Students should recognize
that linear growth patterns change at a constant rate. For example, a
plant may grow one inch every day. They should also begin to see that
if these patterns are graphed, then the graph looks like a straight
line. They may model this line by using a piece of spaghetti and use
their graph to make predictions and answer questions about points that
are not included in their data tables. In contrast, exponential
growth patterns change at an increasingly rapid rate; if you start
with one penny and double that amount each day, you receive more and
more pennies each day as time goes on. Students should note that the
graphs of these situations are not straight lines. At this grade
level, students should also begin to imagine processes that could in
theory continue forever even though they could not be carried out in
practice; for example, although in practice a cake can be repeatedly
divided in half only about ten times, nevertheless it is possible to
imagine continuing to divide it into smaller and smaller pieces.
Many of the examples used should come from other subject areas,
such as science and social studies. Students might look at such linear
relationships as profit as a function of selling price, but they
should also consider nonlinear relationships such as the amount of
rainfall over time. Students should look at functions which have
"holes" or jumps in their graphs. For example, if students
make a table of the parking fees paid for various amounts of time and
then plot the results, they will find that they cannot just connect
the points; instead there are jumps in the graph where the parking fee
goes up. A similar situation exists for graphs of the price of a
postage stamp or the minimum wage over the course of the years. Many
of the situations investigated by students should involve such
changes over time. Students might, for example, consider the
speed of a fly on a spinning disk; as the fly moves away from the
center of the disk, he spins faster and faster. Students might be
asked to write a short narrative about the fly on the disk and draw a
graph of the fly's speed over time that matches their story.
As students begin to explore the decimal equivalents for fractions,
they encounter non-terminating decimals for the first time. Students
should recognize that calculators often use approximations for
fractions such as .33 for 1/3. They should look for patterns
involving decimal representations of fractions, such as recognizing
which fractions have terminating decimal equivalents and which do not.
Students should take care to note that pi is a nonterminating,
nonrepeating decimal; it is not exactly equal to 22/7 or 3.14, but
these approximations are fairly close to the actual value of pi
and can usually be used for computational purposes. The examination
of decimals extends students' understanding of infinity to
very small numbers.
Students in grades 5 and 6 continue to develop a better
understanding of the approximate nature of measurement.
Students are able to measure objects with increasing degrees of
accuracy and begin to consider significant digits by looking at the
range of possible values that might result from computations with
approximate measures. For example, if the length of a rectangle to
the nearest centimeter is 10 cmand its width to the nearest centimeter
is 5 cm, then the area is about 50 square centimeters. However, the
rectangle might really be as small as 9.5 cm x 4.5 cm, in which case
the area would only be 42.75 square centimeters, or it might be almost
as large as 10.5 cm x 5.5 cm, with an area of 57.75 square
centimeters. Students should continue to explore how to determine the
surface area of irregular figures; they might, for example, be asked
to develop a strategy for finding the area of their hand or foot.
They should do similar activities involving volume, perhaps looking
for the volume of air in a car. Most of their work in this area in
fifth and sixth grade will involve using squares or cubes to
approximate these areas or volumes.
Standard 15 - Conceptual Building Blocks of Calculus - Grades 5-6
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 5 and
6.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 5-6 will be such that all students:
4. Recognize and express the difference between
linear and exponential growth.
- Students develop a table showing the sales tax paid on
different amounts of purchases, graph their results, note that the
graph is a straight line, and recognize that this situation represents
a constant rate of change, or linear growth.
- Students make a table showing how much money they
would have at the end of each of eight years if $100 was invested at
the beginning and the investment grew by 10% each year. They note that
the graph of their data is not a straight line; this graph represents
exponential growth.
- Students make a table showing the value of a car as it
depreciates over time. They note that the graph of their data is not
a straight line; this graph represents "exponential
decay."
- Students are presented stories which represent real life
occurrences of linear and exponential growth and decay over time, and
are asked to construct graphs which represent the situation and
indicate whether the change is linear, exponential, or neither.
5. Develop an understanding of infinite
sequences that arise in natural situations.
- Students make equilateral triangles of different
sizes out of small equilateral triangles and record the number of
small triangles used for each larger triangle. These numbers are
called triangular numbers. If the following triangular
pattern is continued indefinitely, then the number of 1s in the first
row, the number of 1s in the first two rows, the number of 1s in the
first three rows, etc. form the sequence of triangular numbers. The
triangular numbers also emerge from the handshake problem: If each
two people in a room shake hands exactly once, how many
handshakes take place altogether? If the answers are listed for
2, 3, 4, 5, 6, 7, ... people, the numbers are again the triangular
numbers 1, 3, 6, 10, 15, 21, ... .
1
1 1
1 1 1
1 1 1 1
1 1 1 1 1
1 1 1 1 1 1
- Students imagine cutting a sheet of paper into half,
cutting the two pieces into half, cutting the four pieces into half,
and continuing this over and over again, for about 25 times. Then they
imagine taking all of the little pieces of paper and stacking them on
top of one another. Finally, they estimate how tall that stack would
be.
- Students describe, analyze, and extend the
Fibonacci sequence (1, 1, 2, 3, 5, 8, ...). They research occurrences
of this sequence in nature, such as sunflower seeds, the fruit of the
pineapple, and the rabbit problem. They create their own
Fibonacci-like sequences, using different starting numbers.
6. Investigate, represent, and use non-terminating
decimals.
- Students use their calculators to find the decimal
equivalent for 2/3 by dividing 2 by 3. Some of the students get an
answer of 0.66667, while others get 0.6666667. They do the problem by
hand to try to understand what is happening. They decide that
different calculators round off the answer after different numbers of
decimal places. The teacher explains that the decimal for 2/3 can be
written exactly as .666... .
- Students have been looking for the number of different
squares that can be made on a 5 x 5 geoboard and have come up with
1x1, 2x2, 3x3, 4x4, and 5x5 squares. One student finds a different
square, however, whose area is 2 square units. The students wonder
how long the side of the square is. Since they know that the area is
the length of the side times itself, they try out different numbers,
multiplying 1.4 x 1.4 on their calculators to get 1.89 and
1.5 x 1.5 to get 2.25. They keep adding decimal places,
trying to get the exact answer of 2, but find that they cannot, no
matter how many places they try!
7. Represent, analyze, and predict relations
between quantities, especially quantities changing over
time.
- Students study which is the better way to cool down a soda,
adding lots of ice at the beginning or adding one cube at a time at
one minute intervals. Each student first makes a prediction and the
class summarizes the predictions. Then the class collects the data,
using probes and graphing calculators or computers and displays the
results in table and graph form on the overhead. The students compare
the graphs and write their conclusions in their math notebooks. They
discuss the reasons for their results in science class.
- Students make a graph that shows the price of mailing a
letter from 1850 through 1995. Some of the students begin by simply
plotting points and connecting them but soon realize that the price of
a stamp is constant for a period of time and then abruptly jumps up.
They decide that parts of this graph are like horizontal lines. The
teacher tells them that mathematicians call this a "step
function"; another name for this kind of graph is a piecewise
linear graph because the graph consists of linear pieces.
- Students review Mark's trip home from school on his
bike. Mark spent the first few minutes after school getting his books
and talking with friends, and left the school grounds about five
minutes after school was over. He raced with Ted to Ted's house
and stopped for ten minutes to talk about their math project. Then he
went straight home. The students draw a graph showing the distance
covered by Mark with respect to time. Then, with the teacher's
help, the class constructs a graph showing the speed at which Mark
traveled with respect to time. The students then write their own
stories and generate graphs of distance vs. time and graphs of speed
vs. time.
8. Approximate quantities with increasing degrees of
accuracy.
- Students find the volume of a cookie jar by first using
Multilink cubes (which are 2 cm on a side) and then by using
centimeter cubes. They realize that the second measurement is more
accurate than the first.
- Students measure the circumference and diameter of a paper
plate to the nearest inch and then divide the circumference by the
diameter. They repeat this process, using more accurate measures each
time (to the nearest half-inch, to the nearest quarter-inch,
etc.). They see that the quotients get closer and closer to
pi.
- Using a ruler, students draw an irregularly shaped
pentagon on square-grid paper, taking care to locate the vertices
of the pentagon at grid points. They estimate the area of the
pentagon by counting the number of squares completely inside the
pentagon and adding to it an estimate of the number of full square
that the partial squares inside the pentagon would add up to.
Then they divide the pentagon into triangles and rectangles and find
the area of the pentagon as a sum of the areas of the triangles and
rectangles. They compare the results and explain any
difference.
9. Understand and use the concept of significant
digits.
- Students measure the length and width of a rectangle in
centimeters and find its area. Then they measure its length and width
in millimeters and find the area. They note the difference between
these two results and discuss the reasons for such a difference. Some
of the students think that, since the original measurements were
correct to the nearest centimeter, then the result would be correct to
the nearest square centimeter, while the second measurements would be
correct to the nearest square millimeter. However, when they
experiment with different rectangles, for example, one whose
dimensions are 3.2 by 5.2 centimeters, they find that the area of 15
square centimeters is not correct to the nearest centimeter.
- Students find the area of a "blob" using a square
grid. First, they count the number of squares that fit entirely
within the blob (no parts hanging outside). They say that this is the
least that the area could be. Then they count the number of squares
that have any part of the blob in them. They say that this is the
most that the area could be. They note that the actual area is
somewhere between these two numbers.
10. Develop informal ways of approximating the surface
area and volume of familiar objects, and discuss whether the
approximations make sense.
- Students trace around their hand on graph paper and count
squares to find an approximate value of the area of their hand. They
use graph paper with smaller squares to find a better
approximation.
- Students work in groups to find the surface area of a
leaf. They describe the different methods they have used to
accomplish this task. Some groups are asked to go back and reexamine
their results. When the class is convinced that all of the results
are reasonably accurate, they consider how the surface area of the
leaf might be related to the growth of the tree and its needs for
carbon dioxide, sunshine, and water.
- Each group of students is given a mixing bowl and asked to
find its volume. One group decides to fill the bowl with centimeter
cubes, packing them as tightly as they can and then to add a
little. Another group decides to turn the bowl upside down and try to
build the same shape next to it by making layers of centimeter cubes.
Still another group decides to fill the hollow 1000-centimeter cube
with water and empty it into the bowl as many times as they can to
fill it; they find that doing this three times almost fills the bowl
and add 24 centimeter cubes to bring the water level up to the top of
the bowl.
11. Express mathematically and explain the impact of
the change of an object's linear dimensions on its
surface area and volume.
- While learning about area, the students became curious
about how many square inches there are in a square foot. Some
students thought it would be 12, while others thought it might be
more. They explore this question using square-inch tiles to make a
square that is one foot on each side. They decide that there are 144
square inches in a square foot; they make the connection with
multiplication, noticing that 144 is 12 x 12 and that
there are 12 inches in a foot. They realize that the square numbers
have that name because they are the areas of squares whose sides are
the whole numbers.
- Having measured the length, width, and height of the
classroom in feet, the students now must find how many cubic yards of
air there are. Some of the students convert their measurements to
yards and then multiply to find the volume. Others multiply first,
but find that dividing by 3 does not give a reasonable answer. They
make a model using cubes that shows that there are 27 cubic feet in a
cubic yard and divide their answer by 27, getting the same result as
the other students.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during Spring
1997. In time, we hope to post additional resources relating to this
standard, such as grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
Standard 15 - Conceptual Building Blocks of Calculus- Grade 7-8
Overview
In grades 7 and 8, students begin to develop a more detailed and
formal notion of the concepts of approximation, rates of change for
various quantities, infinitely repeating processes, and limits.
Activities should continue to emerge from concrete, physical
situations, often involving the collection of data.
Students in grades 7 and 8 continue to develop their understanding
of linear growth, exponential growth, infinity, and
change over time. By collecting data in many different
situations, they come to see the commonalities and differences in
these types of situations. They should recognize that, in linear
situations, the rate of change is constant and the graph is a straight
line, as in plotting distance vs. time at constant speeds or plotting
the height of a candle vs. time as it burns. In situations involving
exponential growth, the graph is not a straight line and the rate of
change increases or decreases over time. For example, in a situation
in which a population of fish triples every year, the number of fish
added each year is more than it was in the previous year. Students
should also have some experience with graphs with holes or jumps
(discontinuities) in them. For example, students may look at how the
price of mailing a letter has changed over the last hundred years,
first making a table and then generating a graph. They should
recognize that plotting points and then connecting them with straight
lines is inappropriate, since the cost of mailing a letter stayed
constant over a period of several years and then abruptly increased.
They should be aware of other examples of "step functions"
whose graphs look like a sequence of steps.
Students in these grades should approximate irrational numbers,
such as square roots, by using decimals; they should recognize the
size of the error when they use these approximations. Students should
take care to note that pi is a nonterminating, nonrepeating
decimal; it is not exactly equal to 22/7 or 3.14, but lies between
these values. These approximations are fairly close to the actual
value of pi and can usually be used for computational purposes.
Students may also consider sequences involving rational numbers such
as 1/2, 2/3, 3/4, 4/5, ... . They should recognize that this sequence
goes on forever, getting very close to a limit of one. Students
should also consider sequences in the context of learning about
fractals. (See Standard 7 or 14. )
Seventh and eighth graders continue to benefit from activities that
physically model the process of approximating measurement
results with increasing accuracy. Students should develop a clearer
understanding of the concept of significant digits as they begin to
use scientific notation. They should be able to use these ideas as
they develop and apply the formulas for finding the areas of such
figures as parallelograms and trapezoids. Students should understand,
for example, that if they are measuring the height and diameter of a
cylinder in order to find its volume, then some error is introduced
from each of these measurements. If they measure the height as 12.2
cm and the diameter as 8.3 cm, then they will get a volume of
pi(8.3/2)2(12.2), which their
calculator may compute as being 660.09417
cm3. They need to understand that this answer
should be rounded off to 660.09 cm3
(five significant digits). They also should understand that the
true volume might be as low as
pi(8.25/2)2(12.15) ÷
649.49 cm3 or almost as high as
pi(8.35/2)2(12.25) ÷
670.81 cm3.
Students in these grades should continue to build a repertoire of
strategies for finding the surface area and volume of irregularly
shaped objects. For example, they might find volume not only by
approximating irregular shapes with familiar solids but also by
submerging objects in water and finding the amount ofwater displaced
by the object. They might find surface areas by first laying out
patterns of the objects called "nets"; for example, the net
of a cube consists of six squares connected in the shape of a cross
- when creased along the edges of the squares, this
"net" can be folded to form a cube. Then they would place a
grid on the net and count the small squares, noting that the finer the
grid the more accurate the estimate of the area.
Explorations developing the conceptual underpinnings of calculus in
grades 7 and 8 should continue to take advantage of students'
intrinsic interest in infinite, iterative patterns. They should also
build connections between number sense, estimation, measurement,
patterns, data analysis, and algebra. More information about
activities related to these areas can be found in the chapters
discussing those standards.
Standard 15 - Conceptual Building Blocks of Calculus- Grades 7-8
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 7 and
8.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 7-8 will be such that all students:
4. Recognize and express the difference between
linear and exponential growth.
- Students measure the height of water in a beaker at five
second intervals as it is being filled, being careful to leave the
faucet on so that the water runs at a constant rate. They make a
table of their results and generate a graph. They note that this is a
linear function.
- Students investigate patterns of exponential growth with
the calculator, such as compound interest or bacterial growth. They
make a table showing how much money is in a savings account after one
quarter, two quarters and so on for ten years, if $1000 is deposited
at 5% interest and there are no further deposits or withdrawals. They
represent their findings graphically, noting that this is not a linear
relationship, although in the case of simple interest, where the
interest does not earn interest, the graph is linear.
- Students obtain a table showing the depreciated value of a
car over time. They graph the data in the table and observe that it
is not a straight line. The value of the car exhibits
"exponential decay."
- Students compare different pay scales, deciding which is a
better deal. For example, is it better to be paid a salary
of $250 per week or to be paid $6 per hour? They realize that the
answer to this question depends on the number of hours worked, so they
create a table comparing the pay for different numbers of hours
worked. They make a graph and decide at what point the hourly rate
becomes a better deal.
- Students predict how many times they will be able to fold a
piece of paper in half. Then they fold a paper in half repeatedly,
recording the number of sections formed each time in a table.
Students find that the number of folds physically possible is
surprisingly small (about 7). The students try different kinds of
paper: tissue paper, foil, etc. They describe in writing any patterns
they discover and try to find a rule for the number of sections after
10, 20, or n folds. They also graph the data on a rectangular
coordinate plane using integral values. They extend this problem to a
new situation by finding the number of ancestors each person had
perhaps ten generations ago and also to the situation of telling a
secret to 2 people who each tell two people, etc.
5. Develop an understanding of infinite
sequences that arise in natural situations.
- Students discuss how the Fibonacci sequence (1, 1, 2, 3, 5,
8, 13, ... ) is related to the following problem: begin with two
rabbits (one male and one female), each adult pair of rabbits produces
two babies (one male and one female) each month, the babies themselves
become adults (and start having their own babies) after one month, and
none of the rabbitsever die. The students decide that the Fibonacci
sequence shows how many pairs of rabbits there are each month. The
students explore other patterns in this sequence, noting that each
term is the sum of the two preceding terms.
- Students look for infinite sequences in Pascal's
triangle. Starting at the top 1 and moving diagonally to the left,
there is a constant infinite sequence 1, 1, 1, 1, ... . Starting at
the next 1 and moving diagonally to the left there is the sequence 1,
2, 3, 4, 5, ... of whole numbers. Starting at the next 1 and moving
diagonally to the left, there is the sequence 1, 3, 6, 10, 15,
... of triangular numbers, which records the solutions to all
handshake problems. Also the sum of the numbers in each row yield the
exponential sequence
1, 2, 4, 8, 16, ... .
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6. Investigate, represent, and use non-terminating
decimals.
- Students investigate using simple equations to iterate
patterns. For example, they use the equation y = x + 1 and
start with any x-value, say 0. The resulting
y-value is 1. Using this as the new x yields
a 2 for y. Using this as the next x gives a
3, and so on, resulting in the sequence 1, 2, 3, 4, ... . Then
students use a slightly different equation, y = .1x +
.6, starting with an x-value of .6 and finding the
resulting yvalue. Repeating this process yields the sequence
of y-values .6, .66, .666, .6666, ..., which
approximates the decimal value of 2/3.
- Students explore the question of which fractions have
terminating decimal equivalents and which have repeating decimal
equivalents. They discover that the only fractions in lowest terms
which correspond to terminating decimals are those whose denominators
have only 2 and 5 as prime factors.
- Students explore the question of which fractions have
decimal equivalents where one digit repeats and learn that these are
the fractions 1/9, 2/9, 3/9, ... . They generalize this to find the
fractions whose decimal equivalents have two digits repeating like
.171717 ... .
7. Represent, analyze, and predict relations between
quantities, especially quantities changing over time.
- Students describe what happens when a ball is tossed into
the air, experimenting with a ball as needed. They make a graph that
shows the height of the ball at different times and discuss what makes
the ball come back down. They also consider the speed of the ball:
when is it going fastest? slowest? With some help from the
teacher, they make a graph showing the speed of the ball over
time.
- Students use probes and graphing calculators or computers
to collect data involving two variables for several different science
experiments (such as measuring the time and distance that a toy car
rolls down an inclined plane or measuring the brightness of a light
bulb as the distance from the light bulb increases or measuring the
temperature of a beaker of water when ice cubes are added). They look
at the data that has been collected intabular form and as a graph on a
coordinate grid. They classify the graphs as straight or curved lines
and as increasing (direct variation), decreasing (inverse variation),
or mixed. For those graphs that are straight lines, the students try
to match the graph by entering and graphing a suitable equation.
- Students measure the temperature of boiling water as it
cools in a cup. They make a table showing the temperature at
five-minute intervals for an hour. Then they graph the results and
make observations about the shape of the graph, such as "the
temperature went down the most in the first few minutes,"
"it cooled more slowly after more time had passed," or
"it's not a linear relationship." The students also
predict what the graph would look like if they continued to collect
data for another twelve hours.
- Students make Ferris wheel models from paper plates (with
notches cut to represent the cars). They use the models to make a
table showing the height above the ground (desk) of a person on a
Ferris wheel at specified time intervals (time needed for next chair
to move to loading position). After collecting data through two or
three complete turns of the wheel, they make a graph of time versus
height. In their math notebooks, they respond to questions about
their graphs: Why doesn't the graph start at zero?
What is the maximum height? Why does the shape of the graph
repeat? The students learn that this graph represents a periodic
function.
- Students compare two ways of cooling a glass of soda,
adding lots of ice at the beginning or adding one cube at a time at
one minute intervals. Each student first makes a prediction about
which cools the soda faster, and the class summarizes the
predictions. Then the teacher collects the data, using probes and
graphing calculators or computers and displays the results in table
and graph form on the overhead. The students compare the graphs and
write their conclusions in their math notebooks. They discuss the
reasons for any difference between these two methods with their
science teacher.
- Students compute the average speed of a toy car as it
travels down a ramp by dividing the length of the ramp by the time the
car takes to travel the ramp. They try different angles for the ramp,
recording their results. They make a graph of average speed vs. angle
and discuss whether this graph is linear.
- Students make a graph that shows the minimum wage from the
time it was first instituted until the present day. Some of the
students begin by simply plotting points and connecting them but soon
realize that the minimum wage was constant for a time and then
abruptly jumped up. They decide that parts of this graph are like
horizontal lines. They look for other examples of "step
functions."
8. Approximate quantities with increasing degrees
of accuracy.
- Students measure the speed of cars
using different strategies and instruments and compare the accuracy of
each. For example, they first determine the speed of a car by using a
stopwatch to find out how long it takes to travel a specific distance.
They note that the speed of the car actually changes over the time
interval, however. They decide that they can get a better idea of how
fast the car is moving at a specific time by shortening the distance.
They collect data for shorter and shorter distances. Finally, they
ask a police officer to bring a radar gun to their class to help them
collect data about the speed of the cars going past the school.
- Students find the area of a "blob" using a
square grid. First, they count the number of squares that fit
entirely within the blob (no parts hanging outside). They say that
this is the least that the area could be. Then they count the number
of squares that have any part of the blob in them. They say that this
is the most that the area could be. They note that the actual area is
somewhere between these two numbers. Finally, the students put
together parts of squares to try to get a more accurate estimate of
the area of the blob.
9. Understand and use the concept of significant
digits.
- Students measure the radius of a circle in centimeters and
find its area. Then they measure its radius in millimeters and find
the area. They note the difference between these two results and
discuss the reasons for such a difference. Some of the students think
that, since the original measurements were correct to the nearest
centimeter, then the result would be correct to the nearest square
centimeter, while using the second measurements would give a value for
the area which is correct to the nearest square millimeter. However,
after experimenting with circles of different sizes, they find that if
the radius is measured to the nearest centimeter.
- Students explore the different answers that they get by
using different values for pi when finding the area of a circle.
They discuss why these answers vary and how to decide what value to
use.
- Students estimate the amount of wallpaper, paint, or carpet
needed for a room, recognizing that measurements that are accurate to
several decimal places are unnecessary for this purpose.
10. Develop informal ways of approximating the surface
area and volume of familiar objects, and discuss whether the
approximations make sense.
- In conjunction with a science project,
students need to find the surface area of their bodies. Some of the
students decide to approximate their bodies with geometric solids; for
example, their head is approximately a sphere, and their neck, arms,
and legs are approximately cylinders. They then take the needed
measurements and compute the surface areas of the relevant solids.
Other students decide to use newspaper to wrap their bodies and then
measure the dimensions of the sheets of newspaper used.
- Students estimate the volume of air in a balloon as a way
of looking at lung capacity. Some of the students decide that the
balloon is approximately the shape of a cylinder, measure its length
and diameter, and compute the volume. Other students think the
balloon is shaped more like a cylinder with cones at the ends; they
measure the diameter of the balloon at its widest part, the length of
the cylinder part, and the height of each cone and then compute the
volume of each shape. Some other students decide that they would like
to check their work another way; they place a large graduated cylinder
in the sink, and fill it with water. They submerge the balloon, and
read off how much water is left after the balloon is taken out. Since
they know that 1 ml of water is 1 cm3, they know that the
volume of the water that was displaced is the same as that of the
balloon.
- Students develop different strategies for finding the
volume of water in a puddle.
11. Express mathematically and explain the impact of
the change of an object's linear dimensions on its
surface area and volume.
- Students analyze cardboard milk containers to determine how
the dimensions of the container affect the volume of milk contained in
the carton and how the amount of cardboard used varies. In addition
to measuring actual cartons, students make their own cartons of
different sizes by varying the length, width, and height one at a
time. They write up their results and share them with the class.
- Placing a number of identical cereal boxes next to or on
top of one another, students learn that doubling one of an
object's length, width, or height doubles its volume, that
doubling two of these dimensions increases the volume by a factor of
4, and that doubling all three dimensions increases the volume by a
factor of 8.
- Students sketch a 3-dimensional object such as a box or a
cylindrical trash can. They then make a sketch twice as large in all
dimensions. How much larger is the volume of the larger
object? How much larger would it be if the dimensions all increased
by a factor of 3? Square grid paper might be helpful for
this exercise.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during Spring
1997. In time, we hope to post additional resources relating to this
standard, such as grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
Standard 15 -Conceptual Building Blocks of Calculus - Grades 9-12
Overview
This standard does not advocate the formal study of calculus
in high school for all students or even for all college-intending
students. Rather, it calls for providing opportunities for all
students to informally investigate the central ideas of calculus:
limit, the rate of change, the area under a curve, and the slope of a
tangent line. Considering these concepts will contribute to a deeper
understanding of the function concept and its usefulness in
representing and answering questions about real-world situations.
Instruction should be highly exploratory, based on numerical and
geometric experiences that capitalize on both calculator and computer
technology. Activities should be aimed at providing students with an
understanding of the underlying concepts of calculus rather that at
developing manipulative techniques.
The development of calculus is one of the great intellectual
achievements in history, especially with respect to its use in
physics. Calculus is also increasingly being used in the social and
biological sciences and in business. As students explore this area,
they should develop an awareness of and appreciation for the
historical origins and cultural contributions of calculus.
Students' earlier study of patterns is extended in high school
to the study of finite and infinite processes. Students
continue to look at linear growth patterns as they develop
procedures for finding the sums of arithmetic series (e.g., the sum of
the numbers from 1 to 100). They may consider this sum in many
different ways, building different types of models. Some students may
look at 1 + 2 + 3 + ... + 100 geometrically by putting together
two "staircases" to form a rectangle that is 100 by 101.
Other students may look at the sum arithmetically by adding 1 + 2 +
3 + ... + 100 to 100 + 99 + 98 + ... + 1 and getting 100 pairs of
numbers that add up to 101. Still others may look at the sum by
finding the limit of the sequence of partial sums. Students also look
at exponential growth as they develop procedures for finding
the sum of finite and infinite geometric series (e.g., 2 + 4 + 8 +
16 + 32 or 6 + 3 + 3/2 + ... or finding the total distance
traveled by a bouncing ball). Students' work with patterns and
infinity also includes elaborating on the intuitive notion of limit
that has been addressed in the earlier grades.
High school students further develop their understanding of
change over time through informal activities that focus on the
understanding of interrelationships. Students should collect data,
generate graphs, and analyze the results for real-world situations
that can be described by linear, quadratic, trigonometric, and
exponential models. Some of the types of situations that should be
analyzed include motion, epidemics, carbon dating, pendulums, and
biological and economic growth. They should use Calculator Based Labs
(CBLs) in conjunction with graphing calculators to gather and analyze
data. Students should recognize the equations of the basic models
(y = mx + b, y = ax2 + bx + c, y=sin
x, and y = 2x) and be able to relate
geometric transformations to the equations of these models. Students
should develop a thorough understanding of the idea of slope; for
example, they need to be able compare the steepness of two graphs at
various points on the graph. They also need to be able explain what
the slope means in terms of the real-world situation described by a
graph. For example, what information does the slope give for a
graph of the levels of medicine in the bloodstream over time?
Students also extend their understanding of the behavior of
functions to include the concept of the continuity of a function,
considering features such as removable discontinuities (holes or
jumps), asymptotes, and corners.
Students in high school apply their understanding of approximation
techniques not only with respect to numbers in the context of using
initial portions of nonrepeating, nonterminating decimals but also
with respect to measurement situations. Students further
develop their understanding of significant digits and the arithmetic
of approximate values. They also use repeated approximations to find
the areas of irregular figures, including experimenting with
situations in which they need to find the area under a curve.
Looking at the conceptual underpinnings of calculus provides an
opportunity for high school students to pull together their
experiences with data analysis, patterns, algebra, measurement, number
sense, and numerical operations. It also provides the opportunity to
apply technology to real-world situations and to gain experience with
mathematics as a dynamic human endeavor.
Standard 15 - Conceptual Underpinnings of Calculus - Grades 9-12
Indicators and Activities
The cumulative progress indicators for grade 12 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 9, 10,
11, and 12.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 9-12 will be such that all students:
12. Develop and use models based on sequences and
series.
- Students work in groups to collect data about a bouncing
ball. They first decide how to measure the height of each bounce and
then record their data in a table. They notice the pattern of the
heights and make two graphs, one of height vs. bounce and the other of
total distance traveled vs. bounce. They note that the distance
traveled involves adding together the heights of each of the bounces,
and so is represented by a series. They describe the general behavior
of each graph and have their graphing calculators compute various
regression lines. In their report, they describe what they did, their
results, and why they think that the type of function they used to
describe each graph is reasonable.
- Students use M&Ms to model exponential decay. They
spill a package of M&Ms on a paper plate and remove those with the
M showing, and record the number of M&Ms removed. They put the
remaining M&Ms in a cup, shake, and repeat the process until all
of the M&Ms are gone. They plot the trial number versus the
number of M&Ms removed and note that the graph represents an
exponential function. Some of the students try out different
exponential functions until they find one that they think fits the
data pretty well.
13. Develop and apply procedures for finding the sum
of finite arithmetic series and finite and infinite geometric
series.
- Students investigate a situation in
which a contractor is fined $400 if he is one day late completing a
project, $475 more if he is two days late, $550 more if he is three
days late, and so on. They want to find out how much he will lose if
he is two weeks late finishing the job. They recognize that this is
an arithmetic series where the first term is $400 and each term is
obtained from the preceding one by adding $75. They draw upon several
techniques they have learned to add up the terms of this series. One
method that they have discussed involves reversing the order of the
terms of the series and adding the two series. Some of the students
thus solve the problem by writing the fourteen terms of the series and
underneath writing the same fourteen terms backwards, a technique
sometimes called Gauss' method because, according to legend, he
discovered it as a child while walking to the back of his class to
perform his punishment of adding together the first hundred numbers.
They obtain the following format:
400 + 475 + 550 + ... + 1300 + 1375
1375 + 1300 + 1225 + ... + 475 + 400
-------------------------------------------------------------
1775 + 1775 + 1775 + ... + 1775 + 1775
They recognize that they have 14 pairs of numbers, each of which
adds up to 1775. This gives them a total of $24,850 which they divide
in half (since they added together both sequences) to find the answer,
$12,425. Another group decides that they can separate out the 14
charges of $400, for a total of 14 x 400 = $5600, and then
deal with the remainders $0 + 75 + 150 + 225 + ... + 975, or $75(0 + 1
+ 2 + 3 + ... + 13); this series they recognize as (13x14)/2, so the
total fine is $5600 + $75x91 or $5600 + $6825, for a grand total of
$12,425. Still another group of students uses a formula for the sum
of a finite arithmetic series.
- Students are asked to find a method similar to Gauss'
method to find the sum of the series 9 + 3 + 1 + 1/3 + 1/9 +
1/27. The students notice that this series is not an arithmetic
series since different amounts have to be added in order to get the
next term. They discover, however, that each term is 1/3 of the
previous term, and they write down 1/3 of the series and arrive
at:
9 + 3 + 1 + 1/3 + 1/9
3 + 1 + 1/3 + 1/9 + 1/27
------------------------------------------------------
They subtract to get 9 - 1/27 or 242/27 . Since they
subtracted 1/3 of the series from itself, this total is 2/3 of the sum
of the series, so the sum is 3x242/2x27 or 121/9. The teacher uses
this technique to motivate the standard formula for the sum of a
finite geometric series, where a is the first term of the series and r
is the common multiple:
Sn = a (1 - r n)/(1 - r).
- After investigating how to find the sum of a finite geometric
sequence, students begin looking at infinite geometric sequences.
They realize that the same technique they used for the finite
geometric series works for the infinite one as well. Thus for example,
if we added the first 100 terms of the series by the method above, the
sum would be 9 - 1/397, which is very close to 9. Since
this sum is again 2/3 of the sum of the original series, the actual
total is 27/2. For those students who are likely to use a formula,
the teacher generalizes this discussion and tells them that the sum
gets closer to a/(1 - r) as the number of terms expand.
They confirm this conclusion by checking out the partial sums of
some sequences.
14. Develop an informal notion of limit.
- After a class discussion of the repeating
decimal .9999 ... , the students are asked to write in their
journals an "explanation to the skeptic" on why .9999 ... is
equal to 1. Among their explanations: There is no room between .9999
... and 1; .9999 ... is 3 times .3333 ... which everyone agrees is
1/3 ; if you take 10 times .9999 ... and subtract .9999 ... , you get
9 and 9 times .9999 ... - so 1 must be .9999 ... ; if you sum a
geometric series whose first time is .9 and whose common multiple is
.1 you get a/(1-r) which amounts to .9/(1-.9), or 9. Given all
these convincing reasons, the class decides that the limit of the
sequence is 1.
- Students consider the sequence 1/2, 1/4, 1/8,
... in different contexts. First, they look at it as representing a
situation in which someone eats half of a pizza, then half of what is
left, then half of what is left, etc. They decide that, while
theoretically there will always be some of pizza left, in the end it
would be all gone. However, in practice, by the end often stages or
so the entire pizza would in effect have disappeared. Similarly, if a
sheet of paper is repeatedly torn in half, then in theory some part is
always left; however, in practice, after about ten tearings the paper
will have disappeared.
15. Use linear, quadratic, trigonometric, and
exponential models to explain growth and change in the natural
world.
- Students use a graphing calculator, together with a light
probe, to examine the relationship between brightness of a light and
distance from it. They do this by collecting data with the probe on
the brightness of a light bulb at increasing distances and then
analyzing the graph generated on the calculator to see what kind of
graph it is. They use other CBL probes to investigate the kinds of
functions used to model a variety of real-world situations.
- Students learn about the Richter Scale for measuring
earthquakes, focusing on its relation to logarithmic and exponential
functions, and why this kind of scale is used.
- Students use recursive definitions of functions in both
geometry and algebra. For example, they define n! recursively as n!
= n (n-1)! They use recursion to generate fractals in
studying geometry. They may use patterns such as spirolaterals, the
Koch snowflake, the monkey's tree curve, the chaos game,
or the Sierpinski triangle. They may use Logo or other computer
programs to iterate patterns, or they may use the graphing calculator.
In studying algebra, students consider the equation y = .1x +
.6, start with an x-value of .6, and find the resulting yvalue.
Using this yvalue as the new xvalue, they then calculate its
corresponding yvalue, and so on. (The resulting values are .6, .66,
.666, .6666, etc. - an approximation to the decimal value of
2/3!) Students investigate using other starting values for the same
function; the results are surprising! They use other equations and
repeat the procedure. They graph the results and investigate the
behavior of the resulting functions, using a calculator to reduce the
computational burden.
- Students work through the Breaking the Mold lesson
described in the Introduction to this Framework. They
grow mold and collect data on the area of a pie plate covered by the
mold. They make a graph showing the percent of increase in the area
vs. the days. The students graph their data and find an equation that
fits the data to their satisfaction.
16. Recognize fundamental mathematical models (such as
polynomial, exponential, and trigonometric functions) and apply
basic translations, reflections, and dilations to their
graphs.
- Students work in groups to investigate what size square to
cut from each corner of a rectangular piece of cardboard in order to
make the largest possible open-top box. They make models, record the
size of the square and the volume for each model, and plot the points
on a graph. They note that the relationship seems to be a polynomial
function and make a conjecture about the maximum volume, based on the
graph. The students also generate a symbolic expression describing
this situation and check to see if it matches their data by using a
graphing calculator.
- Students look at the effects of changing the coefficients
of a trigonometric equation on the graph. For example, how is the
graph of y = 4 sin x different from that of y = sin x? How is y
= .2 sin x different from y = sin x? How are y = sin x + 4, y = sin x
4, y = sin (x 4), and y = sin (x + 4) each
different from y = sin x? Students use graphing calculators to
look at the graphs and summarize their conjectures in writing.
- Students study the behavior of functions of the form y
= axn. They investigate the effect of
"a" on the curve and the characteristics of the graph when n
is even or odd. They use the graphing calculator to assist them and
write a sentence summarizing their discoveries.
- Students begin with the graph of y =
2x. They shift the graph up one unit and try to
find the equation of the resulting curve. They shift the original
graph one unit to the right and try to find the equation of that
curve. They reflect the original graph across the x-axis and try to
find the equation of that curve. Finally, they reflect the original
graph across the y-axis and try to find the equation of the resulting
curve. They describe what they have learned in their journals.
17. Develop the concept of the slope of a curve, apply
slopes to measure the steepness of curves, interpret the
meaning of the slope of a curve for a given graph, and use the slope
to discuss the information contained in the graph.
- Students collect data about the height of a ball that is
thrown in the air and make a scatterplot of their data. They note
that the points lie on a quadratic function and use their graphing
calculators to find the curve of best fit. Then they make some
conjectures about the speed at which the ball is traveling. They
think that the ball is slowing down as it rises, stopping at the
maximum point, and speeding up again as it falls.
- Students take on the role of "forensic
mathematicians," trying to determine how tall a person would be
whose femur is 17 inches long. They measure their own femurs and
their heights, entering this data into a graphing calculator or
computer and creating a scatterplot. They note that the data are
approximately linear, so they find the y-intercept and slope from the
graph and generate an equation that they think will fit the data.
They graph their equation and check its fit. They also use the
built-in linear regression procedure to find the line of best fit and
compare that equation to the one they generated. (An instructional
unit addressing this activity can be found in the Keys to Success in
the Classroom chapter of this Framework.)
- Students plot the data from a table that gives the amount
of alcohol in the bloodstream at various intervals of time after a
person drinks two glasses of beer. Different groups use different
techniques to generate an equation for the graph; after some
discussion, the class decides which equation they think is best. The
students consider the following questions: What information does
the slope give for this situation? Would that be important to know?
Why or why not?
- Students investigate the effect of changing the radius of
a circle upon its circumference by measuring the radius and the
circumference of circular objects. They graph the values they have
generated, notice that it is close to a straight line, and use the
slope to develop an equation that describes that relationship. Then
they discuss the meaning of the slope in this situation.
18. Develop an understanding of the concept of
continuity of a function.
- Students work through the On the Boardwalk lesson
found in the Introduction to this Framework. A quarter is
thrown onto a grid made up of squares, and you win if the quarter does
not touch a line. A grid is drawn on the floor using masking tape,
and a circular paper plate is thrown onto the grid several hundred
times to simulate the game. The activity is repeated several times,
varying each time the size of the squares in the grid. The students
collect data and make a graph of their results (size of squares
vs. number of wins out of 100 tosses). The graph looks like a
straight line, suggesting that as the size of the squares increases
without bound, so does the percentage of "hits". But, of
course, the percentage of hits cannot exceed 100%, so the line is
actually curved, with an asymptote at y=100.
- The school store sells pencils for 15 cents each, but it
has some bulk pricing available if you need more pencils. Ten pencils
sell for $1, and twenty-five pencils sell for $2. The students make a
table showing the cost of different numbers of pencils and then
generate a graph of number of pencils vs. cost. The students note
that the graph has discontinuities at ten and twenty-five, since these
are the jump points for pricing. They also note that if you need at
least seven pencils, it is better to buy the package of ten and if you
need 17 or more, you should get the package of 25.
- Students make a table, plot a graph (number of people
vs. cost), and look for a function to describe a situation in which
the Student Council is sponsoring a Valentine's Day dance and
must pay $300 to the band, no matter how many people come. They also
must pay $4 per person for refreshments, with a minimum of 50 people.
The students note that the cost will be $500 for anywhere from 0-50
people and then increase at a rate of $4 per person. They decide that
this is a function with a corner and needs to be defined in
pieces:
-
f(x) = 500 |
for x < 50 |
f(x) = 500 + 4x |
for x >50 |
19. Understand and apply approximation techniques to
situations involving initial portions of infinite decimals and
measurement.
- Students investigate finding the area under the curve y
= x2 + 1 between -1 and
1. They approximate the area geometrically by dividing it into
rectangles 0.5 units wide. They find the height of each rectangle
that fits under the curve and use it to find the areas. Then they find
the height of each rectangle that contains the curve and use these
measurements to find the areas. They realize that this gives them a
range of values for the area under the curve. They refine this
approximation by using narrower rectangles, such as 0.1.
- After some experience with collecting data about balls
thrown into the air, students are given a table of data about a model
rocket and its height at different times. They plot the data, find an
equation that fits the data, and use the trace functions on their
graphing calculators to find the maximum height.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during Spring
1997. In time, we hope to post additional resources relating to this
standard, such as grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
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