STANDARD 15 - CONCEPTUAL BUILDING BLOCKS OF CALCULUS
All students will develop an understanding of the conceptual
building blocks of calculus and will use them to model and analyze
Standard 15 - Conceptual Building Blocks of Calculus- Grade 7-8
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
649.49 cm3 or almost as high as
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
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
- 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
- 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
1, 2, 4, 8, 16, ... .
6. Investigate, represent, and use non-terminating
- 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
- 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
- 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
- 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
8. Approximate quantities with increasing degrees
- 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
- 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
- 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
- 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
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