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 - Grades K-2
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
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
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
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
- 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
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
- 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
- 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
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