STANDARD 11 - PATTERNS, RELATIONSHIPS, AND FUNCTIONS
All students will develop an understanding of patterns,
relationships, and functions and will use them to represent and
explain real-world phenomena.
Standard 11 - Patterns, Relationships, and Functions - Grades 7-8
The key components of pattern-based thinking, as identified
in the K-12 Overview, involve exploring, analyzing, and
generalizing patterns, and viewing rules and input/output
situations as functions. In grades 7-8, the importance of
studying patterns continues with an emphasis on representing and
describing relationships with tables and graphs and on the development
of rules using variables. Patterns also become important now in the
analysis of statistics and the development of geometric relationships.
Graphing calculators and computers are helpful in illustrating the
usefulness of symbols and in making symbolic relationships more
tangible. Although the symbolism and notation used become more
algebraic at these grade levels (e.g., A = 4s instead of
A = 4 x s), students should still be encouraged
to model many patterns with concrete materials. Engineers,
scientists, architects, and other researchers all build working models
of projects for analysis and demonstration.
Students in these grades should also be given ample opportunity to
analyze patterns, to discover the relevant features of the patterns,
and to construct understandings of the concepts and relationships
involved in the patterns. From these investigations, students should
develop the language necessary to communicate their ideas about the
patterns and should learn to differentiate among the variety of
patterns they have studied (that is, to categorize and to
classify them). They will apply their understanding of patterns
as they learn about such topics as exponents, rational numbers,
measurement, geometry, probability, and functions.
Seventh and eighth graders continue to discover rules for
mathematical relationships and for quantifiable situations from other
subject areas. In particular, students should focus on
relationships involving two variables. Students should
analyze how a change in one quantity results in a change in another.
They further need to develop their understanding of the general
behavior of functions and use these to model a variety of
Students should be encouraged to solve problems by looking for
patterns that involve words, pictures, manipulatives, and number
descriptions. These situations naturally lead to the use of variables
and informal algebra in solving problems.
As seen in prior grades, computers and graphing calculators provide
many benefits for students in investigating mathematical concepts and
problems. These tools make mathematics accessible to more students
because they enable the students to analyze what they can see rather
than requiring them to develop mental images or manipulate situations
symbolically from the outset. Furthermore, technology enables
students to calculate rapidly and to investigate conclusions
immediately, freeing them from the limitations imposed by cumbersome
and timeconsuming computations.
Standard 11 - Patterns, Relationships, and Functions - 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:
7. Represent and describe mathematical relationships with
tables, rules, simple equations, and graphs.
- Students use calculators to investigate which fractions
have decimal equivalents that terminate and which repeat. They
summarize their findings in their math journals.
- Students study patterns made by the units digit in the
expansion of powers of a number. For example, what is the units digit
of 918? The pattern 91, 92,
93, . . . yields either a 1 or a 9 in the units place.
Students record their findings in a table or as a graph on rectangular
coordinate paper. They write a paragraph justifying their answer.
They then similarly investigate the patterns made by the units digit
in the expansion of the powers of other one-digit numbers.
- Students consider what happens if you start with
two bacteria on a kitchen counter and the number of bacteria doubles
every hour. They make a table and graph their results, noting that
the graph is not linear.
8. Understand and describe the relationships among
various representations of patterns and functions.
- Students arrange bowling pins in the shape of
equilateral triangles of various sizes, as shown in the diagram below.
They make a table showing the number n of rows in each triangle
and the number b of bowling pins in each triangle. The numbers
in the second column - 1, 3, 6, 10, ... - are called the
triangular numbers. They find a rule expressing this
relationship p = n(n + 1) /2, by putting two triangles of the
same size side by side, counting the total number of bowling pins, and
dividing by 2.
- Using a 5x5 geoboard or dot paper,
students create various sized parallelograms. For each parallelogram,
they record the length of the base (b), the height of the
parallelogram (h), and the area of the parallelogram (A), found by
counting squares. The students look for a relationship among the
numbers in the three columns of their table, express this relationship
as a verbal rule, and then write the rule in symbolic form.
- Students investigate how many stools with three legs and
how many chairs with four legs can be made using 48 legs. They may
use objects or draw pictures to make models of the solutions. They
look for patterns in the numbers and display their results in a table,
as ordered pairs graphed on the rectangular coordinate plane, as a
rule like 3s + 4c = 48, and as an equation like s = 16
- 4c/3, which gives the number of stools as a function
of the number of chairs. They describe the pattern and how they found
it in writing.
- Students create their own designs using iteration. They
may use patterns such as spirolaterals or write a program in Logo on
the computer. They use 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 value
yields a 2 for y. Using this as the next x gives a 3, and so on. The
related values can be organized in a table and the ordered pairs
graphed on a rectangular coordinate system. Students note that the
graph is a straight line and use this to predict other values. Then
students use a slightly different equation, y = .1x + .3. Again,
starting with an x value of 0 they find the resulting y value of .3.
Using this as the new x value gives a value of .33 for y. Repeating
this process yields the series of y values .3, .33, .333, ... which
get closer and closer to 1/3.
9. Use patterns, relationships, and functions to model
situations and to solve problems in mathematics and in other
- Students analyze a given series of terms and fill in the
missing terms. Patterns include various arithmetic (repeating
patterns) and geometric (growing patterns) sequences and other number
and picture patterns. Students develop an awareness of the
assumptions they are making. For example, given the sequence 0, 10,
20, 30, 40, 50, one might expect 60 to be next; but not on a football
field, where the numbers now decrease!
- 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 create a table comparing the
pay for different numbers of hours worked and decide at what point the
hourly rate becomes a better deal.
- Students supply missing fractions between any two given
numbers on a number line. They might label each of eight intervals
between 1 and 2, or they might label the next 16 intervals from 23 1/2
to 24. They extend this to decimals, labeling each missing number in
increments of .1 or .01. For example, students might label each of
five intervals between 59.34 and 59.35.
- Students decide how many different double-dip ice cream
cones can be made from two flavors, three flavors, and so on up to
Baskin and Robbins' 31 flavors. They arrange the information in
a table. They discuss whether one flavor on top and another on the
bottom is a different arrangement from the other way around, and how
that would change their results. They also discuss a similar problem
(see Standard 14 and the 5-6 Vignette Pizza
Possibilities in the First Four Standards): How many
different types of pizzas can be made using different
- 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. They
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 inwriting any patterns they discover
and generate a rule for finding 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 ten generations ago and also to the problem of telling
a secret to two people who each tell two people, etc.
10. Analyze functional relationships to explain how a
change in one quantity results in a change in another.
- Students investigate how increasing the temperature
measured in degrees Celsius affects the temperature measured in
degrees Fahrenheit and vice versa. They collect data using water,
ice, and a burner. They use their data to develop a formula relating
Celsius to Fahrenheit, summarize the formula in a sentence, and graph
the values they have generated.
- Students investigate how the temperature affects the number
of chirps a cricket makes in a minute.
- Students investigate the effect of changing the radius or
diameter of a circle upon its circumference by measuring the radius
(or diameter) and the circumference of circular objects. They graph
the values they have generated, notice that it is close to a straight
line, and describe the relationship they have found in a paragraph.
Then they develop a symbolic expression that describes that
- Students investigate the effect on the perimeters of given
shapes if each side is doubled or tripled. They summarize their
- Students investigate how the areas of rectangles change as
the length is doubled, or the width is doubled, or both are doubled.
They discuss their findings.
- Students work on problems like this one from the
New Jersey Department of Education's Mathematics Instructional
Guide (p. 7-69): Two of the opposite sides of a square are
increased by 20% and the other two sides are decreased by 10%.
What is the percent of change in the area of the original
square to the area of the newly formed rectangle? Explain the
process you used to solve the problem.
- Students investigate how the areas of triangles change if
the base is kept the same, but the height is repeatedly increased by
- Students stack a given number of unit cubes in various ways
and find the surface areas of the structures they have built. They
sketch their figures and discuss which of the figures has the largest
surface area and which has the smallest, and justify their
- Students make models of cubes using blocks or other
manipulatives, and investigate how the volume changes if the length,
width, and height are all doubled.
- Using a spreadsheet, students investigate how adding (or
subtracting) values to given data can affect the mean, median, mode,
or range of the data. They discuss how various other changes to the
data would affect the mean, the median, the mode, or the range.
11. Understand and describe the general behavior of
- Students investigate graphs without numbers. For example,
they may study a graph that shows how far Olivia has walked on a trip
from home to the store and back, where time isshown on the horizontal
axis and the distance covered is on the vertical axis. Students tell
a story about her trip, noting that where the graph is horizontal, she
has stopped for some reason. In addition, their stories account for
those parts of the graph that are steeper, by explaining why Olivia is
walking faster (e.g., she is running from a dog), and those parts of
the graph that are not as steep, by explaining why Olivia is walking
slower (e.g., she is going up a hill).
- 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 the temperature of a beaker of water
when ice cubes are added). They look at the data that has been
collected in tabular 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.
- Given several nonlinear functions, such as y =
x2, y = 3x2, y =
x2 + 1, y = x3, or
y = 16/x, students create a table of values for each
and use graphing calculators to graph them.
12. Use patterns, relationships, and linear functions
to model situations in mathematics and in other areas.
- Groups of students pretend that they work for construction
companies bidding on a federal project to build a monument. The
monument is to be built from marble cubes, with each cube being one
cubic foot. The monument is to have a "triangular" shape,
with one cube on top, then two cubes in the row below, then three
cubes, four cubes, and so on. The monument is to be 100 feet high.
The students make a chart and look for a pattern to help them predict
how many cubes they will need to buy so that they can include the cost
of the cubes in their bid.
- Students look at the Sierpinski triangle as an example of a
fractal. Stage 0 is an unshaded triangle. To get Stage 1, you take
the three midpoints of the sides of the unshaded triangle, connect
them, and shade the new triangle in the middle. To get Stage 2, you
repeat this process for each of the unshaded triangles in Stage 1.
This process continues an infinite number of times. The students make
a table that records the number of unshaded triangles at each stage,
look for a pattern, and use their results to predict the number of
unshaded triangles there will be at the tenth (39)
and twentieth (319) stages.
- Students use the constant function on the calculator to
determine when an item will be on sale for half price. If the price
goes down by a constant dollar amount each week, then they record
successive prices, such as 95 - 15 = = = . . . (or 15 -
- 95 = = = on other calculators). If the price is reduced by a
certain percent each week, then they use the constant function on the
calculator to obtain successive discounts as percents bymultiplying.
For example, if a $95 item is reduced by 10% each week, they key in 95
x .9 = = = . . . (or as .9 x x 95 = = = . . . on other
- Using a temperature probe and a graphing calculator or
computer, students measure the temperature of boiling water in a cup
as it cools. 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 or 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
- Students use coins to simulate boys (tails) and girls
(heads) in a family with five children. They make a list of all of the
possible combinations, use patterns to help them organize all of the
possibilities, and find the probability that all five children are
girls or that exactly three are girls. As a question on a test, they
are asked to react to an argument between Pam and Jerry, a couple who
want to have four children. Jerry thinks that they will probably end
up with two boys and two girls, while Pam thinks that they will
probably wind up with an unequal number of boys and girls.
- Students make Ferris wheel models from paper plates, with
notches representing the cars. They use the models to make a table
showing the height above the ground of a person on a ferris wheel at
specified time intervals, determined by the time needed for the 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
13. Develop, analyze, and explain arithmetic
- Students use the following chart of postal rate
history to make a graph of the increases and then to try to predict
what the cost will be to mail a oneounce letter in the year
|Cost to Mail a One-Ounce Letter Since 1917
- Students describe, analyze, and extend the Fibonacci
sequence 1, 1, 2, 3, 5, 8, ... , where each term is the sum of the two
preceding terms. They investigate applications 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.
- Students read Isaac Asimov's short story
Endlessness and write book reports to convey their
Asimov, Isaac. Endlessness. (in Literature:
Bronze, 2nd Ed.) Englewood Cliffs, NJ: Prentice Hall,
New Jersey Department of Education. Mathematics Instruction
Guide. D. Varygiannes, Coord. January, 1996.
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