STANDARD 6  NUMBER SENSE
All students will develop number sense and an ability to represent
numbers in a variety of forms and use numbers in diverse
situations.

Standard 6  Number Sense  Grade 78
Overview
Seventh and eighth graders should have a good sense of whole
numbers and their orders of magnitude and should be focusing on
further developing number sense with decimals and fractions. They
should be extending their understanding of whole numbers to negative
numbers, including comparison and ordering. They also should be
working on incorporating ratio, proportion and percent, powers and
roots, and scientific notation into their conception of the number
system. The key components of number sense, as explained in the K12
Overview, are an awareness of the uses of numbers in the world
around us, a good sense of approximation, estimation, and
magnitude, the concept of numeration, and an understanding
of comparisons and the equivalence of different
representations and forms of numbers.
Students at this age are capable of categorizing the ways in which
numbers are used in our society. One interesting activity is to have
them collect data on how numbers appear in a portion of a newspaper,
categorize these uses, and then graph their results.
In their work with numeration, seventh and eighthgraders
should begin to see mathematics as a coherent body of knowledge. They
should begin to see the integers and the rationals as logical and
necessary extensions of the whole number system. Only with these
extensions can expressions like 3 7 and 4
divided by 3 have answers.
In grades 7 and 8, students are focusing on ratios, proportions,
and percents, topics which they were just beginning to consider in
grades 5 and 6. Their work with these concepts and the relationship
of these numbers to fractions, decimals, and whole numbers form the
foundation for a very powerful problem solving skill: proportional
reasoning. New topics at these grade levels are exponents, roots, and
scientific notation. Students should also explore irrational numbers,
such as pi, square roots of numbers which are not perfect squares,
and other decimals which neither end nor repeat.
Students at these grade levels need to continue learning about
equivalence, but there is a vast array of kinds of equivalence
to be considered here. Students focus on the representation of the
same quantity using different types of numbers and the
selection of the appropriate number type given a particular problem
context. It is particularly important that students understand the
difference between the exact value of a fraction, such as 2/3, and its
approximation of .667, especially since they now use calculators
routinely. Relationships among decimals, fractions, ratios, and
percents comprise the largest emphasis, but work with exponents and
roots and their relationship to scientific notation is also a focus in
these grades. Number theory provides a rich context for interesting
problems in this area. Questions about infinity, division by zero,
and primes and composites, combine with discussions about finite and
infinite sequences and series and searches for patterns to open up the
full richness of a mathematical world.
Estimation should be a routine part of mathematics classes.
Students should be regularly engaged in estimating both quantities and
the results of operations. They should respond to questions that
arise naturally during the course of the class with answers which
demonstrate confident and wellconceived use of estimation strategies
and sense of number.
Technology plays an important role in number sense at these
grade levels. Calculators can be wonderful exploration tools when
examining numerical relationships. Many insights about the
relationships between fractions and decimals, for instance, can be
attained by simply dividing the numerators of fractions by their
denominators. Generalizations about what kinds of fractions produce
what kinds of decimals start to flow very freely in such openended
explorations. Computer software can also be very useful.
Spreadsheets, for example, can show a great many ratios on the screen
at the same time. For example, the five ingredients of a waffle
recipe that makes 4 waffles can be listed across the top row of the
spreadsheet, with following rows showing how the recipe changes to
make 2, 8, and 12 waffles (Curriculum and Evaluation Standards,
1989, p. 89).
The topics that should comprise the number sense focus of the
seventh and eighth grade mathematics program are:
 rational numbers (both positive and negative)
 equivalence
 integers
 ratio, proportion, and percent
 exponents, roots, and scientific notation
Standard 6  Number Sense Grades 78
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 78 will be such that all students:
10. Understand money notations, count and compute money,
and recognize the decimal nature of United States
currency.
 Students can solve a variety of realworld money problems
such as: If you make $750.00 a month, would you rather have
a 12% raise or an $85 a month raise? or Which sale is
better on a $17.00 sweater, a 1/3 off sale, a $5.00 discount,
or a 30% discount?
 Students use timecards and pay rates to compute weekly
wages and deductions for various workers. Issues such as,
timeandahalf for overtime, double time for holidays, and
percentages of total wages to be deducted for various taxes all come
into play.
 Students investigate the details and make plans to dispose
of the proceeds of the $27,000,000 lottery which they just won.
11. Extend their understanding of the number system by
constructing meanings for integers, rational numbers, percents,
exponents, roots, absolute values, and numbers represented in
scientific notation.
 Students develop a scientific notation Olympics by
creating events like the 9.144 x 10^{3} centimeter sprint
(100 yard dash) or the 7.272 x 10^{6 } milligram
hurl (shot put).
 Students use a bookkeeping simulation to explore the effect
of bills and credits coming into, or going out of, their business.
These financial activities are recorded as various actions on positive
and negative integers, all affecting the net worth of the
business.
 Students view the Powers of Ten video, developed to
show how one's view of the world is affected by changes in the
order of magnitude of one's position.
 Students construct a hypothetical stock portfolio, using
$10,000 to "buy" shares of stock, and track the performance
of the portfolio and each individual stock day by day.
 Students explore absolute values as the distance between two
points on a number line and compare this to subtraction.
 Students measure the circumference and diameter of 15 to 20
round objects, recording the results in a table. They make a
scatterplot (with diameter on the horizontal axis) and use a piece of
spaghetti to draw the line of best fit. They discuss why pi is
represented in the drawing as the slope of the line.
 Students construct line segments of varying irrational
lengths on a geoboard or dot paper. For example, the diagonal of a
unit square has length sqrt(2), the diagonal of a 1 x 2 rectangle has
length sqrt(5), and the circumference of a circle with unit diameter
has length pi. Through exploring these common irrational numbers
that arise in problem situations, students learn that not all numbers
can be represented as a ratio of two integers.
 Students construct their own graph for the square roots of
the numbers from 1 to 25, using trialanderror to approximate each
root to the nearest tenth. They plot the numbers on the horizontal
scale, and their square roots on the vertical scale.
 Students work on traditional "systems of
equations" problems, involving two unknowns, by devising
nonalgebraic solution strategies for them. Some samples are: Two
numbers have a sum of 32; they have a product of 240. What are
the numbers? or Sally is 22 years younger than her
Dad. In 3 years, her Dad will be 3 times as old as she. How old is
Sally?
12. Develop number sense necessary for estimation.
 Students wrestle with this classic problem: After
spending most of the day looking for her missing pet cat,
Whiskers, the eccentric billionaire, Ms. Money Bags, received a ransom
demand. The caller said she was to bring a suitcase packed
with $1,000,000 in one and fivedollar bills to the bus
station and leave it in Locker #26C. Then her pet would be
returned to her. How did she respond?
 Students describe a scale model of the solar system built on
the premise that the earth is represented by a pingpong ball.
 Students make estimates of the number of times various
events happen in an average lifetime, discuss their strategies for
estimation, and then check their estimates against some reference. A
good reference for this activity is In an Average Lifetime by
Tom Heymann. Among other things, in an average lifetime, an American
consumes 10,231 gallons of beverages, spends $1,331 on homedelivered
food, and spends 911 hours brushing his or her teeth.
13. Expand the sense of magnitudes of different number
types to include integers, rational numbers, and roots.
 Students play Locate the Point. A
number line with end points 5 and 5 is suspended in the
classroom, using a long string with tabs to indicate the positions of
the integers between the two end numbers. Students are given cards
with different types of numbers on them. (For example: 12/3,
1.1, 1.01, sqrt(2), pi, 2^{2},
(2)^{2}, sqrt(3), sqrt(8),
1.999..., 2, the cube root of 8, etc.) They take turns and
attach their card on the appropriate spot on the number line.
Classmates decide whether the position is correct. If more than one
expression is used for the same number, the cards with those numbers
are attached by tape.
 Students use only the multiplication and division functions
on their calculators to perform a series of successive approximations
to find acceptable values for several roots: the square roots of 2, 3,
7, and 10, and the cube roots of 10 and 100.
 Pairs of students play HiLo with decimals as a way
to emphasize the density of the rational numbers. One student thinks
of a number between 0 and 10 with up to 4 decimal places. The other
student tries to guess the number, receiving feedback after each guess
as to whether the guess was too high or too low. Written records of
the guesses and the feedback are kept. The goal is to find the number
using as few guesses as possible.
14. Understand and apply ratios, proportions, and
percents in a variety of situations.
 Students take consumer price data from 10 and 25 years ago
and figure out the percentage increase or decrease in the prices of
various products over those periods of time. They discuss questions
such as: What makes a price go up? What would make it go
down?
 Students predict, and then determine, which body part ratios
are fairly constant from person to person. Some interesting ones are
height/arm span, wrist circumference/hand span, and waist/neck
circumference.
 Students make a threedimensional model of the classroom
with different groups taking responsibility for modeling different
objects in the class. First the desired size of the model is
discussed and a scale factor agreed upon. Then each of the groups
measures and applies that scaling factor to their objects, determines
appropriate materials and means of construction, and builds the
models.
 Students examine whether it is better to take a discount of
20% and then add a 6% sales tax or add the sales tax and then take a
20% discount. (The answer may surprise the students!)
 Students examine different statements involving proportions
and discuss which ones make sense and which do not. For example:
If one girl can mow the yard in 30 minutes, then two girls can
mow the yard in 15 minutes. If one boy can walk to school in 20
minutes, then two boys can walk to school in 10
minutes.
 Students compare magazine subscription prices for 6, 9, and
12 months in order to decide which is the better buy.
 Students estimate what percent of plain M&M's are
red, green, yellow, blue, brown, and orange. They test their guesses
by counting the number of each color in a small bag and finding the
percentages. They also discuss whether they improve their estimates
by combining their data.
 Students simulate running a business using the computer
program The Whatsit Corporation or Survival Math.
15. Develop and use order relations for integers and
rational numbers.
 Students use concrete and pictorial models to develop order
relations among fractions and integers. Using Cuisenaire Rods and
varying the unit, students demonstrate that one fraction is
larger than another. Similar arguments and conclusions are made on a
number line for integers.
 Students' abilities to order rational numbers (both
positive and negative) are assessed by asking them to identify points
on a number line between, say, 3 and 5.
 Students are each given a rational number on a large card
(1.2, 4, 3/4, 2 1/4, 1, 3.14, 22/7, and so on).
They then order themselves from least to greatest along the front or
side of the classroom. They also respond to instructions like:
Hold up your card if it is between 2.5 and +0.7.
16. Recognize and describe patterns in both finite and
infinite number sequences involving whole numbers, rational
numbers, and integers.
 Students formulate a description of the nth row of
Pascal's triangle.
 Students investigate the Fibonacci sequence (1, 1, 2, 3, 5,
8, 13, 21, ....) to see how it is generated and then do library
research to find theories about its startling occurrences in
nature.
 Students explore the golden ratio discovered and used
by the ancient Greeks. They find examples of golden rectangles (whose
sides are in the golden ratio) in everyday objects (3 x 5 cards,
bricks, cereal boxes), and in architecture (the Parthenon).
 Students discuss and predict the sum of this wellknown
series:
17. Develop and apply number theory concepts, such as
primes, factors, and multiples, in realworld and mathematical
problem situations.
 Students write a Logo or a BASIC computer program to find
all the factors of any number that is provided as input. They can
then use the same program to determine if any input number is
prime.
 Students explore Goldbach's conjecture (a mathematical
hunch which has never been proved nor disproved) which states that:
Any even number greater than two can be written as the sum
of two prime numbers. For example: 14 = 11 + 3, 24 = 11 + 13,
and 56 = 3 + 53. Can you find one that cannot be written this
way?
 Students develop rules of divisibility for all onedigit
numbers and explain and apply these rules on a test.
 Students investigate the path of a ball on a billiard table
with sides of whole number length when the ball starts in a corner and
always travels at a 45 degree angle. For example, a ball on the 3 x 5
table in the diagram starts in the lower left corner and takes the
path shown, hitting the perimeter eight times (including the first and
last corners) and going through all 15 squares, before ending at the
top right corners. They make a table which records the length and
width of the billiard table, the number of hits against the perimeter,
and the number of squares passed through, for billiard tables of
various sizes, and look for relationships. The number of hits against
the perimeter of the table, including the first and last corners, is
the sum of the width and the length of the billiard table divided by
their greatest common factor.
18. Investigate the relationships among fractions,
decimals, and percents, and use all of them
appropriately.
 Given a circle graph of some interesting data, students
estimate the size of each section of the graph as a fraction, a
percent, and as a decimal. Students also create their own circle
graphs.
 Students use two differentcolor interlocking paper circles
(each has a cut along a radius so they fit together), each marked off
in wedges that are one hundredth of the circle, to show fractions that
have denominators of 10 or 100, decimals to hundredths, and whole
number percents to 100%.
 Students explore patterns in particular families of decimal
expansions, such as those for the fractions, 1/7, 2/7, 3/7,
... or 1/9, 2/9, 3/9, ... .
19. Identify, derive, and compare properties of
numbers.
 Students work on this problem from the Curriculum and
Evaluation Standards for School Mathematics (p. 93):
Find five examples of numbers that have exactly three
factors. Repeat for four factors, and then again for five
factors. What can you say about the numbers in each of your
lists?
 Students explore perfect numbers, those numbers that
are equal to the sum of all of their factors including 1 but excluding
themselves. Six is the first perfect number, where 6 =
1 + 2 + 3. Interestingly, the next one has 2 digits, the
third has 3 digits and the fourth has 4 digits. The pattern breaks
down there, though, since the fifth perfect number has 8
digits. Students who have worked on a computer program to find all of
the factors of numbers (see Indicator 17 on the previous page) may
want to revise their program to see how many perfect numbers they can
find.
References

Heyman, Tom. On an Average Day. New York: Fawcett
Columbine, 1989.
National Council of Teachers of Mathematics. Curriculum and
Evaluation Standards for School
Mathematics. Reston, VA, 1989.
Software

Logo. Many versions of Logo are commercially
available.
Survival Math. Sunburst Communications.
The Whatsit Corporation. Sunburst Communications.
Video

Powers of Ten. Philip Morrison, Phylis Morrison, and the
office of Charles and Roy Eames. New York: Scientific American
Library, 1982.
OnLine 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 gradespecific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
