STANDARD 13  ALGEBRA
All students will develop an understanding of algebraic concepts
and processes and will use them to represent and analyze relationships
among variable quantities and to solve problems.

Standard 13  Algebra  Grades 56
Overview
Students can develop a strong understanding of algebraic concepts
and processes from consistent experiences in classroom activities
where a variety of manipulatives and technology are used. The key
components of this understanding in algebra, as identified in the K12
Overview, are: patterns, unknown quantities, properties,
functions, modeling realworld situations, evaluating expressions
and solving equations and inequalities.
It is important that students continue to have informal
algebraic experiences in grades 5 and 6. Students have
previously had the opportunity to generalize patterns, work informally
with open sentences, and represent numerical situations using
pictures, symbols, and letters as variables, expressions, equations,
and inequalities. At these grade levels, they will continue to build
on this foundation.
Algebraic topics at this level should be integrated with the
development of other mathematical content to enable students to
recognize that algebra is not a separate branch of mathematics.
Students must understand that algebra is an expansion of the
arithmetic and geometry they have already experienced and a tool to
help them describe situations and solve problems.
Students should use algebraic concepts to investigate situations
and solve interesting mathematical and real world
problems. There should be numerous opportunities for collaborative
work. Since algebra is the language for describing patterns,
students should have regular and consistent opportunities to discuss
and explain their use of these concepts. They should write
generalizations of situations in words as well as in symbols. To
provide such opportunities, the activities should move from a concrete
situation or representation to a more abstract setting. Students at
this level can begin using standard algebraic notation to represent
known and unknown quantities and operations. This should be
developed gradually, moving them from the previous symbols in such a
way that they can appreciate the power and elegance of the new
notation.
Students need to learn how variables are different from numbers (a
variable can represent many numbers simultaneously, it has no place
value, it can be selected arbitrarily) and how they are different from
words (variables can be defined in any way we want and can be changed
without affecting the values they represent). Students need to see
variables (letters) used as names for numbers or other objects, as
unknown numbers in equations, as a range of unknown values in
inequalities, as generalizations in pattern rules or formulas, and as
characteristics to be graphed (independent and dependent
variables).
An algebraic expression involves numbers, variables, and
operations such as 2b, 3x  2, or
c  d. In fifth and sixth grade, students
should begin to become familiar with the common notational shortcut of
omitting the operation sign for multiplication, so that when
b=3, 2b equals 6 and not 23. Thus they
recognize that there are slightly different rules for reading
expressions involving variables than those involving only numbers.
Students in grades 5 and 6 should focus on understanding the role
of the equal sign. Because it is so often used to signal the answer
in arithmetic, students may view it as a kind of operation sign 
a "write the answer" sign. They need to come to see its
role as a relation sign, balancing two equal quantities. Students
should develop the ability to solve simple linear equations
using manipulatives and informal methods. With the appropriate
background, students at grades 5 and 6 have the ability to find the
solution of an equation, such as 7 for x+5=12, whether
they use manipulatives, a graph, or any other method. It is
imperative that in the discussion of the solution of an equation, the
many methods in obtaining that solution are described.
Students in grades 5 and 6 should use concrete materials, such as
algebra tiles, to help them develop a visual, geometric understanding
of algebraic concepts. For example, students can represent the
expression 3x  2 by using three strips and two
units. They should make graphs on a rectangular coordinate system
from data tables, analyze the shape of the graphs, and make
predictions based on the graphs. Students should have opportunities
to plot points, lines, geometric shapes, and pictures. They should
use variables to generalize the formulas they develop in studying
geometry (e.g., p = 4s for a square or A = l×w for a
rectangle). Students should be able to describe movements
of objects in the plane through horizontal and vertical slides
(translations). They should experiment with probes which generate the
graphs of experimental data on computers or graphing calculators. The
majority of this work will be with graphs that are straight lines
(linear functions), but students should have some experience seeing
other shapes of graphs as well; in particular, when dealing with real
data and probes, many times the graph will not be linear.
Standard 13  Algebra  Grades 56
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 56 will be such that all students:
5. Understand and use variables, expressions,
equations, and inequalities.
6. Represent situations and number patterns with
concrete materials, tables, graphs, verbal rules, and standard
algebraic notation.
 Each group of students is given a Mr. or
Mrs. Grasshead (i.e., a sock filled with dirt and grass seed
which sits in a dish of water). They create a name for their
grasshead and begin a diary, recording the number of days that have
passed and the height of the grass. At the end of specified time
periods, they discuss the changes in the height, the average rate of
change over the time period, and the overall behavior of the grass
growth. Each group makes a graph of height versus the number of days.
The students note whether the graph is close to a straight line.
 Students find the number of tiles around the border of a
floor 10 tiles long and 10 tiles wide by looking at smaller square
floors, making a table, and identifying a pattern. They describe
their pattern in words and, with assistance from the teacher, develop
theexpression (4 × n) + 4 for the number of border tiles
needed for an n × n floor.
 Students play Guess My Rule by suggesting
input and having the rulemaker (the teacher or a student) put
the corresponding output into a table like this one:
Input 
Output 
3 
9 
1 
5 
16 
35 
. 
. 
. 
. 
. 
. 
n 
2n+3 
Students should always be challenged to show they understand the
rule by filling in the last row for an input of n. Partially
filledin Guess My Rule tables are also a good assessment
technique to evaluate students' inductive reasoning power and
their ability to use standard algebraic notation to express
relationships.
 Students use play money to act out the following situation
and solve the problem.
A man wishes to purchase a pair of slippers marked $5. He gives
the shoe salesman a $20 bill. The salesman does not have
change for the bill so he goes to the pharmacist next door and
gets a $10 and two $5 bills. He gives the customer his change
and the man leaves. The pharmacist enters shortly after and complains
the $20 was counterfeit. The shoe salesman gives her $20 and
gives the counterfeit bill to the FBI. How much did the shoe
salesman lose?
 Students place 8 twocolor chips in a
paper cup and toss them ten times, recording the number of red and
yellow sides showing on each toss. For each red chip that shows, they
lose $1. For each yellow, they win $1. For each toss, the students
write a number sentence that shows their win or loss for that toss.
For example, after tossing 3 yellows and 5 reds, their sentence would
read 3  5 = 2. Afterwards, the students look for patterns
in the number sentences that they have written. They discuss these
patterns and then write about them in their notebooks.

In both classroom and assessment situations,
students interpret simple nonnumeric graphs and decide what kinds of
relationships they demonstrate. For example: Which of the
following graphs would show the relationship between the height
of a flag and time as a boy scout raised the flag on a
flagpole?
 Students work through the Powers of the Knight
lesson that is described in the Introduction to this Framework.
They learn that doubling the number of coins on consecutive
squares of a chessboard results in a rapidly increasing sequence of
numbers  the powers of 2.
 Students read Anno's Mysterious Multiplying
Jar by Mitsumasa Anno and try to analyze and represent the
numerical patterns shown using variables.
7. Use graphing techniques on a number line to
model both absolute value and arithmetic operations.
 Students use number lines to demonstrate addition of
integers. They point to the number representing the amount currently
in the bank and then slide their finger in the appropriate direction
(right for deposits and left for withdrawals) over the distance
indicated by the second amount. As they slide their finger, they use
arrows to track their movements over the number line, and the teacher
keeps track of the operations using positive and negative integers.
Through this dual representation, students begin to understand the
relationship between the addition of integers and movement along the
number line.
 Students are given a variety of objects whose dimensions
they must determine. They are given a number line marked from 5 to 27
to simulate a broken yardstick. Students work in pairs to develop a
process for determining the lengths of each of the objects to the
nearest unit. After they have a workable method and have written an
explanation of it in their journal, the teacher replaces the tape with
another which is marked from 10 to 10. The students repeat their
effort. This process helps them develop an understanding of
subtraction of integers and the relationship between the operation of
subtraction and distance on the number line.
 Addition and subtraction of signed numbers is explored
using twocolored disks and a number line. Red is used to represent
positive numbers and yellow is used to represent negative numbers.
When given a problem such as 3 + 5 the students place 3
yellows and 5 reds on the table. They pair up as many red and yellow
disks as they can and remove them from the table. In the case of the
example, 3 red and yellow disks would be paired and removed, leaving 2
red disks which represents the sum, +2.
8. Analyze tables and graphs to identify
properties and relationships.
 Students use tables or
twocolor chips to help them solve the following problem: A
classroom has 25 lockers in a row. The first person to enter
the room opened every locker. The second person closed every
other locker beginning with the second locker. The third person
started with the third locker and changed every third locker from open
to closed or closed to open. This continued until 25 people
had passed through the room. Which lockers would be open after
the 25th person walked into the room?

A plastic rectangular shape is exhibited on the overhead. The
lengths of both sides of the image, and the distance from the screen
to the overhead are measured. The overhead is moved and the process
is repeated so that measurements are taken at six to ten different
distances. One group of students is responsible for determining
the relationship between the distance from the screen and the
length of one side of the image. A second group is responsible for
studying the relationship between the distance from the screen and the
area of the image. Each group makes a scatterplot of its data and
eyeballs a line of best fit using a piece of spaghetti. They then use
the graph to answer questions about the relationship between distance
and length or distance and area. They also develop a summary
statement describing the relationship.
 Students make pendulums using strings of length 64, 32,
16, and 8 cm with a washer at one end and a screw eye or ruler at the
other. The strings are swung from a constant height and the number of
swings in 30 seconds is recorded. A graph is made plotting the number
of swings against the string length. Students study the results and
determine if there might be a pattern they could continue. They
attempt to answer questions such as: Will the number of swings ever
reach zero? (This activity is a good one to repeat at later
grades since the relationship appears linear but when very short
lengths and very long lengths are used, it becomes clear that it is
actually a quadratic relationship.)
 Students are given the times of the Olympic 100meter
freestyle swimming winners both in the men's event and the
women's event. Using different colors for the two genders, they
produce a scatterplot and use a piece of spaghetti to eyeball a line
of best fit for each set of data. They use their lines to determine
times in the years not given (when no Olympics were held) and to
predict times in the years beyond those they were given. They also
determine if the data supports the assertion that the women will some
day swim as fast as the men and predict from their lines when that
would happen.
9. Understand and use the rectangular coordinate
system.
 Students are paired to play a game similar to battleship
in which they attempt to determine where the two lines their opponent
has drawn intersect. Both students draw axes which go from 10
to 10 in both the x and ydirections. They sit so that neither can see
the other's paper. The first player draws two lines which
intersect at a point with integer coordinates and colors the four
regions different colors. The second player gives the coordinates of
a point. The first player responds with the color of the region the
point is in, or that the point is on a line, or that it is the
point! The second player keeps a record of his guesses on his
axes and continues guessing until the chosen point is determined.
 Students keep track of the high and low temperatures for a
month in two different colors on a graph. The horizontal axis
represents the day of the month and the vertical axis represents the
temperature. At the end of the month, they connect the points making
two brokenline graphs. They use their graphs to discuss the
temperature variations of that month and to determine the overall
"high" and "low" for the month.
 Students consider what happens if they start with two
bacteria and the number of bacteria doubles every hour. They make a
table showing the number of hours that have passed and the number of
bacteria and then plot their results on a coordinate graph.
 Students draw brokenline pictures in a cartesian plane
and identify the coordinates of critical points in the pictures.
Their partners attempt to recreate the picture using the coordinates
of the critical point and verbal descriptions of how the critical
points should be connected.
10. Solve simple linear equations using
concrete, informal, and graphical methods, as well as
appropriate paperandpencil techniques.
 Students use algebra tiles to solve an equation. For
example, they represent the equation 3x + 2 = 5x by placing
three strips and two units on one side of a picture of a balance beam
and five strips on the other side. They decide that the balance will
stay even if they take the same number of objects off both sides, so
they take three strips off both sides and have two units balanced with
two strips. Then they correctly decide that 1 unit must balance one
strip.
 Students want to use their class fund as a donation to a
town in Missouri devastated by the summer floods. They agree that
each of the 26 students in the class is going to contribute 25 cents a
week. The fund already contains $7. The students develop the
expression $6.50×W+$7 as the amount of money in the
fund at the end of W weeks. The teacher asks them how much they would
like to send to the town, and the students agree on $100. The teacher
then asks them to write an equation which would say that the amount of
money after W weeks was $100. The students write
$6.50×W + $7 = $100. Finally, the students try a
number of different strategies for finding out what W should be. Some
of the students use calculators in a guessandcheck method. Some
students go to the computer and use a spreadsheet to generate the
amounts for different weeks until the total is more than $100. Others
express the relationship as a composite function using function
machines and then use the inverse operations to subtract $7 and then
divide by $6.50. They use their calculators to carry out the division.
The teacher discusses all these methods and introduces the traditional
algebraic shorthand method for solving the problem.
11. Explore linear equations through the use
of calculators, computers, and other technology.
 Using a motion sensor connected to a graphing calculator
or computer, the class experiments with generating graphs which
represent the distance from the sensor against time. They discover
that if they walk at a fixed rate the graph is a straight line. They
try walking away from the sensor at different fixed rates of speed to
determine what effect the speed has on the line. They start at
different distances from the sensor to see the effect that has. They
try walking toward the sensor and standing still. Students discuss
the relationships between the lines they are generating and the
physical activity they do. As an assessment, the teacher has
individual students walk so as to generate a straight line. The
students are then asked to write in their journals what someone would
have to do to produce a line which was less steep. After closing
their journals, individual students are provided an opportunity to
verify their conclusions using the graphing calculator.
 After measuring several students' heights and the
length of the shadows they produce, the data is entered in a
spreadsheet, computerized statistics package, or graphing
calculator. A scatterplot is formed from the data and the students see
that the plot is approximately linear. The technology is used to
produce a line of best fit which the class uses to determine heights
of unknown objects (such as a flagpole) and the length of the shadow
of objects with known heights.
12. Investigate inequalities and nonlinear
equations informally.
 Students explore patterns involving
the sums of the odd integers (1, 1+3, 1+3+5, ...) byusing small
squares to make Ls to represent each odd number and then nesting the
Ls. They make a table that shows how many Ls are nested and the total
number of squares used.
They look for a pattern that will help them predict how many
squares will be needed if 10 Ls are nested (i.e., if the first 10 odd
numbers are added together). They make a prediction and describe the
basis for their prediction (e.g., when you added the first 3 odd
numbers, and placed the three Ls together, they formed a square that
was three units on a side, so when you add the first 10 odd numbers,
that should make a square that is ten units on a side and whose total
area is 100 squares.) They share their solution strategies with each
other and develop an expression that can be used to find the sum of
the first n odd numbers (i.e., n × n).
 Students set up a table listing the length of the sides of
various squares (x) and their areas (y). Some students use the
centimeter blocks to help them find the values. The teacher completes
a table of values in a function graphing computer package on the class
computer which has an LCD panel for overhead projection or on an
overhead version of a graphing calculator. When the students have
finished completing the table, the teacher turns on the overhead and
displays her table. The students check their answers and ask
questions. The teacher graphs the data on the computer or calculator,
and the students use the graph to answer questions such as If the
side was 3.5 cm, what would the area be? and If the
area was 60 square centimeters, what would the side be?
The teacher uses the trace function to identify the points being
discussed.
 Students explore inequality situations such as: I have
$150. How many more weeks would I need to save my $15
allowance to buy a stereo that costs $200? They represent the
relationship as an inequality, both in words and in symbols, and use
play money, base ten blocks, graphs, or trial and error to solve the
problem.
13. Draw freehand sketches of, and interpret,
graphs which model real phenomena.
 Students keep track of how far they are from home during
one specified day. They draw a graph which represents the distance
from home against the time of day and write an explanation of their
graph in relation to their actual activities on that day.
 Students are presented with a graph representing a
student's monthly income from performing lawn care for people
over the past year. The graph shows no income during the months of
November, December, and March. They write a story which explains the
behavior of the graph in terms of the need for services over the
course of the year.
References

Anno, Mitsumasa. Anno's Mysterious Multiplying
Jar. Philomel Books, 1983.
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.
