EdSec 426
Development of Linear Systems Through K-12
Everette Keller
Kindergarten:
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Standard K – 3: The student will demonstrate through the mathematical processes an
emerging sense of repeating and growing patterns and classification based on attributes.
o Indicator K – 3.4: Classify objects according to one or more attributes such as color,
size, shape, and thickness.
o Indicator K – 3.1: Identify simple growing patterns.
Standard K – 2: The student will demonstrate through the mathematical processes an
emerging sense of quantity and numeral relationships, sets, and place values.
Standard K – 5: The student will demonstrate through the mathematical processes an
emerging sense of coin values and the measurement concepts of length, weight, time, and
temperature.
National Standard K2 - NO2: Understand meanings of operations and how they relate to one
another.
National Standard K2 - NO1: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
National Standard K2 – A1: Understand patterns, relations, and functions.
National Standard K2 – A2: Represent and analyze mathematical situations and structures
using algebraic symbols.
First Grade:
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Standard 1-3: The student will demonstrate through the mathematical processes a sense of
numeric patterns, the relationship between addition and subtraction, and change over time.
Standard 1-5: The student will demonstrate through the mathematical processes a sense of
the value of combinations of coins and the measurement of length, weight, time, and
temperature.
Standard 1-2: The student will demonstrate through the mathematical processes a sense of
quantity and numeral relationships; the relationships among addition, subtraction, and
related basic facts; and the connections among numeric, oral, and written-word forms of
whole numbers.
National Standard K2 - 2: Understand meanings of operations and how they relate to one
another.
National Standard K2 - 1: Understand numbers, ways of representing numbers, relationships
among numbers, and number systems.
National Standard K2 – A1: Understand patterns, relations, and functions.
National Standard K2 – A2: Represent and analyze mathematical situations and structures
using algebraic symbols.
EdSec 426
Development of Linear Systems Through K-12
Everette Keller
Second Grade:
 Standard 2-3: The student will demonstrate through the mathematical processes an
understanding of numeric patterns and quantitative change.
 Standard 2-5: The student will demonstrate through the mathematical processes an
understanding of the value of combinations of coins and bills and the measurement of
length, weight, time, and temperature.
 Standard 2-2: The student will demonstrate through the mathematical processes an
understanding of the base-ten numeration system; place values; and accurate, efficient, and
generalizable methods of adding and subtracting whole numbers.
 National Standard K2 - NO2: Understand meanings of operations and how they relate to one
another.
 National Standard K2 - NO1: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
 National Standard K2 – A1: Understand patterns, relations, and functions.
 National Standard K2 – A2: Represent and analyze mathematical situations and structures
using algebraic symbols.
This is the most basic root standards of the entire linear systems unit. Here the students will
learn how to identify objects as they notice the likenesses and differences of the objects they
are working with. Now typically in these grade levels, they will be working a lot more hands
on so their objects will be pertaining more towards concrete objects, but this gets the ball
rolling towards understanding linear systems later on. Students at this age are also required
to demonstrate numeral relationships as well as the mathematical processes that can be
asked when given a data set. However the interesting part of how both standards documents
grouped this age group together. Now there will be progression throughout the standards and
the lessons where the students will gradually be given more advanced work as they continue
to master their previous material. Nonetheless it is rather interesting that these grades were
grouped the way they were.
Third Grade:
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Standard 3-3: The student will demonstrate through the mathematical processes an
understanding of numeric patterns, symbols as representations of unknown quantity, and
situations showing increase over time.
Standard 3-5: The student will demonstrate through the mathematical processes an
understanding of length, time, weight, and liquid volume measurements; the relationships
between systems of measure; accurate, efficient, and generalizable methods of determining
the perimeters of polygons, and the values and combinations of coins required to make
change.
Standard 3-2: The student will demonstrate through the mathematical processes an
understanding of the representation of whole numbers and fractional parts; the addition
EdSec 426
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Development of Linear Systems Through K-12
Everette Keller
and subtraction of whole numbers; accurate, efficient, and generalizable methods of
multiplying whole numbers; and the relationships among multiplication, division, and
related basic facts.
National Standard 35 – NO1: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
National Standard 35 – NO2: Understand meanings of operations and how they relate to
one another.
National Standard 35 – A1: Understand patterns, relations, and functions.
National Standard 35 – A2: Represent and analyze mathematical situations and structures
using algebraic symbols.
Fourth Grade:
 Standard 4-3: The student will demonstrate through the mathematical processes an
understanding of numeric and nonnumeric patterns, the representation of simple
mathematical relationships, and the application of procedures to find the value of the
unknown.
 Standard 4-2: The student will demonstrate through the mathematical processes an
understanding of decimal notation as an extension of the place-value system; the
relationship between fractions and decimals; the multiplication of whole numbers; and
accurate, efficient, and generalizable methods of dividing whole numbers, adding decimals,
and subtracting decimals.
 National Standard 35 – NO1: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
 National Standard 35 – NO2: Understand meanings of operations and how they relate to
one another.
 National Standard 35 – A1: Understand patterns, relations, and functions.
 National Standard 35 – A2: Represent and analyze mathematical situations and structures
using algebraic symbols.
Fifth Grade:
 Standard 5-3: The student will demonstrate through the mathematical processes an
understanding of the use of patterns, relations, functions, models, structures, and algebraic
symbols to represent quantitative relationships and will analyze change in various contexts.
 Standard 5-2: The student will demonstrate through the mathematical processes an
understanding of the place value system; the division of whole numbers; the addition and
subtraction of decimals; the relationships among whole numbers, fractions, and decimals;
and accurate, efficient, and generalizable methods of adding and subtracting fractions.
 National Standard 35 – NO1: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
 National Standard 35 – NO2: Understand meanings of operations and how they relate to
one another.
EdSec 426
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Development of Linear Systems Through K-12
Everette Keller
National Standard 35 – A1: Understand patterns, relations, and functions.
National Standard 35 – A2: Represent and analyze mathematical situations and structures
using algebraic symbols.
Much to the same degree of the standards for the kindergarten through second grade
standards, the standards between third grade through fifth grade are grouped together
though only truly in the national standards. The state standards do dive a bit more in depth
with each grade regarding each of the standards that the students must master in their grades
however they do not differ too much. Granted this is partly due to the fact that the students
are still progressing through the material, and it’s difficult to give one an advanced concept
and to expect them to understand it without building up the foundations of that material first.
In these grades, the students work more with understanding their numerical operations as
well as mastering the relationships between different types of numbers. This will do wonders
for the foundations of linear systems as this will help the students understand how to solve
the systems of linear equations such as substitution or elimination. They will also work with
functions and equations in these grades. Although they might not necessarily refer to the
problems that they work as equations, they will be able to build off what they are learning to
help establish the concept of equations and functions in the later grades.
Sixth Grade:
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Standard: 6-3: The student will demonstrate through the mathematical processes an
understanding of writing, interpreting, and using mathematical expressions, equations, and
inequalities.
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Standard 6-2: The student will demonstrate through mathematical processes an
understanding of the concepts of whole-number percentages, integers, and ratio and rate;
the addition and subtraction of fractions; accurate, efficient, and generalizable methods of
multiplying and dividing fractions and decimals; and the use of exponential notation to
represent whole numbers.
National Standard 68 – NO1: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
National Standard 68 – NO2: Understand meanings of operations and how they relate to
one another.
National Standard 68 – A1: Understand patterns, relations, and functions.
National Standard 68 – A2: Represent and analyze mathematical situations and structures
using algebraic symbols.
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Seventh Grade:
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Standard 7-3: The student will demonstrate through the mathematical processes an
understanding of proportional relationships.
EdSec 426
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Development of Linear Systems Through K-12
Everette Keller
Standard 7-2: The student will demonstrate through the mathematical processes an
understanding of the representation of rational numbers, percentages, and square roots of
perfect squares; the application of ratios, rates, and proportions to solve problems;
accurate, efficient, and generalizable methods for operations with integers; the
multiplication and division of fractions and decimals; and the inverse relationship between
squaring and finding the square roots of perfect squares.
National Standard 68 – NO1: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
National Standard 68 – NO2: Understand meanings of operations and how they relate to
one another.
National Standard 68 – A1: Understand patterns, relations, and functions.
National Standard 68 – A2: Represent and analyze mathematical situations and structures
using algebraic symbols.
Eighth Grade:
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Standard 8-3: The student will demonstrate through the mathematical processes an
understanding of equations, inequalities, and linear functions.
Standard 8-2: The student will demonstrate through the mathematical processes an
understanding of operations with integers, the effects of multiplying and dividing with
rational numbers, the comparative magnitude of rational and irrational numbers, the
approximation of cube and square roots, and the application of proportional reasoning.
National Standard 68 – NO1: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
National Standard 68 – NO2: Understand meanings of operations and how they relate to
one another.
National Standard 68 – A1: Understand patterns, relations, and functions.
National Standard 68 – A2: Represent and analyze mathematical situations and structures
using algebraic symbols.
Again much like the previous sections of the standards where they were grouped according to
certain grades such as the grades of kindergarten through second as well as the grades of third
through fifth, the national standards groups the sixth grade through the eighth grade the
same. However the big difference from the previous two groupings is that the national
standards actually stay the same going into high school. However that might be misleading
some so allow me to clarify. Although the standards stay the same, the expectations that
come with each standard differ in that as the standards progress through the grades, the
expectations seek a deeper understanding of the standards. This is beneficial for teachers in
that these standards are aided with the state standards that do state a more clearer point of
view of what is expected. Now in these grades the students will be building on their
knowledge about functions. Here they will be expected to learn about relationships between
EdSec 426
Development of Linear Systems Through K-12
Everette Keller
equations and inequalities as well as linear functions. Once they have learned about linear
functions and how to solve for the variable(s) in the linear functions, then they will be halfway
there to understanding systems of linear equations.
High School:
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Standard EA-2: The student will demonstrate through the mathematical processes an
understanding of the real number system and operations involving exponents, matrices, and
algebraic expressions.
Standard EA-3: The student will demonstrate through the mathematical processes an
understanding of relationships and functions.
Standard EA-4: The student will demonstrate through the mathematical processes an
understanding of the procedures for writing and solving linear equations and inequalities.
Standard EA-5: The student will demonstrate through the mathematical processes an
understanding of the graphs and characteristics of linear equations and inequalities.
Standard IA-2: The student will demonstrate through the mathematical processes an
understanding of functions, systems of equations, and systems of linear inequalities.
Standard IA-4: The student will demonstrate through the mathematical processes an
understanding of algebraic expressions and nonlinear functions.
National Standard HS – NO1: Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
National Standard HS – NO2: Understand meanings of operations and how they relate to
one another.
National Standard HS – A1: Understand patterns, relations, and functions.
National Standard HS – A2: Represent and analyze mathematical situations and structures
using algebraic symbols.
Finally we have reached the high school standards. These standards are not possible without
the building of the previous standards all the way from kindergarten. Here the students will
add the final layers of their knowledge to the foundation that they’ve made. Once they have
mastered all that previous knowledge regarding numerical operations, functions, linear
equations, etc., then they will be able to put the pieces together to comprehend linear
systems. If the students are missing some of the previous standards and expectations, then it
will take a bit longer for the concept of linear systems to set in as the teacher would have to
reteach what the student does not understand. However this is the culmination of those many
standards from kindergarten.