School Curriculum: Patterns and Algebra
This page is designed to enable parents to understand what their child should be learning, when they should be learning it, and what degree of mastery the child should have attained (at a median level) by a certain grade level. For Homeschoolers, we hope that this page will serve as a valuable asset in establishing a baseline curriculum. For parents whose children attend public or private schools (or for the inquisitive student) this page should give some guidance as to whether or not the school curriculum and methods are providing students with an adequate standard of education.
What is meant by "Patterns and Algebra," why is it important, and how is it approached ? Below is a description of the core discipline and its components, and the answers to why-how-when these components are taught. Patterns and Algebra components have median level goals to be attained by the end of Grade 2, by the end of Grade 3, by the end of Grade 4, by the end of Grade 5, by the end of Grade 6, by the end of Grade 7, by the end of Grade 8, and by the end of Grade 12.
This page does not contain articles for education in this discipline. For educational articles, go to "Mathematics"
Patterns and Algebra: A. Patterns and Relationships, B. Functions, C. Modeling, D. Procedures
Descriptive Statement: Algebra is a symbolic language used to express mathematical relationships. Students need to understand how quantities are related to one another, and how algebra can be used to concisely express and analyze those relationships. Modern technology provides tools for supplementing the traditional focus on algebraic procedures, such as solving equations, with a more visual perspective, with graphs of equations displayed on a screen. Students can then focus on understanding the relationship between the equation and the graph, and on what the graph represents in a real-life situation.
Patterns. Algebra provides the language through which we communicate the patterns in mathematics. From the earliest age, students should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and, by doing so, to make mathematical discoveries. They should have opportunities to analyze, extend, and create a variety of patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena.
Functions and Relationships. The function concept is one of the most fundamental unifying ideas of modern mathematics. Students begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships.
Modeling. Algebra is used to model real situations and answer questions about them. This use of algebra requires the ability to represent data in tables, pictures, graphs, equations or inequalities, and rules. Modeling ranges from writing simple number sentences to help solve story problems in the primary grades to using functions to describe the relationship between two variables, such as the height of a pitched ball over time. Modeling also includes some of the conceptual building blocks of calculus, such as how quantities change over time and what happens in the long run (limits).
Procedures. Techniques for manipulating algebraic expressions – procedures – remain important, especially for students who may continue their study of mathematics in a calculus program. Utilization of algebraic procedures includes understanding and applying properties of numbers and operations, using symbols and variables appropriately, working with expressions, equations, and inequalities, and solving equations and inequalities.
Algebra is a gatekeeper for the future study of mathematics, science, the social sciences, business, and a host of other areas. In the past, algebra has served as a filter, screening people out of these opportunities.
Fostering respect for the power of mathematics. All students should learn that mathematics is integral to the development of all cultures and civilizations, and in particular to the advances in our own society. They should be aware that the adults in their world (parents, relatives, mentors, community members, role models) use mathematics on a daily basis. And they should know that success in mathematics may be a critical gateway to success in their careers, citizenship, and lives.
Setting high expectations. All students should have high expectations of themselves. These high expectations should be fostered by their teachers, administrators, and parents all of whom should themselves believe that all students can and will succeed in mathematics. This belief in his or her abilities often makes it possible for a child to succeed.
Providing opportunities for success. High expectations can only be achieved if students are provided with the appropriate opportunities. At all grade levels, students should receive instruction by teachers who have had the training and professional development appropriate for their grade level. Students should receive prompt and appropriate services essential to ensure that they can learn the mathematical skills and concepts included in the core curriculum, and to ensure that their weaknesses do not result in trapping them in a cycle of failure. Students should receive equitable treatment without regard to gender or ethnicity, and should not be conditioned to fail by predetermined low expectations.
Encouraging all students to go beyond the standards. Teachers should help students develop a positive attitude about mathematics by engaging them in exploring and solving interesting mathematical problems, by using mathematics in meaningful ways, by focusing on concepts and understanding as well as on rules and procedures, and by consistently expecting them to go beyond repetition and memorization to problem solving and understanding. Every effort should be made to ensure that all students are continuously encouraged, nurtured, and challenged to maximize their potential at all grade levels and to become prepared for college-level mathematics. Students who have achieved the standards should be encouraged to go beyond the standards. If schools challenge all students at lower grade levels, they will attain the goal of having advanced mathematics classrooms whose students reflect the diversity of the school’s total population.
Strands and Cumulative Progress Indicators
By the end of Grade 2, students will:
A. Patterns
1. Recognize, describe, extend, and create patterns.Using concrete materials (manipulatives), pictures, rhythms, & whole numbers
Descriptions using words and symbols (e.g., "add two" or "+ 2")
Repeating patterns
Whole number patterns that grow or shrink as a result of repeatedly adding or subtracting a fixed number (e.g., skip counting forward or backward)
B. Functions and Relationships
1. Use concrete and pictorial models of function machines to explore the basic concept of a function.
C. Modeling
1. Recognize and describe changes over time (e.g., temperature, height).
2. Construct and solve simple open sentences involving addition or subtraction.Result unknown (e.g., 6 – 2 = __ or n = 3 + 5)
Part unknown (e.g., 3 + ÿ = 8)
D. Procedures
1. Understand and apply (but don’t name) the following properties of addition:Commutative (e.g., 5 + 3 = 3 + 5)
Zero as the identity element (e.g., 7 + 0 = 7)
Associative (e.g., 7 + 3 + 2 can be found by first adding either 7 + 3 or 3 + 2)
Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:
A. Patterns
1. Recognize, describe, extend, and create patterns.Descriptions using words and number sentences/expressions
Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11, . . . or 800, 400, 200, . . .)
B. Functions and Relationships
1. Use concrete and pictorial models to explore the basic concept of a function.Input/output tables, T-charts
C. Modeling
1. Recognize and describe change in quantities.Graphs representing change over time (e.g., temperature, height)2. Construct and solve simple open sentences involving addition or subtraction(e.g., 3 + 6 = __, n = 15 – 3, 3 + __ = 3, 16 – c = 7).
D. Procedures
1. Understand and apply the properties of operations and numbers.Commutative (e.g., 3 x 7 = 7 x 3)2. Understand and use the concepts of equals, less than, and greater than to describe relations between numbers.
Identity element for multiplication is 1 (e.g., 1 x 8 = 8)
Any number multiplied by zero is zeroSymbols ( = , < , > )
Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:
A. Patterns
1. Recognize, describe, extend, and create patterns.Descriptions using words, number sentences/expressions, graphs, tables, variables (e.g., shape, blank, or letter)
Sequences that stop or that continue infinitely
Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11, . . . or 800, 400, 200, . . .)
Sequences can often be extended in more than one way (e.g., the next term after 1, 2, 4, . . . could be 8, or 7, or … )
B. Functions and Relationships
1. Use concrete and pictorial models to explore the basic concept of a function.
Input/output tables, T-charts
Combining two function machines
Reversing a function machine
C. Modeling
1. Recognize and describe change in quantities.
Graphs representing change over time (e.g., temperature, height)2. Construct and solve simple open sentences involving any one operation (e.g., 3 x 6 = __, n = 15 ¸ 3, 3 x __ = 0, 16 – c = 7).
How change in one physical quantity can produce a corresponding change in another (e.g., pitch of a sound depends on the rate of vibration)
D. Procedures
1. Understand, name, and apply the properties of operations and numbers.
Commutative (e.g., 3 x 7 = 7 x 3)2. Understand and use the concepts of equals, less than, and greater than in simple number sentences.
Identity element for multiplication is 1 (e.g., 1 x 8 = 8)
Associative (e.g., 2 x 4 x 25 can be found by first multiplying either 2 x 4 or 4 x 25)
Division by zero is undefined
Any number multiplied by zero is zero.
Symbols ( = , < , > )
Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:
A. Patterns
1. Recognize, describe, extend, and create patterns involving whole numbers.
Descriptions using tables, verbal rules, simple equations, and graphs
B. Functions & Relationships
1. Describe arithmetic operations as functions, including combining operations and reversing them.
2. Graph points satisfying a function from T-charts, from verbal rules, and from simple equations.
C. Modeling
1. Use number sentences to model situations.
Using variables to represent unknown quantities2. Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.
Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations
Changes over time
Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)
D. Procedures
1. Solve simple linear equations with manipulatives and informally
Whole-number coefficients only, answers also whole numbers
Variables on one side of equation
Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:
A. Patterns
1. Recognize, describe, extend, and create patterns involving whole numbers and rational numbers.Descriptions using tables, verbal rules, simple equations, and graphs
Formal iterative formulas (e.g., NEXT = NOW * 3)
Recursive patterns, including Pascal’s Triangle (where each entry is the sum of the entries above it) and the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, . . . (where NEXT = NOW + PREVIOUS)
B. Functions and Relationships
1. Describe the general behavior of functions given by formulas or verbal rules (e.g., graph to determine whether increasing or decreasing, linear or not).
C. Modeling
1. Use patterns, relations, and linear functions to model situations.Using variables to represent unknown quantities2. Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.
Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations/inequalitiesChanges over time
Relations between quantities
Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)
D. Procedures
1. Solve simple linear equations with manipulatives and informally.
Whole-number coefficients only, answers also whole numbers2. Understand and apply the properties of operations and numbers.
Variables on one or both sides of equation
Distributive property3. Evaluate numerical expressions.
The product of a number and its reciprocal is 1
4. Extend understanding and use of inequality.
Symbols ( ³ , ¹ , £ )
Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:
A. Patterns
1. Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.
Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions
Finite and infinite sequences
Generating sequences by using calculators to repeatedly apply a formula
B. Functions and Relationships
1. Graph functions, and understand and describe their general behavior.
Equations involving two variables
C. Modeling
1. Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations.
2. Use patterns, relations, symbolic algebra, and linear functions to model situations.
Using manipulatives, tables, graphs, verbal rules, algebraic expressions/equations/inequalities
Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)
D. Procedures
1. Use graphing techniques on a number line.
Absolute value2. Solve simple linear equations informally and graphically.
Arithmetic operations represented by vectors (arrows) (e.g., "-3 + 6" is "left 3, right 6")
Multi-step, integer coefficients only (although answers may not be integers)3. Create, evaluate, and simplify algebraic expressions involving variables.
Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology
Order of operations, including appropriate use of parentheses4. Understand and apply the properties of operations, numbers, equations, and inequalities.
Substitution of a number for a variable
Additive inverse
Multiplicative inverse
Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:
A. Patterns
1. Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.
Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions
Finite and infinite sequences
Arithmetic sequences (i.e., sequences generated by repeated addition of a fixed number, positive or negative)
Geometric sequences (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1)
Generating sequences by using calculators to repeatedly apply a formula
B. Functions and Relationships
1. Graph functions, and understand and describe their general behavior.
Equations involving two variables2. Recognize and describe the difference between linear and exponential growth, using tables, graphs, and equations.
Rates of change (informal notion of slope)
C. Modeling
1. Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations.
2. Use patterns, relations, symbolic algebra, and linear functions to model situations.
Using concrete materials (manipulatives), tables, graphs, verbal rules, algebraic expressions/equations/inequalities
Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)
D. Procedures
1. Use graphing techniques on a number line.
Absolute value2. Solve simple linear equations informally, graphically, and using formal algebraic methods.
Arithmetic operations represented by vectors (arrows) (e.g., "-3 + 6" is "left 3, right 6")
Multi-step, integer coefficients only (although answers may not be integers)3. Solve simple linear inequalities.
Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology
4. Create, evaluate, and simplify algebraic expressions involving variables.
Order of operations, including appropriate use of parentheses5. Understand and apply the properties of operations, numbers, equations, and inequalities.
Distributive property
Substitution of a number for a variable
Translation of a verbal phrase or sentence into an algebraic expression, equation, or inequality, and vice versa
Additive inverse
Multiplicative inverse
Addition and multiplication properties of equality
Addition and multiplication properties of inequalities
Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:
A. Patterns
1. Use models and algebraic formulas to represent and analyze sequences and series.
Explicit formulas for nth terms2. Develop an informal notion of limit.
Sums of finite arithmetic series
Sums of finite and infinite geometric series
3. Use inductive reasoning to form generalizations.
B. Functions and Relationships
1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs.
2. Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.
Slope of a line or curve3. Understand and perform transformations on commonly-used functions.
Domain and range
Intercepts
Continuity
Maximum/minimum
Estimating roots of equations
Intersecting points as solutions of systems of equations
Rates of change
Translations, reflections, dilations4. Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions.
Effects on linear and quadratic graphs of parameter changes in equations
Using graphing calculators or computers for more complex functions
Linear vs. non-linear
Symmetry
Increasing/decreasing on an interval
C. Modeling
1. Use functions to model real-world phenomena and solve problems that involve varying quantities.
Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years)2. Analyze and describe how a change in an independent variable leads to change in a dependent one.
Direct and inverse variation
Absolute value
Expressions, equations and inequalities
Same function can model variety of phenomena
Growth/decay and change in the natural world
Applications in mathematics, biology, and economics (including compound interest)
3. Convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling).
D. Procedures
1. Evaluate and simplify expressions.
Add and subtract polynomials2. Select and use appropriate methods to solve equations and inequalities.
Multiply a polynomial by a monomial or binomial
Divide a polynomial by a monomial
Linear equations – algebraically3. Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.
Quadratic equations – factoring (when the coefficient of x2 is 1) and using the quadratic formula
All types of equations using graphing, computer, and graphing calculator techniques
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