Core Ideas

• Understand patterns and use mathematical models to represent and understand qualitative and quantitative relationships.
• Describe and extend patterns of sound, shape, or number and translate from one representation to another.
• Describe and extend growing as well as repeating patterns.
• Use the general principles and properties of operations, such as commutativity, with specific numbers.
• Model problem situations using objects, pictures, and symbols
• Describe change qualitatively, such as students growing taller or the weather turning colder.

Core Ideas

• Understand patterns and use mathematical models to represent and understand qualitative and quantitative relationships.
• Describe, extend, and create patterns of sound, shape, and number and translate from one representation to another.
• Describe, extend, and create growing as well as repeating patterns.
• Compare principles and properties of operations, such as commutativity, between addition and subtraction.
• Use concrete, pictorial, and verbal representations to develop an understanding of symbolic notations.
• Describe change quantitatively such as a student's growing two inches in one year.

Core Ideas

• Understand patterns and use mathematical models to represent and to understand qualitative and quantitative relationships.
• Describe and extend geometric and numeric patterns.
• Represent and analyze patterns using words and/or tables.
• Illustrate general principles and properties of operations, such as commutativity, using specific numbers.
• Use concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations.
• Model problem situations with objects and use representations such as graphs and tables to draw conclusions.
• Describe qualitative change (such as students growing taller).
• Describe quantitative change (such as a student's growing two inches in one year).
• Solve simple problems involving a functional relationship (two quantities which vary together, like finding the total cost of multiple items when you know the cost of one).

Core Ideas

• Understand patterns and use mathematical models to represent and to understand qualitative and quantitative relationships.
• Represent and analyze patterns and functions using words, tables, and graphs.
• Find the results of a rule for a specific value.
• Use inverse operations to solve multi-step problems.
• Use concrete, pictorial, and verbal representations to solve problems involving unknowns.
• Understand and use the concept of equality.

Core Ideas

• Understand patterns and use mathematical models such as algebraic symbols and graphs to represent and understand quantitative relationships.
• Represent the idea of a variable as an unknown quantity using a letter or a symbol.
• Express mathematical relationships using equations and graph them on a coordinate grid.
• Investigate how a change in one variable relates to a change in a second variable.
• Identify and describe situations with constant and varying rates of change and compare them.
• Understand and use properties of operations, such as the distributive property of multiplication over addition.

Core Ideas

• Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change.
• Represent, analyze, and generalize a variety of relations and functions with tables, graphs, and words.
• Use symbolic algebra to represent situations and to solve linear equations.
• Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

Core Ideas

• Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change.
• Represent, analyze, and generalize a variety of functions including linear and simple exponential relationships.
• Relate and compare different forms of representation for a relationship including words, tables, graphs in the coordinate plane, and symbols.
• Express mathematical relationships using expressions and equations.
• Develop conceptual understanding of different uses of variables.
• Use symbolic algebra to represent situations to solve problems.
• Identify and describe situations with constant or varying rates of change and compare them.

Core Ideas

• Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change.
• Identify functions as linear or nonlinear, and contrast their properties from tables, graphs, or equations.
• Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.
• Recognize and generate equivalent forms of simple algebraic expressions and solve linear equations.
• Model and solve contextualized problems involving inequalities.
• Use graphs to analyze the nature of changes on quantities in linear relationships
• Recognize, represent and solve contextualized problems involving polynomials and their factors.

Core Ideas

• Understand patterns, relations, and functions.
• Generalize patterns using explicitly defined functions.
• Understand relations and functions and select, convert flexibly among, and use various representations for them.
• Analyze functions of one variable by investigating local and global behavior, including slopes as rates of change, intercepts, and zeros.

MARS Tasks

Sorting Functions

• Task
• Rubric
• Core Mathematical Ideas and Challenges
• Questions for Teacher Reflection
• Discussion of Successful Examples of Student Work
• Discussion of Student Misconceptions
• Graph and Analysis of the MARS Task Data
• Summary of Student Understandings and Misunderstandings
• Implications for Instruction
• Entire "Sorting Functions" Toolkit Packet (large file)

Sidewalk Patterns

• Task
  
• Rubric
  
• Core Mathematical Ideas and Challenges
  
• Questions for Teacher Reflection
  
• Discussion of Successful Examples of Student Work
  
• Discussion of Student Misconceptions
  
• Graph and Analysis of the MARS Task Data
  
• Summary of Student Understandings and Misunderstandings
  
• Implications for Instruction
  
• Entire "Sidewalk Patterns" Toolkit Packet (large file)

Functions

• Task
• Rubric
• Core Mathematical Ideas and Challenges
• Questions for Teacher Reflection
• Discussion of Successful Examples of Student Work
• Discussion of Student Misconceptions
• Graph and Analysis of the MARS Task Data
• Summary of Student Understandings and Misunderstandings
• Implications for Instruction
• Entire "Functions" Toolkit Packet (large file)

Core Ideas

• Understand patterns, relations, and functions.
• Understand and perform transformations on functions.
• Understand and compare the properties of classes of functions, including linear, quadratic, reciprocal, and exponential functions.
• Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships.