Long Descriptions for Chapter Six

Figure 6.1: The Why, How, and What of Learning Mathematics (accessible version)

Figure 6.2: Content Connections, Mathematical Practices, and Drivers of Investigation

A spiral graphic shows how the Drivers of Investigation (DIs), Standards for Mathematical Practice (SMPs), and Content Connections (CCs) interact. The DIs are listed under “In order to ... ”: Make Sense of the World (Understand and Explain); Predict What Could Happen (Predict); Impact the Future (Affect). The SMPs are listed under “Students will ... ”: Make sense of problems and persevere in solving them; Reason abstractly and quantitatively; Construct viable arguments and critique the reasoning of others; Model with mathematics; Use appropriate tools strategically; Attend to precision; Look for and make use of structure; Look for and express regularity in repeated reasoning. Finally, the CCs are listed under “While ... ”: Communicating Stories with Data; Exploring Changing Quantities; Taking Wholes Apart, Putting Parts Together; Discovering Shape and Space.

Figure 6.8: Transitional Kindergarten Big Ideas

The graphic illustrates the connections and relationships of some transitional kindergarten mathematics concepts. Direct connections include:

• Look for Patterns directly connects to Create Patterns, Count to 10, Measure & Order, See & Use Shapes, Make & Measure Shapes
  
  
• Make & Measure Shapes directly connects to Look for Patterns, Create Patterns, Measure & Order, Shapes in Space, See & Use Shapes
  
  
• See & Use Shapes directly connects to Make & Measure Shapes, Look for Patterns, Measure & Order, Create Patterns, Count to 10, Shapes in Space
  
  
• Shapes in Space directly connects to See & Use Shapes, Make & Measure Shapes, Measure & Order, Create Patterns, Count to 10
  
  
• Count to 10 directly connects to Shapes in Space, See & Use Shapes, Measure & Order, Look for Patterns
  
  
• Create Patterns directly connects to Look for Patterns, Make & Measure Shapes, See & Use Shapes, Measure & Order, Shapes in Space
  
  
• Measure & Order directly connects to Look for Patterns, Make & Measure Shapes, See & Use Shapes, Shapes in Space, Count to 10, Create Patterns

Figure 6.10: Kindergarten Big Ideas

The graphic illustrates the connections and relationships of some kindergarten mathematics concepts. Direct connections include the following:

• How Many directly connects to Being Flexible within 10, Shapes in the World, Sort and Describe Data, Bigger or Equal, Place and Position of Numbers
  
  
• Model with Numbers directly connects to Being Flexible within 10, Sort and Describe Data, Place and Position of Numbers
  
  
• Being Flexible within 10 directly connects to Model with Numbers, How Many, Making Shapes from Parts, Shapes in the World
  
  
• Shapes in the World directly connects to Being Flexible within 10, How Many, Sort and Describe Data, Bigger or Equal, Making Shapes from Parts
  
  
• Making Shapes from Parts directly connects to Shapes in the World, Being Flexible within 10, Sort and Describe Data, Bigger or Equal
  
  
• Bigger or Equal directly connects to Making Shapes from Parts, Shapes in the World, Sort and Describe Data, How Many
  
  
• Place and Position of Numbers directly connects to How Many, Model with Numbers, Sort and Describe Data
  
  
• Sort and Describe Data directly connects to How Many, Model with Numbers, Shapes in the World, Making Shapes from Parts, Bigger or Equal, Place and Position of Numbers.

Figure 6.12: Grade One Big Ideas

The graphic illustrates the connections and relationships of some first-grade mathematics concepts. Direct connections include the following:

• Clocks & Time directly connects to Equal Parts Inside Shapes, Reasoning About Equality, Make Sense of Data, Tens & Ones
  
  
• Equal Expressions directly connects to Reasoning About Equality, Make Sense of Data, Tens & Ones, Measuring with Objects
  
  
• Reasoning About Equality directly connects to Equal Expressions, Clocks & Time, Make Sense of Data, Tens & Ones
  
  
• Tens & Ones directly connects to Reasoning About Equality, Make Sense of Data, Equal Expressions, Clocks & Time
  
  
• Measuring with Objects directly connects to Equal Expressions, Make Sense of Data
  
  
• Equal Parts Inside Shapes directly connects to Clocks & Time, Make Sense of Data
  
  
• Make Sense of Data directly connects to Reasoning About Equality, Tens & Ones, Measuring with Objects, Clocks & Time, Equal Expressions, Equal Parts Inside Shapes

Figure 6.14: Grade Two Big Ideas

The graphic illustrates the connections and relationships of some second-grade mathematics concepts. Direct connections include the following:

• Dollars & Cents directly connects to Problem Solving with Measure, Skip Counting to 100, Number Strategies, Represent Data
  
  
• Problem Solving with Measure directly connects to Skip Counting to 100, Number Strategies, Represent Data, Measure and Compare Objects, Dollars & Cents
  
  
• Skip Counting to 100 directly connects to Number Strategies, Seeing Fractions in Shapes, Squares in an Array, Represent Data, Dollars & Cents, Problem Solving with Measure
  
  
• Number Strategies directly connects to Skip Counting to 100, Problem Solving with Measure, Dollars & Cents, Represent Data
  
  
• Seeing Fractions in Shapes directly connects to Skip Counting to 100, Represent Data, Squares in an Array
  
  
• Squares in an Array directly connects to Seeing Fractions in Shapes, Skip Counting to 100, Represent Data, Measure and Compare Objects
  
  
• Measure and Compare Objects directly connects to Squares in an Array, Represent Data, Problem Solving with Measure
  
  
• Represent Data directly connects to Measure and Compare Objects, Dollar & Cents, Problem Solving with Measure, Skip Counting to 100, Number Strategies, Seeing Fractions in Shapes, Squares in an Array.

Figure 6.45: Model for Finding Part of a Part – Example 1

There are two rectangles. The rectangle on the left is divided vertically into three equal parts. The two parts on the right are marked with an indicator.

The rectangle on the right is the same rectangle divided vertically into three equal parts and horizontally into four equal parts. The two rightmost vertical parts and the three uppermost horizontal parts are marked with indicators and shaded.

Figure 6.53: Grade Three Big Ideas

The graphic illustrates the connections and relationships among some third-grade mathematics concepts. Direct connections include the following:

• Fractions of Shape & Time directly connects to Square Tiles, Fractions as Relationships, Unit Fractions Models, Represent Multivariable Data
  
  
• Measuring directly connects to Number Flexibility to 100 for All Four Operations, Analyze Quadrilaterals, Represent Multivariable Data
  
  
• Patterns in Four Operations directly connects to Number Flexibility to 100 for All Four Operations, Unit Fraction Models, Analyze Quadrilaterals, Represent Multivariable Data
  
  
• Square Tiles directly connects to Fractions as Relationships, Number Flexibility to 100 for All Four Operations, Fractions of Shape & Time
  
  
• Fractions as Relationships directly connects to Square Tiles, Fractions of Shape & Time, Unit Fraction Models
  
  
• Unit Fraction Models directly connects to Fractions as Relationships, Patterns in Four Operations, Fractions of Shape & Time, Represent Multivariable Data
  
  
• Analyze Quadrilaterals directly connects to Number Flexibility to 100 for All Four Operations, Patterns in Four Operations, Measuring
  
  
• Represent Multivariable Data directly connects to Unit Fraction Models, Number Flexibility to 100 for All Four Operations, Patterns in Four Operations, Measuring, Fractions of Shape & Time
  
  
• Number Flexibility to 100 for All Four Operations directly connects to Square Tiles, Analyze Quadrilaterals, Represent Multivariable Data, Measuring, Patterns in Four Operations.
  

Figure 6.55: Grade Four Big Ideas

The graphic illustrates the connections and relationships of some fourth-grade mathematics concepts. Direct connections include the following:

• Number & Shape Patterns directly connects to Shapes & Symmetries, Connected Problem Solving, Circles Fractions & Decimals, Factors & Area Models, Fraction Flexibility, Multi-Digit Numbers
  
  
• Shapes & Symmetries directly connects to Connected Problem Solving, Circles Fractions & Decimals, Multi-Digit Numbers, Number & Shape Patterns
  
  
• Rectangle Investigations directly connects to Connected Problem Solving, Measuring & Plotting, Circles Fractions & Decimals
  
  
• Connected Problem Solving directly connects to Rectangle Investigations, Shapes & Symmetries, Number & Shapes Patterns, Multi-Digit Numbers, Circles Fractions & Decimals, Factors & Area Models, Measuring & Plotting
  
  
• Measuring & Plotting directly connects to Connected Problem Solving, Rectangle Investigations, Visual Fraction Models
  
  
• Visual Fraction Models directly connects to Measuring & Plotting, Circles Fractions & Decimals, Fraction Flexibility
  
  
• Factors & Area Models directly connects to Connected Problem Solving, Circles Fractions & Decimals, Number & Shape Patterns, Multi-Digit Numbers, Fraction Flexibility
  
  
• Fraction Flexibility directly connects to Factors & Area Models, Circles Fractions & Decimals, Number & Shape Patterns, Multi-Digit Numbers
  
  
• Multi-Digit Numbers directly connects to Number & Shape Patterns, Shapes & Symmetries, Connected Problem Solving, Circles Fractions & Decimals, Factors & Area Models, Fraction Flexibility
  
  
• Circles Fractions & Decimals directly connects to Multi-Digit Numbers, Number & Shape Patterns, Shapes & Symmetries, Rectangle Investigations, Connected Problem Solving, Visual Fraction Models, Factors & Area Models, Fraction Flexibility.

Figure 6.57: Grade Five Big Ideas

The graphic illustrates the connections and relationships of some fifth-grade mathematics concepts. Direct connections include the following:

• Factors & Groups directly connects to Powers & Place Values, Layers of Cubes, Modeling, Seeing Division
  
  
• Shapes on a Plane directly connects to Telling a Data Story, Modeling, Plotting Patterns
  
  
• Powers & Place Value directly connects to Layers of Cubes, Fraction Connections, Modeling, Factors & Groups
  
  
• Layers of Cubes directly connects to Powers & Place Value, Factors & Groups, Modeling, Seeing Division
  
  
• Telling a Data Story directly connects to Shapes on a Plane, Modeling, Plotting Patterns
  
  
• Seeing Division directly connects to Layers of Cubes, Modeling, Factors & Groups
  
  
• Plotting Patterns directly connects to Telling a Data Story, Modeling, Fraction Connections, Shapes on a Plane
  
  
• Fraction Connections directly connects to Powers & Place Value, Modeling, Plotting Patterns
  
  
• Modeling directly connects to Plotting Patterns, Factors & Groups, Shapes on a Plane, Powers & Place Value, Fraction Connections, Layers of Cubes, Telling a Data Story, Seeing Division.