Long Descriptions for Chapter Seven

Figure 7.1: The Why, How, and What of Mathematics

Figure 7.2: Drivers of Investigation, Mathematical Practices, and Content Connections

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); and 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; and
• Look for and express regularity in repeated reasoning.

Finally, the CCs are listed under, “While ...”:

• Reasoning with Data;
• Exploring Changing Quantities;
• Taking Wholes Apart, Putting Parts Together; and
• Discovering Shape and Space.

Figure 7.5: Environment-Oriented Data Visualization

A bar graphs display with the left axis representing the number of endangered species on the IUCN Red List, by class, labeled from 0 to 15,000. The bottom axis is in years, 2007 to 2019, with one bar every three years, so five bars total. The bars increase in height to indicate that the number of endangered species has risen from 2007 to 2019. The height of the first bar, for year 2007, is 7,851, the second bar is 9,618 for year 2010, the third bar is 11,212 for year 2013, the fourth bar is 12,630 for year 2016, and the height of the last bar, for year 2019, is 14,234. The bars are also color coded to indicate more specificity for eight types of species : mammals, reptiles, amphibians, birds, insects, mollusks, fish and other (including other invertebrate [spineless] animals, such as crustaceans, corals, and arachnids [spiders, scorpions]).

Figure 7.6: Squares with Borders for Use in the Border Problem

Six rectangles include two squares each. Squares include borders comprised of various shadings. Rectangle one includes two squares shaded to indicate 10 + 10 + 8 + 8 and n + (n – 2) + n + n (n – 2). Rectangle two includes two squares shaded 10 + 9 + 9 + 8 and n + 2(n – 1) + (n – 2). Rectangle three includes two squares shaded 4 × 8 + 4 and 4(n – 2) + 4. Rectangle four includes two squares shaded 9 + 9 + 9 + 9 = 9 × 4 and (n – 1) × 4. Rectangle five includes two squares shaded 4 × 10 – 4 and 4n – 4. Rectangle six includes two squares shaded (10 × 10) – (8 × 8) and n squared – (n – 2) squared.

Figure 7.7: Big Ideas for Sixth Grade

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

• Variability in Data directly connects to: The Shape of Distributions, Relationships Between Variables
• The Shape of Distributions directly connects to: Relationships Between Variables, Variability in Data
• Fraction Relationships directly connects to: Patterns Inside Numbers, Generalizing with Multiple Representations, Model the World, Relationships Between Variables
• Patterns Inside Numbers directly connects to: Fraction Relationships, Generalizing with Multiple Representations, Model the World, Relationships Between Variables
• Generalizing with Multiple Representations directly connects to: Patterns Inside Numbers, Fraction Relationships, Model the World, Relationships Between Variables, Nets & Surface Area, Graphing Shapes
• Model the World directly connects to: Fraction Relationships, Relationships Between Variables, Patterns Inside Numbers, Generalizing with Multiple Representations, Graphing Shapes
• Graphing Shapes directly connects to: Model the World, Generalizing with Multiple Representations, Relationships Between Variables, Distance & Direction, Nets & Surface
• Nets & Surface directly connects to: Graphing Shapes, Generalizing with Multiple Representations, Distance & Direction
• Distance & Direction directly connects to: Graphing Shapes, Nets & Surface Area
• Relationships Between Variables directly connects to: Variability in Data, The Shape of Distributions, Fraction Relationships, Patterns Inside Numbers, Generalizing with Multiple Representations, Model the World, Graphing Shapes

Note: The sizes of the circles vary to indicate the relative importance of the topics. The connecting lines between circles show links among topics and suggest ways to design instruction so that multiple topics are addressed simultaneously.

The size of the circles, from largest to smallest, is as follows:

1. Relationships Between Variables (largest)
2. Generalizing with Multiple Representations
3. Graphing Shapes
   Model the World
4. Fraction Relationships
   Patterns Inside Numbers
5. Nets and Surface Area
6. Distance and Direction (smallest and same size as the two that follow)
   Variability in Data
   The Shape of Distributions

Figure 7.9: Big Ideas for Seventh Grade

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

• Angle Relationships directly connects to: Scale Drawings, 2D & 3D Connections, Populations & Samples, Proportional Relationships, Shapes in the World, Visualize Populations, Probability Models
• Scale Drawings directly connects to: 2D & 3D Connections, Graphing Relationships, Populations & Samples, Unit Rates in the World, Proportional Relationships, Visualize Populations, Probability Models, Angle Relationships
• Graphing Relationships directly connects to: Populations & Samples, Unit Rates in the World, Proportional Relationships, Probability Models, Scale Drawings
• 2D & 3D Connections directly connects to: Scale Drawings, Angle Relationships, Probability Models, Proportional Relationships, Visualize Populations, Shapes in the World, Populations & Samples
• Populations & Samples directly connects to: 2D & 3D Connections, Scale Drawings, Angle Relationships, Probability Models, Proportional Relationships, Visualize Populations, Shapes in the World, Unit Rates in the World, Graphing Relationships
• Unit Rates in the World directly connects to: Populations & Samples, Graphing Relationships, Scale Drawings, Proportional Relationships, Probability Models, Visualize Populations
• Shapes in the World directly connects to: Populations & Samples, 2D & 3D Connections, Proportional Relationships, Scale Drawings, Angle Relationships, Probability Models, Visualize Populations
• Visualize Populations directly connects to: 2D & 3D Connections, Scale Drawings, Angle Relationships, Probability Models, Proportional Relationships, Populations & Samples, Shapes in the World, Unit Rates in the World
• Probability Models directly connects to: 2D & 3D Connections, Scale Drawings, Angle Relationships, Proportional Relationships, Visualize Populations, Shapes in the World, Unit Rates in the World, Graphing Relationships, Populations & Samples
• Proportional Relationships directly connects to: 2D & 3D Connections, Scale Drawings, Angle Relationships, Probability Models, Populations & Samples, Visualize Populations, Shapes in the World, Unit Rates in the World, Graphing Relationships

Note: The sizes of the circles vary to indicate the relative importance of the topics. The connecting lines between circles show links among topics and suggest ways to design instruction so that multiple topics are addressed simultaneously.

The size of the circles, from largest to smallest, is as follows:

1. Probability Models (largest and same size as the two that follow)
   Populations and Samples
   Scale Drawings
   
2. Visualize Populations
   
3. 2D and 3D Connections
   Unit Rates in the World
   Shapes in the World
   Angle Relationships
   Proportional Relationships
   
4. Graphing Relationships (smallest)

Figure 7.11: Big Ideas for Eighth Grade

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

• Data Explorations directly connects to: Slopes & Intercepts, Linear Equations, Multiple Representations of Functions, Data Graphs & Tables, Interpret Scatter plots, Big & Small Numbers
• Slopes & Intercepts directly connects to: Linear Equations, Multiple Representations of Functions, Data Graphs & Tables, Interpret Scatter plots, Data Explorations
• Linear Equations directly connects to: Slopes & Intercepts, Data Explorations, Multiple Representations of Functions, Data Graphs & Tables, Interpret Scatter plots
• Multiple Representations of Functions directly connects to: Data Graphs & Tables, Interpret Scatter plots, Data Explorations, Slopes & Intercepts, Linear Equations
• Data Graphs & Tables directly connects to: Multiple Representations of Functions, Linear Equations, Slopes & Intercepts, Data Explorations, Interpret Scatter plots, Shape Number & Expressions, Big & Small Numbers, Pythagorean Explorations
• Pythagorean Explorations directly connects to: Data Graphs & Tables, Interpret Scatter plots, Cylindrical Investigations, Transformational Geometry, Shape Number & Expressions, Big & Small Numbers
• Big & Small Numbers directly connects to: Pythagorean Explorations, Data Graphs & Tables, Interpret Scatter plots, Data Explorations, Cylindrical Investigations, Transformational Geometry, Shape Number & Expressions
• Shape Number & Expressions directly connects to: Big & Small Numbers, Pythagorean Explorations, Data Graphs & Tables, Interpret Scatter plots, Cylindrical Investigations
• Transformational Geometry directly connects to: Big & Small Numbers, Pythagorean Explorations, Cylindrical Investigations
• Cylindrical Investigations directly connects to: Big & Small Numbers, Pythagorean Explorations, Shape Number & Expressions, Transformational Geometry
• Interpret Scatter plots directly connects to: Data Explorations, Slopes & Intercepts, Linear Equations, Multiple Representations of Functions, Data Graphs & Tables, Pythagorean Explorations, Big & Small Numbers, Shape Number & Expressions

Note: The sizes of the circles vary to indicate the relative importance of the topics. The connecting lines between circles show links among topics and suggest ways to design instruction so that multiple topics are addressed simultaneously.

The size of the circles, from largest to smallest, is as follows:

1. Interpret Scatterplots (largest and same size as the following circle)
   Data, Graphs, and Tables
2. Big and Small Numbers
3. Slopes and Intercepts
   Linear Equations
   Shape, Number, and Expressions
   Pythagorean Explorations
4. Data Explorations
   Multiple Representations of Functions
5. Cylindrical Investigations
6. Transformational Geometry (smallest)