Long Descriptions for Chapter Eight

Figure 8.1: The Why, How, and What of Learning Mathematics

Figure 8.2: Drivers of Investigation, Standards for 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 the “Why,” described as, “In order to ...”: DI1, Make Sense of the World (Understand and Explain); DI2, Predict What Could Happen (Predict); DI3, Impact the Future (Affect). The SMPs are the “How,” listed under “Students will ...”: SMP1, Make sense of problems and persevere in solving them; SMP2, Reason abstractly and quantitatively; SMP3, Construct viable arguments and critique the reasoning of others; SMP4, Model with mathematics; SMP5, Use appropriate tools strategically; SMP6, Attend to precision; SMP7, Look for and make use of structure; SMP8, Look for and express regularity in repeated reasoning. Finally, the CCs are the “What,” listed under, “While ...”: CC1, Reasoning with Data; CC2, Exploring Changing Quantities; CC3, Taking Wholes Apart, Putting Parts Together; CC4, Discovering Shape and Space.

Figure 8.3: The Statistical Problem-Solving Process (GAISE II)

The statistical problem-solving process is represented as a series of ovals connected by large arrows pointing to the next one on the right, with smaller arrows leading back from the right ovals to the earlier ones. From left to right, the ovals include the following text: 1. Formulate statistical investigative questions; 2. Collect/consider the data; 3. Analyze the data; 4. Interpret the results.

Figure 8.4: High School Pathways to STEM and Non-STEM Careers

Diagram indicating two pathways of courses indicating a variety of course offerings for Years 3 and 4 in high school. The preparatory courses are Algebra I* and Mathematics I*, followed by Geometry and Mathematics II. The later course options include Algebra II, Mathematics III, Computer Science, Statistics, Data Science I, Data Science II, Precalculus, Calculus**, Discrete Math, Financial Algebra, and Other Math. All of these options lead to STEM and Non-STEM Majors and Careers.

* Students may take Algebra I or Mathematics I in middle school.

** Calculus, which can be taken during or after high school, is an important course to support student selection of a STEM career.

Note: Many of the third- and fourth-year high school courses included in the figure such as financial algebra, data science, statistics with algebra, or other math will require prerequisite knowledge of Mathematics I and Mathematics II, or Algebra I and Geometry, depending on district policy. See the section that follows figure 8.4, “Third- and Fourth-Year Courses,” for more details.

Figure 8.5: Big Ideas Map for Algebra I

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

• Model with Functions directly connects to: Features of Functions, Growth & Decay, Investigate Data, Systems of Equations, Function Investigations
• Features of Functions directly connects to: Growth & Decay, Systems of Equations, Function Investigations, Model with Functions
• Growth & Decay directly connects to: Features of Functions, Model with Functions, Function Investigations, Systems of Equations
• Systems of Equations directly connects to: Growth & Decay, Features of Functions, Model with Functions, Function Investigations
• Function Investigations directly connects to: Model with Functions, Features of Functions, Growth & Decay, Investigate Data, Systems of Equations
• Investigate Data directly connects to: Model with Functions, Function Investigations

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. Function Investigations (largest and same size as the one that follows)
   Model with Functions
2. Systems of Equations
   Features of Functions
   Growth and Decay
3. Investigate Data (smallest)
   

Figure 8.8: Big Ideas Map for Geometry

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

• Probability Modeling directly connects to: Fairness in Data
• Fairness in Data directly connects to: Probability Modeling
• Trig Explorations directly connects to: Triangle Congruence, Geometric Models, Triangle Problems, Geospatial Data, Circle Relationships, Points & Shapes
• Triangle Congruence directly connects to: Geometric Models, Triangle Problems, Transformations, Geospatial Data, Circle Relationships, Points & Shapes, Trig Explorations
• Geometric Models directly connects to: Triangle Problems, Transformations, Circle Relationships, Points & Shapes, Trig Explorations, Triangle Congruence
• Triangle Problems directly connects to: Geometric Models, Triangle Congruence, Transformations, Geospatial Data, Circle Relationships, Points & Shapes, Trig Explorations
• Transformations directly connects to: Geometric Models, Triangle Problems, Triangle Congruence, Geospatial Data, Circle Relationships, Points & Shapes
• Circle Relationships directly connects to: Geometric Models, Triangle Problems, Transformations, Geospatial Data, Triangle Congruence, Points & Shapes, Trig Explorations
• Points & Shapes directly connects to: Geometric Models, Triangle Problems, Transformations, Geospatial Data, Circle Relationships, Triangle Congruence, Trig Explorations
• Geospatial Data: Triangle Problems, Transformations, Triangle Congruence, Circle Relationships, Points & Shapes, Trig Explorations

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. Points and Shapes (largest and same size as the four that follow)
   Circle Relationships
   Triangle Congruence
   Geometric Models
   Triangle Problems
2. Geospatial Data
3. Trig Explorations
   Transformations
4. Probability Modeling (smallest and same size as the one that follows)
   Fairness in Data
   

Figure 8.11: Geometric Transformations

The image illustrates the effects of translations, rotations, and reflections on two-dimensional figures using coordinates––part of an eighth-grade geometry standard. The image illustrates the reasoning that corresponding parts being congruent implies triangle congruence, in which point A is translated (i.e., shifted to the right) to D, the resulting image of ΔABC is rotated at point D so as to place B onto E, and finally (as shown by an arrow), the image is then reflected along line segment DE to match point C to F.

Figure 8.13: Big Ideas Map for Mathematics I

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

• Systems of Equations directly connects to: Variability, Comparing Models, Modeling with Functions
• Correlation & Causation directly connects to: Variability, Comparing Models
• Variability directly connects to: Correlation & Causation, Comparing Models, Systems of Equations, Modeling with Functions, Building with Triangles
• Building with Triangles directly connects to: Variability, Comparing Models, Transformations & Congruence, Shapes in Structures, Modeling with Functions
• Composing Functions directly connects to: Transformations & Congruence, Shapes in Structures
• Modeling with Functions directly connects to: Building with Triangles, Variability, Comparing Models, Systems of Equations
• Shapes in Structures directly connects to: Transformations & Congruence, Building with Triangles, Composing Functions
• Transformations & Congruence directly connects to: Building with Triangles, Composing Functions, Shapes in Structures
• Comparing Models directly connects to: Correlation & Causation, Variability, Building with Triangles, Modeling with Functions, Systems of Equations

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. Comparing Models (largest and same size as the two that follow)
   Building with Triangles
   Variability
2. Systems of Equations
   Shapes in Structures
   Modeling with Functions
   Transformations and Congruence
3. Correlation and Causation
4. Composing Functions (smallest)
   

Figure 8.16: Big Ideas Map for Mathematics II

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

• Function Representations directly connects to: Equations to Predict & Model, Polynomial Identities, Circle Relationships, Functions in the World, Trig Functions, Experimental Models & Functions
• Equations to Predict & Model directly connects to: Polynomial Identities, Circle Relationships, Trig Functions, Functions in the World, Transformations & Similarity, Experimental Models & Functions, Function Representations
• Polynomial Identities directly connects to: Geospatial Data, Circle Relationships, Trig Functions, Transformations & Similarity, Functions in the World, Experimental Models & Functions, Function Representations, Equations to Predict & Model
• Geospatial Data directly connects to: Polynomial Identities, Functions in the World, Transformations & Similarity, Trig Functions, Circle Relationships
• Circle Relationships directly connects to: Geospatial Data, Polynomial Identities, Trig Functions, Transformations & Similarity, Functions in the World, Experimental Models & Functions, Function Representations, Equations to Predict & Model
• Trig Functions directly connects to: Geospatial Data, Circle Relationships, Polynomial Identities, Transformations & Similarity, Experimental Models & Functions, Function Representations, Equations to Predict & Model
• Transformations & Similarities directly connects to: Geospatial Data, Circle Relationships, Trig Functions, Polynomial Identities, Experimental Models & Functions, Equations to Predict & Model
• Experimental Models & Functions directly connects to: Circle Relationships, Trig Functions, Transformations & Similarity, Polynomial Identities, Function

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. Polynomial Identities (largest)
2. Experimental Models and Functions
   Circle Relationships
3. Equations to Predict and Model
   Trig Functions
4. Function Representations
   Transformations and Similarity
5. Functions in the World
   Geospatial Data
6. Probability Modeling (smallest and same size as the one that follows)
   The Shape of Distributions