Section 32.1 Yet Another Pause for Reflection
Congratulations! You have just completed the first two "branches" of the course.
In the third branch, took a long look at some basic arithmetic concepts that you have probably known for the vast majority of your life. But we looked at these things in ways that you may have never seen before in your life. This is an important process in our intellectual growth as humans. There is value to periodically returning to some basic ideas to explore them again with the added knowledge and insights that can be gained only through years of experience. The discovery of new insights from old ideas drives a wide swath of mathematical thinking.
The big emphasis of this branch was to expand your mental toolbox for working with numbers. Students that can only work with numbers as manipulations of digits tend to be less prepared to make conceptual connections than those that have multiple ways of thinking about numbers. In particular, being able to relate arithmetic to geometric concepts is useful in practical applications. For example, when working with information on a coordinate grid, knowing that the distance between two points is the difference between them brings insight into certain mathematical formulas, such as the point-slope form of a line, the distance formula, and the equations of circles.
In this portion of the course, we have covered the following topics:
The Number Line and Base-10 blocks as Visualizations of the Integers
Visualizations of Addition and the Addition Algorithm
Visualizations of Subtraction and the Subtraction Algorithm
Integer Chips as a Representation of Negative Numbers in Addition and Subtraction Calculations
Movement on the Number Line for Negative Numbers in Addition and Subtraction Calculations
Visualizations of Multiplication as "A Groups of B" and Area
Visualizations of Division as Making Groupings and Equal Distribution