1.
Were there any topics that you had seen before, but you understand better as a result of working through it again?
Chapter 32 Reflection
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
Section 32.2 Worksheets
PDF Version of these Worksheets
1
external/worksheets/32-Worksheets.pdfWorksheet Worksheet 1: Questions about the Content
2.
Were there any ideas that you had never seen before?
3.
Based on your experience, which of these ideas seems the most important to understand well?
4.
Did any part of the presentation make you curious about math in a way that went beyond the material? Are there questions or ideas that you would like to explore?
Worksheet Worksheet 2: Questions about You
1.
How has your mathematical writing continued to evolve from the previous pause for reflection? Do you find yourself thinking in different ways?
2.
Did you have any "Aha!" moments where you had an insight into something that you had not noticed before?
3.
What is the biggest mathematical connection that you made?
4.
What is your level of confidence in mathematics? Is this higher, lower, or about the same as it was at the beginning of class?
