Section 23.1 Another Pause for Reflection
Congratulations! You have just completed the first two "branches" of the course.
In the first branch, we looked at some of the ideas of the coordinate plane, which will be important for anything you do in the future that involves graphs. We also looked at the equations of lines, which are important for both analytical mathematical calculations as well as applied mathematical calculations. It turns out that lines are one of the core conceptual models we use in the development of calculus (approximating curves with lines) and lines are one of the core applied models we use in applications (including in the development of artificial intelligence).
In the second branch, we broadened our numerical experiences to include fractions and decimals. Fractions represent a division concept and can be interpreted using the concept of parts of a whole. We saw that addition and subtraction of fractions require us to work with a common denominator, and that the common denominator is simply a way of allowing us to work with pieces that are all the same size, rather than pieces that are of different sizes.
With that framework in mind, we saw that fraction multiplication is just a bookkeeping exercise, where multiplying the numerators is tracking the number of wedges and multiplying the denominators is tracking the size of the wedges. And this idea is what makes "multiply straight across" a logical and non-arbitrary computational method. Next, we explored the relationship between multiplication and division, and focused on the idea that these operations are inverses of each other. We also developed more computational methods for handling increasingly complex fraction division calculations.
Finally, we looked at decimals as yet another representation of parts of a whole. By thinking about common denominators, we saw that addition and subtraction was easiest to perform if we simply viewed the numbers as having the same number of digits after the decimal point. We also saw that moving the decimal around in multiplication calculations is the exact same concept we used in fraction multiplication, but presented in a different way.
At this point, we're going to take another pause in order to let you practice your metacognitive skills by thinking about what you've learned in order to continue to better understand our the development of your own thinking.
In this portion of the course, we have covered the following topics:
Lines and the Coordinate Plane
Slope-Intercept Form
Solving Systems of Equations by Substitution
Solving Systems of Equations by Elimination
Fraction Basics
Fraction Addition and Subtraction
Fraction Multiplication
Fraction Division
Decimal Addition and Subtraction
Multiplying Decimals and Percents