Chapter 1
The Language of Algebra
Think of a number, double it, add six, halve the result, then subtract the number you started with. You land on three. Try a different starting number and you land on three again. Checking numbers never tells you what happens for every number; there are too many to check. Letters solve that, standing for a number before you know which one it is. This chapter builds the toolkit letters need: powers and roots that are not whole numbers, expressions you can shorten or expand, factoring that reverses multiplication, and fractions that carry a letter of their own.
What You'll Explore
5 lessons.
- From Arithmetic to Algebra
A letter can name one unknown number, or it can appear in a statement that holds true no matter which number replaces it, and those are two very different jobs. Here you will look at what it takes for two expressions to count as the same thing, and at how far testing with actual numbers can settle a question like that.
- Fractional Exponents and Radicals
Raising a number to the power three means multiplying three copies of it together, but half a copy is not something you can multiply. The exponent laws you already trust still apply to that half, and this lesson follows them to find out what such a power has to mean.
- Arithmetic with Expressions
Add two boxes and three books to one more box, and the total takes fewer words than the list you started with. Expressions can shorten the same way, though not every part will join with every other. You will work out which parts combine, and how to be sure the shorter version says the same thing.
- Expanding and Factoring Expressions
Buying five identical bags of groceries, you could total one bag and multiply by five, or add up each kind of item across all five bags. The bill comes out the same. Expressions offer that same choice, and the lesson goes both ways: turning a product into a sum, and finding a product hiding inside a sum.
- Algebraic Fractions
Once a letter can appear underneath a fraction bar, the fraction stops being a fixed amount and starts depending on what the letter is. Some choices make it behave strangely. What carries over from ordinary fractions, what has to be handled with care, and which values are off limits are the questions here.