Section 17.4 Closing Ideas
Things get a bit more complicated when we start introducing decimals into fractions. The basic concept of thinking about fractions as division still applies, but it becomes more difficult to create diagrams.
Consider the fraction \(\frac{6.37}{1.32}\) This is asking "how many groups of size 1.32 can be made if you have 6.37 items?" Conceptually, it makes sense, and we can come up with some basic analogies to help us think through it, such as "How many items can you buy for \$1.32 if you only have \$6.37 to spend?"
While we really can't draw an effective parts of a whole diagram for this problem, it's possible to set this up with a number line diagram to represent division. And we can push forward with manipulating the symbols the same way we did for fractions.
In real life, we usually reach for a calculator for fractions involving decimals. We invented the technology precisely to help us with those situations. But this doesn't negate the importance of having a conceptual foundation. While calculators can give us numerical results, it does not have the ability to conceptualize the idea of a fraction.
In this section, we used the intuition of "parts of a whole" with integer values to develop a pattern for manipulating fractions with variables. But we did not use any diagrams to try to represent those fractions. The reason is that our diagrams basically fall apart on us. Here are how different the pictures can look depending on the value of the variable:
The absence of concrete images makes fraction manipulations very abstract when we start involving variables. And this is where algebraic fluency really needs to kick in. As we go through the next few sections, we will be continuing to explore fractions. While we will be primarily working in the context of integer fractions, it is important to keep in mind that the algebraic processes that we're developing can also be applied to variable fractions, and you will have to trust your algebraic fluency more and more as you get further along.