Function Transformations as Composition
(A new question of the week) We have discussed transformations of functions and their graphs at length, but a recent question suggested a slightly different way to think about them.
Algebra
Why Are Functions Defined as They Are?
Last week we looked at what functions are; but many students wonder why it all matters. What makes them useful? What makes functions worth distinguishing from non-functions? Why do we make the distinction we do? We love “why” questions, because they make us think more deeply!
What is a Function, Really?
A recent question, from Anindita, touched on the relationship of functions, relations, and rules. I referred to several answers we’ve given, which I’d long planned to put into a post (or two). This is it! We’ll start with a set of questions about what functions are.
A Hen and a Half …
Last week, we looked at problems involving some number of people making some number of things in some amount of time. In a classic twist on this problem, we’ll now examine several variants starting with “If a hen and a half can lay an egg and a half …”. Can we make sense of half-eggs …
How Many A’s Can Make This Many B’s in This Much Time?
A popular kind of word problem tells us how many people (or cats, or hens, …) it takes to make some number of houses (or kill some number of mice, or lay some number of eggs) in some amount of time, and then asks us to fill in one of the blanks for a different …
How Many A’s Can Make This Many B’s in This Much Time? Read More »
One-Variable Equations in a Two-Variable World
(A new question of the week) This week we have a short discussion of a question that takes a basic concept one step further: How do you graph an equation on the plane, that contains only one variable? It’s a simple question when applied to linear equations, but takes on new dimensions when we generalize …
Parallel Vectors: Missing a Solution
(A new question of the week) We were recently asked to check work on an interesting little question about parallel vectors, and I was almost convinced that there was no solution … until I realized there was one! How was it missed? How can we avoid doing that? That’s our goal today.
Why Properties Matter: Beyond Addition and Multiplication
We’ve been looking at the commutative, associative, and distributive properties of operations, starting at an introductory level. But why are these properties important? Why do they have names in the first place? And what other operations have them?
Commutative, Associative, and Distributive Properties
In working on last week’s post, I realized that I haven’t yet covered the general idea of properties, such as commutative, associative, and distributive. Here I’ll collect some introductory answers on that topic. Next week, we’ll dig a little deeper.
Parentheses and the Associative and Distributive Properties
(A new question of the week) This week, we’ll look at two recent questions about how parentheses (brackets) are used, how they relate to the properties we use in algebra that let us add or drop them, and the related concept of factoring a polynomial. They are examples of how student questions can touch on …
Parentheses and the Associative and Distributive Properties Read More »