Thanks to David Ginsburg for inspiring this post.
Do you struggle explaining to your students what “simplify” means? Do you wonder why they won’t reduce fractions to their lowest terms?
As in “Where do the lines intersect?” or “Where is the highest point of the graph?”
“Where” in math almost always means “at what x-value?” Every now and then “where” can mean “at what point?” In the first case, “Where do the lines intersect?” can be just as easily satisfied by a point (x, y) as the x-value only.
When you get into calculus, the word “where” is used everywhere (pardon the pun). “Where is the slope of the tangent line equal to zero?” and “Where are there discontinuities?” among others.
Keep your eyes peeled in the textbooks for this tricky term.
Typically this means to reduce to lowest terms. Which isn’t really helpful, since you still need the definition of “terms.”
Numerators and denominators (tops and bottoms of fractions) that have one or more common factors are not officially “reduced.” Once they have no common factors at all (except 1), they are said to be reduced or reduced to lowest terms.
A reduced fraction helps in many ways. You can “feel” the value more easily: 2/5 is a lot easier to picture than 14/35. Also, when you add fractions, starting out with all the numbers in their reduced form will often help to find common denominators.
Zeros, Real Zeros and Roots
Zeros, real zeros and roots all mean the same thing. It is where a graph passes through the x-axis. We can write “where are the zeros” meaning at what x-value (“where” from above) does the graph pass through the x-axis? We also will write “what are the zeros” and mean the same thing.
To find the zeros or roots, we set a function (or any equation solved for y that involves x) equal to zero. Then we solve for x. This will give the result that we want – the x-values where the graph will go through the x-axis.
We call them zeros because they are where y = 0. We call them real zeros because there can be complex or imaginary zeros that don’t exist on the graph.
Finding the zeros or roots of a function help when graphing it. If you know all the places it goes through the x-axis, you can then determine where it’s above or below the x-axis. Then you can roughly sketch the graph.
Of all the terms in mathematics, this one is the most slippery. This meaning of this term is always in context to the book or situation. What it means is “manipulate the expression.” The end result may or may not be “simpler” than the starting expression.
Better textbooks use phrases that are clearer, like “multiply the polynomials” and “combine like terms.”
If “simplify” is the instruction to a problem set in a textbook, consult the examples to see what type of manipulation to do.
In later courses, students have to recognize when a situation requires a different form of the expression. Practicing various forms of manipulation (or “simplifying”) gives them tools to manipulate the expression to satisfy their needs.
Do you run into other terms in mathematics that are hard to explain to students? Hard to understand yourself? Share them in the comments!
- Is Zero Positive or Negative?
- What’s the Domain, Why You Need It and How You Get It
- What’s a Function?
- How Calculators Inhibit Learning the Distributive Property in Algebra
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