Exploration of fractions as ratios, division problems, and the inverse of multiplication. What they look like, what they mean, and their lack of algebraic properties (associative, commutative).
Category: Concepts
Algebra Intro 10: Fractions and Multiplication
Exploration of ways to think about, and approach, fractions that are useful when working with them. What does a product of fractions mean, and what alternatives are there for evaluating it?
Algebra Intro 11: Dividing Fractions, Equivalent Fractions
Once a person is comfortable with multiplying fractions, dividing one fraction by another becomes fairly straightforward. Dividing Fractions An alternative to division by any number (not just a fraction) is “multiplying by the reciprocal”. Dividing by two has the same effect as multiplying by one half. Multiplying by the reciprocal of a number will always produce…… Continue reading Algebra Intro 11: Dividing Fractions, Equivalent Fractions
Algebra Intro 12: Adding and Subtracting Fractions
Once someone knows how to multiply fractions, and is comfortable creating equivalent fractions by multiplying by a fraction that equals 1, they have to tools needed to add and subtract fractions. Why Can’t I Just Add Two Fractions As Written? Consider the fraction “two thirds.” The phrase as written can be represented in two ways:…… Continue reading Algebra Intro 12: Adding and Subtracting Fractions
Function Dilations: How to recognize and analyze them
How to recognize vertical and horizontal dilations in both graphs and equations.
Function Translations: How to recognize and analyze them
How to determine both vertical and horizontal translation factors (relative to a “parent function”) when looking the definition of a function.
Inverse Musings: * and /
Inverse operations and functions are wonderful things. Without them, solving equations would be much more challenging. Yet inverse operations can also be odd beasts. My previous “inverses” post pondered addition and subtraction, which led us (as young students) to expand our initial universe of counting numbers into the integers. Addition and subtraction are operations that…… Continue reading Inverse Musings: * and /
Interactive Graphs for Exponential and Logarithmic Functions
A few more interactive GeoGebra applets have been added to my collection. Each of these graphs the function indicated in the name, with parameters that you can adjust using sliders. As you move a slider, you can watch how that parameter affects the graph of the function, and see what the resulting function definition looks…… Continue reading Interactive Graphs for Exponential and Logarithmic Functions
Multiplication
What does multiplication really represent? An exploration of alternative conceptual models for multiplication, along with a dimensional interpretation of multiplication.
Sigma and Pi Notation (Summation and Product Notation)
Explanation of how Sigma notation describes a sum of terms, with examples. Pi notation, which describes the product of a series of factors, is also introduced. Sum symbol, sum notation, summation notation, product symbol, product notation.
Summary: Algebra
When faced with an algebraic expression or equation, there are only two types of things you can do to it without changing the quantitative relationship that it describes. Re-write one or more terms in an equivalent form This can be done to any expression (no equal sign) or equation (with an equal sign) at any…… Continue reading Summary: Algebra
Inverse Musings: + and –
Inverse operations and functions are wonderful things. Without them, solving equations would be much more challenging. Yet inverse operations can also be odd beasts. This is the first of several postings on operations/functions and their inverses. Addition The first arithmetic operation we all learned was addition. It seems to arise fairly naturally from the counting…… Continue reading Inverse Musings: + and –
Function Notation
Function notation: why it is needed, how to interpret it, and what some common operations would look like if defined using function notation.
Notation & Concept
I use a lot of notation when working a math problem. It is familiar to me, comfortable even. So it is all too easy to forget that this same notation can get in the way for others. Math teachers have many goals for their students, mostly revolving around work and thought habits that help them…… Continue reading Notation & Concept
Multiplication Notation
Exploration of multiplication notations, and ways to help students better understand some of the notation used in algebra.
MathMaine 