Explanation of why a negative sign can be placed, or moved to, in front of a fraction, in front of its numerator, or in front of its denominator, without changing the value of the fraction.
I have started a separate blog devoted to helping students learn to find mistakes in worked problems (their own, or someone else’s). If this is of interest, check it out: http://mathmistakes.wordpress.com/ 7/17/11 Update: There can be great value in work that contains mistakes. Learning to catch your own mistakes is a critical life skill, as is…… Continue reading Where’s the mistake?
This post begins a series intended to help introduce or re-introduce some of the core concepts of Algebra. It is often very helpful to re-visit these concepts with students who may have memorized their way through previous math courses without slowing down to contemplate some of the concepts behind Algebra. Numbers Numbers are used in…… Continue reading Algebra Intro 1: Numbers and Variables
Mathematical thinking probably started with addition. Someone may have combined two piles of bricks, and wondered how many were in the single large pile. Addition is the mathematical term that describes “joining quantities together”. Properties of Addition Looking at the patterns that can occur when quantities are joined together, you might have noticed that it does…… Continue reading Algebra Intro 2: Addition
Once addition has been explored a bit, it leads pretty naturally to a new question: if there are three bricks in a pile, how many bricks do I need to add to it so that there will be five in the pile? Our addition problems were all phrased using a pattern like number + number…… Continue reading Algebra Intro 3: Subtraction
Negative Numbers Negative numbers are something new and interesting to think about. What do they mean? They arose from changing the order in which we subtracted two numbers. While we usually think of a “difference” by starting our thought process with the larger number, when we fail to do that and try to “take away”…… Continue reading Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites
Explanation of why subtracting has the same effect as adding a negative, and how rewriting addition as subtraction allows terms to be easily re-arranged. Introduction to the use of the word “term” in algebra, along with how terms may be re-arranged.
We have all known our multiplication tables for years, and have successfully answered questions like “what is 6 times 7?”, but do we really understand what multiplication represents? Repeated Addition One interpretation of multiplication, which only works when multiplying by an integer, is “repeated addition”. From this perspective, “6 times 7” is a compact way to…… Continue reading Algebra Intro 6: Multiplication
Properties Of Multiplication Do the patterns that applied to addition also apply to multiplication… do the following all produce the same result? $latex 3 \cdot 5 \cdot 7 \\*~\\*3 \cdot (5 \cdot 7) \\*~\\*(3 \cdot 5) \cdot 7&s=2&bg=ffffff&fg=000000$ After carefully following the order of operations, we see that they all result in a value of 105.…… Continue reading Algebra Intro 7: Properties of Multiplication
The last “arithmetic” operation introduced in school is usually division. While multiplication allows us to calculate the total needed for a group when a fixed quantity is required for each person, division allows us to determine how much each person will get when a fixed quantity is divided equally among all in a group. …… Continue reading Algebra Intro 8: Division
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).
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?
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
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
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 /