Negative Fractions

Question: Where should I put the negative sign when I am writing a fraction like negative two thirds?

Answer: As long as you write only one negative sign, it does not matter where you put it.

Two ideas are useful to keep in mind during the explanation that follows:
– Subtraction is the same thing as the addition of a negative.
– The negative of a number can be created by multiplying the number by negative one.

These principles apply to fractions as well, so:


Placing the negative sign before the entire fraction (subtracting the fraction) is equivalent to adding the same fraction, but with a negative numerator. But this is not the only option…

Continue reading Negative Fractions

Where’s the mistake?

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:

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 learning to review other people’s work while seeking to understand it fully (the best way to do this is by looking for mistakes).

Along these lines, I came across an interesting blog posting by Kelly O’Shea.  She came up with the idea of insisting that each group who is presenting try to sneak a mistake in their work past their peers.  Brilliant!

Algebra Intro 1: Numbers and Variables

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 are used in the English language as both nouns and adjectives. However, the only physical instance of a “number” in our world is a symbol or group of symbols such as “23”. Most of the time, we seem to use numbers as adjectives: “Look at the two trees.” Continue reading Algebra Intro 1: Numbers and Variables

Algebra Intro 2: Addition

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 not matter whether 2 bricks are added to a pile of 3, or 3 bricks are added to a pile of 2… either way, we still end up with a pile of 5. So, the order in which we add two quantities does not change the result. This probably makes intuitive sense to you when you visualize the situation above and the final pile that results. Continue reading Algebra Intro 2: Addition

Algebra Intro 3: Subtraction

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 = what?

and the question above rearranges it a bit:

number + what? = number
what? + number = number

This question is usually asked as “What is the difference between Continue reading Algebra Intro 3: Subtraction

Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites

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 Continue reading Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites

Algebra Intro 5: Addition, Subtraction, and Terms

What happens when problems involve both addition and subtraction? Addition is both associative and commutative, and subtraction is neither…

One solution is to follow the order of operations (parentheses, exponents, multiplication, division, addition, subtraction) working from left to right in the event of a “tie”. This will always produce the intended result.

However, the order of operations does not provide any guidance about how Continue reading Algebra Intro 5: Addition, Subtraction, and Terms

Algebra Intro 6: Multiplication

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

Algebra Intro 7: Properties of Multiplication

Properties Of Multiplication

Do the patterns that applied to addition also apply to multiplication… do the following all produce the same result?

3 \cdot 5 \cdot 7 \\*3 \cdot (5 \cdot 7) \\*(3 \cdot 5) \cdot 7

After carefully following the order of operations, we see that they all result in a value of 105. Therefore, it is reasonable to conclude that multiplication is associative, similar to addition.

Continue reading Algebra Intro 7: Properties of Multiplication

Algebra Intro 8: Division

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.   Division is the inverse of multiplication.

But how does one go about dividing?  Should we use our knowledge of Continue reading Algebra Intro 8: Division

Algebra Intro 9: Fractions, Reciprocals, and Properties of Division

Now that fractions and rational numbers have been introduced, let’s explore what they do for us a bit

Fractions And Rational Numbers

A rational number is one that can be represented by the ratio of two integers.

A fraction is a number that is written Continue reading Algebra Intro 9: Fractions, Reciprocals, and Properties of Division

Algebra Intro 10: Fractions and Multiplication

Many people seem a bit phobic about “fractions”. This anxiety likely has two sources: not really understanding what a fraction represents, and having memorized a bunch of rules way back in elementary school without understanding why they work.

Revisiting fractions using variables as well as constants, with Continue reading Algebra Intro 10: Fractions and Multiplication

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 the exact same result as dividing by the original number.

Using this approach, any division problem can be 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 Continue reading Algebra Intro 12: Adding and Subtracting Fractions

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 only make sense when acting on “like” quantities, or within a single spacial dimension.

Continue reading Inverse Musings: * and /