Are quadratics really that important

If I was a student who had just finished a course in Algebra I in most schools across America I think that I would think that the following were the most important things that we had done. I would think this because we had spent so much of our efforts on it.

• linear equations
• distributive law
• quadratic equations
• factoring

The problem with this list isn't so much what is on it, it is what isn't on it. It seems to me that we simply must include exponential functions and therefore also logarithms. I guess my question is, "Are quadratics really that important?"

The only way to add these topics in is to take other topics away. So what can go? Here is a partial list of topics that might be cut or reduced in the standard curriculum.

• completing the square
• conics
• rational root theorem
• long division of polynomials

Exponentials seem like such a natural topic, and one that could be made to support and build on linear equations so beautifully. I will give more of this idea when I write later about "schlope".

Determinants as proportions

This is a continuation of a discussion of determinants started here. And there is more here.

Proportions

So if determinants are important how important are they and when can they be introduced reasonably. One place that they could possibly come up is in reference to proportions. As a reminder, a proportion is equation where two ratios are equal. One method to solve equations such as this is the Means-Extremes Property. This is more commonly known as “Cross-Multiplying”. I think I speak for many teachers that cross-multiplying is a “bane of existence”. Cross-multiplying is a rule that is often overused. It seems to quickly rise to the top of all students list of favorite methods so that whenever in doubt about how to proceed in a problem with fractions teachers often here the idea put forth that the correct method might include cross-multiplying. This is probably because few students really understand what this method does or why it works

Connections between determinants and proportions

So what is the connection between determinants and cross multiplying. Well
it can be seen from a variety of ways. First is in the formulas themselves. A
proportion has the form:

after cross-multiplying we know that

this last equation can be rewritten in terms of determinants as

In a proportion we are given that two fractions are the same. Each of those fractions can be thought of as vectors. Similar to the definition of slope as a fraction or as a vector. With this definition of the fraction we see that the two fractions will be equivalent if their vectors point in the same direction. If they point in the same direction then the area given by the determinant will be zero.

Advantages of the Determinant Formulation

The advantage of this determinant method to solving proportions is that we eliminate the fractions from the problem. Cross-multiplying would only exist in determinants, where the rightly do play a role. Students would be less likely to misapply the idea of cross-multiplication to every situation with fractions. Determinants would be introduced earlier and their presentation of area would be well supported. Students would also have to have a clearer understanding of slope as a primary way of looking at fractions.

Bear hunting Algebra

I am finding myself more of a shameless shill for mathematics in my 8th grade Algebra class. Want me to jump around like an idiot? I will do it. Want me to make a snowball and bring it in to the classroom? Good as done. So you can see I am interested doing anything that will meet my students where they are...

One more crazy thing I have been doing is trying to connect the kids books I read to my math class. And I think I found out one that actually worked a little bit. It is Going on a Bear Hunt...

You may the know story, you may have been to camp, you may have never heard of it, which may mean that you have never had a child, been to any place where teenagers are asked to supervise young children. If that is you click here to get a sense of what it means to go on a bear hunt.

I use this to remind/reinforce the idea about doing and undoing to solve an algebraic equation. So with the equation:

3x+1=10

I say we are going on bear hunt we are going catch a big one we are not scared. Oh no a times 3!. Can't go over it can't go under it... Oh no a plus 1, etc. And then I said a 10! AAAAAAHHHHHH! Back through the 1 (minus 1), back through the 3 (divide by three). These are the steps (minus 1 and divide by 3) that you need to solve the problem.

Now as many of surely know, much of this is not "necessary", but my students need the background of reading through the problem the first time to understand the order of operations. In fact, many of my students still don't understand that the 3x cannot be undone by subtracting 3.

I hope that this helps them see not only what to do, but why they are doing it.

At the very least I am exposing them to classic literature, well literature, well words.