How i can teach my students how to factor numbers and variables
I teach them the distributive law
and the master it I tried to tell them that factoring is the anti of distributive law
any ideas
My teacher just said it like this:
Take the example of:
$\displaystyle x^2+6x+9$
Now, $\displaystyle 6$ is the "target number"
We need to split $\displaystyle 9$ into it's factors,
$\displaystyle 1,9$ and $\displaystyle 3,3$
and try to find a combination that will make 6. This is $\displaystyle 3,3$, so the factors are $\displaystyle (x+3)(x+3)$
The next example he explained were of the type $\displaystyle x^2-bx+c$, ie. $\displaystyle x^2-5x+4$ because these factored into things of the form $\displaystyle (x-a)(x-d)$ so they were easy to get our head around. Now, $\displaystyle 5$ is our target number and we look at the factors of $\displaystyle 4$:
$\displaystyle 4,1$
$\displaystyle 2,2$ and we can get the target number with $\displaystyle 4$, $\displaystyle 1$ so the factors are $\displaystyle (x-4)(x-1)$
Then came $\displaystyle x^2-ax-b$, ie $\displaystyle x^2-2x-8$. Again, $\displaystyle 2$ is our "target number" and again we look at the factors of $\displaystyle 8$ which are $\displaystyle 1,8$ and $\displaystyle 4,2$
clearly $\displaystyle 4-2=2$ so the factors are $\displaystyle (x\pm 2)(x\mp 4)$ and then we could just adjust the signs as necessary to get $\displaystyle -2$.
Etc, etc.
When it came to things of the form $\displaystyle ax^2+bx+c$, it would be the same thing, except this time you'd have to use both the factors of $\displaystyle a$ and the factors of $\displaystyle c$ to get the target number.
For cubics, quartics, etc, which we didn't cover until 2 years later, we just learned the factor, remainder and rational roots theorems, which made things very straightforward.
Then, consider this approach:
Split $\displaystyle 2x^2$ into its factors:
$\displaystyle 2\times{x}\times{x}$
Split $\displaystyle 4x$ into its factors:
$\displaystyle 2\times 2\times x$
Now look for factors which are common to both, which here are $\displaystyle 2\times x$ and isolate these:
$\displaystyle 2x[\cdots]$
Then, just list the remaining factors.
$\displaystyle 2x(x-2)$
Another example, $\displaystyle 6x^2-3x$
$\displaystyle 6x^2=3\times 2\times x\times x$
$\displaystyle 3x=3\times{x}$
Common are $\displaystyle 3\times{x}$
$\displaystyle =3x[\cdots]$
Then, filling in the remaining factors:
$\displaystyle =3x(2x-1)$
sort of a tangent, but one way i have always thought of polynomials with integer coefficients, is "integers in base x". that is, we use the same rules in multiplying, adding, subtracting and dividing, that we do with ordinary integers, it's just that base x is so big, we never get from the "1's" place to the "x's" place or from the "x's" place to the "x^2's" place.
the distributive law is....well, you just cannot stress enough how important it is. it splits AND collects, it slices, it dices, it's a whole kitchen in one!
since I am new in teaching field so I always try to learn from others how to teach
I think I was suppose that my students should learn from one example which is not true
I found out that giving examples for the way or the techniques of something with different ways they will learn it without trying to teach them the basic things
Thanks