Fourth Grade Math in Ninth Grade Algebra

The effective teacher develops old schemata to new situations. Ninth grade algebra students come equipped with a schemata for adding, subtracting, multiplying and dividing large numbers that is perfectly suitable to the adding, subtracting, multiplying and dividing of polynomials. It's easier to extend an existing schemata than create a whole new one.

The only numbers in the base ten number system are 0 through 9. Every "number" in the base ten numbering system is a numerical expression of sums of multiples of powers of ten. For example, 432 is actually the numerical expression 4×10²+3×10¹+2×10⁰. When we add "numbers" in the base ten number system, or in any number system for that matter, we stack the summands and add the powers of ten. If a column of a power of ten goes over that power of ten we create a carry to the next power of ten to the left.
4×10²+3×10¹+2×10⁰
+2×10²+8×10¹+1×10⁰
7×10²+1×10¹+3×10⁰
Adding polynomials is adding numbers base x. The only difference between adding numbers base 10 and adding numbers base x is you don't create a carry in numbers base x. The carry is determined by the base in use and base x is a generic, abstract base without a carry. The same addition problem base x would be
4x²+3x+2
+2x²+8x+1
6x²+11x+3
Subtraction base ten involves subtracting the subtrahend (the number after the minus sign) from the minuend. If the number in the subtrahend is bigger than the number in the minuend we borrow ten from the power to the left. Subtraction base x works the same way except we don't have to borrow because there was no carry in addition. We simply let the power of x in a particular column go negative. When subtracting numbers base 10 we implicitly change the signs of all the numbers in the numeric expression by stipulating subtraction as the operation. To avoid confusion in base x numbers we can change the signs of all the terms and add the subtrahend to the minuend.

Multiplication base ten involves getting a product for each of the powers of ten in the multiplier by the multiplicand and then adding the products. If a product produces a multiple of the next higher power of ten, that multiple is carried and then added to the product of the multiplication of the next higher power. As the products progress across the multiplier lesser powers of ten are held open by zeros, so each product starts one column further to the left than the previous product. After all the terms in the multiplier have been multiplied the products are then added by the addition rules. Multiplication base x works the same way except there is no carry. The distributive property is automatically implemented by this multiplication process. And like terms in the products are already lined up in columns waiting for the addition step.

Most algebra teachers revert to fourth grade long division techniques to teach division of polynomials.

I am a constructivist so I always look for ways to expand on prior knowledge. The fourth grade schemata for fundamental operations on large base ten numbers works just fine if you can get the students to abstract that schemata to base x numbers.