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.

Module: Quadratic Equations and Parabolas

I begin by asking the students to consider what happens when we multiply two linear equations together. Start by looking at the graph of (x+1)(x-1). We make a chart of x-values and calculate the values for (x+1),(x-1) and (x+1)(x-1). Then we plot the values for (x,(x+1)(x-1)). The question is, what kind of curve is graphed when the points are connected? What is the y-intercept? How could you find the y-intercept from the two linear factors? What are the x-intercepts? How could you find them from the two linear factors?

Next we actually multiply two linear equations together to see what the product looks like. Multiply (x-2)(x+1). What is this kind of equation called? How is this equation different from a linear equation? What is the degree of this equation? How is it different from a linear equation? Make a chart and pick some x-values and calculate the values for the quadratic equation. Plot the points and sketch the curve. What is the y-intercept? How could you find it from the quadratic equation? What are the x-intercepts? Can you find them from the quadratic equation? What do you need to do to the quadratic equation to find the x-intercepts? Identify the lowest point on the graph as the vertex. What is the x-coordinate of the vertex? From the quadratic equation, how could you find the x-coordinate of the vertex?

Multiply (x+3)(x+2). How did you get the middle term? How did you get the third term? How could you work backwards from the quadratic equation to get the linear factors?

Look at the quadratic equation x²-4. How is this quadratic equation different from the quadratic equations you've seen so far? How could you find the linear factors of this type of quadratic equation?

I finish with some practice on quadratic equations with real solutions. Have them identify the y-intercept and the vertex of the parabola from the quadratic equation, then factor the quadratic equation and find the x-intercepts. Make sure to throw in a difference of two squares quadratic. Give the students a challenging problem like 2x²-3x-2.

The questions in this module are all level 2 and level 3 questions from Costa's Levels of Thinking and Questioning. They require the students to analyze, contrast and compare, generalize, hypothesize and speculate.