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: The Properties of Real Numbers

The order in which mathematical operations are performed makes a difference in the outcome of the operations. To ensure that all mathematicians, and students, get the same answers to mathematical equations, a particular order of operations has been agreed upon. Operations inside parenthesis are always performed from the inside out. This means, operations inside parenthesis are performed before any other operations. Exponentiation (raising numbers to powers) is performed next. Multipication and division are done next in order. Addition and subtraction are always done last.

A mnemonic is an abbreviation of a series of words or phrases that is itself a word or phrase that can be more easily remembered than the series of words or phrases. The mnemonic for the order of operations is a nonsense word, PEMDAS, representing the first letter of the operations in the correct order. A phrase that helps keeps the operations in order, and is a better mnemonic, is Please Excuse My Dear Aunt Sally. Apparently Dear Aunt Sally needs excusing for some reason, possibly having to do with her mental state, but the first letters of the words in the phrase are the first letters of the operations in the correct order.

Now that we have a standard for the order of operations that we can all follow, let's look at some equations and see what properties these equations lead us to.

The Commutative Property
1. What is the sum of 3 and 4? What is the sum of 4 and 3? Is there a difference? Can you state a principle that would describe this property of real numbers? This property is called the commutative property of addition. Does this property hold if you take a difference of two numbers?

2. What is the product of 3 and 4? What is the product of 4 and 3? is there a difference? Can you state a principle that would describe this property of real numbers? This property is called the commutative property of multiplication. Does this property hold for the division of two numbers?

The Associative Property
3. What is the sum of 2, 3, and 4? We can only add two numbers at a time so we must pick two numbers of the three to add first before adding the third. This requires us to group two of the numbers together. Does it matter which two numbers we group together first? Can you state a principle that would describe this property of real numbers? This property is called the associative property of addition. Does this property work if one or more of the operations is a difference?

4. What is the product of 2, 3, and 4? We can only multiply two numbers at a time so we must pick two numbers of the three to multiply first before multiplying the third. This requires us to group two of the numbers together. Does it matter which two numbers we group together first? Can you state a principle that would describe this property of real numbers? This property is called the associative property of multiplication. Does this property work if one or more of the operations is a division?

The Distributive Property
5. Calculate the result of the following equation: 2×(3+4). Calculate the result of the following equation: 6+8. How are these two equations related? What happened to the multiplication and the addition between the two equations? This property of real numbers is called the distributive property and is the agreed upon method for using multiplication and addition together. Can you undo the number 27 so that you have the product of a number and a sum like the first equation? This is the process of undoing a distribution and is called factoring. We will make extensive use of this property to combine a product over a sum and to undo a product over a sum.

6. Can you use the distributive property to properly distribute (2+3)×(4+5)? How would this look if you wrote it in distributed form like the second equation above? Hint: Multiplication of large numbers uses a method that stacks large numbers and keeps powers of ten in the same columns. A large number is actually a sum of powers of ten, a multiple of a hundred plus a multiple of ten plus a multiple of one for example. Stacking the numbers and multiplying in columns helps properly perform the distributive property. Try doing something similar with (2+3)×(4+5) and see if you get the right answer.

The Identities
7. Can you think of a number that when added to any other number doesn't change the value of that second number? This number is called the additive identity because adding this number doesn't change a number's identity.

8. Can you think of a number that when multiplied to any other number doesn't change the value of the second number? This number is called the multiplicative identity because multiplying this number doesn't change a number's identity. We will use this property to change the appearance of rational numbers without changing the rational numbers' values. Any rational number whose numerator and denominator are the same are equivalent to the multiplicative identity. Multiplying a rational number by a rational representation of the multiplicative identity changes the appearance of a rational number but doesn't change its value.

The Inverses
9. Can you think of an integer that when added to 3 will give you the additive identity? This integer is 3's additive inverse. Can you think of an integer that when added to −4 will give you the additive identity? This number is −4's additive inverse. Can you state a principle that will allow you to find any real number's additive inverse? Adding a number and its additive inverse to a number or equation adds the additive identity to the number or equation. This is called adding a well chosen zero and we will find many situations when this will become advantageous.

10. Can you think of a rational number that when multiplied to ³/₄ will give you the multiplicative identity? This rational number is ³/₄'s multiplicative inverse. Can you think of a rational number that when multiplied to ⁵/₄ gives you the multiplicative identity? This rational number is ⁵/₄'s multiplicative inverse. Can you state a principle that would allow you to find the multiplicative inverse of any real number? The multiplicative inverse is called the reciprocal.

The Laws of Exponents
Exponentiation, or raising a number to a power, is multiple multiplications. In the number 2³, 2 is the base and 3 is the exponent. The exponent tells you how many times to multiply the base to itself. A number raised to the 2^nd power is said to be squared. A number raised to the 3^rd power is said to be cubed. Any number without an exponent is understood to have an exponent of 1. Any number raised to the 0 power equals the multiplicative identity.

Let's look at some consequences of this description of exponentiation.

11. Using the multiple multiplications of exponentiation, can you find a way to rewrite the product of 2² and 2³ as a single exponent? Can you state a principle that would allow you to describe this product of two exponentiated numbers? Does it work for the product of 2³ and 3²? Can you make your principle more specific?

12. Using the multiple multiplications of exponentiation, can you find a way to rewrite 2² cubed using a single exponent? Can you state a principle that would allow you to describe this exponentiation of an exponent? Why do you get the same answer for 2³ squared?

13. Using the multiple multiplications of exponentiation, can you find a way to rewrite (2×3)³ as a product of two numbers each raised to a power? Can you state a principle that would allow you to express a product raised to a power as a product of two numbers each raised to a power?

14. Use the multiple multiplications of exponentiation and the rational representation of the multiplicative identity to simplify 2 cubed over 2 squared. Can you state a principle that would allow you to rewrite this rational expression using a single exponent? Does your principle work for the rational number 2 sqaured over 2 cubed? Can you state a principle for negative exponents?

15. Using the multiple multiplications of exponentiation, can you rewrite 2 cubed over 3 cubed as a rational number raised to a power? Can you state a principle that would allow you to rewrite any rational number in which the numerator and denominator are raised to a power as a rational number raised to a power? Can you apply a previous principle and this principle to see what happens to 2 cubed over 3 to the sixth?

3 is the 3^rd root of 27 if and only if 3³=27. In ⁴√2³, 4 is the index, 2 is the base and 3 is still an exponent. The 2^nd root is called the square root. Any radical without an index is under stood to be the square root. The 3^rd root is called the cube root.

To reduce a radical, factor the base into its prime number factors, then, using the index, remove groups of factors by the index until you can't remove any more. Each time you remove a group, you count the base once on the outside of the radical. When you can remove no more groups, the remaining factors are multiplied back together again and the bases on the outside are multiplied together. Any radical with no base left inside it is a whole integer number. Any radical with a base left inside it is an irrational number. Most radicals are irrational numbers.

Using this description of radicals as a basis, let's look at some consequences.

16. Simplify ³√2³. Simplify ³√2⁶. What would you do with ³√2⁸? Can you state a principle for the case where the index and the exponent on the base are the same?

17. Simplify ³√2³×³√3³. Using the definition of a root, rewrite these two numbers a product under a single radical. Can you state a principle that would describe this radical. Do the bases have to be the same? Do the indexes have to be the same?

A radical can be written as a rational exponent. The exponent of the number under the radical is the numerator, and the index of the radical is the denominator. Express ⁵√2⁶ as a rational exponent. Using rational exponents for 17, do the indexes still have to be the same?

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.

Linear to Quadratic Equations

This is an idea I picked up from an article in Mathematics Teacher.

After students get used to linear equations you have to ask what interesting things can we do with linear equations besides draw lines? What if we add two linear equations? Well, obviously you just get another line. Not too interesting. But what if we multiply two linear equations together?

Make a chart
x|y₁|y₂|y₁×y₂

(x,y₁) plots the first linear equation
(x,y₂) plots the second linear equation
(x,y₁×y₂) plots a new function

Now you can introduce the quadratic equation and look at how the y-intercept is the product of the b's in the linear equations and the x-intercepts are the opposite sign of each b in the linear equations.

After you've done all your quadratic work, try dividing a linear equation by another linear equation or try multiplying three linear equations. Both are interesting and essential topics in Algebra.

Geometer's Sketchpad makes a perfect platform for investigating these equations.

Welcome to Math 2.0

I have started this blog to document and share my journey into teaching math in a 2.0 world. I am reading ISTE's Reinventing Project-Based Learning and will be posting ideas as they come to me. I'm looking for ways to get the students more involved in the discovery of math and ways to incorporate technology into the classroom. For example, I teach my students to complete the square on quadratic equations to find the roots then, after they're comfortable with the procedure, I set them to completing the square on the standard form of a quadratic equation. They end up with the quadratic formula. This is how math was discovered in the first place. Rather than handing them the information prepackaged, I lead them through the same discovery processes mathematicians went through. I feel it gives the students a personal ownership of the math.

Please feel free to comment and collaborate as I can use all the ideas I can get.