Math Dynamics

Hands-on Math Program

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Mathematics is the study of numbers. Arithmetic and algebra are two parts of that study. In algebra we use letters to represent the numbers. Following is a demonstration of how we teach algebra with manipulatives. You can make your own paper manipulatives and color them to use with this demonstration. If you have a color printer you can make colorful manipulatives.

Every number has two parts:

bulletHow many?
bulletWhat kind?

This is one unit. How many? .. 1 .. What kind? .. units.


Here we have how many? .. 5 .. What kind? .. units.


Here is one blue bar. We cannot count any units, so we don't know how long it is. Since it is not an exact number then its value can change or vary, so we call it a variable. We use letters to name variables in algebra. We'll call this length x because we can't count it, so it is one x. How many? .. 1 .. What kind? .. x.


Here we have several bars. How many? .. 3 .. What kind? .. x.


We can put nine units together so we have a rectangle that is 3 across and 3 up. We call that 3 two ways or 3-squared which is also written as 32.


When we have a rectangle that is x across and x up, then it looks like this. It is x two ways and we call it x-squared which is written as x2. How many? .. 1 .. What kind? .. x2


Now let's build some numbers. Here is 2x + 4 units or simply 2x + 4.


Here is 1x2 + 3x + 2 or usually written x2 + 3x + 2.


Here is 2x2 + 5x + 4.


In arithmetic and algebra we often need to find the factors of a number to solve a problem. The factors of a number are two or more numbers which can be multiplied together to get that number. For example 3 and 2 are factors of 6 because 3 times 2 equals 6.

Using manipulatives we can find the factors of a number by forming a rectangle with the blocks. Here are the factors of 6.


Since the expression x2 + 3x + 2 is really just one number with several parts, we can find the factors of this number by reshaping it into a rectangle like this:


The factors are the dimensions of the rectangle, x + 2 and x + 1, so we write:

x2 + 3x + 2 = (x + 2) (x + 1)

Let's try factoring one more expression, say x2 + 5x + 6. First we see the blocks to represent the parts of the number. Then we reshape these blocks into a rectangle and read the dimensions of the rectangle which are the factors of the number.


We can write it like this:

x2 + 5x + 6 = (x + 3) (x + 2)

In this demonstration you have learned how we factor an algebraic expression using manipulatives. We use the blocks to do many other operations in algebra such as addition, subtraction, multiplication and division.

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