Foundations of Algebra

Algebraic thinking involves finding and describing patterns, making generalizations about numbers, using symbols and models to represent patterns, quantitative relationships, and changes over time.
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Function Machines

Using the concept of a "function machine" is an excellent way to demonstrate algebraic concepts to students. A function machine is something that takes in a value, does something to it, and generates a new value based on a rule or operation. A function machine shows students how different values for variables change the results of a situation, and can be any series of operations that demonstrate a pattern. For example, suppose you are having friends over for dinner, and estimate that each friend will eat two hot dogs. You can figure out how many hot dogs you need by constructing a function machine.

Patterns and Relationships in Tables

A data table is another common way of studying algebraic relationships. For example, imagine a "mystery" function machine in which students input a number of friends and the function machine outputs the number of hot dogs.

Number of Friends Number of Dogs

What is the relationship between the numbers in the table? What pattern do you see? How many hotdogs will be needed if 155 friends attend the party?

Careful study of the table shows that the number of hot dogs is equal to the number of friends multiplied by two. This relationship can be represented by an expression such as:

numHotdogs = numFriends x 2

If 115 friends are expected at the party, then

numHotdogs =115 x 2
numHotdogs= 232


Another important concept for students to understand is algebraic equality. Young students may be accustomed to having the numbers and characters on the left side of the equal sign represent an arithmetic operation, the equal sign mean, "Here comes the answer," and the numbers and characters on the right side of the equal sign represent the answer. Students should be taught to view the equals sign as a symbol of equivalence and balance. The equal sign simply expresses that both mathematical phrases on either side of it are equal, not necessarily that either phrase is an answer.

In other words,

numHotdogs = numFriends x 2

is mathematically equivalent to

numFriends x 2 = numHotdogs

To further illustrate the idea of algebraic equality, consider a balanced scale with an apple on the left side of the scale and two bananas on the right side of the scale. In this example, we assume the weight of the apple is equal to the weight of two bananas. To verbalize this to students, say, "The weight of one apple is equal to the weight of two bananas."

The following expression can be used to represent this equality algebraically:

apple weight = banana weight + banana weight

Assuming the two bananas are of equal weight, then

apple weight = 2 x banana weight

To find the value of the weight of a banana, we can divide. When an operation is performed on one side of an equal sign, the same operation must be performed on the other side of the equal sign to keep the equation balanced. The results of dividing both sides of the equation by two are shown below.

apple weight = 2 × banana weight

apple weight / 2 = 2 × banana weight / 2

apple weight / 2 = 1 × banana weight

apple weight / 2 = banana weight


A constant is a value that does not change. For example, in the expression

Fahrenheit = Celsius x 1.8 + 32

Fahrenheit and Celsius are variables, while 1.8 and 32 are constants.

In the hot dog example, one constant is the number of hot dogs you will buy for each invited friend, or two hot dogs. If your brother eats three hot dogs, then you will always need one extra hot dog, regardless of the number of friends you invite. The number of hot dogs for your brother is a constant, because that number will not change. Based on this, you can construct a new data table that shows the total number of hot dogs you need.

Number of Friends Number of Hot Dogs for Friends Number of Hot Dogs for Brother Total number of Hot Dogs
2 4 1 5
3 6 1 7
4 8 1 9

The first column describes the number of friends invited-a variable. The second column is the number of friends multiplied by two (the number of hot dogs per friend- a constant), the third column is the number of hot dogs for your brother, one (a constant), and the fourth column is the number of hot dogs for your friends plus one (the number needed for your brother). This can be stated as, "The number of hot dogs needed is equal to two times the number of friends plus one."

This is the algebraic equation that explains the rule:

totalHotdogs = (2 x numFriends) + 1


Algebra is also used to study change in different situations. Algebraic equations can be derived to describe how quickly or slowly something changes such as the change in a student's weight or height over the years. For young students, having the ability to interpret graphs that describe change, either at a variable rate or constant rate, is an important skill that will prepare them to both derive and interpret change situations in higher-level algebra. By having students create and analyze change represented by tables or graphs, they can determine patterns for the rate of change.

The following data table:

Height of Jack's Beanstalk

DayHeight (feet)
1 2
2 5
3 10
4 17
5 26
6 37

can also be represented as a line graph or as an equation:

stalkHeight = day x day + 1
stalkHeight = day2 + 1
or as a function machine.
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