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ScienceFusion Intro to Science & Technology: Online Textbook Help13 chapters | 63 lessons

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Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Mathematical models can be used to represent just about everything in the real world. This lesson will explain what mathematical models are, in addition to describing and providing examples of the various types of models.

Suppose you are building a rectangular sandbox for your neighbor's toddler to play in, and you have two options available based on the building materials you have. The sandbox can have a length of 8 feet and a width of 5 feet, or a length of 7 feet and a width of 6 feet, and you want the sandbox to have as large an area as possible. In other words, you want to determine which dimensions will result in the larger area of a rectangle. Thankfully, in mathematics, we have a formula for the area (*A*) of a rectangle based on its length (*l*) and width (*w*).

*A*=*l*×*w*

Awesome! We can use this formula to figure out which dimensions will make a bigger sandbox!

We can calculate the two areas by plugging in our lengths and widths for each choice:

*A*1 = 8 × 5 = 40 square feet*A*2 = 7 × 6 = 42 square feet

We see that a length of 7 feet and a width of 6 feet will result in the larger area of 42 square feet. Problem solved!

Here's something neat! The formula we used to solve this area problem is an example of a mathematical model. A **mathematical model** is a tool we can use to replicate real-world situations and solve problems or analyze behavior and predict future behavior in real world scenarios.

The mathematical model we just used was in the form of a formula, or **equation**. Equations are the most common type of mathematical model.

Here's another example of an equation as a mathematical model. Suppose that a store is having a closeout sale where everything in the store is 15% off. That is, if an item is *x* dollars, then the discount is 15% of *x*, or 0.15*x*. The sale price can be found by subtracting the discount (0.15*x*) from the original price (*x*), giving the following equation that models the sale price of any item in the store with the original price *x*:

*S*=*x*- 0.15*x*

We can also combine like terms and write this equation as:

*S*= 0.85*x*

Both of these equations are mathematical models, because they represent a real-world scenario by using a formula to find the sale price of anything in the store. For instance, if something is originally $5, then the sale price can be found using our model by plugging in *x* = 5:

*S*= 0.85(5) = 4.25

We see that the sale price is $4.25.

Another type of mathematical model is a **graph**. As we just said, most mathematical models are expressed in the form of an equation. Equations can be graphed, so it makes sense that another type of mathematical model would be a graph. For example, we could illustrate the sale prices of store items on a graph, where the *y*-axis is the sale price, and the *x*-axis is the original price of an item.

We can determine the sale price of an item by locating its original price along the *x*-axis and then finding the corresponding *y*-value, or sale price, on the graph. As shown on the graph, if an item has an original sale price of $5, then the corresponding sale price is $4.25, which is what we expected based on our findings from the equation. A graph is another tool, or mathematical model, that we can use to understand real-world scenarios.

Though equations and graphs are the most common types of mathematical models, there are other types that fall into this category. Some of these include pie charts, tables, line graphs, chemical formulas, or diagrams.

A **chemical formula** is an expression that specifies the types and numbers of atoms present in a molecule. It can take on the following form:

- (Atom abbreviation)
*N*, where*N*is the number of the atoms in the molecule.

A common example of a chemical formula would be the one for water, or *H*2*O*. As the oxygen atom has no subscript, there is only one of that type of atom. Therefore, this formula indicates that one molecule of water contains two hydrogen atoms (*H*2) and one oxygen atom (*O*). Some other common chemical formulas are shown in the following table:

Chemical Name | Common Name | Number and Type of Atom |
---|---|---|

NaCl | Table salt | 1 sodium (Na), 1 chlorine (Cl) |

C2 H6 O | Alcohol | 2 carbon (C2), 6 hydrogen (H6), 1 oxygen (O) |

NH3 | Ammonia | 1 nitrogen (N), 3 hydrogen (H3) |

CO2 | Carbon dioxide | 1 carbon (C), 2 oxygen (O2) |

A chemical formula is a good example of a mathematical model that might have escaped consideration. The formulas themselves don't look like what we're used to seeing in mathematics, but they are still a mathematical way of modeling molecules.

**Mathematical models** are tools we can use to approach real-world situations mathematically. While there are many types of mathematical models, the most common one is the equation. We can model a given scenario with an **equation**, or formula, and use that equation to answer different questions about the scenario. Another common mathematical model is a **graph**, which can be used to model different scenarios in the same way we use equations.

Some lesser-known mathematical models, but still equally important, are pie charts, diagrams, line graphs, chemical formulas, or tables, just to name a few. We can use a **chemical formula** to specify the types and numbers of atoms present in molecules we find in our everyday lives, such as *H*2*O*.

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