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Explorations in Core Math - Grade 6: Online Textbook Help10 chapters | 118 lessons

Instructor:
*Michael Eckert*

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

We can make an expression (or rule) from a table of ordered pairs. Given a sequence of numbers as ordered pairs, we can find an expression to represent this sequence. Furthermore, we can use such expressions to represent numerical relationships in real-world problems.

Let's say that we have 2 runners. Let's say that they run at the same pace or speed. If the 1st runner gets a 1-mile head start on the other, can we create an **expression** or *rule* to represent their positions in relation to each other through any number of miles?

Before we can find an expression to represent the 2 runners' positions in relation to one another, we should define what an **ordered pair** is. An ordered pair is essentially two numbers that are paired together in an ordered **sequence** with other pairs of numbers. Note that we call it a sequence, because these pairs follow a set pattern or rule. For example, we might have a sequence of ordered pairs as follows:

In this table, the 1st column gives us our *position* of a *n term* in a sequence of numbers. The 2nd column represents a *value* in another sequence of numbers, dependent on the numbers in the 1st column. Each row of two numbers in this table is an ordered pair. Looking at the sequence, we see that for every increase of 1 in the 1st column, there is an increase of 4 in the 2nd column.

We can develop an expression to represent the relationship between these rows or ordered pairs. Again, for every increase of 1 in the 1st position column, there is an increase of 4 in the 2nd value column, which means that whatever the number is in the 1st column, it gets multiplied by 4; therefore, we can say that whatever number *n* represents, it will be multiplied by 4.

For example, in the 5th row, we can say that *4n* equals 20 or *4n = 20*. Note our expression is *4n*. To find out what *n* equals in the 5th row down, we can divide both sides of *4n = 20* by 4: *4n / 4 = 20 / 4*, which is *1n = 5* or *n = 5*. In other words, the value of *n* in the 1st column of the 5th row is 5. If we wished to check to see if our expression *4n* is true for any position, we can place any number in the 1st column into n and we will get its match for its ordered pair on the right.

For instance, when *n = 1*, 4(1) = 4. When *n = 2*, 4(2) = 8, etc. In other words, in row 1, we have an ordered pair of 1 and 4. In row 2, we have an ordered pair of 2 and 8. Now we know that our expression is true!

These pairs of numbers in the last table might represent some real-world relationship -now refer to the table below. For instance, the numbers in the 1st column might represent some number of cars, while the numbers in the 2nd column might represent the total number of wheels on those total number of cars. Remember that each row in the table below represents an ordered pair or a relationship.

1 car has 4 wheels. 2 cars have 8 wheels and so on. Using the expression *4n* that we found earlier, we have a rule to represent the total number of tires that we need for a total n number of cars. We can say that for every *n* cars, there are 4 wheels; therefore, if we have *n = 5* cars and *4n = 20*, then 4 wheels (5 cars) = 20 wheels. In other words, 5 cars have 20 wheels. Note that this expression *4n* will hold true for any number we put in for *n*. So, for any number of cars, we can determine a total amount of tires!

Since we have learned what an expression is and how it may be used for a real-world problem, we can look back at our very first problem with the runners:

Say that we have 2 runners. Suppose that they run at the same pace or speed. Suppose that the 1st gets a mile head start over the other. We would have the following sequence of ordered pairs to represent where the runners are with respect to one another:

The 1st column represents the position of the 1st runner, and the 2nd column represents the value of the 2nd runner's position in relation to the 1st runner. For instance, as the 1st runner is at mile 1, the 2nd is at mile 0. As the 1st is at 2 miles, the 2nd is at 1, and so on. We note again that the 2nd runner is always behind the 1st by 1 mile, so when the 1st runner is at *n* miles, the 2nd is at *n - 1* miles; therefore, *n - 1* gives us the expression or rule to represent the value of the position of the 2nd runner in relation to the 1st.

So, wherever the 1st runner is at position *n*, the 2nd runner is always at *n - 1*. Notice that when the 1st runner is at *n = 5* miles, the 2nd runner is at *n - 1* miles = 4 miles.

With this expression, we can determine both the runners' positions at any point! When the 1st runner has hit the *n = 10*-mile mark, the position of the 2nd will be n - 1 miles = 10 miles - 1 miles = 9 miles.

Like their name suggests, ordered pairs are ordered, as they are part of a larger sequence of numbers. For any sequence or list of numbers that follow a set pattern in a table, we can form an expression that gives the value of a number in the 2nd column in relation to the position of a number in the 1st column. This 1st and 2nd number form an ordered pair in the overall sequence. Furthermore, we can create an expression so that whenever we are given a number positioned in the 1st column, we can find its corresponding match for / in its ordered pair in the 2nd column. Lastly, we can create such expressions to solve real world problems.

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Explorations in Core Math - Grade 6: Online Textbook Help10 chapters | 118 lessons

- What is a Variable in Algebra? 5:26
- Evaluating Simple Algebraic Expressions 7:27
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- Solving One-Step Linear Inequalities 7:08
- Go to Explorations in Core Math Grade 6 - Chapter 2: Introduction to Algebra

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