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NY Regents Exam - Integrated Algebra: Help and Review25 chapters | 272 lessons | 12 flashcard sets

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Lesson Transcript

Instructor:
*Karin Gonzalez*

Karin has taught middle and high school Health and has a master's degree in social work.

In this lesson, you will learn the definition of an open sentence and will be given many examples to illustrate this concept. Following this lesson will be a quiz to test your new knowledge and skills.

You were probably quite young when a teacher taught you about sentences and you learned that you needed a subject and a verb to make a sentence, like in these examples:

'The baby laughed.'

'The ocean is blue.'

In math, there are similar sentences with similar rules. A **mathematical sentence** makes a statement about two expressions. The two expressions either use numbers, variables, or a combination of both. A mathematical sentence can also use symbols or words like equals, greater than, or less than.

An **open sentence** in math means that it uses variables, meaning that it is not known whether or not the mathematical sentence is true or false.

A **closed sentence**, on the other hand, is a mathematical sentence that is known to be either true or false.

Before we look at open sentences, let's look at some examples of true closed sentences in math:

1. 9 is an odd number.

2. 4 + 4 = 8

3. 10 - 1 = 9

4. 6 - 6 = 0

5. The square root of 4 is 2.

If you notice, all of these sentences are true. Because we know they are true, they are closed sentences!

Let's look at some examples of false closed sentences in math:

1. 9 is an even number.

2. 4 + 4 = 5

3. 10 - 1 = 8

4. 6 - 6 = -1

5. The square root of 4 is 1.

Wow, it definitely did not feel right saying those mathematical sentences. Why? Because these mathematical sentences are completely false. We know that they are wrong. Therefore, they are false closed mathematical sentences.

We get open sentences when we use variables in mathematical sentences.

Here are some examples:

1. The obtuse angle is N degrees.

2. X + Y = 5

3. 4 - 3 = X

4. A square has N sides.

5. 5 + Y = X

In the examples, the mathematical sentences or expressions may or may not be true depending on the values of the variables X, Y and N. These mathematical sentences simply lack the proper knowledge needed to know whether or not the expressions are true or false.

Let's take each example one at a time.

Example 1: The obtuse angle is 'n' degrees.

We have no way of knowing whether or not this sentence is true without more information. Even if we are told the value of N = 110 degrees, we need more information. In this case, a diagram would be sufficient as a means of more information.

Because it is a common geometric fact that the three angles of a triangle equal 180 degrees, seeing that the other two angles of 25 and 45 equal 70 degrees, an obtuse angle of 110 degrees makes complete sense because it equals 180 degrees when you add the three angles together.

Example 2: x + y = 5

If we are told that X = 4 and Y = 2, would this open sentence be true?

X + Y = 5

4 + 2 = 5 ?

No, it's false!

The second statement is considered a false closed sentence.

Example 3: 4 - 3 = x

If we are given x = 1, is this mathematical sentence true?

4 - 3 = X

4 - 3 = 1 ?

Yes, it is true!

Example 4: a square has 'n' sides

Since we know that a square has to have 4 sides, any value of N other than 4 would be false.

Example 5: 5 + y = x

If we are given x = 7 and y = 2, is this open sentence true or false?

5 + y = X

5 + 2 = 7

Yes, this mathematical sentence is true!

Remember the following important facts from the lesson:

- A
**mathematical sentence**makes a statement about two expressions. - A
**closed sentence**is a mathematical sentence that is known to be either true or false. - An
**open sentence**in math means that it uses variables and is not known whether or not the mathematical sentence is true or false. - When given the values of the variables, it is possible to plug them in to determine if the open sentence is true or false.

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NY Regents Exam - Integrated Algebra: Help and Review25 chapters | 272 lessons | 12 flashcard sets

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