Section 1.5

Open Sentences

 


What you should learn

To solve open sentences by performing arithmetic operations

NCTM Curriculm Standards 2, 6 - 9

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Open Sentences

Solution

Replacement set

Set

Element

Solution set

Equation

Inequality

 

 

 

Introduction: The grams of fat found in cheeseburgers sold by various fast food chains are listed in the stem-and-leaf plot below.

 

Obviously, some of the fast food chains are selling burgers with more than the usual amount of fat. Some students at Middletown High School have formed a consumer awareness clud and are concerned about the high content of fat in this particular cheeseburger. They decide to write a letter to the company recommending a way to reduce the fat of their cheeseburger to 37 grams.

 

The students know that a one-onuce slice of American cheese has about 7 grams of fat and that ground beef has about 6 grams of fat per ounce. The students need to determine the number of ounces of ground beef that can be used to make a cheeseburger with a fat content of 37 grams.

 

Let b represent the number of ounces of beef that will go into the cheeseburger. This problem can be represented by the equation below.

6b + 7 = 37

The number 6 represents the amount of fat in one ounce of ground beef. The 7 represents the number of grams of fat in one slice of cheese. The 37 represents the total number of grams of fat for a cheeseburger.

 

Mathematical statements with one or more varibables, or unknown numbers, are called open sentences. An open sentence is neither true nor false until the variable has been replaced by a value. Finding a replacement for the variable that results in a true sentence is called solving the open sentence. This replacement is called a solution of the open sentence.

 

Replace b in 6b + 7 = 37 with the values 3, 4, 5, and 6. Then see whether each replacement results in a true or false sentence.

 Replace b with:  6b + 7 = 37  True or False?
 3  6(3) + 7 = 18 + 7 = 25  False
 4  6(4) + 7 = 24 + 7 = 31  False
 5  6(5) + 7 = 30 + 7 = 37  True
 6  6(6) + 7 = 36 + 7 = 43  False

Since b = 5 makes the sentence 6b + 7 = 37 true, the solution for 6b + 7 = 37 is 5. The students can write a letter suggesting that the chain reduce the amount of fat to 37 grams by using 5 ounces of ground beef in their cheeseburger.

 

A set of numbers form which replacements for a variable may be chosen is called a replacement set. A set is a collection of objects or numbers. Sets are often shown by using braces { }. Each object or number in a set is called an element, or member. Sets are usually named by capital letters. Set A has three elemetns; they are 1, 3, and 5.

A = {1, 3, 5}

B = {2, 4, 5}

C = {1, 2, 3, 4, 5}

The solution set of an open sentence is the set of all replacements for the variable that make the sentence true.

 

 

 

Exercise 1: Find the solution set for y + 5 7 if the replacemetn set is {0, 1, 2, 3, 4}.

 Replace y with: y + 57   True or False?
 0 0 + 5 = 5   True
 1  1 + 5 = 6  True
 2  2 + 5 = 7  True
3  3 + 5 = 8   False
4   4 + 5 = 9  False

Therefore, the solution set for y + 5 7 is {0, 1, 2}.

 

A sentence that contains an equals sign =, is called an equation. A sentence having the symbols <, >, or is called an inequality. Which of the open sentences below are equations? Which ones are inequalities?

 Mathematical Sentence  Equation or Inequality?
 2x + 10 = 50  Equation
 3a 43 Inequality 
 y - 6 > 12  Inequality

Sometimes you can solve an equation by simply applying the order of operations.

 

 

 

Exercise 2: Solve

 

 

 

Exercise 3: Refer to the application at the beginning of the lesson. solve for the number of grams of fat f in a cheeseburger that has one slice of cheese and found ounces of ground beef. Assume that the bun and toppings contain no fat.

 

 

 

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

 

 

 

Homework: The homework to be assigned for tonight would be: 13 - 33 odd, 35 - 40

 

Alternative Homework: Enriched: 14 - 30 even, 31 - 40

 

Extra Practice: Students book page 757 Lesson 1-5

 

Extra Practice Worksheet: Click Here.

 

 

 


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