Multiplication Of Real Numbers

Algebra 1 Ch 2.5 – Multiplication of Real Numbers

Objective 

Students will multiply real numbers

Rules for Multiplying 

The rules for multiplying real numbers are:  

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If the signs are the same the answer is positive If the signs are different the answer is negative.

Let’s look at some examples…

Example #1 – Same Signs 

Multiply

3●3

In this instance both factors are positive so your answer will be positive

3●3 = 9

Example #2 – Same Signs 

Multiply

- 3 ● (- 5)

In this instance both factors are negative so your answer will be positive

- 3 ● (- 5) = 15

Example #3 – Different Signs 

Multiply

5 ● (- 8)

In this instance the factors have different signs. The 5 is positive and the 8 is negative – your answer will be negative

5 ● (- 8) = - 40

Example #4 – Different Signs 

Multiply

-6 ● 3

In this instance the factors have different signs. The 6 is negative and the 3 is positive – your answer will be negative

- 6 ● 3 = - 18

Comment 

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Of course the examples that were used were pretty simple and you could do the math in your head… A strategy that you can use here to make you life easy is to chunk the problem…that is 1. 2.

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Multiply the numbers first Then evaluate for the sign

Let’s look at an example of what I mean…

Chunking 

Multiply

- 10 ● 4

In this instance the factors have different signs. 1. Multiply 10 ● 4 first to get the total of 40

10 ● 4

= 40

2. Then evaluate for the signs – in this instance the signs are different so your answer will be negative

- ● +

= -

The solution to the problem is: 40

-10 ● 4 = -

Comment 

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Make no mistake about it…we will use this concept throughout this course… It is expected that you master the rules of multiplying real numbers… If the sign is wrong…the whole problem is wrong….As a rule I do not give partial credit…

Properties of Multiplication 

There are 5 properties of multiplication     

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Commutative Property Associative Property Identity Property Property of Zero Property of Opposites

You are probably already familiar with some of these properties…However, in this lesson we will give you the appropriate mathematical name for the property and show you what it looks like… Let’s look at each one individually…

Commutative Property The commutative property states: “ The order in which two numbers are multiplied does not change the product” This property can be expressed as follows: Algebraically: a●b=b●a Example: 3 ● (-2) = (-2) ● 3 -6 = -6 

Associative Property The associative property states: “ The way you group three numbers when multiplying does not change the product” This property can be expressed as follows: Algebraically: (a ● b) ● c = a ● (b ● c) Example: (-6 ● 2) ● 3 = -6 ● (2 ● 3) -12 ● 3 = -6 ● 6 - 36 = - 36 

Identity Property The identity property states: “The product of a number and 1 is that number” This property can be expressed as follows: Algebraically: a●1 =a Example: - 4 ● 1 = -4 

Property of Zero The property of zero states: “The product of a number zero is zero” This property can be expressed as follows: Algebraically: a●0=0 Example: 5●0=0 

Property of Opposites The property of opposites states: “The product of a number and – 1 is the opposite of the number” This property can be expressed as follows: Algebraically: a ● (-1) = -a Example: -1 ● (-3) = 3 

Using the rules of multiplication 

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The purpose for reviewing the rules and properties of multiplication is so that you know how to use them when solving problems… When solving problems you will use the process that we learned in an earlier lesson… That is…   

Write the problem Substitute Simplify

Example 

Evaluate the expression when x = -3 5(x – 4) 1. Write the problem 2. Substitute

5(x – 4) 5(-3 – 4)

3. Simplify

5(-7) -35

Analysis 

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In the previous example you should have analyzed the problem first to know what to do to solve it… Let’s take a look at what you were expected to know…

Analysis 

Evaluate the expression when x = -3 5(x – 4)

In this problem you were expected to know: • To substitute -3 for x to get 5(-3 – 4) • The order of operations tells you to evaluate the parenthesis first • The rules for subtracting real numbers states that you add the opposite and follow the rules for adding • In this case, since the signs are the same you add and keep the signs • After simplifying the parenthesis you were to multiply • The rules for multiplying state that if the signs are different the answer is negative

Comments 

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As you can see in the previous example, solving the problem involved more then the concepts that we just reviewed… Often it is not explicitly stated what you need to do…throughout this course you will be expected to analyze problems and then solve them based upon what you have learned… That is why I require that you show your work…so that we can go back and problem solve when you don’t get the correct solution…

Comments 

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On the next couple of slides are some practice problems…The answers are on the last slide… Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error… If you cannot find the error bring your work to me and I will help…

Your Turn  2. 3. 4.

 7. 8.

Find the product (-8)(3) (20)(-65) 3I (-15) F GJ

H5K

Simplify the variable expression (-3)(-y) 5(-a)(-a)(-a)

Your Turn      

Evaluate the expression: -8x when x = 6 3x2 when x = -2 -4(|y – 12|) when y = 5 -2x2 + 3x – 7 when x = 4 9r3 – (- 2r) when r = 2

Your Turn Solutions     

-24 -1300 -9 3y -5a3

1. 2. 3. 4. 5.

-48 12 -28 -27 76

Summary 

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A key tool in making learning effective is being able to summarize what you learned in a lesson in your own words… In this lesson we talked about multiplying real numbers and the properties of multiplication… Therefore, in your own words summarize this lesson…be sure to include key concepts that the lesson covered as well as any points that are still not clear to you… I will give you credit for doing this lesson…please see the next slide…

Credit 

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I will add 25 points as an assignment grade for you working on this lesson… To receive the full 25 points you must do the following: 

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Have your name, date and period as well a lesson number as a heading. Do each of the your turn problems showing all work Have a 1 paragraph summary of the lesson in your own words

Please be advised – I will not give any credit for work submitted:   

Without a complete heading Without showing work for the your turn problems Without a summary in your own words…