Algebra 1

1
Algebra 1

• Ch 2.5 Multiplication of Real Numbers

2
Objective

• Students will multiply real numbers

3
Rules for Multiplying

• The rules for multiplying real numbers are
• If the signs are the same the answer is positive
• If the signs are different the answer is
  negative.
• Lets look at some examples

4
Example 1 Same Signs

• Multiply 3 ? 3

In this instance both factors are positive so
your answer will be positive
3 ? 3
9

5
Example 2 Same Signs

• Multiply - 3 ? (- 5)

In this instance both factors are negative so
your answer will be positive
- 3 ? (- 5)
15

6
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

7
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

8
Comment

• 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 problemthat is
• Multiply the numbers first
• Then evaluate for the sign
• Lets look at an example of what I mean

9
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 -10 ? 4 -
40
10
Comment

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

11
Properties of Multiplication

• There are 5 properties of multiplication
• Commutative Property
• Associative Property
• Identity Property
• Property of Zero
• Property of Opposites
• You are probably already familiar with some of
  these propertiesHowever, in this lesson we will
  give you the appropriate mathematical name for
  the property and show you what it looks like
• Lets look at each one individually

12
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

13
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

14
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

15
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

16
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

17
Using the rules of multiplication

• 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

18
Example

• Evaluate the expression when x -3
• 5(x 4)

1. Write the problem
5(x 4)
5(-3 4)
2. Substitute
3. Simplify
5(-7)
-35
19
Analysis

• In the previous example you should have analyzed
  the problem first to know what to do to solve it
• Lets take a look at what you were expected to
  know

20
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

21
Comments

• 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 dothroughout 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 workso
  that we can go back and problem solve when you
  dont get the correct solution

22
Comments

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

23
Your Turn

• Find the product
• (-8)(3)
• (20)(-65)
• (-15)
• Simplify the variable expression
• (-3)(-y)
• 5(-a)(-a)(-a)

24
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

25
Your Turn Solutions

1. -24
2. -1300
3. -9
4. 3y
5. -5a3

• -48
• 12
• -28
• -27
• 76

26
Summary

• 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
  lessonbe 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
  lessonplease see the next slide

27
Credit

• 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
• 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