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PROPERTIES OF NUMBERS

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					FSD Lesson Plan Template Lesson Title Properties of Numbers 7th Grade Pre-Algebra Grade Level/Cours e Standard Algebra and Functions 1.3 Addressed Simplify numerical expressions by applying properties of rational numbers (e.g. identity inverse, distributive, associative, commutative) and justify the process used. Textbook Chapter Goal Chapter 3 Holt Mathematics Course 2 Pre-Algebra Multi-Step Equations and Inequalities Learn number properties to include commutative property of addition/multiplication, associative property of addition/multiplication, Identity of addition/ multiplication, inverse of addition/ multiplication. OH of lesson handouts, white boards, markers, overhead tiles

Materials for Teacher Materials Lesson handouts, white boards, markers, square tiles for Students Description Lesson Description 1. Opening Activity Teacher begins with examples leading to the development of basic properties of numbers, beginning with identity element of addition. Students will be encouraged to develop the words for the properties themselves rather than just being offered the definitions. 2. Student Problem Solving Students will first develop basic number properties using number examples and later may explore some non-numerical examples in order to focus on and gain better understanding of “the properties”. 3. Teacher as Facilitator The teacher will direct students using traditional numerical examples encouraging students to develop the definitions of the properties and then some non-traditional examples may be explored. Students and teachers may find these non-traditional examples challenging. Teachers will determine whether to incorporate this part of the lesson into the following week’s lesson or to consider them merely as personal enlightenment! Assessment Eight-question “fill in” assignment at end of class. Reflection/ This lesson will focus on a clear and solid foundation of basic number properties. This will pave the way for future discussions that will later Looking include more properties such as distributive and closure. Ahead

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I am thinking of a number that I can add to 12 without changing the identity of the number. What number am I thinking of?

12 +

= 12

and

+ 12 = 12

Let’s give this number a name. What shall we call this number?

I am thinking of a number that I can multiply 12 by without changing the identity of the number. What number am I thinking of?

12 ·

= 12

and

· 12

= 12

What shall I call this number?

Identity Property of Addition and Multiplication
a + 0 = a and 0 + a = a so 0 is called the identity element of addition a  1 = a and 1  a = a so 1 is called the identity element of multiplication

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Is there an identity element for subtraction? Please explain.

I am thinking of a number that I can add to 17 to get the identity element of addition. Can you help me to find such a number?

17

+

=

0

Do the same for - 10: (-10) + = 0

Additive Opposites The sum of an integer and its opposite is zero. Numbers that are opposites of each other are sometimes called additive inverses. x + (-x) = 0 (-x) + x = 0

Note: later we will try to find the multiplicative inverse for any number!
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When I get up in the morning, I put on my socks and then my shoes. Do any of you put on your shoes and then your socks? Why not?

There is a property of addition that does not have this problem! When I add 2 + 4, I get 6. How about when I add 4 + 2? Isn’t it nice addition is not like putting on your socks and shoes? Does anyone remember what this property of addition is called? Can you think of another operation that has this type of behavior? Commutative Property of Addition and Multiplication
a+b=b+a ab=ba

This property is obvious when using numbers like 2+3=3+2. Let’s try it with a new operation which I will define for you. Suppose I have an operation called star,  ,and it works like this:  means “select the first of the two numbers.” For example: 4*8=4 (-3) * (-10) = -3 a*4=a Can you tell me if the operation star has the commutative property? Please explain your answer.

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You are familiar with the multiplication table. Fill out the multiplication table below:

x 0 1 2 3 4 5 0 1 2 3 4 5

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Here is a table using an operation that you may not be familiar with. I am pretty sure since I am making it up! The operation is called spiral and here is the table for spiral:

QuickTime™ and a TIFF (Uncompressed) decompressor are need ed to see this picture.

Can you interpret this table and operation spiral. Try the following problems for fun!

QuickTime™ and a TIFF (Uncompressed) decompressor are need ed to see this picture.

QuickTime™ and a TIFF (Uncompressed) decompressor are need ed to see this picture.

= =
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Can you tell me if operation spiral is commutative? Please explain!

Let’s go back to addition. Draw a line segment connecting expressions in the left hand column with those in the right hand column that make use of the commutative property for addition.

x · (y + z) (y · z) +x (y + z) + x

x + (y · z) (z + y) + x x · (z + y)

Now show the connectors for the expressions linked using the commutative property for multiplication.

x · (y + z) (y + z) · x (y · z) + x

x · (y + z) (y + z) · x (z · y) + x

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Recall the “order of operations.” Let’s try some problems. In the first we will “associate” the first two numbers, the 2 and the 3. This will be indicated by the parenthesis. Which two numbers are “associated” in the second problem? Show the steps for solving these problems. (2 + 3) + 5 2 + (3 + 5)

Notice the final result is 10 for both cases. Do you think this will always be the case? Please explain. What operation is being used? What shall we call this property? If we consider the English language some word triples nicely illustrate the associative property. Which ones below do you think illustrate or make you think of the associative property? BIG RED APPLE HIGH SCHOOL STUDENT BROWN SUEDE JACKET BARE FACTS PERSON Of course not all of you may agree on which ones are good illustrations of the associative property but with math there is no argument!
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Challenge: Go back to the star operation and tell me if operation “star” is associative. Next go back to the spiral table and tell me if “spiral” is associative. How do we check this out?

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Assessment

Name

State which property of numbers is being illustrated. There may be more than one property used! Be sure to state the property and the operation being used! 1] (a + b) + (c + d) = (c + d) + (a + b) 1]____________

2] (a x b) + (c x d) = (a x b) + (d x c)

2]____________

3] (c + d) + (a + b) = c + (d + (a + b)) 4] a  b + w = w + a  b
c c

3]____________ 4]____________ 5]____________

5] (a + b) + c = a + (b + c)

6] x + y + -(x + y) = 0 z z 7] x + y + -(y + x) = 0 z z 8] (a (b + c)) (f) = a((c + b) (f))

6]____________

7]____________

8]____________
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