Unit 1

Classifying numbers; identifying opposites; absolute values;  locating & comparing integers; add, subtract, multiply, and divide integers; modeling integer operations; intro to coordinate grids.

In the review of this unit, we will be discussing sets of numbers, integers operations, and integers on the coordinate grid.

Natural Numbers:      {  1, 2,  3,  4,  5, …} or Counting Numbers
Whole Numbers:         {0, 1, 2,  3,  4,  5, …}
Integers:   {… -4, -3, -2, -1,  0, 1,  2, 3,  4,  }
Includes positive numbers and negative numbers.
Also the neutral number zero

   Real Numbers. The set of all numbers that can be represented by points on a number line.

   Rational Number.  Is a real number that can be expressed as the ratio of two integers {n/m}

(Rational numbers are whole numbers, fractions, and decimals )

    Irrational Number.     Is a real number that is not a rational number (it cannot be expressed as the ratio of two integers)

System of Real Numbers from Ms Garcia on Vimeo.

Number Line

On a number line, positive numbers are to the right of zero. Negative numbers are to the left of zero.  Zero is neither positive nor negative.

Opposites. Two integers that lie the same distance from the origin in opposite directions.

For example, “-3” (negative 3) is the opposite of “-3” (positive 3).

   Absolute Value:  A number’s distance from zero on a number line

We can use the number line to compare and order positive and negative numbers.

Going from left to right, numbers increase in value.

Going from right to left, numbers decrease in value.

 

 

 

 

 

 

 

The Number Line from Ms Garcia on Vimeo.

     
Comparing numbers

 

 

Compare Numbers on a Number Line from Ms Garcia on Vimeo.

 

Compare and order Integers

We can compare different integers by looking at their positions on the number line. For any two different places on the number line, the integer on the right is greater than the integer on the left.
Ex. Order 3, 5, -3, 1, -5

Compare

 

 

 

 


Operations with Signed Numbers


Addition. 
  • If the signs of the numbers are the same, ADD. The answer has the same sign.
  • If the signs of numbers are different, SUBTRACT. The answer has the sign of the larger number.
Subtraction. Change subtraction to addition of the opposite number.
Multiplication and Division.   Multiply or  Divide.   
If the numbers have the same sign the answer is POSITIVE.         
If the numbers have different signs the answer is NEGATIVE.

Factors.    
Numbers that are multiplied.
Ex:   
2 * 3 = 6            the 2 and the 3 are factors of 6   
prime number is a whole number greater than 1 whose only factors are 1 and itself. 
 
Prime Factorization.     Do the factor tree
(A diagram used to show the prime factors of a number.)
                                     
 The prime factorization of 24 is  2 * 2 * 2 * 3,
because they are all prime numbers.
Exponent.     A little number that tells you how many times you multiply a number by itself.
                       Ex.        8 = 23          
        
Greatest Common Factor GCF.    Do the factor tree for the given numbers and find what they have in common. 
Ex. Find the GCF of (8,12)
                                    8 = ②*②* 2
                                  12 = ②*②* 3
              
The GCF of (8,12) is 4
Least Common Multiple LCM.    Is the lowest number that can be divided both. Use the answers of the multiplication tables of the given numbers and find the smallest number they have in common.   Ex. Find the LCM of (8, 12)
                                                   8, 16, 24, 32, 40, 48
                                                 12, 24, 36, 48, 60, 72             
the LCM of (8, 12) is 24

Prime Numbers.    Numbers that can only be divided by 1 and itself and have no remainders.
Prime Numbers   =  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 
Composite Numbers.  Numbers that can be divided by more than two factors and have no remainders.    
Composite Numbers =  4, 6, 8, 10, 12, 14, 16, 18, 20, .
Neutral Numbers  These numbers are not prime nor composite:       0   and   1

  

Divisibility Rules

  
Fraction.  A part of a whole     n/m
Decimal. Fraction with a denominator of 10, 100, 1000, …
Repeating Decimal.  Is a decimal fraction in which the digits endlessly repeat a pattern, such as 1/3 = 0.3333333333333  


Terminating Decimal. Is a fraction whose decimal representation contains a finite number of digits.                                         

Place Value.  It’s determined by its position with respect to the decimal point.

Positive, negative and zero exponents with a base of ten                    

10000 = 1 x 10
1000 = 1 x 10³    
100 = 1 x 10²       
10 = 1 x 10         
1 = 10  
1/10 = 0.1 = 1 x 10⁻ᴵ         
1/100 = 0.01 = 1 x 10⁻²    
1/1000 = 0.001 = 1 x 10⁻³

1/10000 = 0.0001 = 1 x 10⁻⁴

Scientific notation

For example, instead of writing 65,000,000, we write 6.5 x 10

Comparing and Ordering decimals.     

Align decimal points and place values,  write a place keeper if needed.    (Pretend it’s money)

The Coordinate Plane from Ms Garcia on Vimeo.

Cartesian_Hlp



You can find more information on the following pages:

Integers

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Understandings:

  • Know the Real Number System (know what relationships exist between the various subsets of rational numbers).
  • Plotting numbers on number lines.
  • Comparing and ordering numbers.
  • Fluently, add, subtract, multiply, and divide rational numbers.

Vocabulary:

  • Divisor, dividend, quotient, factor, product, sum, difference, numerator, denominator, subsets, sets, real numbers, rational numbers, irrational numbers, whole numbers, integers, terminating decimals, nonterminating decimals, repeating decimal, Venn diagram.
  • Bar notation, common denominator, least common denominator, like fractions, rational numbers, repeating decimal, terminating decimal, unlike fractions.

Learning Targets:

  • The Number System.
  • Compute fluently with multi-digit numbers and find common factors and multiples.
  • Fluently add, subtract, multiply and divide multidigit decimals
  • Locate and plot rational numbers on a number line (horizontal and vertical) and a coordinate plane.
  • Graph points in all four quadrants of a coordinate plane
  • Find distances between points using my knowledge of coordinates and absolute value.
  • Draw polygons in the coordinate plane.
  • Identify the length of a side using coordinates.
  • Solve real-world problems involving coordinate planes.
  • Graph points in all four quadrants of a coordinate plane.
  • Find distances between points using my knowledge of coordinates and absolute value.
  • Find the greatest common factors of two whole numbers (up to 100)
  • Find the least common multiple of two whole numbers (less than or equal to 12)