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Lower East Side International Community School
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HARLEM INTERNATIONAL COMMUNITY SCHOOL
Founded 1976, Wallie Simpson, Founder and Principal
6TH Grade Mathematics [Arithmetic 6]

TABLE OF CONTENTS
  • MAIN OBJECTIVES
  • DIVISIBILITY, FACTORIZATION
    |--Multiples and factors
    |--LCM and GCF
  • RATIO AND PROPORTION
    |--Solving Problems Involving ratios
    |--Percentage Problems
    |--Similar Polygons
  • PROBABILITY
    |--Evaluating Simple Probability
    |--Counting Principle and Sample Space
    |--Probabilities With Two or More Activities
  • ALGEBRA, EQUATIONS AND RELATIONSHIPS
    |--Algebraic Expressions
    |--Solving Simple Equations
    |--Real-Life Relationships
  • GEOMETRY
    |--Geometric Constructions
    |--Perimeter
    |--Solid Figures
    |--Faces, Edges, and Vertices
  • REFERENCES
    MAIN OBJECTIVES

    Required Textbook: A Beka Arithmetic 6, Work-Text, Pensacola, Florida, current edition.

    [b] Make sense of numbers.
    [i] Compare and order positive and negative fractions, decimals, and mixed numbers. Solve problems involving fractions, ratios, proportions, and percentages.
    [ii] Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
    [iii] Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a/b, a to b, a:b ).
    [iv] Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/ 21, find the length of a side of a polygon similar to a known polygon).
    [v] Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips.
    [vi] Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain the meaning of multiplication and division of positive fractions.
    [vii] Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).

    [c] Learn to write verbal expressions and sentences as algebraic expressions and equations, evaluate algebraic expressions, solve simple linear equations, and graph and interpret their results.
    [i] Write and solve one-step linear equations in one variable.
    [ii] Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process.
    [iii] Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches).
    [iv] Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity.
    [v] Solve problems involving rates, average speed, distance, and time.

    [d] Algebraic description of geometric patterns.
    [i] Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = 1/2bh, C = πd - the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively).
    [ii] Deepen your understanding of the measurement of plane and solid shapes and use this understanding to find their circumference and area.
    [iii] Learn to use the formulas for the volume of triangular prisms and cylinders (area of base x height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.
    [iv] Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms.
    [v] Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.

    [e] Mathematical Reasoning: making decisions about how to approach problems.
    [i] Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.
    [ii] Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.
    [iii] Determine when and how to break a problem into simpler parts.
    [iv] Use estimation to verify the reasonableness of calculated results.
    [v] Use words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
    [vi] Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.


    1 DIVISIBILITY, FACTORIZATION

    1.1 - Multiples and factors
    Suppose you have to clean your room every 4TH day of the month. To figure out which days you have to clean your room, you multiply 4 by the counting number,
                        |---4---8---12---16---20---24---28---|
                           1x4 2x4 3x4  4x4  5x4  6x4  7x4
    
    You would clean your room on the 4TH, 8TH, 12TH, 16TH, 20TH, 24TH and the 28TH of the month.
    Thus when you multiply, you are using factors.
              3x6 = 18  <-- 3 and 6 are factors of 18
              5x7 = 35  <-- 5 and 7 are factors of 18
    
    Some numbers have only two factors: 1 and the number itself. Numbers with only two factors are called prime numbers.
               5 = 5x1  <-- the factors of 5 are 1 and 5
               7 = 7x1  <-- the factors of 7 are 7 and 1
    
    Numbers that have more than two factors are called composite numbers.
               6 = 6x1     
               6 = 3x2  <-- the factors of 6 are 1, 2, 3 and 6
    
    The numbers 0 and 1 are neither prime nor composite.
    A prime factor is a factor which is a prime number. According to the Fundamental Theorem of Arithmetic all composite numbers can be factored into one particular group of prime numbers. This group of prime numbers can be found by various methods.
    Prime Factoring
    2 | 48                 [1] Divide by the smallest prime number, which is 2
      ----|                    until the result is not divisible by 2.
        2 | 24
          ----|            [2] When the numbers is not divisible by two, try another number 
            2 | 12
              ----|        [3] Stop dividing when the quotient is prime
                2 | 6
                  ----|
                    3 | 
     
    The factoring tree
                36         36 can be factored into 6x6       
               /  \
              /    \
             6      6       6 can be factored into 2x3
            / \    / \
           2   3  2   3     all bottom numbers are prime
    The factors of 36 are 2,2,3,3
    
    The factoring tree is made more compact by using factors that are closer together. However, the same factors will be found no matter which factors start the tree.
                36         36 can be factored into 3x12       
               /  \
              /    \
             3     12       3 is prime, 12 can be factored into 2x6
                   / \
                  2   6     2 is prime, 6 can be factored into 2x3
                     / \
                    2   3 
    The factors of 36 are still 2,2,3,3
    
    The answer can be written in exponent form. You might remember that 3 x 3 = 32, the superscript 2 means that you have 2 numbers 3 multiplying each other. If we have 3 x 3 x 3, then the number becomes 33.
    Then we could write,
    2,2,3,3 as the exponents 22, 32



    [1.1.a] James put $50.75 in his bank account every seventh day in June, starting on June 7. How much money did he put in the account during the month of June?

    First let's find out how many times 7 goes into a month, most months have 30 days.
                        |----7---14---21---28---35---|
                            1x7  2x7  3x7  4x7  5x7  
    
    The last multiple of 7 cannot possibly be a day of any month (months do not have 35 days). So the number of days when deposits where made is 4, these days are June 7, 14, 21, and 28.
    Now we multiply the amount of money that went in each day by the number of days when deposits where made,
    ($50.75 per day) x (4 days) = $203.00


    [1.1.b] The prime factors of 64 can be written as 26. This means that the base 2 is used as a factor six times. The superscript 6 is called an exponent. Write the prime factors of 81 using an exponent.

    Let's use the factoring tree to get an idea of what 64 = 26 means.
                64         
               /  \
              /    \
             2      32       
                    / \
                   /   \
                  2     16
                        / \
                       /   \
                      2     8
                           / \
                          2   4
                             / \
                            2   2
     
    
    The factors of 64 are 2,2,2,2,2,2 and we can write this as 26.
    Now let's do the same with 81
                81         
               /  \
              /    \
             9      9       
            / \    / \
           3   3  3   3        
               
    
    The factors of 81 are 3,3,3,3 and we can write this as 34.


    1.2 - LCM and GCF

    The smallest common multiple of two or more numbers is called the lowest common multiple (LCM).
    For example;
    Multiples of 8 are 8, 16, 24, 32, ...

    Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, ...
    Thus, the LCM of 3 and 8 is 24.

    In general:
    To find the lowest common multiple (LCM) of two or more numbers, list the multiples of the larger number and stop when you find a multiple of the other number. This is the LCM.


    Example 1

    Find the lowest common multiple of 6 and 9.


    Solution:

    List the multiples of 9 and stop when you find a multiple of 6.

    Multiples of 9 are 9, 18, ...
    Multiples of 6 are 6, 12, 18, ...

    The LCM of 6 and 9 is 18.

    Example 2

    Find the lowest common multiple of 5, 6 and 8.


    Solution:

    List the multiples of 8 and stop when you find a multiple of both 5 and 6.

    Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, ...

    Stop at 120 as it is a multiple of both 5 and 6.

    So, the LCM of 5, 6 and 8 is 120.

    The Greatest Common Factor (G.C.F.) of two numbers is the largest number that is a divisor of both. It is sometimes called the Greatest Common Divisor. It can be used to simplify (or reduce) fractions.
    For example:
    Find the Greatest Common Factor (G.C.F.) of 6 and 10.

        6 = 2 * 3 You can divide 6 by 2 or by 3

        6 = 1 * 6 You can divide 6 by 1 or by 6

    Therefore 1, 2, 3, and 6 are all factors of six.

        10 = 2 * 5 You can divide 10 by 2 or by 5

        10 = 1 * 10 You can divide 10 by 1 or by 10

    Therefore 1, 2, 5, and 10 are all factors of ten.

    Both 6 and 10 can be divided by 1 and by 2; 2 is greater than 1, so 2 is the Greatest Common Factor (G.C.F.) of 6 and 10.



    2 RATIO AND PROPORTION

    2.1 - Solving Problems Involving ratios
    ...

    3 PROBABILITY

    3.1 - Evaluating Simple Probability
    ...

    4 ALGEBRA, EQUATIONS AND RELATIONSHIPS

    4.1 - Algebraic Expressions
    ...

    5 GEOMETRY

    5.1 - Geometric Constructions

    ...
    5.2 - Perimeter

    The perimeter of a polygon is the distance around the polygon. Here is how to use formulas to find the perimeters of some polygons,

    More Examples:

    Find the perimeters,

          Square A                 Rectangle B                        Triangle C
            
         ..........          .......................                 .
         .        .          .                     .                 . .
         .        . 14 cm    .                     . 17 cm           .   .  5 cm
         .        .          .                     .           4 cm  .     .
         ..........          .                     .                 .       .
            14 cm            .......................                 ...........         
                                     26.5 cm                             3 cm         
    
    Square A:    P = 4s
                 P = 4(14)
                 P = 56 cm
    Rectangle B: P = 2(L + W)
                 P = 2(26.5 + 17)
                 P = 2(43.5)
                 P = 87 cm
    Triangle C:  P = A + B + C
                 P = 4 + 3 + 5
                 P = 12 cm         
    

    ...


    5.3 - Solid Figures
    ...

    REFERENCES

    [1] A Beka Arithmetic 6, Work-Text, Pensacola, Florida.
    [2] Integrated Mathematics Second Edition, Dressler I, Keenan E., AMSCO, New York, NY.
    [3] Foundations of Sequential Mathematics, Lewry D, Ockenga E, Rucker W., HEATH, Toronto, Ontario.