Welcome to BADM 51 -- Ms. Abbott -- sabbott@bcconline.com -- Barstow College

Lesson #1

Whole Numbers: How to Dissect and Solve Word Problems

Fractions

1. Read: Chapters 1&2

2. Homework :

Chapter 1 pp. 21 - 30 even problems

Chapter 2 pp. 53 – 61 even problems

3. Review key terms and end of chapter study guide

4. Post solution for one problem from each chapter (see syllabus for example)

5. Click here to take the Quiz

 Chapter 1

Whole Numbers: How to Dissect and Solve Word Problems

Learning Objectives:

Students should be able to:

  • Use place values to read and write numeric and verbal whole numbers.
  • Round whole numbers to the indicated position.
  • Add whole numbers; check and estimate additions computations.
  • Subtract whole numbers; check and estimate subtraction computations.
  • Multiply whole numbers; check and estimate multiplication computations.
  • Divide whole numbers; check and estimate division computations.
    •

    Lecture Notes:

    Reading Whole Numbers

    The value of each digit in a number is determined by using a place value chart. Most of us by this point have this chart committed to memory. Key things to remember when reading whole numbers:

  • Commas separate every three digit moving from the right to the left

  • Do not read zeros or use the word "and" when reading or writing whole numbers

  • Hyphenate numbers twenty-one to ninety-nine

  • Read numbers from left to right reading each group of three digits as if they were alone and then add the group name (trillion, billion, million, or thousand) at the end.

    • For example: 1,606,456,333,679 is read

    One trillion, six hundred six billion, four hundred fifty-six million, three hundred thirty-three thousand, six hundred seventy-nine.
  • Convert part of million, billion, etc….to regular whole number by dropping the decimal, insert comma and add zeros until desired group is reached

    • For example:

    12.3 million 12,300,000

    2.5 billion 2,500,000,000

    Rounding Whole Numbers

    Many of the business numbers you will see on financial reports and government documents are rounded numbers. Rounded numbers are used because they are easier to use to estimate, to change/update and to remember.

  • Step 1. Identify the digit you want to round

  • Step 2. If the number directly to the right of the number you want to round is five or greater, increase the number you are rounding by one (round up). If the digit to the right is less than five do not change identified digit (round down).

  • Step 3. Change all digits to the right of the rounded digit to zeros.

    •

    For example: Round 3,456 to the nearest thousand

    Step 1. Identify digit 3,456

    Step 2. Round down in this case 3,456

    Step 3. Change all digits to right to zeros 3,000 ANSWER

    To round all the way, just round to the first digit of the number and have only one non-zero digit in number.

    Adding Whole Numbers

  • Align numbers according to their place value

  • Add units columns. Write the sum below the column. If sum is more than nine, write the units digit and carry the tens digit.

  • Move left, repeating step above until all place values are added

  • Check all your answers starting at right column, add up and then add down for next column ( see page 10 for examples)

    For example:

    2 1 2   

    1,456
    707
    + 5,998
    8,161

     

    • Check bottom to top and note the 2 1 2 are the numbers carried.

    Subtracting Whole Numbers

  • Align the minuend (larger # in subtraction) and the subtrahend (smaller # in subtraction) by place values.

  • Subtract unit digits from right to left. If necessary borrow one from tens digit in minuend

  • Moving left, repeat above step until all place values are subtracted.

  • Examples on page 12

    •

    Multiplying Whole Numbers

  • Step 1. Align multiplicand (top # we want to multiply) and multiplier (# doing the multiplying) at the right.

  • Step 2. Begin at right and keep multiplying. Align first partial product at right with multiplicand and multiplier.

  • Step 3. Move left through multiplier and continue multiplying multiplicand. Partial product right digit or first digit is placed directly below second digit in multiplier.

  • Step 4. Repeat Steps 2 and 3 until multiplication is complete. Add partial products to get final product.

  • Examples on page 12-14

    ***Remember*** multiplication is a short cut to addition.

    • Dividing Whole Numbers

    **Remember** division is the reverse of multiplication.

    Dissecting Word Problems

    For some students, the mention of word problems creates immediate anxiety. The key to becoming proficient at word problem includes persistence and having a plan of attack. Your book uses a tool they refer to as a Blueprint Ad (p. 8). This tool may help you to organize you thoughts, which is the first step when solving word problems. The second is practice, practice, practice and more practice.

    Things to think about when solving word problems:

     Chapter 2

    Fractions

    Learning Objectives:

    Students should be able to:
  • Recognize the three types of fractions.
  • Convert improper fractions to whole or mixed numbers and mixed numbers to improper fractions.
  • Convert fractions to lowest and highest terms.
  • Add and subtract like and unlike fractions.
  • Find the least common denominator by inspection and prime numbers.
  • Add and subtract mixed numbers with the same or different denominators.
  • Multiply and divide proper fractions and mixed numbers.
  • Use the cancellation method in the multiplication and division of fractions.
    •

    Lecture Notes:

    Fraction Anatomy

    3 Numerator – top number of the fraction (part of the whole)
    4 Denominator – bottom number of the fraction (the whole)

    Types of Fraction
  • Proper Fraction: with numerator smaller than the denominator.

    Examples: 1/3, 3/7, 1/5
  • Improper Fraction: fraction with numerator larger than denominator.

    Examples: 9/4, 13/10, 5/3
  • Mixed Number: consists of a whole number and proper fraction
    •

    Examples: 3 1/8, 56 ¾, 12 ½

    Converting Fractions
  • Improper Fractions to Mixed Numbers
  • Divide numerator by denominator
  • Whole number is the quotient and the remainder place over original denominator

    Examples:

    10/10 = 1

    17/7 = 2 3/7  (7 divides evenly into 17 two times with three remaining out of 7

    23/14 = 1 9/14  (14 divides evenly into 23 one time with 9 remaining out of 14)

     
  • Mixed Numbers to Improper Fractions

    (Denominator x Whole # ) + Numerator
    Original denominator

     

    Examples:

    •
    1. 7 1/3 = (7 x 3) + 1  = 22
        3  3
    2.  3 5/7 = (3 x7) + 5 = 26
        7 7

    3. 14 3/9 = (14 x 9) + 3 = 129
        9 9

    Reducing Fractions to Lowest Terms

    Mastering reduction of fractions takes some trial and error. This is an important skill to master. It helps to know some of the divisibility tests such as those listed on p.41 to make the learning curve smaller.

    Steps to reducing fractions:

  • Divide numerator and denominator by largest possible divisor.
  • When reduced to lowest term, no number besides 1 will divide evenly into both numerator and denominator.
  • It may be necessary to reduce more than once if largest possible divisor was not originally selected.

    • Examples:

    20  = 20 ÷ 20  =  1
    40 40 ÷ 20  2

     

    12  = 12 ÷ 6  = 2
    18 18 ÷ 6 3

    Raising Fractions to Higher Terms

    It is sometimes necessary to raise fractions to higher terms (the opposite of reducing fraction) when adding and subtracting fractions. Multiply the numerator and denominator by the same whole number.

    Examples:

    1  x 5 = 5
    3 5 15

     

    4  x 3 = 12
    7 3 21

    Finding Least Common Denominator (LCD)

    Before we can add and subtract fraction it is necessary to know how to find the LCD. This is the smallest nonzero whole number into which all denominators will divide evenly. Prime numbers or inspection can be used to find LCD.

    A prime number is a whole number greater than 1 that is only divisible by itself and 1 (1 is not a prime number).

    Examples: 2,3,5,7,11

    When you have more than two denominators follow these steps to get the LCD:

    Step 1: Copy denominators and arrange them in a separate row. ¼ + 1/6 + 2/9

                So 4 6 9

    Step 2: Divide denominator by smallest prime number that will divide evenly into at least two numbers
    2 / 4 6 9
      2 3 9

    Step 3: Continue until no prime numbers divides evenly into at least two numbers.
    2 / 4 6 9
    3 / 2 3 9
      2 1 3

    Step 4: Multiply all the numbers in the divisors and the last row to find LCD

    2 x 3 x 2 x 1 x 3 = 36 = LCM

    Step 5: Raise fractions so each has a common denominator and complete computation

    1/4 becomes 9/36

    1/6 becomes 6/36

    2/9 becomes 8/36

    Adding and Subtracting Fractions With the Same Denominator
  • Add or subtract numerators and keep denominator the same. Remember to make sure your fraction answers are in reduced form.

    Example: ¼ + 2/4 = ¾

    5/9 + 2/9 = 7/9

    2/8 + 6/8 = 8/8 = 1

    4/7 – 2/7 = 2/7

    2 3/9 – 1/9 = 2 2/9

    5/6 – 1/6 = 4/6 = 2/3

    • Adding and Subtracting Fractions with Different Denominators

    Example: 7/12 – 9/16 =

    Find LCD

    2 / 12   16
    2 / 6 8
      3 4

    LCD = 2 x 2 x 3 x 4 = 48

    Raise fractions: 7/ 12 ( 4/4) = 28 / 48

    9/16 x (3/3) = 27/ 48

    Add: 28/48 + 27/48 = 55/48

    Change to proper fraction: 1 7/48

    Adding and Subtracting Mixed Numbers:
    1. Add or subtract fractions
    2. Add whole numbers
    3. Combine totals of Step 1 and Step 2

      If denominators are different, a common denominator must be found. Make sure that you change improper fraction into a proper fraction for final answer.

    Multiply Fractions / Mixed Numbers
  • If there are mixed numbers change to improper fractions
  • Multiply numerator by numerator and denominator by denominator.
  • Reduce answer to lowest term, use cross cancellation method when possible.
    •

    Divide Fractions / Mixed Numbers
  • Convert any mixed numbers to improper fractions
  • Invert (flip) divisor and then follow step for multiplication
  • Always reduce answer to lowest terms, use cross cancellation when possible.
    •

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