Math 80 Lecture - Chapter 1                                                           Merced College; Don Power   

 

1.1  Intro to Whole Numbers

 

Obj:  Order relations

 

Defs:  natural numbers, whole numbers, number line

 

      Define whole numbers:  a set of counting numbers, including 0.

      Ex:  3.2,  -5,  3/5,  0,  8/3,  5 2/3,  43,601,  -0.2   which are whole numbers?

 

Concept:  the graph of a number is its location on the number line

 

Inequalities:  > and < --- relate to left and right on a number line

      Why not to larger and smaller? 

 

Obj:  Place Value:  Ident the place value of a digit through hundred billions

      Mark off digits in groups of 3

      Know names of the groups, "periods" (count from small to large; say nothing for units group)

      Know the position within the group (hundred, ten, units[say nothing])

      Ex  506,430,000,019,827  Ident where each digit is.

 

Read/Write a whole nr in words

      Determine the name of the largest group [on the left]

      Read the 3-digit nr in the group as though it were the entire nr…(unless it's all 000)

      Name the group (unless it's all 000)

      Hyphen in numbers from 21-99; commas between groups

      Don't say "and" after "hundred" -- "and" refers to a decimal (ambiguity in decimal fractions)

            Ex:  0.142 vs 100.042

Write a whole nr in digits:

      write each group, separated by commas

      use 000 for each missing group

      Ex:  X34, 36   One ex without missing groups, one ex with missing groups.

      Ex (app)  ~X37  Given digits 0, 3, 5, 6, 7, 9, 0, 3, write largest & smallest possible number.

 

Obj:  Rounding

 

Locate the place to which the nr is to be rounded.

      Draw a line under (textbook approach) that place or draw a cutoff line after that place

      Look at the next digit to the right;

            5 or above, increase the underlined digit (text);

                  Or add 5 to the digit to the right; if there's a carry, you've rounded up.

            4 or below, don't change

      Change all the digits to the right to zeros  [throw out if they are after a decimal points]

      Don't lose the sign of your number.

 

 

Obj:  Applications and statistical graphs

 

Accidents on US Roadways

 

 

 

Month

Crashes

Vehicles

7

3459

5210

8

3557

5473

9

3239

4912

10

3344

5060

11

3162

4816

12

3303

5242

Favorite Sports

 

 

Football

80

Basketball

75

Baseball

50

Tennis

45

Hockey

30

Golf

20

 

1.2  Add and Subtract Whole Numbers

 

Hopefully, this is all review.  What might be new?

 

1.  Terminology: 

      addends and sum

      Key phrases for addition:  added to, more than, increased by, the total of, plus...

            Be careful:  "3 more than 8" is 11,

but the answer of "how much more than 3 is 5" calls for subtraction

      For subtraction: 

            minuend - subtrahend = difference

            Phrases minus, less, difference, decreased by, subtract ... from

            Note subtract 5 from 8 means 8-5  The key is position, not which number is bigger

2.  Adding when numbers are arranged horizontally

3.  Substitution technique when we have a phrase like "Find a+b-c and numbers representing a, b, c

4.  Subst technique:  for "is 13 a solution of 2+h=16?"

5.  Properties and their names:

      Addition property of zero             (for mult, you have properties of 0 and 1)

      Commutative property of addition  (also exists for mult)

      Associative property of addition (also exists for mult)

6.  Geometry terminology

      See pg 29-30

7.  Substitution into geometry formulas

 

Use front end rounding to estimate an answer:

      How to do front end rounding

      Round both numbers to the same place, if the are different lengths.

Rounding vs estimating
      Rounding answers:  Add/subt first, then round.

      Estimating answers:  Round first, then estimate.

 

1.3  Multiplication and Division

What's new?

1.  Definitions

Mult = repeated additon

Div = repeated subt:  2 answers:  1.  How many times could we subtract (quotient),  2. What is left over (remainder)

2.  Vocab

      Mult:  product, times, twice

 

3.  Prime Factorization

      Successive division by primes -- keep trying the smallest prime until it doesn't work any more

      or, Factor Tree

 

4.  Listing All Factors of a Number

      Start small (with 1)

      If division works, you get a factor pair

      Continue until the number you are testing, when squared, is larger than the target number.

 

1.4  Solving Equations with Whole Numbers

 

Def:  Solution

 

Subtraction property:

Subtract the same number from both sides of an equation:  result is equivalent

      Principle:  do the same thing to both sides.  If you start with equal amts, you get equal amts.

      Goal:  "isolate" the term with the variable (find out what it equals)

            Strategy:  Whatever is added to the variable in the equation has to be subtracted to isolate the var

 

      Ex:  X10

 

Division property

      Divide both sides of an equation by the same number:  result is equivalent

      Same principle of doing the same thing to two equal amounts.

      Goal:  "isolate" the variable (find out what it equals)

            Strategy:  Whatever multiplies the variable in the equation has to be divided to isolate the var

 

      Ex:  X8

 

Applications:

      Translate word picture to an equation, then solve it

      You may take several steps to do the translation:  this helps when problem is wordy

            For the example on pg 69, first sentence simplifies as "last mo. = 3 times this mo."

 

      Ex:  X30

 

Formulas:

      Substitute for the appropriate letter.

      Then solve using the subt or division property [more extensive strategy later].

 

      Ex:  X40

 

1.5  Order of Operations

 

PEMDAS - Please Excuse My Dear (or Demented) Aunt Sally

 

P          Refers to operations inside parentheses

            Parentheses can also mean multiplication - but we don't multiply until later

            Other symbols can also be used as grouping symbols:

                  Brackets [ 4+3 ]

                  Braces    { 5 − 2  }

                  Absolute value      | 14 − 7 |

                  Parentheses with exponents    (1+1)2 or (5x2)3

                  Square roots   

                  Fraction bars:  means (30 − 6) / (5 + 1)

 

                  Regardless of the grouping symbol, always get the contents down to a single term before clearing the grouping symbol

                  Rarely, there are special rules that will let us clear a grouping symbol without first getting down to one term)

 

E          Apply exponents

                  Only to the thing immediately in front of the exponent:  Contrast 3x2 versus (3x)2

 

MD      Mult and Div:

                  These have the same priority:

                  Do them in the same order as you read them (from left to right)

 

AS       Addition and Subtraction - Same idea as Mult and Div:

                  These have the same priority:

                  Do them in the same order as you read them (from left to right)

 

Second half of exercises require substitition first, then simplification:

 

      Replace the letter by the number, or by the number in parentheses

      Later, when we get negative numbers, we will always replace using the number in parentheses

      Implied mult (e.g. 6x, meaning 6 times x):  Show multiplication by parentheses or a raised dot

 

 

Return to:  Merced College; Don Power               Updated 10/05/06 by Don Power