Algebra 1

1
Algebra 1
Click me for video gt

• Lecture 1
• Video Lectures and Notes
• by
• David V. Anderson

2
Introduction
Click me for video. gt

• Welcome to the Stellar Schools course Algebra 1.
  As in our other courses, Stellar Schools follow a
  curriculum very similar to the one used by the
  Hillsdale Academy of Hillsdale, Michigan. In the
  case of Algebra 1, Hillsdale uses the text
• Algebra 1 An Incremental Development, 3rd Ed.
• John H. Saxon, Jr.
• Saxon Publishers, Inc.
• At Stellar Schools we assign this book as a
  secondary but required text. You are requested to
  obtain the Saxon book before continuing with this
  course.
• The primary text for Algebra 1 is this set of
  lecture notes which is provided to you both in
  hardcopy and digital formats.
• Our curriculum for Algebra 1 is mostly based on
  the Saxon text but also includes some additional
  items that we believe adds clarity to the
  presentation. In fact, the curriculum for this
  course (in accordance with the Stellar Schools
  format we apply to all courses) is defined as the
  universe of all possible examination questions
  and answers. All students are expected to master
  these knowledge items by demonstrating
  examination scores of 95 and above. In this text
  we specify these knowledge items in what we call
  learning concept statements or LCS.

3
About MasteringThe LCS
Click me for video gt

• Examinations you will take to demonstrate your
  mastery of this course will be based only on the
  collection of learning concept statements (LCS)
  presented in this course and on those from
  prerequisite mathematics courses. For an example,
  we show LCS 149 in the display below.

4
More About the LCS
Click me for video gt

• Mastering Algebra 1
• There are well over two-thousand LCS in Algebra 1
  but any examination, for practical reasons, will
  only employ a small subset of questions
  addressing the subject matter. That subset will
  be determined by applying a random number
  generator to each subtopic in this course. Thus
  if and when you retake the courses examination,
  the questions will almost always be different
  than those encountered before- though they will
  be covering similar or related ground.

• Do not be afraid. We expect you to achieve a
  mastery level of 95 before we will certify you
  as having mastered Algebra 1. You will not likely
  achieve this result the first time you attempt to
  take the course examination. But after review and
  further practice doing problems you will find
  your subsequent scores higher and higher until
  you exceed the proficiency level of 95.
• Instructional Philosophy
• The approach taken by Saxon math is what they
  call incremental development. New subject
  material is introduced in short chapters that
  are called lessons in the Saxon book. This
  approach also puts emphasis on continual review
  of content learned in earlier years. Thus it
  should not be surprising to find this Algebra 1
  book begins with a review of fractions.
• Structure of Lecture Notes
• These notes are built around the Learning
  Concept Statements (LCS). We make this quite
  explicit by directly quoting each LCS as we begin
  its description. The LCS are currently stored in
  a database and we simply paste the relevant row
  of the data array into the text of these notes to
  provide the precise LCS text.

5
Segment 1
Click me for video gt

• The first nine lessons from Saxons Algebra 1
  comprise Segment One. We chose this length for
  purposes of demonstrating the capabilities of the
  Stellar Schools instructional model. It has no
  mathematical or pedagogical relevance.
• Lecture 1 Combining Fractions and Combining Line
  Segments
• Adding and Subtracting Fractions

We begin by reviewing the mechanics of adding and
subtracting fractions. Before either of these
operations can be completed it is necessary to
put all of the terms to be combined to have the
same denominators. For any pair of terms we need
to find a common denominator. Since multiplying
the numerator and denominator of any fraction by
the same number does not alter the value of the
fraction we can then represent the sum n/k
m/l as ln/lk km/kl.
6
Finding A Common Denominator
In the step taken here, the first term is
multiplied by 1 l/l and the second term is
multiplied by 1 k/k. This produces the common
denominator lkkl. We also notice that the
numerators have now changed to ln and km.

• So, for example, lets consider adding two
  fractions as follows
• 8/9 3/4 ?
• We multiply the terms by 1 4/4 and 1 9/9
  respectively to get
• (4/4)(8/9) (9/9)(3/4) (48)/(49)
  (93)/(94) 32/36 27/36
• Here we have carried out the indicated
  multiplications of numerator factors and of
  denominator factors. We postpone, for the minute,
  carrying out the indicated addition because that
  is the subject of the next LCS.

Click me for video
7
Adding and Subtracting FractionsIf there are
more than two terms in the expression it may be
necessary to find new common denominators that
the 3rd, 4th, and other terms will share with the
first two terms.
Click me for video gt

• Another point is that this denominator, lk, may
  not be the smallest common denominator of the two
  fractions. The smallest such denominator is
  called the lowest common denominator. In a
  subsequent lesson we shall explore this
  mathematical topic further. But for the purposes
  of adding and subtracting fractions there is no
  need to find the lowest common denominator the
  common denominator as obtained above will be
  sufficient.
• Given the close similarity of LCS numbers 2 and
  3, we have combined their presentations. Once the
  terms of the addition or subtraction problem are
  converted to have equal denominators, it is quite
  easy to perform the addition or subtraction
  simply by adding or subtracting the numerators
  and keeping the converted denominator. This is
  shown above in the LCS numbers 2 and 3.

8
Proper and Improper Fractions

• We can continue with the example we developed for
  LCS 1
• 32/36 27/36 (32 27)/36 59/36
• This is a correct answer but it is in the form of
  an improper fraction. We will soon address
  converting this to a mixed format in LCS 6,
  below.

Click me for video gt
________________________________________________
It should be recalled that proper
fractions (fractions of value less than one) can
be combined with whole numbers to form what is
called a mixed number. Unlike the usual format in
algebra for indicating multiplication, the mixed
number places the whole number on the left next
to the fraction to its right (with no symbols
between) and no multiplication is indicated by
this placement. In fact, it is addition of the
whole number plus the fraction that is indicated
by this arrangement. You may recall that mixed
numbers can result from improper fractions. An
improper fraction indicates division of a larger
numerator by a smaller denominator. When we carry
out that division incompletely to form a whole
number quotient plus a remainder portion of the
numerator (yet to be divided) we have produced
the mixed number (whole number plus a proper
fraction).
9
Adding Mixed Numbers
Click me for video gt

• To add mixed numbers we keep in mind that each
  mixed number is itself two numbers a whole
  number and a fraction to which it is added. Thus
  adding mixed numbers generally means we are
  adding four numbers two are whole numbers and
  two are fractions. Since addition can proceed in
  any order we can separately add the fractions
  together and the whole numbers together to give
  us the procedure for adding mixed numbers. The
  result is then a mixed number. Sometimes the
  fraction is an improper fraction whose value
  exceeds one in which case further simplification
  is required. We cover that next.

__________________________________________________
__________________________________________________
_
A mixed number that has an improper fraction is
not really following the rule for a mixed number
in which the fractional part is preferred to be a
proper fraction- less than one. When the mixed
number resulting from the addition of mixed
numbers produces an improper fraction, we convert
the improper fraction to its own mixed notation.
In that conversion a whole number and a proper
fraction result. The whole number generated from
the improper fraction is then added to the
original whole number to produce the sought after
mixed number result.
10
More on Mixed Numbers

• So for example, lets consider the sum of
• 2 7/8 3 5/9 .
• After writing these with common denominators, we
  have instead
• 2 63/72 3 40/72
• Performing the indicated additions gives us
• 5 103/72
• Then reducing the improper fraction to its own
  mixed form yields,
• 5 1 31/72 6 31/72,
• where we have shown the final mixed result
  containing the whole number 6 and the proper
  fraction 31/72.

Click me for video
11
Subtracting Mixed Numbers
Click me for video gt

• Subtracting mixed numbers is very similar to the
  addition of them. Here we subtract the fractional
  parts and the whole parts separately. If the two
  numbers each had proper fractional parts, the
  resulting fractional part will also be a proper
  fraction. Here another difficulty can arise when
  the subtracted fraction is larger than the other
  fraction because it will produce a negative
  fraction. LCS 8 describes the treatment of this
  case.
• __________________________________________________
  __________________________________
• When the fractional difference l/k i/k is
  negative we simply borrow 1 from m.. and add
  it to l/k prior to performing the fractional
  subtraction. The whole number part of the mixed
  fractional result is then decreased by 1 to be
  m 1. The fractional difference then has the
  1 k/k added to the prior negative fraction to
  produce a positive fraction (lk-i)/k. These two
  operations have the effect of adding and
  subtracting 1 from the expression thus leaving
  its value the same. It helps to consider an
  example.

12
Example Subtracting Fractions Where BorrowingIs
Required
Click me for video gt

• Lets compute
• 7 2/7 3 5/7
• for which we notice the fractional part will
  beless than zero. So we do the
  indicatedborrowing by writing,
• 7 - 1 -3 7/7 2/7 -5/7
• where the terms - 1 and 7/7 indicate the
  borrowing process. Notice that -1 7/7 0,
  so introducing these terms does not change the
  value of the expression. We finish this by
  performing the indicated operations to yield
• 3 4/7
• This is the desired mixed number result including
  a positive and proper fractional part.

13
Mathematical Linesand Line Segments
Click me for video gt
Their Connection to Fractions From the
consideration of combining fractions we now move
to the study of lines and line segments. In a
practical sense the two topics are related That
is the measurement of line segments is often
expressed in mixed number notation- particularly
in the English system where inches, feet and
yards measure the lengths of them. In that
context it is often necessary to add or subtract
the mixed numbers representing these lengths.

• The idealized mathematical straight line is
  described more precisely in advanced mathematics
  but for our purposes it is useful to give its
  (rather obvious) properties. For one, lines are
  very narrow- in fact they have zero width. And
  they are very long. They are extending in both
  directions without end. They are infinite,
  meaning that there is no finite bound where they
  end. Any such line can be defined by giving two
  points through which the line passes. In our
  description here we shall say that points A and
  B determine this line.
• __________________________________________________
  _______________________________

14
Lengths of Lines Mathematical versus Segments
Click me for video gt

• It follows that the length of a mathematical line
  is without limit that is to say it is infinite.
• __________________________________________________
  _________________________________
• A more practical type of straight line is the
  line segment. It is simply the line between any
  two points on a mathematical line. In the
  previous example the points A and B laid on a
  mathematical line. Here they can be considered
  the endpoints of a line segment.
• __________________________________________________
  __________________________________

15
Double Arrowed Overbar Notation for Mathematical
Line

• Symbolically, a mathematical straight line can be
  represented by labeling any two points on such a
  line with letters or other symbols. We then
  construct a symbol for a line by putting the two
  letters together and then indicate it is the
  idealized mathematical line by putting a
  two-arrowed overbar over these letters. Thus if a
  line passes through points A and B we
  indicate it by
• We can draw a portion of such a line as
• 
  A.
• B.
• The arrows are suggestive of the idea that the
  line does not end, but continues indefinitely.
• _____________________________________

Click me for video
16
Simple Overbar Notation for Line Segment

• The notation we use to name a line segment is
  very similar to that used for the mathematical
  line differing only in the form of the overbar.
  Thus the line segment between A and B is
  given by
• We can draw a line segment, again from point A
  to point B, as follows
• 
  B .
• A .

Click me for video
17
Combining Line Segments, End-to-End

• Again, an important property of a line segment is
  that it begins and ends. In the drawing the line
  begins at point A and ends at point B. Also
  there is the related line segment that
  begins at point B and ends at point A.
• __________________________________________________
  __________________________________
• Sometimes it is desired to combine two
  neighboring line segments that share an endpoint
  and that both lie on the same mathematical
  straight line. If the second line segment begins
  at point B (the one shared with segment )
  we use C to designate the endpoint of the
  second segment. The combined line segment then
  begins at A and ends at C and we name it
  . Drawn as a graph this is shown as
• 
  
  C .
• 
  B .
• A .
• __________________________________________________
  _____________

Click me for video gt
17
18
Length of Combined Line Segment

• A property of a straight line is that the length
  of a combined line segment is the sum of the
  component lengths. We also note that the concepts
  of distance and length are the same for straight
  lines. However, when a line is curved the length
  (measured along the line) and the distance
  (measured along a straight line connecting the
  two end-points are generally different with the
  length greater than the distance.
• Homework LCS 1 15
• Please complete all of the problems in Problem
  Set 1 in the Saxon text on page 3. Do numbers 1
  through 30.

Click me for video gt