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Click me for video > Algebra 1 Lecture 1 Video Lectures and Notes by David V. Anderson
Click me for video. > Introduction 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.
Click me for video > About MasteringThe LCS 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.
Click me for video > More About the LCS • 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 course’s 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.
Click me for video > Segment 1 The first nine lessons from Saxon’s 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.
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 lk=kl. We also notice that the numerators have now changed to ln and km. So, for example, let’s 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) = (4*8)/(4*9) + (9*3)/(9*4) = 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 ^
Click me for video > 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. 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.
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 > ________________________________________________ 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).
Adding Mixed Numbers 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. Click me for video > _____________________________________________________________________________________________________ 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.
More on Mixed Numbers So for example, let’s 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 ^
Subtracting Mixed Numbers Click me for video > 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 (l+k-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.
Example: Subtracting Fractions Where BorrowingIs Required Let’s 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. Click me for video >
Click me for video > Mathematical Linesand Line Segments: 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. _________________________________________________________________________________
Lengths of Lines: “Mathematical” versus Segments 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. ____________________________________________________________________________________ Click me for video >
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 ^
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 ^
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 > 17
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 >