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YOU are better than YOU think. Show yourself how:
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-/[]\- Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite. Logic
Mastery Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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-/[]\- What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts. Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice. |
Dilatations - Coordinate ViewA dilatation in the plane depends on a non-zero scale factor k, and is given by a mapping [X,Y] = fk(x,y) = (kx, ky) When k is larger than 1 in magnitude, that is, when |k| > 1, the dilatation, is an expansion. It moves points further away from the origin and from each other. Exercise: show why.
Example: For k =2,
and fk(-3,1) = [2*(-3), 2*1] = [-6,2] When k is smaller than 1 in magnitude, when 0< |k| < 1, the dilatation, the dilatation is a contraction. It moves points closer to the origin and to each other. Exercise: show why Example: For k =¾,
and fk(-3,1) = [(¾)*(-3), (¾)*1] = [-9/4,¾] When k equals 1 in magnitude, when |k| = 1, the dilatation is reflection in the case k = -1 or the identify map in the case k=1. The distance between the images of a pair of points equal the distance between the points. Example: For k =1,
and fk(-3,1) = [(1)*(-3), (1)*1] = [-9/4,-1] Example: For k =-1,
and fk(-3,1) = [(-1)*(-3), (-1)*1] = [3,-1] Distances and Lengths are Multiplied by |k|The distance c between the points [x1,y1] and [x2,y2] is given by
Therefore the distance between the images of these point under a dilatation fk(x,y) = (kx, ky) that is the distance d between the points [X1,Y1] =[kx1,ky1] and [X2,Y2] = [kx2,ky2] is given by
In the discussion of square roots, we learn that root of a product of non-negative terms is a product of the square root of the terms. Lines go into LinesTheorem: If [x,y] satisfying ax+by=c then [X,Y}=[kx,ky] satisfy the equation aX+bY = c/k Proof: c = ax+by = a (X/k) + b (Y/k) = aX (1/k) + bY (1/k) = (aX+bY)(1/k) since division by k gives the same result as multiplication by 1/k, and since multiplication by a real number, here 1/k is distributive. The above theorem say straight lines are taken into straight lines by dilatations. We observe the image line is parallel to the original line. Theorem: If [x,y] satisfying ax+by=c then [X,Y}=[kx,ky] satisfy the equation aX+bY = c/k Line Segments Go Into Line Segments The line segment between the points [x1,y1] and [x2,y2] can be described algebraically by the parameter-based equations
where 0 < t < 1, that is t varies between 0 and 1.
If we let the points [X1,Y1] =[kx1,ky1] and [X2,Y2] = [kx2,ky2] be the image of the two end points [x1,y1] and [x2,y2] with respect to a dilatation of scale factor k then the image of
is
That is because
So the image of
is
From above the length of the image line segment between point [X1,Y1] =[kx1,ky1] to point [X2,Y2] = [kx2,ky2] is absolute value of the scale factor |k| times the distance of the original line segment between points [x1,y1] and [x2,y2]. So a dilatation with scale factor k mutliples lengths by |k|
Preservation of AnglesAn angle is formed when two line segments meeting at a common endpoint. The image of those line segments are parallel line segments. The foregoing suggests the angle of image line segments equals the angle of formed by the original. Similar Polygons and Similar TrianglesTwo polygons in the plane are said to be similar when and only when corresponding angles are equal and corresponding sides are proportional. To be more precise, there is a scale factor K for which length of a side in the second polygon are K times the length of the corresponding side in the first. Now the image of a first polygon under a dilatation scale factor k is a second polygon with the following properties:
So the image of a first polygon under a dilatation is a similar polygon, a similar polygon with corresponding sides parallel. Maps, Plans and SimilarityA dilatation [X,Y] = fk(x,y) = (kx, ky) represents a contraction, expansion or identify map in a plane if [X,Y} and [x,y] determine points in the same coordinate system. But a A dilitation [X,Y] = fk(x,y) = (kx, ky) may also represent a function between two different planes, the original and the image. The image may be regarded as a map (the kind used for roads, cities and geographic regions in a flat earth approximation society). Many human constructions and roads are based on polygons or can approximated by polygons. In a map, corresponding figures are similar and perhaps parallel. Corresponding similar figures can be made parallel if you rotate the map. Producing a map of a flat region preserves angles while it multiples lengths by a scale factor k >0. The ratio 1: 100 corresponds to a map with scale factor k = 1/100. That is lengths in the map are 1/100 lengths of the original. Scale Factors for lengths, areas and volumes.Plans and models, if not maps, can be 3 dimensional. Curves, Regions and Solids (or their surfaces) can be drawn with scale factor k (a ratio). In such plans and models, angles are preserved (the angle of the image or model equals the angle in the original or intended original). While lengths are multiplied by k, areas are multiplied by a = k2 and volumes are multiplied by b = k3. The foregoing has implications for the amounts proportional to lengths, surface area and volume. If you know any one of the three scale factors k, a and b, the other two can be computed. |
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www.whyslopes.com
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