Geometry - Measurement of Angles and Lengths

Page Contents:

1. Measurement Matters

2. Drawing Skills and Geometric Terms

3. More Measurement Matters

4. A 3D Construction Exercise [Do with carboard first]

5. Navigation and Treasure Hunting with Maps

6. What is Similarity - Optional, unless you have to teach it.

A. Measurement Matters

1. Length. Students should be able to measure lengths and angles with the aid of rulers and protractors. Students should learn that the zero point on a ruler need not be an end of the ruler or tape measure.
   
   
   
   
   Students should know to measure from the origin - zero point or mark - of a ruler when the origin or zero mark is not at the very end - hope but verify.

   Slogan: When the zero point of a ruler is not at the initial end of a ruler, do not measure from the end, measure from the zero mark.
   

2. Measurement Skills and Sense - Numerically Perspective: With the aid of rulers and tape measures, and in particular the use of unit distance for a divisor, lengths or line segments can described as proper fractions, whole numbers, improper fractions and mixed numbers multiples of the chosen or implied unit length, say 1 cm - one centimeter.

   Practice: All lengths can be described as whole number, fractional and decimal multiples of the unit length - an assumption with consequences.

   Show how length comparison, which is longer or shorter, corresponds numerical coefficient comparison in the description of lengths by numbers or numerical coefficients of the chosen unit length. Moreover, the physical or geometric addition, subtraction, multiplication and division of lengths implies and defines operations on the measures or numerical coefficients associated with the unit length. That is, numerical methods for addition and subtraction of fractions can thus be introduced or reviewed as means to compute the length of the products apart from physical measurement. The issue of irrational lengths is postponed - not mentioned due to the use of mixed number approximations for measures.

3. Measuring the shortest distance between two points. Take a chord or a piece of thread or string, and hold it taut between the two points. Next measure the length of the string. A taut string gives the shortest path between the two points. Illustrate the foregoing physically in a room and on maps and plans.

4. Angle Measurements with Protractors: Show students how to measure convex angles - angles < 180 degrees - and convex angles - angles between 180 and 360 degrees. Show students how to measure and recognize acute, right, obtuse and straight angles. Show how to compare angles physically by superposition and by angle measurement.

   

   Show how to measure angles with a protractor. By examples involving measurement, show how the sum of angles in triangles in the plane add up to 180 degrees. By measurement, show how the sum of angles in a rectangle add up to 360 degrees.

5. Map and Plan Usage: Show students how to draw maps and plans to full scale, and a proper or improper fractional scale - the same in all directions. Then show or confirm angles on maps and plans equal corresponding angles in real life or other maps and plans when the same scale is used horizontally and vertically on the map or plan in question. Before or after, show how lengths in the maps or plans are often proportional to corresponding lengths in real life, or other maps and plans.

6. Map and Plan Usage:Shows what happens to corresponding angles and lengths when maps and plans do not have the same scale horizontally and vertically.

7. Map and Plan Usage: Show or confirm the number of map unit lengths needed to cover a straight or curved line segment equals the number of unit squares needed to cover the same straight or curved lined in reality, or drawn on another map.

8. Map and Plan Usage: Show or confirm the number of map unit squares needed to cover a rectangle with integral sides drawn on map equals the number of unit squares needed to cover the same rectangle in reality, or drawn on another map.

9. Map and Plan Usage: Introduce map contour lines or curves. Draw Arrows to indicate direction of steepest ascent. Approximate the slope of the direction of steepest ascent between two contour lines. Give Real life examples of slopes for steep roads in the neighbourhood.

   Associated Slope Sense Experiment: Go the gym and have students walk 3 meter [10 foot] planks with varying slopes. Measure the slope. Measure the angle of inclination - if possible. Observe that walking uphill along the plank becomes harder and then near impossible as the slope increases. During this exercise, make sure students arms are held by fellow students to prevent or limit falls. Finally, have students calculated the slope [rise over run] of steps.

10. State the Pythagorean Theorem. Next illustrate it exactly with 3-4-5 right triangle and with the 5-12-13 right triangle. If you are calculating square roots of 2 and 3 with the aid of a calculator, show how an isoceles right triangle with legs of length 1 will have a hypotenuse of length $\sqrt{2}.$ Also show how a right triangle with legs of length 1 and $\sqrt{3}$ will have a hypotenuse of length 2. Possible Exercise: In the latter case, students may be invited to draw right triangles using a large unit length and decimal approximations to the $\sqrt{3}$ and to measure in them, the length of the hypotenuse. The difference from length 2 units might lead to a discussion of how many digits are needed in practice for good enough accuracy. Please report any difficulties with the exercise imagined here.

Additional Measurement Skills

1. Measure to the nearest eighth of an inch or millimeter

2. Use units of length

3. use units of weight or mass

4. measure capacity [volume]

B. Drawing Skills and Geometric Terms

1. Ruler and Compass Constructions: Students should learn the Side-Side-Side, Side-Angle-Side and Angle-Side-Angle methods to construct triangles from given data and to duplicate other triangles. They may see that the duplicated triangles are isometric to the original via a correspondence - a matching, pairing or mapping that associates vertices and hence measures in different triangles. In isometry, corresponding corresponding sides have equal length measure and corresponding angles have equal angle measure. Following that they may see two triangles constructed from the same data with the Side-Angle-Side, Angle-Side-Angle or Side-Side-Side methods can be considered duplicates of each other, and so are isometric.

2. Triangle Inequality: Observe the sum of lengths of two sides of a triangle is greater than a third. Illustrate this by joining two ends of a string together and then forming triangles with it. This suggests the shortest distance between two points is a straight line. - The word linear in mathematics comes from line. On a flat plane, a taught cord or line between two points defines a straight path. On a curved surface, a taught string between two points may provide the shortest path between those points.
   

3. Parallel Lines and Transversals:
   
   The concept of interior, alternating and corresponding angles should be taught for a line or line segment transversal to two others lines in cases where the other two may be close to parallel, but are not.
   
   Comments in site pages about when SSS, ASA and SAS methods fail or work in unexpected ways point to a context for a later study of Euclidean geometry and a context for the discussion of when two lines will intersect or be parallel. Note that whenever a line cuts two others, interior angles, alternating angles and corresponding angles are formed.
   

4. Angle and Line Segment Bisection, etc. Students may also meet ruler and compass methods with justifications included for bisecting angles and line segments, and for dropping or drawing a perpendicular to a line from a point for [i] the point off line and for [ii] point in line. Methods may given by rote - here are the constructions and apply them, or explanations of why the methods work may be based on the postulates.
   
   Verification: Check by measurement that angles and line segments are bisected.
   
   Extension: Show how division of angle measures and length measures into thirds, quarters and fifths may be used with the aid of protractors and rulers to divide angles and lengths. State as a curiousity that some divisions are possible with ruler and compass constructions but not all.
   

5. Vocabulary and Notation: Introduce a point as the center of a circular dot or disk - explain how that implies a point has no breadth nor width. Introduce the use of capital letters as point labels, identification and names.
   
   
   

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   Show or suggest how pairs of distinct points in the plane determine a finite line segment
   
   
   

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   Show or suggest how pairs of distinct points in the plane determine a line
   
   
   

   Explain how the points R and S may be moved without changing the direction of the line.

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   Show or suggest how pairs of distinct points in the plane determine a ray
   
   
   Explain how the point D may be moved without changing the direction of the ray.

   Note: Some course may use the latter notation not for a ray, but for the arrow that starts at E and ends at D.

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   Remark: Explain how we may need to say distinct or different when using letters [etc] to identify points as the same point may have more than one identifier or label in much the same way a person may have more than one name - In algebra, there is or will be a similar need to say letters denote different numbers or quantities, especially in situations where division by y-x appears.
   

6. More Vocabulary and Notation: When two distinct rays emanate [start at] the same point, they may form a pair of angles, one convex and one concave, or both straight. Examples follow.
   
   
   

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7. Rigid Body Motions: With the aid of graph paper, if not coordinate systems in the plane, students may see how to translate, rotate and reflect points, triangles, circles and further figures in the plane. [i] The notion that two triangles are isometric if one is the image of the other under a translation, rotation or reflection may be suggested. [ii] The notion that two circles have the same radius if one is the image of the other under a translation, rotation or reflection may appear. The two notions [i] and [ii], or [i] alone, supports labeling translations, rotations and reflections being as rigid body motions.
   
   Mechanical Point [1]: Show that quadrilaterals where the angles are not fixed are flexible. Examples of that are provided by rods joined at that their end points by pivots [is there a better word for that] to form a polygon with variable or changeable angles. Parallelograms [opposite sides of equal length] and Rhombuses [four sides of equal length] are special examples. What happens when angles are fixed by [a] specifying their measurement or [b] bracing via a line segment between the arms of the angle to form a triangle.
   
   Mechanical Point [2]: In the plane, show students that triangles and some "connected" figures composed of triangles are rigid in the sense that corresponding angles and lengths do not change when the triangle or the figure is moved or drawn in different positions following a rotations, translations, reflections and combinations there of.
   
   Mechanical Point [3]: Show the rotation of a triangle about a vertex moves the midpoint [mark it with a dot] of the opposite side into the midpoint of the opposite side for the triangle in any rotated position. That empirical observation essentially implies the location or calculation of midpoints of a line segment commutes with rotations in the plane. The site exposition of complex numbers depends on this point.
   
   All the foregoing patterns may be implied or confirmed by physical or geometric expiriments.

C. Coordinates for Maps and Plans

Maps -- use of coordinates. Points in the plane can located by identifying the square to which they belong, but coordinates provide the location more precisely. Coordinates may be introduced in two steps.

• Maps Coordinates -- basic concepts. Take a map of a location - say your town or region. Explain how letters and numbers are used to located grid squares/regions. Include explanation of scaling. Identify North-South, East and West. Standard Convention North at top -- explain exceptions are possible. The floor plan of a house for instance need not have North at the top.

• Maps Coordinates -- Battle Ships. Variants of the game of battle ship with a mix of letters and numbers to identify squares or grid points may be played to develop and check location of points or squares with unsigned and even signed coordinates
• Maps Coordinates -- Join the dots activities. Earlier students may have learnt to join dots to develop knowledge of the alphabet and show skill in joining numbers in sequence. Joining the dots usually completes a picture or figure. Here a sequence of dots that traces a shape - animal, robots, flowers, houses, or other figures - can with dot location being provided by coordinates. The coordinates may given by a mix of integers, proper and improper fractions, decimals and mixed numbers prefixed by signs or without signs. This activity will develop and check the ability to locate points with coordinates.

Unsigned Coordinates

Tutors: This lesson and the next offers motivation for the introduction of signs. In elementary school, people learn about whole numbers n and fractions p/q before the use of signs. Ordered pairs of unsigned numbers may be introduced as coordinates in a first quadrant. Introducing signs + and - gives ordered pairs of numbers with signs as prefixes to provide coordinates for four quadrants.

Ordered pairs of numbers without signs such [1,4] or [3,2] may be used to locate points on a map.

when the origin or reference point is at the bottom left corner. On such maps there is no need for signs. More generally, you use coordinates such as [1.5, 3.27] or [a, b] to locate points on the map -- provide their rectangular coordinates. Here a and b stand for any pair of unsigned numbers including zero that may be used as coordinates.

In the above map, the left edge of the map region give the vertical coordinate axis while the bottom edge gives a horizontal vertical axis for coordinate use.

The word rectangular is used above as "polar coordinates" will be introduced later. Rectangular coordinates are also called Cartesian Coordinates.

Still more

Unsigned or First Quadrant Coordinates. Positive Rectangular Coordinates: locate origin [0,0] at bottom-left corner and then use ordered pairs [a,b] of nonnegative numbers to locate points in the plane. [Descartes when he introduced coordinates only employed them in the first quadrant. Negative numbers were thus not needed.] Also employ polar coordinates [r, theta] where r is distance to the origin and theta is between 0 and 90 degrees, to show a second way of locating points. The origin [0,0] is at distance 0 from itself, and traveling 0 units in any direction from the origin, represents the zero displacement. Optionally, By measurement, show how to go back and forth between polar coordinates and rectangular coordinates. Apply the Pythagorean theorem, if it understood, to show $r^2= a^2 + b^2$

Signed Coordinates

Tutors: This lesson and the previous one offers motivation for the introduction of signs. In elementary school, people learn about whole numbers n and fractions p/q before the use of signs. Ordered pairs of unsigned numbers may be introduced as coordinates in the first quadrant. Introducing signs + and - gives ordered pairs of numbers with signs as prefixes to provide coordinates for four quadrants.

If our first map extends to the left and/or below the origin, the horizontal and vertical coordinate axis's may be extended. These extensions divide the map into four regions call quadrants. To get coordinates for all four regions or quadrants we may place signs in front of numbers. See the diagram below.

In the above map, identify the points with coordinates [+2,+1], with coordinates [+2,-4], with coordinates [-2.5, -3] and lastly with coordinates [-4, +3]. By convention, + signs in front of numbers are optional. So +2 = 2 and +1 = 1.

Still more

Positive and Negative Rectangular Coordinates: Locate origin [0,0] in the map interior. Use coordinates [a,b] to indicate position relative to this origin. [Negative numbers need to be understood first.] Also employ polar coordinates [r, theta] where r is distance to the origin and theta is between 0 and 360 degrees. With polar coordinates, a comprehension of negative numbers is not required. By measurement, show how to go back and forth between polar coordinates and rectangular coordinates.

D. Drawing or Describing Movements on Maps

Map -- their use in navigation [describing journeys or movements]. The aim is to explain and describe Navigation in the Plane. Use vectors to represent movements, one at a time or one after another on a map. Description of these operations is left for later.

The use of maps for navigation involves plotting of actual or intended paths over land or sea.

1. Navigation with Arrows or Vector

If you pull a string or line taut between two points A and B in or on a plane, you get a straight line. On a flat lake or small sea, boats and ships try to go in straight lines. The edge of a ruler may also give a straight line. The concept of a taut or straight line may suggests the mathematical idea. or extrapolation.

On a map, a sequence of straight line motions may be used to precisely or approximately represent the path of an object [ship, plane or person] over land or sea. These motions and their directions may be represented by arrows with tail at the starting point of a motion and head at the other end or last point in that motion. Here is Motivation and a context for the use of arrows, or vectors, in navigation.

Directed line segments initial points and terminal points, or arrows or vectors with heads and tails may be used to describe or show straight line movements and the direction of motion

In the next figure, the path of the sailboat takes it from A to B, then B to D, then D to G, then G to C, then C to H and then H to M. Think of this as the head-to-tail map addition of movement or vectors.

2. Resultant of Movements - Net Movement

A straight line arrow from one point to another may summarize the movement of an object. The object itself may follow a curved path between the tail or initial point of the arrow and the head or terminal point. Similarly when a sequence of straight line motions is followed, one after another, the arrow joining the initial point of the first motion to the terminal point of the last motion summarizes or gives the sum or resultant of the intermediate motions. Here is a context and motivation for the head to tail addition of arrows or vectors in navigation.

As the Crow Flies: On a map pick an initial point and a terminal point for a motion. Now draw an arrow, head at the destination [terminal point] and tail at the starting point [initial point]. This arrow represents the straight or taut line motion between the two points -- the path that a Crow could fly. [Land-based animals may not be able to go in a straight line.] Observe that the result of this single straight line motion can be given by a horizontal motion followed by a vertical motion, or by a vertical motion plus a horizontal motion. These motions are called the horizontal and vertical components of the original motion. They too can be represented by horizontal and vertical arrows.

5. Plotting actual or intended movements in plane [ Sea or on land

There are three basic ways to define a path on map

1. Define a sequence of Plot paths points A, B, D, G, C, H and M etc on a map [as in the above figure]. Then join them by arrows [directed line segments] to show the piece-wise linear approximation to an intended or actual.

2. Describe a succession of movements [arrows] using [i] direction and length in a polar coordinate like manner. North-South, East West compass bearing may be used for this.

   Method B is assumed to be possible along a line, in a plane, or in space, and may be done with signed coordinates

3. Describe a succession of movements [arrows] by describing the pairs of horizontal and vertical displacements in which the sum or net result of each pair is the movement.

Combination of each way are possible if different segments of the path employ different methods. Giving direction for finding buried treasure in a field or on map provides examples.

Review and Extension

1. Basic Navigation Ideas. Find a real map of a lake district, or draw on graph paper an imaginary lake with several islands in it, and give a scale Water is crossed via a boat. Land is crossed by carrying the boat over it -- a portage. Direction and Distance Displacement/Movement -- From a location, specify a direction [e.g. at angle 37 degrees above horizontal axis, 5 steps] or [North-West, 13 steps], etc. Note how result of a rectangular displacements is almost equivalent to a Direction-Distance movements, and vice-versa, when they each movement has the same starting point and finishing point. But direction-distance movement as the crow flies is more efficient: the hypotenuse of a right triangle is shorter than sum of the lengths of the other two sides. In consequence, may want to replace a two-step rectangular displacement by more singe step direct distance-direction movement. Again, the latter can represented by a single arrow joining the initial point to them movement destination. [Rectangular displacements can be represented by two arrows, each parallel to the maps sides -- the map is assumed to be rectangular.]
   

2. Pirate or Hidden Treasure Activities may give directions to buried treasure in terms of direction and steps. As a pirate treasure alternative, in your kitchen, you may give children a tape measure, and then give directions [movements to do] in terms of distance and angles, horizontal and vertical movements. E.g. From the door go 5 feet south, then 3 feet north, then open the draw 24 inches off the ground to find the buried treasure. The treasure could be some item of value to the child, or another set of directions :

   More generally, Hidden treasure activities may be done in a physical room or space or in a map. The clues represents a sequence of steps or displacements that may be drawn as or represented by on a map by arrows added or joined in a head-to-tail manner. The ability to plot the steps or arrows precisely or not illustrates the domino effects of care and errors in this activity. The activity can be disguised as a game with perhaps rewards points based on how close the last step or arrow head is to the true location of the hidden treasure.

3. Rectangular Displacements and Movement. Examples: from a location, illustrate movements 3 units rightward, 4 units upward; or from another location, illustrate a movement, 12 units north and 5 units East. On the map or a piece of paper represent each sideward or upward motion by an arrow. The head of each arrow or vector should be at the destination [terminal point] of each motion. The tail of each arrow should be at the initial point of each motion.
   

4. Arrow and Vector Operations: Coordinate Free Perspective. Island hopping can be represented by a chain of arrows joining the center of one island to the center of the next. Successive straight-line motions in general can be represented by sequence of direction distance movements, each represented by an arrow. [Pirate buried or lost treasures provide an examples --- see above]. Arrows or displacement or movements can be added together graphically by placing the tail of the first at the journey starting point, or at the head of another. [For each arrow, tail = starting point, head = destination]. Alternatively, each arrow can be viewed as the sum of horizontal and vertical displacement arrows, and the rectangular equivalent of the sum of several displacement, can be obtained by their horizontal and vertical displacements separately [several groupings of these displacement is possible]. The latter gives an arithmetic means to add arrows together. See the next item. See Volume 3, Why Slopes and More Math, as well.
   

5. Navigation games are possible. Offer rewards for following a sequence of displacements [distance-direction, rectangular or a mixture of both], for computing or measuring the one-step distance-direction between the origin and finish of all the displacements, and also for giving the equivalent two-step rectangular representation. Rewards might be points, cookies, pennies, privileges, or just a smile.

6. Arrow and Vector Operations: Rectangular and Polar Coordinate Perspective. Each arrow is the sum of a rectangular displacements, one in the up and down directions -- the vertical [or North-South] component, and one in the left and right directions -- the horizontal [or East-West]component. Each arrow may be represented, recorded or written down as an ordered pairs [a,b]. Here a and b are signed numbers or distances. Adding arrows can done nongraphically by adding the ordered pairs together. The equivalence or interchangeability of the graphical and arithmetic methods for adding vectors and their ordered pairs representation should be illustrated via examples. [Their magnitudes |a| and |b| can be obtained by dropping the signs or replacing a negative sign, if present, by a positive one. The Pythagorean theorem r =sqrt [|a|***2 + |b|**2] gives a means to compute the distance r in the polar coordinate or distance-angle [r, theta] description of the equivalent distance-angle description. The angle theta can be measured [or obtained from trigonometry].
   

Using Polar Coordinates [length and direction] to Describe Movements:

Again, on the map or plan the movement is described by a vector [or arrow] with tail at the initial endpoint A and head at the terminal endpoint B. The terminal point of the movement is determined by its initial point A with the direction and length of the arrow or vector leaving the initial point. The direction may be given using an angle with respect to a horizontal ray in a plan or or using points of a compass on map.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science