Lesson 1: Introduction

Transcription

1 Lesson 1: Introduction Geometry curriculum in primary school How is it?, How is it applied? (LOMCE-Block 4) How should be? Tradition says that at the door of Plato s Acadamy was engraved Let no one ignorant of geometry enter

2 Introduction: the value of geometry The geometry should be the entrance door to the logic thinking and reasoning. It is a useful tool as a support and understanding of other concepts. The great advantage of elementary geometry is that their subject are objects to build, play with, visualize.

3 First example: laying floor tiles, tiling An example: It is not difficult tilling the plane or a flat surface with square shapes (tessellation). Can you use other regular polygons to tesellate the plane? And, why not consider more than one type of polygon? Can we design a mosaic based on equilateral triangles, squares and regular hexagons? (The sizes of the polygons may vary)

4 Magnitude and measurement Learning to measure and calculate measurements is an essential human problems, it is on mathematics origin. Measurement is a vital part of elementary mathematics and is a prerequisite for middle and high scholl science classes and real life. The elementary school curriculum includes seven types of measurement (magnitude). Each is developed over several years, in cronological sequence: Length and time. Weight and capacity. Angles. Area. Volume.

5 Points,Lines and Segments We will stress and focus in precise definition However, we can not give a precise, accuarate definition of what is a point, or what is a line. To know more about math definition: Points and lines are the basic objects of geometry and are easily to explained to children. Understanding the concept with examples. Definition por ostensión.

6 Basic principles: Points, Lines and Segments (1) Through any two different points there is one and only one line. (2) If two different lines intersect, their intresection is a single point. A segment is the part of a line between two points on the line. A ray is a part of a line that has only one endpoint and extends indefinitely in one direction.

7 Points, Lines and Segments Primary books EGT, Parker s book Line Ray A B Segment AB A B Drawing a line using a ruler seems an easy task, but fine motor skills (dexterity) are required at must be practiced carefully.

8 Lenght and distance The lenght of a segment AB is the distance between the two point A and B. We will denote it by AB. Basic distance propierty (triangle inequality): If A, B and C are any three points of the plane, then AC AB + BC. And the equality occurs if and only if B is in the segment AC. (collinear) A B C In any triangle, the lenght of the longests side is less than the sum of the lenghts ot the other two sides

9 Length measurement It is important to understand that measuring something is comparing the object with some predefined unit. The study of measurement should start in K-2 choosing non-standard units (hands, feet, pens) in order to compare different lengths. In that way, we can observe that we need a common unit, and the meter can be introduced as the basic unit for lenght. When measuring different objects, we realize that we need bigger and smaller units. Measuring lenghts with the ruler is another activity that requires some practice.

10 International system of units - Changing units The standard approach (stairs are also popular): Km Hm dam meter dm cm mm And then we change units moving left and right (or up and down), adding or removing zeroes. A better alternative (related to reduction to the unit ): 1 Hm are 100 meters, therefore... Didactics comment: in most contries, units are introduced gradually. In the first steps, only m, cm and km are studied.

11 Weight and capacity As in every measurement process, we consider two steps: a) we choose a unit. b) we compare an object with that unit. Draft of proposal (common to all measures): 1) a) Direct comparation of objects. b) Measurement with non-standard units. c) Definition of standard unit and measurement with that unit. 2) Bigger and smaller units are introduced, and change of units is studied. 3) In each step, we should propose problems with various levels of difficulty.

12 A game: Weight and capacity

13 Angles Def: two rays with the same endpoint separete the plane into two infinitive regions, each of this is an angle. For primary students: An angle is made up of two rays (sides) that start at a common endpoint (vertex). Notation: R concave angle (reflex angle) convex angle x x Q P QR P

14 From a Spanish Primary 3 book Pedro Ramos. Mathematics II. Primary Education (bilingual track). Universidad de Alcala.

15 Right angle- perpendicular lines How can we define perpendicular lines? Two intersecting lines form four angles. Two lines are perpendicular if they form four equal angles. A right angle is the angle defined by two penpendicular lines.

16 Measuring angles 1st- a unit must be chosen. 2nd- other angles are expressed in term of that unit angle One entire circle can be defined as the unit of angular measure. A comple turn is one unit. one complete turn More commonly: a 1 degree. 360 right angle 1/4 of a turn of a complete turn is defined as one Angles are measured with a protractor. Careful instruction is required!

17 Angle classification 1 degree acute angle right angle obtuse angle straight angle reflex angle full rotation < between 90 and between 180 o and

18 Measuring angles with a protractor Learning to use a protractor correctly requires instruction and practice. What is the size of each angle?

19 Parallel and penpendicular lines Elementary Geometry is the study of relationships and properties of figures, measurements as lengths, angles, areas and volumes, and solve problems involving this aspects. We must focus on clear reasoning and simple geometry facts, there are no proofs, but facts are studied with clear and convincing arguments Precise definition are required.

20 Parallel and perpendicular lines ( page 29) Two lines are perpendicular if they intersect to form four equal angles (so each of them measure 90 ). Two segments or two rays are perpendicular if the lines containing them are perpendicular. Notation: AB CD. A C B D two lines are parallel if they do not intersect. Obs: this definition could be confuse in some cases, and does not suggest a way of drawing parallel lines. r Are r and s parallel? s

21 Parallel lines An alternative definition: two lines are parallel if both are penpendicular to third line. Notation: if r and s are parallel lines, then we write as r s. Children must draw, starting in third-fourth year parallel and perpendicular lines with set-square and a ruler. Exercise: using a set-square and a ruler draw a line parallel to the line l through the point R. draw a line perpendicular to the line l through the point S. S l R

22 Angles: basic definitions (p. 23) Two angles that share a vertex and have a side in common, but their interiors do not intersect are adjacent angles. Two angles are congruent if they have equal measure. Two angles, the sum of whose measures is 90, are called complementary angles. Two angles, the sum of whose measures is 180 are called suplementary angles C O B A complementary C O suplementary The bisector of an angle is the line that divide the angle into two congruent angles. B A C O P C O Q B A B A

23 A first property (p. 28) Two intersecting lines form four angles. Any two of the angles that are not adjacent are called vertical angles. c a b Property: two vertical angles are congruents (they have the same measure). a + c = 180 b + c = 180

24 Angles associated with parallel lines (p. 62) Let us consider two lines r and s that are intersected by a third one, t (t is called transversal). b a t r s then the sum of angles a and b determines how r and s are. More precise: a + b = 180 r and s are parallel a + b < 180 r and s intersect at the left a + b > 180 r and s interset at the right

25 Angles associated with parallel lines In figure (r and s are parallel, t transversal), there are a total of eight angles. d g e c b h f t a r s a = b = c = d e = f = g = h

26 Angles associated with parallel lines b a Angles a and b are called corresponding angles. Corresponding angles are congruents (they have the same measure). c d Angles c and d are colled alternate-interior angles. Alternate interior angles are congruent (same measure)

27 Exercises Prove that the sum of the measures of the three vertex angles in a triangle is 180. Help: draw a parallel line to any triangle side through the opposite vertex. Based on previous exercise and knowing that lines r and s are parallel, give the measure of angle a in the figure. r 65 a 47 s

28 Angles and perpendicular lines If the rays that defined two angles are mutually perpendicular, then the angles have the same measure (they are equal). a a = b b Problem: knowing that r and s are parallel, what is the measure of angle a in the figure? a? 110 r s