Math-in-CTE Lesson Plan Template

Transcription

1 Lesson Development Math-in-CTE Lesson Plan Template Lesson Title: Basic Geometric Concepts Lesson # Author(s): Phone Number(s): Address(es): Juan Carlos Martínez jcmartinez@dadeschoolsnet Bergman Jose Arias Occupational Area: Technology Education bergmanjarias@dadeschoolsnet CTE Concept(s): Basic Geometry Math Concepts: Basic Geometry Terms 1- The students will be able to identify basic geometric shapes 2- The students will become familiar with basic geometric terms Lesson Objective: 3- The students will be able to identify the basic instruments 45 triangle, triangle, compass, straight edge, scale Supplies Needed: Paper and pencil THE "7 ELEMENTS" 1 Introduce the CTE lesson Whether we realize it or not, we see geometry in use every day The design of the aircraft flying overhead and the design of the automobile that passes on the street is based on various geometric shapes Buildings and bridges utilize squares, rectangles, triangles, circles, and arcs in their design and construction Weather hand drafting or using computer aided design software; geometry is the basis that makes it all possible 2 Assess students math awareness as it relates to the CTE lesson 1- What are the basic tools used in drafting? 2- What are the basic shapes in geometry? TEACHER NOTES (and answer key) 1- Triangles: 45, 45, 90 triangle; 30, 60, 90 triangle; protractor; straight edge: T-square, triangles 2- Triangles: right triangle, isosceles triangle, scalene triangle Quadrilaterals: rectangle, rhomboid, trapezoid Regular Polygons: equilateral triangles, National Research Center for Career and Technical Education

2 Lesson Development square, pentagon, hexagon, heptagon, octagon 3 Work through the math example embedded in the CTE lesson Lesson on Basic Geometry Objective: Identify Basic Terms in Geometry Lines, Points, Intersection, Line segments, Rays, Endpoints, Parallel lines Lines A line is one of the basic terms in geometry We may think of a line as a "straight" line that we might draw with a ruler on a piece of paper, except that in geometry, a line extends forever in both directions We write the name of a line passing through two different points A and B as "line AB" or as, the two-headed arrow over AB signifying a line passing through points A and B Example: The following is a diagram of two lines: line AB and line HG This may need to be taught over a time frame of two to three days depending on the length of your periods The arrows signify that the lines drawn extend indefinitely in each direction Points A point is one of the basic terms in geometry We may think of a point as a "dot" on a piece of paper We identify this point with a number or letter A National Research Center for Career and Technical Education

3 Lesson Development point has no length or width, it just specifies an exact location Example: The following is a diagram of points A, B, C, and Q: Intersection The term intersect is used when lines, rays, line segments or figures meet, that is, they share a common point The point they share is called the point of intersection We say that these figures intersect Example: In the diagram below, line AB and line GH intersect at point D: Example: In the diagram below, line 1 intersects the square in points M and N: National Research Center for Career and Technical Education

4 Lesson Development Example: In the diagram below, line 2 intersects the circle at point P: Line Segments A line segment is one of the basic terms in geometry We may think of a line segment as a "straight" line that we might draw with a ruler on a piece of paper A line segment does not extend forever, but has two distinct endpoints We write the name of a line segment with endpoints A and B as "line segment AB" or as Note how there are no arrow heads on the line over AB such as when we denote a line or a ray Example: The following is a diagram of two line segments: line segment CD and line segment PN, or simply segment CD and segment PN National Research Center for Career and Technical Education

5 Lesson Development Rays A ray is one of the basic terms in geometry We may think of a ray as a "straight" line that begins at a certain point and extends forever in one direction The point where the ray begins is known as its endpoint We write the name of a ray with endpoint A and passing through a point B as "ray AB" or as Note how the arrow heads denotes the direction the ray extends in: there is no arrow head over the endpoint Example: The following is a diagram of two rays: ray HG and ray AB Endpoints An endpoint is a point used to define a line segment or ray A line segment has two endpoints; a ray has one National Research Center for Career and Technical Education

6 Lesson Development Example: The endpoints of line segment DC below are points D and C, and the endpoint of ray MN is point M below: Parallel Lines Two lines in the same plane which never intersect are called parallel lines We say that two line segments are parallel if the lines that they lie on are parallel If line 1 is parallel to line 2, we write this as line 1 line 2 When two line segments DC and AB lie on parallel lines, we write this as segment DC segment AB Example: Lines 1 and 2 below are parallel Example: The opposite sides of the rectangle below are parallel The lines passing through them never meet National Research Center for Career and Technical Education

7 Lesson Development Objective: Identify and measure angles using a protractor See PDF: Lesson on Protractors (click here for the PDF) Objective: Identify figures and polygons Polygon A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others The following are examples of polygons: The figure below is not a polygon, since it is not a closed figure: The figure below is not a polygon, since it is not made of line segments: National Research Center for Career and Technical Education

8 Lesson Development The figure below is not a polygon, since its sides do not intersect in exactly two places each: Regular Polygon A regular polygon is a polygon whose sides are all the same length, and whose angles are all the same The sum of the angles of a polygon with n sides, where n is 3 or more, is 180 (n - 2) degrees The following are examples of regular polygons: The following are not examples of regular polygons: Vertex National Research Center for Career and Technical Education

9 Lesson Development 1) The vertex of an angle is the point where the two rays that form the angle intersect 2) The vertices of a polygon are the points where its sides intersect Triangle A three-sided polygon The sum of the angles of a triangle is 180 degrees Equilateral Triangle or Equiangular Triangle A triangle having all three sides of equal length The angles of an equilateral triangle all measure 60 degrees National Research Center for Career and Technical Education

10 Lesson Development Isosceles Triangle A triangle having two sides of equal length Scalene Triangle A triangle having three sides of different lengths National Research Center for Career and Technical Education

11 Lesson Development Acute Triangle A triangle having three acute angles Obtuse Triangle A triangle having an obtuse angle One of the angles of the triangle measures more than 90 degrees Right Triangle A triangle having a right angle One of the angles of the triangle measures 90 degrees The side opposite the right angle is called the hypotenuse The two sides that form the right angle are called the legs A right triangle has the special property that the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse This is known as the National Research Center for Career and Technical Education

12 Lesson Development Pythagorean Theorem Example: For the right triangle above, the lengths of the legs are A and B, and the hypotenuse has length C Using the Pythagorean Theorem, we know that A 2 + B 2 = C 2 Example: In the right triangle above, the hypotenuse has length 5, and we see that = 5 2 according to the Pythagorean Theorem National Research Center for Career and Technical Education

13 Lesson Development Quadrilateral A four-sided polygon The sum of the angles of a quadrilateral is 360 degrees Rectangle A four-sided polygon having all right angles The sum of the angles of a rectangle is 360 degrees Square A four-sided polygon having equal-length sides meeting at right angles The sum of the angles of a square is 360 degrees National Research Center for Career and Technical Education

14 Lesson Development Parallelogram A four-sided polygon with two pairs of parallel sides The sum of the angles of a parallelogram is 360 degrees Rhombus A four-sided polygon having all four sides of equal length The sum of the angles of a rhombus is 360 degrees National Research Center for Career and Technical Education

15 Lesson Development Trapezoid A four-sided polygon having exactly one pair of parallel sides The two sides that are parallel are called the bases of the trapezoid The sum of the angles of a trapezoid is 360 degrees Pentagon A five-sided polygon The sum of the angles of a pentagon is 540 degrees A regular pentagon: An irregular pentagon: Hexagon A six-sided polygon The sum of the angles of a hexagon is 720 degrees National Research Center for Career and Technical Education

16 Lesson Development A regular hexagon: An irregular hexagon: Heptagon A seven-sided polygon The sum of the angles of a heptagon is 900 degrees A regular heptagon: An irregular heptagon: Octagon An eight-sided polygon The sum of the angles of an octagon is 1080 degrees A regular octagon: An irregular octagon: National Research Center for Career and Technical Education

17 Lesson Development Nonagon A nine-sided polygon The sum of the angles of a nonagon is 1260 degrees A regular nonagon: An irregular nonagon: Decagon A ten-sided polygon The sum of the angles of a decagon is 1440 degrees A regular decagon: An irregular decagon: National Research Center for Career and Technical Education

18 Lesson Development Circle A circle is the collection of points in a plane that are all the same distance from a fixed point The fixed point is called the center A line segment joining the center to any point on the circle is called a radius Example: The blue line is the radius r, and the collection of red points is the circle Convex A figure is convex if every line segment drawn between any two points inside the figure lies entirely inside the figure A figure that is not convex is called a concave figure Example: The following figures are convex The following figures are concave Note the red line segment drawn between two points inside the figure that also passes outside of the figure National Research Center for Career and Technical Education

19 Lesson Development 4 Work through related, contextual math-in-cte examples Name Date Period Basic Geometry Worksheet Name each figure using letters and symbols Here is a link to the worksheet You may use this file to print using Microsoft Word Basic Geometry Worksheetdoc Here are the answers Basic Geometry Worksheet Solutionsdoc Draw and label an example of each 4point N 5line segment LD 6 National Research Center for Career and Technical Education

20 Lesson Development HP Classify each pair of lines as parallel, intersecting, or perpendicular Classify each angle as acute, obtuse, straight, or right Write polygon or not a polygon For a polygon, name the type of polygon and also write regular or not National Research Center for Career and Technical Education

21 Lesson Development regular Classify each triangle as isosceles, scalene, or equilateral and as right, acute, or obtuse Classify each triangle as isosceles, scalene, or equilateral by the lengths of its sides 211 mm, 11 mm, 215 in, 2 ft, 22 ft 220 cm, 20 cm, mm 4 5cm National Research Center for Career and Technical Education

22 Lesson Development 5 Work through traditional math examples The math worksheets for this purpose would be the same type of material as that of the CTE 6 Students demonstrate their understanding 1-Prepare a bulletin board display that illustrates the use of geometric shapes in buildings and bridges 2- Develop a list of everyday items that make use of geometric shapes For example, nut and bolt heads are round, square and hexagonal 7 Formal assessment Class discussion/ assessment You may visit for additional worksheets/assessments National Research Center for Career and Technical Education

23 Name Date Period Basic Geometry Assessment (Solutions) Name each figure using letters and symbols Point N Segment BV Ray ZJ Draw and label an example of each 4 point N N 5 line segment LD 6 HP Classify each pair of lines as parallel, intersecting, or perpendicular intersecting parallel parallel perpendicular

24 Classify each angle as acute, obtuse, straight, or right acute right obtuse acute Write polygon or not a polygon For a polygon, name the type of polygon and also write regular or not regular Polygon- Octagon Polygonquadrilateral not a polygon not a polygon Classify each triangle as isosceles, scalene, or equilateral and as right, acute, or obtuse isosceles, acute equilateral, acute right, acute isosceles, acute

25 Classify each triangle as isosceles, scalene, or equilateral by the lengths of its sides mm, 11 mm, 11mm in, 2 ft, 22 ft scalene cm, 20 cm, 20 cm equilateral equilateral

26 Name Date Period Basic Geometry Assessment Name each figure using letters and symbols Draw and label an example of each 4 point N 5 line segment LD 6 HP Classify each pair of lines as parallel, intersecting, or perpendicular Classify each angle as acute, obtuse, straight, or right

27 Write polygon or not a polygon For a polygon, name the type of polygon and also write regular or not regular Classify each triangle as isosceles, scalene, or equilateral and as right, acute, or obtuse Classify each triangle as isosceles, scalene, or equilateral by the lengths of its sides mm, 11 mm, 11mm in, 2 ft, 22 ft cm, 20 cm, 20 cm

28 Objective: Identify Basic Terms in Geometry Lines, Points, Intersection, Line segments, Rays, Endpoints, Parallel lines Lines A line is one of the basic terms in geometry We may think of a line as a "straight" line that we might draw with a ruler on a piece of paper, except that in geometry, a line extends forever in both directions We write the name of a line passing through two different points A and B as "line AB" or as, the two-headed arrow over AB signifying a line passing through points A and B Example: The following is a diagram of two lines: line AB and line HG The arrows signify that the lines drawn extend indefinitely in each direction Points A point is one of the basic terms in geometry We may think of a point as a "dot" on a piece of paper We identify this point with a number or letter A point has no length or width, it just specifies an exact location Example: The following is a diagram of points A, B, C, and Q:

29 Intersection The term intersect is used when lines, rays, line segments or figures meet, that is, they share a common point The point they share is called the point of intersection We say that these figures intersect Example: In the diagram below, line AB and line GH intersect at point D: Example: In the diagram below, line 1 intersects the square in points M and N: Example: In the diagram below, line 2 intersects the circle at point P:

30 Line Segments A line segment is one of the basic terms in geometry We may think of a line segment as a "straight" line that we might draw with a ruler on a piece of paper A line segment does not extend forever, but has two distinct endpoints We write the name of a line segment with endpoints A and B as "line segment AB" or as Note how there are no arrow heads on the line over AB such as when we denote a line or a ray Example: The following is a diagram of two line segments: line segment CD and line segment PN, or simply segment CD and segment PN Rays A ray is one of the basic terms in geometry We may think of a ray as a "straight" line that begins at a certain point and extends forever in one direction The point where the

31 ray begins is known as its endpoint We write the name of a ray with endpoint A and passing through a point B as "ray AB" or as Note how the arrow heads denotes the direction the ray extends in: there is no arrow head over the endpoint Example: The following is a diagram of two rays: ray HG and ray AB Endpoints An endpoint is a point used to define a line segment or ray A line segment has two endpoints; a ray has one Example: The endpoints of line segment DC below are points D and C, and the endpoint of ray MN is point M below: Parallel Lines Two lines in the same plane which never intersect are called parallel lines We say that two line segments are parallel if the lines that they lie on are parallel If line 1 is parallel to line 2, we write this as

32 line 1 line 2 When two line segments DC and AB lie on parallel lines, we write this as segment DC segment AB Example: Lines 1 and 2 below are parallel Example: The opposite sides of the rectangle below are parallel The lines passing through them never meet

33 Geometry Objective: Identify figures and polygons Polygon A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others The following are examples of polygons: The figure below is not a polygon, since it is not a closed figure: The figure below is not a polygon, since it is not made of line segments: The figure below is not a polygon, since its sides do not intersect in exactly two places each:

34 Geometry Regular Polygon A regular polygon is a polygon whose sides are all the same length, and whose angles are all the same The sum of the angles of a polygon with n sides, where n is 3 or more, is 180 (n - 2) degrees The following are examples of regular polygons: The following are not examples of regular polygons: Vertex 1) The vertex of an angle is the point where the two rays that form the angle intersect 2) The vertices of a polygon are the points where its sides intersect

35 Geometry Triangle A three-sided polygon The sum of the angles of a triangle is 180 degrees Equilateral Triangle or Equiangular Triangle A triangle having all three sides of equal length The angles of an equilateral triangle all measure 60 degrees Isosceles Triangle A triangle having two sides of equal length

36 Geometry Scalene Triangle A triangle having three sides of different lengths Acute Triangle A triangle having three acute angles Obtuse Triangle A triangle having an obtuse angle One of the angles of the triangle measures more than 90 degrees

37 Geometry Right Triangle A triangle having a right angle One of the angles of the triangle measures 90 degrees The side opposite the right angle is called the hypotenuse The two sides that form the right angle are called the legs A right triangle has the special property that the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse This is known as the Pythagorean Theorem Example: For the right triangle above, the lengths of the legs are A and B, and the hypotenuse has length C Using the Pythagorean Theorem, we know that A 2 + B 2 = C 2 Example:

38 Geometry In the right triangle above, the hypotenuse has length 5, and we see that = 5 2 according to the Pythagorean Theorem Quadrilateral A four-sided polygon The sum of the angles of a quadrilateral is 360 degrees Rectangle A four-sided polygon having all right angles The sum of the angles of a rectangle is 360 degrees

39 Geometry Square A four-sided polygon having equal-length sides meeting at right angles The sum of the angles of a square is 360 degrees Parallelogram A four-sided polygon with two pairs of parallel sides The sum of the angles of a parallelogram is 360 degrees Rhombus A four-sided polygon having all four sides of equal length The sum of the angles of a rhombus is 360 degrees

40 Geometry Trapezoid A four-sided polygon having exactly one pair of parallel sides The two sides that are parallel are called the bases of the trapezoid The sum of the angles of a trapezoid is 360 degrees Pentagon A five-sided polygon The sum of the angles of a pentagon is 540 degrees A regular pentagon: An irregular pentagon: Hexagon A six-sided polygon The sum of the angles of a hexagon is 720 degrees A regular hexagon: An irregular hexagon:

41 Geometry Heptagon A seven-sided polygon The sum of the angles of a heptagon is 900 degrees A regular heptagon: An irregular heptagon: Octagon An eight-sided polygon The sum of the angles of an octagon is 1080 degrees A regular octagon: An irregular octagon: Nonagon A nine-sided polygon The sum of the angles of a nonagon is 1260 degrees

42 Geometry A regular nonagon: An irregular nonagon: Decagon A ten-sided polygon The sum of the angles of a decagon is 1440 degrees A regular decagon: An irregular decagon: Circle A circle is the collection of points in a plane that are all the same distance from a fixed point The fixed point is called the center A line segment joining the center to any point on the circle is called a radius Example:

43 Geometry The blue line is the radius r, and the collection of red points is the circle Convex A figure is convex if every line segment drawn between any two points inside the figure lies entirely inside the figure A figure that is not convex is called a concave figure Example: The following figures are convex The following figures are concave Note the red line segment drawn between two points inside the figure that also passes outside of the figure

44 Lesson 9-3 Example 1 Measure Angles a Use a protractor to measure MNO Step 1 Place the center point of the protractor s base on vertex N Align the straight side with side NO so that the marker for 0 is on one of the rays Step 2 Use the scale that begins at 0 at NO Read where the other side of the angle MN, crosses this scale The measure of angle MNO is 50 Using symbols, m MNO = 50 b Find the measures of BCE, DCE, and ACB m BCE = 155 m DCE = 70 m ACB = 25 CE is at 0 on the right CE is at 0 on the right CA is at 0 on the left Example 2 Draw Angles Draw G having a measure of 110 Step 1 Draw a ray with endpoint G Step 2 Place the center point of the protractor on G Align the mark labeled 0 with the ray Step 3 Use the scale that begins with 0 Locate the mark labeled 110 Then draw the other side of the angle Example 3 Classify Angles Classify each angle as acute, obtuse, right, or straight a b c m ABC < 90 m DEF > 90 m GHI = 90 So, ABC is acute So, DEF is obtuse So, GHI is right

45 Example 4 HIKING Use Angles to Solve a Problem The map of a hiking trail at a state park indicates a 31 hill Classify this angle Since 31 is greater than 0 and less than 90, the angle is acute