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3D Shapes Names of 3D solids Prisms Volume Surface Area Plans, elevations and nets Isometric Drawing Units of measure Length, area and volume Capacity / Mass Metric and imperial measure Conversion / conversion graphs Real-life graphs Speed, distance time Parts of Circumference Area Circle geometry Equation of a circle Right-angled triangles Finding missing side lengths Angles and sine, cosine and tangent ratios KS3 Shape, Space and Measure Constructions Triangles Similar & Congruent Shapes Bisectors Loci Angles Angles on a straight line Angles at a point Parallel lines and transversals alternating / corresponding /opposite angles Supplementary angles Polygons: interior & exterior angles Extension work: Bearings Lines & Angles Line Segments Vertical / Horizontal Perpendicular / Parallel Types of angle Estimating measuring and drawing Direction of turn Compass directions Describing angles (90o= ¼ of turn) Pythagoras and Trigonometry Perimeter and Area Dimensions (including volume) Units of measure (including volume) Counting squares Intrinsic and Extrinsic information Rectangles, triangles and compound shapes Using Formulae (Rectangle, Triangle, Trapezium, Parallelogram) Circles Properties of 2D Shapes Names of polygons up to 10 sides Special Quadrilaterals and Triangles Geometric Properties Tessellation Transformations Basic congruent & similar shapes Coordinate geometry Reflection (include lines of symmetry) Rotation (include rotational symmetry) Translation (including vector notation) Enlargement All topics can be covered by the end of year 8. Shape 5: Constructions and Loci Constructions and Loci Must Should Understand that a construction requires the use of correct geometrical equipment including a ruler, pair of compasses and angle measurer or protractor Construct a triangle accurately given; Construct a triangle accurately The length of two sides and the angle given the lengths of all three sides formed at the intersection of those two using a pair of compasses and ruler sides or; The length of one side and the two angles at the end of that side Understand the term, ‘congruent’ and Know that congruent shapes are identify congruent shapes formed after transformations involving rotations, reflections and enlargements Understand the term, ‘similar’ and Know that ‘similar’ shapes have identify similar shapes corresponding angles of the same size and lengths in the same ratio Know the terms, ‘perpendicular’ and ‘bisect’ Construct the perpendicular bisector of a line segment Construct the bisector of an angle Know that a rhombus can be constructed by constructing the perpendicular of a line segment Understand the terms, ‘locus’ and ‘loci’ Construct the locus of all points equidistant from a point or a line segment Construct an angle of 60o Could Construct tessellations using appropriate congruent shapes (such as regular hexagons) Construct scale factor enlargement of given shapes (such as triangles) Understand that the area of a similar shape will be equivalent to n2 × the area of the original shape, where the original shape has been enlarged by scale factor, n Construct the perpendicular to a line which intersects the line segment at a given point Know that an equilateral triangle can be constructed by constructing an angle of 60o Construct the locus of all points equidistant from two given points Key Words: construct, sketch, ruler, protractor, angle measurer, pair of compasses, vertex, vertices, side, line segment, congruent, similar, enlarge, scale factor, perpendicular, bisect / bisector, locus, loci, equidistant Starters: Sorting activity with shapes that have been constructed and shapes that have been sketched or drawn Identify congruent shapes Identify similar shapes Practise measuring and drawing line segments and angles What shape is this? How do we know? eg recapping properties of shapes (such as rhombus or kite) Activities: Construct congruent / similar shapes Loci work: eg goat problems as in 10 ticks worksheets Construct a diagram involving bearings / scale drawings Shape 5: Constructions and Loci Plenaries: Learning framework questions: - What is the difference between a construction and a sketch or drawing? - Following an enlargement, how many times does the original shape fit inside the enlargement? Is this what you expected? - What does ‘bisect’ actually mean? - What equipment would you expect to use for a construction? Resources: Pair of compasses / angle measurer / protractor / ruler 10 ticks worksheets Possible Homeworks: Investigate the golden ratio and shapes / patterns which follow the ratio By re-examining the properties of shapes, construct common triangles, quadrilaterals and other polygons Try to construct a 45 degree / 30 degree / 15 degree angle Investigate the pattern for the number of parts that are formed by bisecting a line segment, then bisecting each bisected part and so on. Describe the locus for an everyday moving object. Does it follow a predictable pattern or is it random / chaotic? Teaching Methods/Points: CONSTRUCTION Students should know that most constructions require the use of a pencil, a ruler, a pair of compasses and an angle measurer or protractor. A sketch or drawing will still require the use of a pencil and ruler in Maths, but the important difference between a sketch and a construction is the accuracy of the lengths and angles in the diagram. Construction lines should remain clear, the geometric equivalent of ‘showing your workings / process’. The topic of construction, in particular, provides an opportunity to remind students of some important 2D shapes and of their important properties. For example; A rhombus has diagonals which perpendicularly bisect each other. A kite has just one diagonal which perpendicularly bisects the other. An equilateral triangle has three angles of 60o. It is also important to ensure students are familiar and conversant with the following terms; Perpendicular lines are two lines that meet each other at right angles (90 degrees) Bisect means to divide or cut something into 2 equal parts, so a bisector is a line which cuts another line into 2 equal parts. Construct means use appropriate geometric equipment to make an accurate drawing. Locus (plural loci) is the pathway of a moving object CONSTRUCTING A TRIANGLE The simplest and most familiar task in terms of using prior skills is that of constructing a triangle requiring precise measurement of length and measurement of angle. This is covered as part of ‘Shape 1’ as a means to consolidate skills drawing line segments and angles. In order to develop construction skills, tasks can be structured as follows; Shape 5: Constructions and Loci 1) To construct a triangle given the length of line segment, AB, the size of angle BAC and the length of AC; Required: Ruler, angle measurer or protractor Ensure students are required to measure and draw line segments in both centimetres (eg 3.4 centimetres) and millimetres (eg 34 millimetres) and stretch them to make judgements on even more precise measurements such as (34.5 millimetres). The accuracy of the angle is important and this provides an opportunity to practise using angle measurers and protractors. 2) To construct a triangle given the length of line segment, AB and the size of angles BAC and ABC; Required: Ruler, angle measurer or protractor Students should be encouraged to draw line segments AC and BC faintly at first (and beyond the length that they are likely to need in order to ensure the line segments intersect) once line segment AB has been accurately drawn and the angles have been measured. 3) To construct a triangle given the length of three line segments, AB, BC, and AC. Required: Ruler, pair of compasses This construction relies on the understanding that the intersection of the arcs drawn from centre points A and B respectively (once line segment AB has been drawn accurately), with radii equal to the lengths BC and AC respectively, provides the location of point C. They must avoid the temptation to simply estimate the location of point C by using a ruler and a trial and error approach! Construction of triangle ABC such that AB = 4.3 cm, AC = 3.9cm and BC = 2.3 cm C 3.9 cm 2.3 cm A 1. Draw line segment AB, length 4.3 cm 2. Draw arc with centre A, radius 3.9 cm 3. Draw arc with centre B, radius 2.3 cm B 4.3 cm OTHER CONSTRUCTIONS Constructing a perpendicular bisector of a line Draw line segment AB. Set the radius of the pair of compasses to be more than half of length AB. Put the point on A and draw an arc above and below the line. Put the point on B and draw an arc above and below the line. Make sure the two arcs cross each other. Label the points where the two arcs cross, C and D then draw line segment CD. CD is the perpendicular bisector of AB. D A B C Shape 5: Constructions and Loci Note: Shape ADBC is a rhombus. Constructing a perpendicular line, MN, which meets line segment, AB at a specific point, P Draw line segment AB. Identify point P which is along the length of line segment AB. Measure and mark an equal distance away from point P in either direction along line segment AB. Ensure the radius of the pair of compasses is sufficient to allow for the arcs to intersect. Put the point on the marks (placed an equal distance away from point P) and draw arcs above and below the line. Make sure the two arcs cross each other. Label the points where the two arcs cross, M and N then draw line segment MN. MN is a line perpendicular to AB which meets AB at a specific point, P. M A P B Note: Shape AMBN is a kite. N Constructing a bisector of an angle Draw angle ABC. Measure and mark an equal distance along line segments BA and BC from corner point B. Set the radius of the pair of compasses to ensure that the arcs will intersect. Draw arcs from the marked points (an equal distance away from point B) so that the arcs intersect. Draw a line which passes through B and D, the point of intersection of the two arcs. BD is the bisector of angle ABC. A D B C Constructing a 60 degree angle Draw line segment AB. Set the radius of the pair of compasses to exactly the same length as the length of line segment AB. Draw an arc from point A. Draw an arc from point B, such that it intersects the first arc at point C C Draw line segment AB. Note: Shape ABC is an equilateral triangle A Shape 5: Constructions and Loci 60o B SIMILARITY AND CONGRUENCE Congruent shapes have exactly the same size and shape (as explained in Shape 3: Transformations). Students must understand that shapes that have been translated, reflected and rotated are always congruent. Similar shapes have corresponding angles of the same size, and corresponding lengths in the same ratio. Any scale factor enlargement of a shape creates a similar shape. For example; scale factor 2 enlargement Here are examples of other shapes that are similar (with corresponding angles of the same size and corresponding sides in the same ratio.) These examples are also given in the notes for enlargement (Shape 4: Transformations) eg A E D B C F Shapes A and B are similar because the corresponding angles are the same size and all of the lengths of Shape B are exactly twice the size of the lengths of Shape A. The ratio of the lengths is 1 : 2 Shapes C and D are similar because the corresponding angles are the same size and all of the lengths of Shape D are three times the size of the lengths of Shape C. The ratio of the lengths is 1 : 3 Shapes E and F are similar because the corresponding angles are the same size and all of the lengths of Shape F are 2.5 times the size of the lengths of Shape E. The ratio of the lengths is 1 : 2.5 [this can be written as 2 : 5] Every length is twice as long. It is useful to investigate the resulting increase in area, leading to the conclusion that for a scale factor enlargement of n, the area will be n2 times as big. Similarly, applying the same investigative approach to volume will lead to the conclusions that a scale factor enlargement of n for a solid will lead to a volume which is n3 times as big. Incorporate activities which require students to identify similar shapes (by considering the ratio of their lengths. While work on constructions is not directly linked to ‘congruent’ and ‘similar’ shapes (which are actually dealt with as part of the topic of ‘Transformations’), this is an opportunity to consolidate student understanding and extend the nature of the tasks. For example, given a triangle ABC, such that AB = 3cm, BC = 4 cm and angle ABC is 60 degrees, construct the enlargement of triangle ABC by scale factor 2. Shape 5: Constructions and Loci LOCI A locus (or loci in the plural) is defined as the path of a moving point. Students must understand the term; equidistant meaning “equal distance away” You have to be able to draw a locus given a description of the rule that the locus follows, and you have to be able to describe the rule that a locus follows. Locus of points a fixed distance from point A A Locus of points a fixed distance from line AB A B B Construction for locus of points equidistant from points A and B A B Construction of locus of points equidistant from AB and AC A C The topic of loci provides an opportunity to apply construction skills to a range of real-life problems. While the above examples demonstrate specific skills that students should learn when drawing a locus, and illustrate the type of descriptions that students need to give in words given a diagram of a locus, the topic can be used to deal with other real-life problems typified by a goat in a field. These problems generally require students to use their geometric equipment to construct specific scenarios, such as a goat tethered at a particular point with a rope of a certain length. These problems are important in developing geometric reasoning skills! Shape 5: Constructions and Loci Constructions: Help Sheet Construct means use appropriate geometric equipment to make an accurate drawing. Perpendicular lines are two lines that meet each other at right angles (90 degrees) Bisect means to divide or cut something into 2 equal parts Constructing a triangle given the length of three line segments, AB, BC, and AC. Construction of triangle ABC such that AB = 4.3 cm, AC = 3.9cm and BC = 2.3 cm C 3.9 cm 4. Draw line segment AB, length 4.3 cm 5. Draw arc with centre A, radius 3.9 cm 6. Draw arc with centre B, radius 2.3 cm 2.3 cm A B 4.3 cm Draw line segment AB. Set the radius of the pair of compasses to be more than half of length AB. Put the point on A and draw an arc above and below the line, then do the same at B so that the arcs intersect each other Label the points where the two arcs cross, C and D then draw line segment CD. CD is the perpendicular bisector of AB. Constructing a perpendicular bisector of a line D A Note: Shape ADBC is a rhombus. B C Constructing a perpendicular line, MN, which meets line segment, AB at a specific point, P M A P Note: Shape AMBN is a kite. B N Draw line segment AB and locate point P on the line segment. Measure and mark an equal distance away from point P in either direction along line segment AB. Put the point on the marks (placed an equal distance away from point P) and draw arcs above and below the line. Make sure the two arcs cross each other. Label the points where the two arcs cross, M and N. MN is a line perpendicular to AB which meets AB at point, P. Constructing a bisector of an angle A Draw angle ABC. Measure and mark an equal distance along line segments BA and BC from corner point B. Draw arcs from the marked points (an equal distance away from point B) so that the arcs intersect. Label the point of intersection, D. BD is the bisector of angle ABC. D B C Constructing a 60 degree angle C Draw line segment AB. Make sure the radius of the pair of compasses is equal to the length of line segment AB. Draw arcs using both points A and B as centres so that the arcs intersect. Label the point of intersection, C. Angle CAB is 60o. Note: Shape ABC is an equilateral triangle A Shape 5: Constructions and Loci 60o B Loci: Help Sheet A locus (or loci in the plural) is the pathway of a moving point. Equidistant meaning “equal distance away” Examples of common loci: Locus of points a fixed distance from point A A Locus of points a fixed distance from line AB A B B Construction for locus of points equidistant from points A and B A B Construction of locus of points equidistant from AB and AC A C LOCI AND GRAPHS (equation of a circle) 10 Using a coordinate grid, you may be asked to construct the locus of a circle given an ‘equation’, or describe the locus of a circle by writing the equation. The equation of a circle is given by the equation x2 + y2 = r2 So, for example; x2 + y2 = 25 is a circle with centre on the origin, r2 = 25, so radius is 5. The locus is shown here Shape 5: Constructions and Loci 5 y 5 x -10 -5 5 -5 5 -5 -5 -10 10