s er ap eP m e tr .X w w w om .c Scheme of work – Cambridge IGCSE® Mathematics (US) 0444 Unit 7: Co-ordinate geometry (Extended) Recommended prior knowledge All of Core and particularly Core 7. Only those parts of the learning objectives or notes and exemplars not included in the core units are itemised, so this document should be read alongside the core document. As there are links to it Extended 1 should be completed too. Context There are five Core geometry units and this is the fourth of five Extended geometry units. Once Core 7 and the other prior experience for Core 7 and Extended 1 are completed this unit can be slotted in at any point. It is probably best taught as a whole but used to revise some of the Core 7. Outline The unit extends the knowledge of Core 7 so be aware that examination questions that relate to aspects of Core 7 not listed here may have a greater degree of challenge as they combine with other areas of mathematics. This unit covers how to find a point on a line split in a given ratio, looking at linear equations of the form ax + by = d and the slope of a perpendicular to a line passing through a given point. Syllabus ref Learning objectives Suggested teaching activities 7.2 As Core curriculum Notes and exemplars e.g. use co-ordinates to compute the perimeters of polygons and areas of triangles using the distance formula. CCSS: G-GPE6 7.3 CCSS: G-GPE7 v1 2Y01 Find the point on a directed line segment between two given points that partitions the segment in a given ratio General guidance In the core unit the skill was explored and linked to Pythagoras. It should now be linked to work with radicals - Unit 1 Extended (1.8) when summing the perimeter of polygons and finding areas of triangles. General guidance Students need to understand this in geometry before they understand this in coordinates. i.e. if two lines in a triangle are divided in the same ratio then the line joining the two points is parallel to the third side of the triangle. So if a right angled triangle is formed either the ratio on the hypotenuse of the triangle is the same as on the x or y height. Cambridge IGCSE Mathematics (US) 0444 Learning resources http://rpmullen.com/standards/geometry /oncore/geounit8_3.PDF 1 Syllabus ref Learning objectives Suggested teaching activities Learning resources 10 8 splitting AB in ratio 1:2 is the same as splitting AD or BD in the ratio 1:2 6 x coordinate of p AD is 14-2 4 = 12 1:2 = 4:8 so x coordinate is 4 more than 2 = 6 2 p 5 5 2 4 B A E 10 15 y coordinate of p BD is 5- 1 = 6 1:2 = 2:4 so y coordinate of p is 2 more than 20 D C 1 = 1 6 The only remaining idea is for students to know whether to start at A or B when working out the split. e.g. If the line had to be split the other way it would have been called BA and the x and y distances subtracted from the B coordinates. 7.5 Interpret and obtain the equation of a straight line as ax + by = d (a, b, and d are integers) Notes and exemplars e.g. obtain the equation of a straight line graph given a pair of coordinates on the line. General guidance Obtaining the equation of a line and plotting them when b = 0 and y is alone on one side of the equation has been tackled in Core 7. Past Paper 41 June 2011 Q9 (syllabus 0580) Past Paper 23 June 2011 Q14 (syllabus 0580) www.purplemath.com/modules/linprog4 .htm The most common way b≠ 0 comes about when the equation is multiplied by the denominator of a fractional gradient and terms are rearranged so that there are no negatives. Students need some practice to see this connection first, and obtaining lines for this form can be tackled this way. To draw a line given in this form challenges students who want to create a table of values for x and y and have difficulty rearranging the equation. However, at this level they should know that only two points are needed to draw a line, but that it is better to plot 3 so that there is a check for errors. Therefore students also need to realise that this should be a v1 2Y01 Cambridge IGCSE Mathematics (US) 0444 2 Syllabus ref Learning objectives Suggested teaching activities Learning resources straight line. The two obvious pairs of coordinates to plot are when x = 0 to obtain the point the line crosses the y axis, and when y = 0 to find the point where the line crosses the x-axis. The third point is more problematic. Using x=1 works for most cases, but when all three points are close they need to choose a value for x of 5 or 10. This last step is the one that requires practice so that students develop sufficient experience to choose a realistic value. Teaching activities Give students three lines to draw and ask them to find the coordinates of the intersections (tie to solving simultaneous equations Core 2 (2.6)). 7.6 CCSS: G-GPE5 Slope of perpendicular line. Find the equation of a line perpendicular to a given line that passes through a given point Look at problems where the solution is in the space enclosed by the three lines by also looking at inequalities. General guidance This has been introduced in the Core unit and simply requires some formalization. http://nrich.maths.org/763 http://nrich.maths.org/7031 Teaching activities To combine several of the parts of this unit ask students to plot a rectangle, given one line, one vertex off the line, and the opposite vertex on the line. They must give the equations of the other three lines. They will have to pull together knowledge about parallel and perpendicular lines (gradients) going through a given point, even if they manage to find the fourth point by eye. Similar problems with Kites and Rhombi can be produced given the equation of the diagonal and some of the vertices. v1 2Y01 Cambridge IGCSE Mathematics (US) 0444 3