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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 7: Co-ordinate geometry (Core)
Recommended prior knowledge
Students should know how to plot points in all four quadrants.
Context
This is the fourth and fifth geometry units but it could equally be the third of three algebra units. There are clearly overlaps between this unit and Unit 2 (2.13), and
Units 3, 4, 5, 6, 8 and 10. There are links between Unit 8 and 7 and the order is a choice. However, if Unit 7 is taught before Unit 8 then a return to Unit 7 to link
Pythagoras and tangent to 7.3, 7.4, 7.5 is a possibility. 7.2 and 7.3 can be used to revise aspects of Units, 4, 5, and 6. 7.4 should be linked to the correlation name
types used for scatter graphs in Unit 10. The skills here need to be taught but could be taught in the linked units rather than treating this as a unit in its own right, or
this unit could be used to revisit those other skills in the other units. If it is broken into blocks
• Block 1 – 7.1 should be taught early on in the course probably after Unit 1 to make the skill available for all other units
• Block 2 - 7.2 and 7.3 could be taught separately from the rest of the unit after Unit 8
• Block 3 - 7.4 to 7.6 could be taught after Unit 2 or 3
Students who are following the extended syllabus will move through this faster but need to have all these skills in place before working on the extended units, or
applying them in other areas of mathematics.
Outline
This unit deals with the technical skills of
• plotting and reading coordinates in all four quadrants,
• finding the distance between points,
• gradient of a line,
• midpoint of a line
• finding the equations of a line by y= mx + c
• understanding lines that are parallel or at right angles to a given line
To facilitate the modelling graphic calculators or graphing packages should be used. The latter skills are delivered through observation of patterns in sets of graphs
so the use of graphics calculators or graphing packages is expected. Within the suggested teaching activities ideas are listed to identify and remediate
misconceptions and to pull learning through to the required standard. The learning resources give both teaching ideas, summaries of the skills and investigative
problems to develop the problem solving skills and a depth of understanding of the mathematics, through exploration and discussion.
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
7.1
Plotting of points and reading
from a graph in the Cartesian
plane
General guidance
Students should be able to plot points in all four quadrants but there may
be some who still plot points the wrong way around. The two resources
will remediate.
www.mathplayground.com/spaceboyres
cue.html
www.mathsisfun.com/t_rex.html
Reading coordinates can be practiced with any graphing activity.
Students need to realise that though a graph may have been plotted
using a range of values for x that are integers x= 1.5 or 5.7 can also be
found from the graph. Link to unit 3 (3.3) functions.
7.2
Distance between two points
CCSS:
G-GPE7
7.3
CCSS:
G-GPE6
Teaching activities
Give 3 pairs of coordinates for a quadrilateral and ask students to find
and name the missing one. This can be a review of past knowledge of
quadrilaterals and does not require students to have completed Unit 4.
Notes and exemplars
Questions on this topic would be structured via diagrams.
General guidance
There are several different possible skills involved in this.
Horizontal and vertical distances can be dealt with as subtraction
Diagonal distances can be measured and linked to work on scale
This can be linked to Pythagoras by considering horizontal and vertical
changes and the difference between the x-coordinates and the
y-coordinates. This therefore needs to be linked to Pythagoras in Unit 8.
Midpoint of a line segment
Teaching activities
Give students quadrilaterals to plot and ask for the areas of shapes
either as a whole or as two triangles (revision of areas Unit 6).
Notes and exemplars
Questions on this topic would be structured via diagrams.
General guidance
Students should explore this by finding the midway points and then
examining the coordinates so that they deduce they find the midway (or
average) of the x-coordinates and then the y-coordinates.
Teaching activities
Give students quadrilaterals to plot and ask them to find the coordinates
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
of the points where diagonals cross (revision of properties of
quadrilaterals Unit 4) or give coordinates to plot regular polygons and
look at where the lines of symmetry cross edges revision of Unit 5).
Record the co-ordinates of the vertices and intersection to establish a
pattern for finding the midpoint. Similarly look at the ways diagonals
intersect for some quadrilaterals and again record the coordinates
vertices and intersection of diagonals and establish a pattern for the
midpoint.
7.4
7.5
Slope of a line segment
Interpret and obtain the equation
of a straight line as y = mx + c
General guidance
This should be defined as the horizontal distance divided by the vertical
distance and can be linked to tangent (trigonometry unit 8).
Students need to appreciate the difference between positive and
negative gradients and to see the link to the descriptions used for
correlation in scatter graphs (Unit 10).
Notes and exemplars
e.g., obtain the equation of a straight line graph given a pair of coordinates on the line.
http://nrich.maths.org/5725
Past Paper 31 June 2011 Q12
(syllabus 0580)
General guidance
This is best developed using either graphics calculators or graphing
packages – there are some free ones that can be downloaded.
Teaching activities
Ask students to plot sets of graphs on a graphics calculator or in a
graphing package that have no constant and a positive gradient. Link
back to 7.4 and ask students to read off values and to realise that for
each increase of 1 in the x-coordinate the y-coordinate increases by the
coefficient of x in the equation of the line- link to Unit 2 (2.13). If different
groups of students are given different sets it makes the conclusion more
powerful.
Get students to plot pairs of graphs like y=2x and y=-2x on a graphics
calculator or in a graphing package and to realise the y axis is the line of
symmetry.
Get students to plot sets of graphs on a graphics calculator or in a
graphing package with varying gradients but the same constant so that
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
they deduce that c is the intercept with the y axis.
Encourage students to deduce the rules for y=mx + c for themselves.
Give students sets of plotted graphs and equations and ask them to
match them.
Ask them to find the equations of lines from graphs.
7.6
Slope of parallel line
CCSS:
G-GPE5
Find the equation of a line
parallel to a given line that
passes through a given point
General guidance
Students need to realise that parallel lines have the same gradient
Students need to realise the link between the gradients of perpendicular
lines.
http://nrich.maths.org/5610
Students need to be able to find an equation of a line that is parallel to
another line or perpendicular to it that goes through a particular point, by
deducing the gradient and substituting the point to find the constant.
Teaching activities
Ask students to plot sets of graphs on a graphics calculator or in a
graphing package with the same gradient but different intercepts /
constants. Ask students to deduce that parallel lines have the same
gradient.
Give students a pairs of lines that intersect and ask them to suggest two
other lines that would enclose a parallelogram. Collect in results from
students to get a number of different possible answers, but show that all
the results contain two sets of parallel lines.
Ask students to plot a number of rhombi and kites and to find the
equations of the diagonals – these will need to be extended to cross the
y axis. Ask them to deduce what happens to the equations of lines that
are perpendicular to one another to establish the rule.
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