The document discusses how mathematics is often compartmentalized into distinct subject areas that students struggle to connect. It then provides details on analytic geometry, including how it connects algebra and geometry by using algebraic thinking to aid geometric understanding. Finally, it discusses various connections between algebra and geometry, such as how geometric concepts have algebraic counterparts and how algebraic and geometric results can be demonstrated in both domains.

1. The compartmentalization of mathematics into
distinct subjects areas, such as geometry,
arithmetic, algebra, calculus, functional analysis,
etc, often fails to present mathematics as a
coherent whole to students, a great deal of
whom find it quite difficult to make links
between these areas.

2. Analytic geometry
Referred to as Algebraic Geometry ; Coordinate Geometry
branch of algebra that is used to model geometric objects - points,
(straight) lines, and circles being the most basic of these.
Analytic geometry is a great invention of Descartes and Fermat: Connections
between algebra and geometry are not new but have their roots several
centuries ago.
It shows that the cognitive processes (symbolization, relations, and
abstraction), which underlie algebraic thinking, can be used to aid any of the
cognitive processes which constitute geometrical understanding (visualization
processes, construction processes, and reasoning), either separately or jointly.

3. Rationale for learning analytical geometry
• Links algebra and geometry into new mathematical structures and
techniques which are need for subsequent work in mathematics and
sciences
• As a tool to develop problem-solving skills
• Geometric problem can be translated into algebraic problem (straight
line equation represented as a graph)

4. Connections between algebra and geometry.
• The use of literal symbols in the form of variables, constants, labels, parameters
and so on abounds in algebra. Symbols abound in school geometry as well.
• Students work with variables and unknowns when generalizing results or solving
problems such as finding side or angle measures.
• Variables are used for making general statements, characterizing general
procedures, investigating the generality of mathematical issues, and handling
finitely or infinitely many cases at once (Schoenfeld & Arcavi, 1988).
• The idea of a variable is also used in geometry using a variable point as in
problems involving loci. Other simple uses of algebra in geometry as far as
symbols are concerned involve labelling points or vertices, sides, and angles of
figures.
• Some other connections between algebra and geometry in the high school
curriculum arise in problem solving and modelling, and in the various modes of
representations – graphical, algebraic, and numeric.

5. Many of the concepts in geometry have their
counterpart in algebra.
For example,
• a point in geometry corresponds to an ordered pair (x, y) of numbers in
algebra,
• a line corresponds to a set of ordered pairs satisfying an equation of the
form ax + by = c (a, b, c 𝜖 R),
• the intersection of two lines to the set of ordered pairs that satisfy the
corresponding equations, and
• a transformation corresponds to a function in algebra (National Council of
Teachers of Mathematics [NCTM], 1989).
• Algebraic results can be achieved geometrically, and geometrical results
can be demonstrated using algebra. For example, Pythagoras’ theorem can
be represented algebraically using the formula a2 + b2 = c 2.

6. Task
You are planning to teach the
• (i) Cartesian coordinate system,
• (ii) points, and
• (iii) plotting points for the first time.
Describe these three concepts to a Grade 8 learner. Your response
must illustrate a deep understanding of the theories of geometry
teaching and learning (Duval, 1995)

7. what are the associated conceptual difficulties
that learners have in the use algebraic thinking in
geometry?
• Many students have difficulties working with variables and unknowns
which they come across in mathematical problems. Students have not
only to identify key components of the problems but also the
underlying relationships.
• The symbolic representations pose problems for the students. Duval
(2002) has claimed that there is no direct access to mathematical
objects other than through their representations. In geometry, this
implies working in different registers (natural language, symbolic, and
figurative) and moving in between registers.