S5 Mathematics Coordinate Geometry

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S5 Mathematics Coordinate Geometry. Equation of straight line. Lam Shek Ki (Po Leung Kuk Mrs. Ma Kam Ming-Cheung Foon Sien College). Main ideas. Abstraction through nominalisation Making meaning in mathematics through: language, visuals & the symbolic The Teaching Learning Cycle.

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S5 MathematicsCoordinate Geometry

Equation of straight line

Lam Shek Ki

(Po Leung Kuk Mrs. Ma Kam Ming-Cheung Foon Sien College)

Main ideas
• Abstraction through nominalisation
• Making meaning in mathematics through: language, visuals & the symbolic
• The Teaching Learning Cycle
Content(According to CG)

S1 to S3

• Distance between two points.
• Coordinates of mid-point.
• Internal division of a line segment.
• Polar Coordinates.
• Slope of a straight line.
Content(According to CG)

S5

• Equation of a straight line
• Finding the slope and intercepts from the equation of a straight line
• Intersection of straight lines
• Equation of a circle
• Coordinates of centre and length of radius
Direct instruction

Given any straight line, there is an equation so that the points lying on the straight line must satisfy this equation, this equation is called the equation of the straight line. …

Why?

How?

What?

A point lying on the line

A point not lying on the line

Pack in

Pointslying on the straight line

(x, y) : symbolic representation of a point

x-coordinate

y-coordinate

3x+2y=5

(Equation of a straight line)

nominal group

Problems

Some students :

- do not understand “x” means “x-coordinate”

- cannot accept “x = 2” represents a straight line.

- don’t know why the point-slope form can help to find the equation

- ……

A point lying on the line

A point not lying on the line

Unpack

Pointslying on the straight line

(x, y) : symbolic representation of a point

x-coordinate

y-coordinate

3x+2y=5

(Equation of a straight line)

nominal group

LANGUAGE

language & visual

language & symbolic

LANGUAGE

VISUAL &

SYMBOLIC

VISUAL

SYMBOLIC

visual & symbolic

A point

(x, y)

Unpack the meaning of Equation of straight line

by

guessing the common feature of the points lying on the straight line.

y

5

L1

4

3

2

1

x

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-1

-2

-3

-4

-5

x  y

(3,3)

(-2,3)

x = y

(-5,2)

(1,1)

(-2,-2)

(x,y)

x-coordinate = y-coordinate

y

5

L1

4

3

2

1

x

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-1

-2

-3

-4

-5

x  y

(3,3)

x = y

(-2,3)

(-5,2)

(1,1)

(-2,-2)

(x,y)

x-coordinate = y-coordinate

y

5

L1

4

3

2

1

x

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-1

-2

-3

-4

-5

x  y

(3,3)

(-2,3)

x = y

(-5,2)

(1,1)

(-2,-2)

(x,y)

x-coordinate = y-coordinate

Visual representation of “lying …” and “not lying…”

y

5

L2

4

3

(-1,3)

2

(-3,2)

1

(1,1)

x

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-1

x+y 2

(x,y)

-2

(4,-2)

-3

The sum of x-coordinate and y-coordinate is 2

-4

-5

x + y=2

Mathematical concepts

Setting the context

Teacher modelling and deconstructing

Students constructing independently

Teacher and students constructing jointly

Developing a mathematical concepts

y

5

4

L3

3

2

1

x

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-1

-2

-3

-4

-5

x - y=3

(x,y)

(4,1)

(2,-1)

(-1,-4)

Findings
• For every straight line, the coordinates of the points on the straight line have a common feature.

Express that feature mathematically

Equation of the straight line

• Moreover, the coordinates of the points that do not lie on the straight line do not have that feature.
Abstraction through nominalisation

x-coordinate  x

common feature  Equation of

of straight line straight line

A point having  The coordinates

the feature satisfy the equation

Abstraction

Vertical lines

(-3 , 2)

The x-coordinate of any point lying on the straight line is -3.

Equation:

x = -3

y

5

L5

4

3

2

(-3,2)

1

x

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

(-3,0)

-1

-2

-3

(-3,-3)

-4

-5

x =-3

The x-coordinate is -3

y

5

4

3

2

(-3,2)

(1,2)

(3,2)

L4

1

(x,y)

x

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-1

The y-coordinate is 2

-2

-3

-4

-5

y =2

Horizontal line

(3, 2)

The y-coordinate of any point lying on the straight line is 2

Equation:

y = 2

Conclusion

Indentify and unpack the nominal groups

Experience the process of abstraction

 Make use of the meaning-making system

in mathematics

 Scaffolding : The teaching learning cycle