Topicm7b2069a43b818d79_1528449000663_0Topic

Introduction to cartesian geometry. The area of a triangletriangletriangle in the Cartesian planecartesian planeCartesian plane

Levelm7b2069a43b818d79_1528449084556_0Level

Third

Core curriculumm7b2069a43b818d79_1528449076687_0Core curriculum

IX. Analytical geometry on Cartesian planecartesian planeCartesian plane. The student:

3) calculates the distance between two points in the coordinate systemcoordinate systemcoordinate system.

Timingm7b2069a43b818d79_1528449068082_0Timing

45 minutes

General objectivem7b2069a43b818d79_1528449523725_0General objective

Using mathematical objects and manipulating them, interpreting mathematical concepts.

Specific objectivesm7b2069a43b818d79_1528449552113_0Specific objectives

1. Reading the coordinates of grid points marked in the coordinate system on Cartesian planecartesian planeCartesian plane. Marking point with given coordinates in the coordinate systemcoordinate systemcoordinate system.

2. Calculating the areas of triangles and polygons whose vertices are grid points.

3. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

Learning outcomesm7b2069a43b818d79_1528450430307_0Learning outcomes

The student:

- reads, writes down and names the coordinates of grid points marked in the coordinate systemcoordinate systemcoordinate system on Cartesian plane,

- marks grid points with given coordinates in the coordinate system on Cartesian planecartesian planeCartesian plane,

- calculates the areas of triangles and polygons whose vertices are grid points.

Methodsm7b2069a43b818d79_1528449534267_0Methods

1. Situational analysis.

2. Controlled discussion.

Forms of workm7b2069a43b818d79_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm7b2069a43b818d79_1528450127855_0Introduction

The teacher informs the students that the lesson is an introduction to analytical geometry. Analytical geometry is one of the most important divisions of mathematics. The students will revise their knowledge about the coordinate systemcoordinate systemcoordinate system and practice calculating the areas of triangles in the coordinate system.

The students recollect the concepts connected with the coordinate system.

Then, working individually on their computers, the students check their skills of reading the coordinates of a point in the coordinate systemcoordinate systemcoordinate system.

Procedurem7b2069a43b818d79_1528446435040_0Procedure

Task
Open Geogebra applet: „Point in the coordinate systemcoordinate systemcoordinate system”:

a) Select option “read the coordinates of a point” Use the applet to check if you can properly read the coordinates of a randomly generated point. Repeat the task until you get 5 consecutive correct answers.

b) Select option “set a point”. Use the applet to check if you can give the correct location of the randomly generated point. Repeat the task until you get 5 consecutive correct answers.

[Geogebra applet 1]

Having finished this activity, the teacher asks several students to do the task in front of the class. In the meantime the teacher asks the other students questions concerning the names referring to the coordinates of a point (abscissax‑axis / abscissaabscissa, ordinate). The teacher can modify the task by asking the student to mark another point with the same abscissa, opposite ordinatey‑axis / ordinateordinate, etc.

The students do the following task in pairs.

Task
Mark all the points in the coordinate systemcoordinate systemcoordinate system which satisfy the given property:

a) Both coordinates are positive numbers and their sum equals 6.

b) Both coordinates are integers and their product equals 12.

c) Both coordinates are positive numbers and the sum of their squares equals 13.

Having finished this activity the teacher starts a discussion, in which he asks the students to report on the methods of solving given points of this activity. The teacher corrects mistakes or gaps in reasoning.

The students solve the tasks individually.

Task
The coordinates of the vertices of triangletriangletriangle ABC are A = (-3, -1), B = (5, -1), C = (2, 5). Calculate the area of this triangle.

The teacher observes the students’ work and asks to demonstrate the solution. The teacher observes if among the students there the ones who completed triangletriangletriangle $ABC$ to become rectangle $A_{BMN}$, where $M=(5,5)$ and $N=(-3,5)$. The teacher asks a student who used the formula $P_{ABC}=\frac{1}{2}·a·h$ for calculations to present the solution to the other students. So, the solution will simply be $P_{ABC}=12·8·6=24$.

The students work individually using their computers. They practise their skills of calculating the areas of triangles in the coordinate systemcoordinate systemcoordinate system.

Task
Open Geogebra applet: „Area of a triangletriangletriangle in the coordinate system”.

[Geogebra applet 2]

Mark triangle ABC, in which A = (-3, -1), B = (5, 1), C = (2, 5) in the coordinate systemcoordinate systemcoordinate system.

To make the calculation of the area of the trianglearea of the trianglearea of the triangle easier select option Complete the triangletriangletriangle to form a rectangle in the applet.

You can check the correctness of your calculations by selecting option „The area of triangle ABC equals” in the applet.
Change the position of the vertices of the triangle and observe how the method of completing to form a rectangle works with various positions of the vertices.m7b2069a43b818d79_1527752263647_0You can check the correctness of your calculations by selecting option „The area of triangle ABC equals” in the applet.
Change the position of the vertices of the triangle and observe how the method of completing to form a rectangle works with various positions of the vertices.

Having finished the task, the students discuss their results while the teacher leads the discussion.

Discussion
Think - Why does drawn rectangle make calculating the area of the triangle easier? Can any triangle be completed to form a rectangle in the same way?m7b2069a43b818d79_1527752256679_0Think - Why does drawn rectangle make calculating the area of the triangle easier? Can any triangle be completed to form a rectangle in the same way?

The students solve the tasks using the information.

Task
Calculate the area of triangletriangletriangle $ABC$, whose coordinates of vertices are $A=(0,0),B=(x_1,y_1),C=(x_2,y_2)$ and $x_1>0,x_2>0,y_1>0,y_2>0$.

Task
Calculate the area of a quadrangle $ABCD$, having the coordinates of its vertices:

a) $A=(-3,-1),B=(5,-2),C=(6,3),D=(1,5)$

b) $A=(1,-1),B=(3,-2),C=(5,5),D=(-2,4)$

c) $A=(-2,-1),B=(6,-1),C=(3,5),D=(-3,6)$

d) $A=(-3,-1),B=(7,1),C=(1,2),D=(-1,5)$

Hint:
Divide the quadrangle into triangles with diagonals. The method of dividing a figure into triangles is called triangulation.

An extra task:
TriangletriangleTriangle $ABC$ is given. The coordinates of its vertices are $A=(n,m),B=(n+1,m-2),C=(4n,3m)$, where $n$ and $m$ are positive integers. Prove that the area of this triangletriangletriangle equals: $P_{ABC}=3n+m$.

Lesson summarym7b2069a43b818d79_1528450119332_0Lesson summary

The students do the consolidation tasks and summarize the class.

- They recollect the used names: the coordinate system, the coordinates of a point.
- They discuss the methods of calculating the area of a triangle.
- They discuss the methods of calculating the area of a polygon as a sum of areas of triangles, which are formed by dividing the polygon with its diagonals.m7b2069a43b818d79_1527712094602_0- They recollect the used names: the coordinate system, the coordinates of a point.
- They discuss the methods of calculating the area of a triangle.
- They discuss the methods of calculating the area of a polygon as a sum of areas of triangles, which are formed by dividing the polygon with its diagonals.

Selected words and expressions used in the lesson plan

area of the trianglearea of the trianglearea of the triangle

cartesian planecartesian planecartesian plane

coordinate systemcoordinate systemcoordinate system

coordinates of pointcoordinates of pointcoordinates of point

triangletriangletriangle

x‑axis / abscissax‑axis / abscissax‑axis / abscissa

y‑axis / ordinatey‑axis / ordinatey‑axis / ordinate