Topicm5ec60bedacc6f368_1528449000663_0Topic

Equation of a line in the standard form

Levelm5ec60bedacc6f368_1528449084556_0Level

Third

Core curriculumm5ec60bedacc6f368_1528449076687_0Core curriculum

X. Analytic geometry on the Cartesian plane. The student:

2) uses the equations of straight lines on a plane in slope‑intercept and standard form, also defines an equation of a lineequation of a lineequation of a line with given properties (such as two intercept, gradient, parallelism or perpendicularity to another straight linelineline, tangent to a circle, etc.);

Timingm5ec60bedacc6f368_1528449068082_0Timing

45 minutes

General objectivem5ec60bedacc6f368_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesm5ec60bedacc6f368_1528449552113_0Specific objectives

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

2. Getting to know the equation of a straight linelineline in the standard form.

3. Converting from the standard form to the slope‑intercept form.

Learning outcomesm5ec60bedacc6f368_1528450430307_0Learning outcomes

The student:

- Gets to know the equation of a straight linelineline in the standard form.

- Converts from the standard form of the equation of the straight linelineline to the slope‑intercept form.

Methodsm5ec60bedacc6f368_1528449534267_0Methods

1. Open ear.

2. Situational analysis.

Forms of workm5ec60bedacc6f368_1528449514617_0Forms of work

1. Individual.

2. Group work.

Lesson stages

Introductionm5ec60bedacc6f368_1528450127855_0Introduction

The students use the „open ear” technique to get the information about the slope‑intercept equation in order.

Procedurem5ec60bedacc6f368_1528446435040_0Procedure

The teacher informs the students that the aim of the class is getting to know the standard equation of the straight linelineline.

Discussion – can any linelineline in the coordinate systemcoordinate systemcoordinate system be described with the equation in the slope‑intercept form? The students formulate hypotheses, check them and formulate the conclusion.

The conclusion

A linelineline perpendicular to OX axis cannot be described with the slope‑intercept equation.

The teacher informs the students that there are other equations that can be used to describe lines in the coordinate systemcoordinate systemcoordinate system. One of them is the standard equation of a lineequation of a lineequation of a line.

The definition

The standard equation of a lineequation of a lineequation of a line is denoted as follows $Ax+By+C=0$, where $A^2+B^2\ne 0$. Condition $A^2+B^2\ne 0$ means that coefficients A and B cannot equal zero at the same time.

The students work in pairs and analyse the INTERACTIVE PRESENTATION which shows the method of converting from the slope‑intercept equation of a lineequation of a lineequation of a line to the standard equation and back again. They formulate their conclusions.

[Interactive illustration]

The conclusions

1. In order to convert from the slope‑intercept equation of a line to the standard equation, you need to move all terms of the equation onto one side and multiply both sides of the equation by the same number, so that all the obtained coefficients are integers.m5ec60bedacc6f368_1527752256679_01. In order to convert from the slope‑intercept equation of a line to the standard equation, you need to move all terms of the equation onto one side and multiply both sides of the equation by the same number, so that all the obtained coefficients are integers.

2. In order to convert from the standard equation of a line $Ax+By+C=0$ to the slope‑intercept equation, you need to move terms Ax and C onto the right side of the equation. If $B\ne 0$,  you need to divide both sides of the equation by B.m5ec60bedacc6f368_1527752263647_02. In order to convert from the standard equation of a line $Ax+By+C=0$ to the slope‑intercept equation, you need to move terms Ax and C onto the right side of the equation. If $B\ne 0$,  you need to divide both sides of the equation by B.

The students individually work out how to define the standard equation of a lineequation of a lineequation of a line intercepting at two points.

Task 1
Define the standard equation of a lineequation of a lineequation of a line intercepting at two points $A(x_A;y_A)$ and $B(x_B;y_B)$, where $x_A\ne x_B$. Formulate your conclusion.

The conclusion

The standard equation of a lineequation of a lineequation of a line intercepting at two points $A(x_A;y_A)$ and $B(x_B;y_B)$, where $x_A\ne x_B$ is expressed as follows $(y-y_A)(x_A-x_B)-(y_A-y_B)(x-x_A)=0$.

Use the new information to solve the tasks.

Task 2
Rewrite the equation of linelineline $k:y+\frac{x-3}{2}=\frac{x+1}{5}$ in the standard form.

Answer: $3x+10y-17=0$.

Task 3
Find the standard equation of linelineline k, on which lie two points $A(-3;-4)$ and $B(4;7)$.

Answer: $7x-11y+49=0$.

Task 4
Write the standard equation of a lineequation of a lineequation of a line intersecting at two points $A(2;6)$ and $B(4;8)$. Check if point $C(-1;-3)$ lies on this linelineline.

Answer: $x-y+4=0$, doesn’t lie on the linelineline.

Task 5
Point $A(a;3)$ lies on linelineline $2x+y-5=0$. Calculate a.

Answer: $a=1$.

Having solved all the tasks, the students present their results. The teacher assesses their work and explains the doubts.

An extra task:
For what value of number m lines $k:x-my+4+m=0$ and $l:2mx+y-m-1=0$ intersect on the OY axis?

Answer: $m\in \{-2,2\}$.

Lesson summarym5ec60bedacc6f368_1528450119332_0Lesson summary

The students do the consolidation tasks. They cooperate to formulate the conclusion to memorize.

- A linelineline perpendicular to OX axis cannot be described with the slope‑intercept equation.

- In order to convert from the slope‑intercept equation of a lineequation of a lineequation of a line to the standard equation, you need to move all terms of the equation onto one side and multiply both sides of the equation by the same number, so that all the obtained coefficients are integers.

- In order to convert from the standard equation of a lineequation of a lineequation of a line $Ax+By+C=0$ to the slope‑intercept equation, you need to move terms Ax and C onto the right side of the equation. If $B\ne 0$, you need to divide both sides of the equation by B.

- The standard equation of a lineequation of a lineequation of a line intercepting at two points $A(x_A;y_A)$ and $B(x_B;y_B)$ , where $x_A\ne x_B$ is expressed as follows $(y-y_A)(x_A-x_B)-(y_A-y_B)(x-x_A)=0$.

Selected words and expressions used in the lesson plan

cooeficientscooeficientscooeficients

coordinate systemcoordinate systemcoordinate system

equation of a lineequation of a lineequation of a line

linelineline

slope‑intercept equation of a lineslope‑intercept equation of a lineslope‑intercept equation of a line

standard equation of a linestandard equation of a linestandard equation of a line

y‑axisy‑axisy‑axis