Topicm1283e18919d9e240_1528449000663_0Topic

Equation of a line in the standard formstandard formstandard form and in the slope‑intercept formslope‑intercept formslope‑intercept form

Levelm1283e18919d9e240_1528449084556_0Level

Third

Core curriculumm1283e18919d9e240_1528449076687_0Core curriculum

IX. Cartesian geometry. Basic level.

The student:

2) uses equations of a line on a plane in the standard form and in the slope‑intercept form, finds the equation of a line with given properties (e.g. interception at two points, known slopeslopeslope, parallelism or perpendicularity to another line, tangency to a circle, etc.)

Timingm1283e18919d9e240_1528449068082_0Timing

45 minutes

General objectivem1283e18919d9e240_1528449523725_0General objective

Using mathematical objects and manipulating them, interpreting mathematical concepts.

Selecting and creating mathematical models to solve practical and theoretical problems.

Specific objectivesm1283e18919d9e240_1528449552113_0Specific objectives

1. Finding the slope of a line intercepting two different points with different z‑axis coordinates.

2. Finding the equation of a line passing through two points in slope‑intercept formslope‑intercept formslope‑intercept form or vertex form if possible.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm1283e18919d9e240_1528450430307_0Learning outcomes

The student:

- finds the slope of a line if it exists,

- finds the equation of a line if it exists,

- finds the equation of a line passing through two different points in the standard formstandard formstandard form,

- converts the equation of a line from the slope‑intercept formslope‑intercept formslope‑intercept form to the standard form and back again if it is possible.

Methodsm1283e18919d9e240_1528449534267_0Methods

1. Situational analysis.

2. Discussion.

Forms of workm1283e18919d9e240_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm1283e18919d9e240_1528450127855_0Introduction

The aim of the lead‑in phase is revision of mathematical concepts which will be used during the class:

- the linear function formula: $f\left(x\right)=ax+b$,

- formula $a=\frac{y_B-y_A}{x_B-x_A}$ for the slopeslopeslope of a line passing through two different points $A=\left(x_A,y_A\right)$ and
$B=\left(x_B,y_B\right)$ so than $x_A\ne x_B$,

- geometrical interpretation of coefficient $b$ in the formula for a linear function.

The students work in pairs. Each pair of students gets a worksheet.

Worksheet:

1. Function $f$ determined by any real number $x$ with formula $f\left(x\right)=3x+2$ is given.

a) What is this type of a function called?

b) What is the graph of this function?

c) What are coefficients 3 and 2 called in the formula for function $f$?

d) What is the geometric sense of coefficients 3 and 2?

e) Which diagram presents this function?

[Illustration 1]

2. Write down the equation of a line presented in diagrams A – D in Task 1 in the slope‑intercept form.

The students demonstrate their results and explain the doubts.

Procedurem1283e18919d9e240_1528446435040_0Procedure

The students discuss another example writing down the equation of a line presented in the diagram
$y=0·x+3$ in the slope‑intercept formslope‑intercept formslope‑intercept form.

[Illustration 2]

Discussion:

How to describe any point located on this line with coordinates?

The teacher directs the discussion so that the students produce the following statement: line $AB$ is a set of all points whose Y‑axis is the same and equals 3, so $y=3$. It indicates that any pair of numbers $(x,3)$ satisfies equation $y=0·x+3$.

How to write down the equation of a line passing through points $A=(4,-1)$and $B=(4,5)$? Which other points are located on line $AB$?

The teacher directs the discussion so that the students suggest the following equation $x=4$ and notice that any point with 4 X‑axis is located on this line. The teacher asks to write down this equation in such a form that both variables $x$ and $y$ appear. After getting answer $x=0·y+4$ the teacher summarizes by formulating the conclusion and the definition of an equation of a line in the standard formstandard formstandard form.

The conclusion:

Any line on the Cartesian plane can be described with equation $Ax+By+C=0$ where coefficients $A$ and $B$ cannot equal 0 at the same time. Yet, if $A=0$, the line is paralllel to X‑axis and if $B=0$, the line is parallel to Y‑axis.

The definition – the equation of a line in the standard form:

Equation $Ax+By+C=0$ when $A$ and $B$ don’t equal 0 at the same time is called the equation of a line in the standard form.

The students solve the tasks individually.

Task 1

Triangle $\mathrm{ABC}$, in which $A=(-3,1)$, $B=(3,5)$, $C=(3,1)$ is given.

Write down the equations of lines including sides $\mathrm{AB}$ and $\mathrm{AC}$ of triangle $\mathrm{ABC}$.

Use formula $y=\frac{y_B-y_A}{x_B-x_A}\left(x-x_A\right)+y_A$ for the equation of a line passing through two different points
$A=(x_A,y_A)$ and $B=(x_B,y_B)$ so that $x_A\ne x_B$.

A selected student presents the solution and the teacher initiates a discussion:

- Can the presented formula be used to write down the equation of a line including side $\mathrm{BC}$ of triangle
$\mathrm{ABC}$? If not, why?

- How to convert the equation so that it can be used to find the equation of line $\mathrm{BC}$?

The teacher directs the discussion so that the students:

- indicate the reason why it isn’t possible. The reason is the equality of X‑axis for points $B$ and $C$ so $x_B=x_C$, because the denominator of the fraction $\frac{y_C-y_B}{x_C-x_B}$ will equal 0.

- suggest multiplying both sides of the equation $y=\frac{y_B-y_A}{x_B-x_A}\left(x-x_A\right)+y_A$ by $x_B-x_A$ and writing it down in the equivalent form $(y_B-y_A)(x-x_A)-(x_B-x_A)(y-y_A)=0$.

The teacher stresses that any line passing through two different points $A=(x_A,y_A)$ and $B=(x_B,y_B)$ can be described with this equation, which is called the standard equation in the vertex form.  

The students work in pairs using their computers. They find the equations for set points $A$ and $B$.

Task 2

Open Geogebra applet „Equations of a line passing through two set points”. Change the position of points $A$ and $B$. Find the slope of line $\mathrm{AB}$ and the equation of this line in the slope‑intercept and the standard form.m1283e18919d9e240_1527752256679_0Open Geogebra applet „Equations of a line passing through two set points”. Change the position of points $A$ and $B$. Find the slope of line $\mathrm{AB}$ and the equation of this line in the slope‑intercept and the standard form.

Check your results.

The students solve the tasks individually. Then, the tasks are discussed.

Task 3

The standard equation of a line is given. Convert it into the slope‑intercept form if possibile.m1283e18919d9e240_1527752263647_0The standard equation of a line is given. Convert it into the slope‑intercept form if possibile.

a) $2x-3y+9=0$,

b) $-x+2y+4=0$,

c) $2x+3y=0$,

d) $x-0·y+9=0$.

An extra task:

Prove that any line with the equation $(m^2+1)x+(m^2-1)y-2=0$ intercepts point $M=(1,-1)$, but none of the lines intercepts point $N=(-1,-1)$.

Lesson summarym1283e18919d9e240_1528450119332_0Lesson summary

The students do the consolidation tasks and summarize the class by formulating the most import ant information to memorize.

The slope‑intercept equation of a line: $y=ax+b$.

The standard equation of a line: $Ax+By+C=0$, where coefficients $A$ and $B$ cannot equal 0 at the same time. Yet, if $A=0$ , the line is parallel to X‑axis and if $B=0$, the line is parallel to Y‑axis.

The equation of a line passing through two different points:

so that $A=(x_A,y_A)$ and $B=(x_B,y_B)$.

Selected words and expressions used in the lesson plan

point‑slope formpoint‑slope formpoint‑slope form

slopeslopeslope

slope‑intercept formslope‑intercept formslope‑intercept form

standard formstandard formstandard form

two‑point formtwo‑point formtwo‑point form

y‑intercepty‑intercepty‑intercept