Topicm3bc5b95181566da8_1528449000663_0Topic

Length of a line segment. Midpoint of a line segment

Levelm3bc5b95181566da8_1528449084556_0Level

Third

Core curriculumm3bc5b95181566da8_1528449076687_0Core curriculum

IX. Cartesian geometry. Basic level.

The Student:

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

Timingm3bc5b95181566da8_1528449068082_0Timing

45 minutes

General objectivem3bc5b95181566da8_1528449523725_0General objective

Using and interpreting the representation. Using mathematical objects and manipulating them, interpreting mathematical concepts. Selecting and creating mathematical models to solve practical
and theoretical problems.

Specific objectivesm3bc5b95181566da8_1528449552113_0Specific objectives

1. Applying formulas for coordinatescoordinatescoordinates of the midpointmidpointmidpoint of the line segment to:

a) calculating coordinates of the midpoint of the line segment when we have coordinates of both its endpointsendpointsendpoints given,

b) calculating coordinates of one of endpoints of the line segment when we have coordinates of the midpoint and the other end given.

2. Calculating lengths of line segments.

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

Learning outcomesm3bc5b95181566da8_1528450430307_0Learning outcomes

The Student:

- calculates coordinatescoordinatescoordinates of the midpointmidpointmidpoint of the line segment there are coordinates of both its endpointsendpointsendpoints given,

- calculates coordinates of one of endpoints of the line segment there are coordinates of the midpoint
and the other end given,

- applies the formula for the distance between two points.

Methodsm3bc5b95181566da8_1528449534267_0Methods

1. Situational analysis.

2. Discussion.

Forms of workm3bc5b95181566da8_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm3bc5b95181566da8_1528450127855_0Introduction

The aim of the introduction part is to revise mathematical concepts that will be used during the lesson:

- the number, being the arithmetic mean of two numbers and its geometric interpretation on the number line,
- calculating distance between tow numbers on the number line,
- the Pythagorean theorem.

The teacher divides students for six 4 - 5 people groups. Two groups get the worksheet number 1, another two worksheet number 2 and last two – worksheet number 3.

Worksheet number 1

Calculate the arithmetic mean of two grades: 2 and 5 that Janek got from two last tests from mathematics. Mark this number and the obtained mean exactly on the number line. What is the position of the obtained number in relation to numbers 2 and 5? Do the same exercise for a few pairs of numbers, including positive numbers, negative numbers and numbers of various signs. Write your observations.

Worksheet number 2

Mark two given numbers a and b as exactly as possible on the number line and calculate the distance between these numbers on the number line. Do this exercise for the following pairs of numbers:

a) a = 2 and b = 5

b) a = 7 and b = 1

c) a = $\frac{2}{3}$ and b = $\frac{1}{2}$

d) a = -2 and b = -6 

Propose such way of calculating the distance that allows to calculate this distance in every case.

Worksheet number 3

Revise the Pythagorean theorem. Calculate lengths of the hypotenuse of the triangle while lengths of the catheti are a and b.

a) a = 12 and b = 5

b) a = 7 and b = 1

c) a = $\frac{\sqrt{3}}{2}$ and b = $\frac{1}{2}$

d) a = $\sqrt{7}-1$ and b = $\sqrt{7}+1$

Propose the formula for the length of the hypotenuse AB of the right‑angled triangle ABC.

Representatives of three chosen groups that do exercises from various worksheets present results of their groups’ work. The teacher adds missing conclusions, if necessary, so that following phrases are said:

- the number, being the arithmetic mean of two numbers is located exactly in the middle of these points on the number line,
- the distance between these numbers on the number line is equal to the difference between the greater and smaller number, that is equal to the absolute value,
- the length of the hypotenuse of a right‑angled triangle is equal to the root of the sum of squares of its catheti.m3bc5b95181566da8_1527752263647_0- the number, being the arithmetic mean of two numbers is located exactly in the middle of these points on the number line,
- the distance between these numbers on the number line is equal to the difference between the greater and smaller number, that is equal to the absolute value,
- the length of the hypotenuse of a right‑angled triangle is equal to the root of the sum of squares of its catheti.

Procedurem3bc5b95181566da8_1528446435040_0Procedure

Students work individually or in pairs, using computers. They investigate relations between coordinatescoordinatescoordinatesof endpointsendpointsendpoints of line segments and coordinates if its midpointmidpointmidpoint and the way of calculating the length
of the line segment.

[Geogebra applet]

Tasks to do in the Geogebra Applet

Open the Geogebra Applet The midpoint and the length of line segment in the coordinate system
on a plane.

1. Choose the option „the midpoint”. Observe how coordinates of midpoint change depending on coordinates of endpoints A and B. Propose formulas to calculate coordinates of the midpoint of the line segment while having the endpoints given: A = (xIndeks dolny A_A, yIndeks dolny A_A) and B = (xIndeks dolny B_B, yIndeks dolny B_B).

After having completed the exercise, students guided by the teacher formulate the conclusion:

- coordinates of the midpoint of the line segment are arithmetic means of proper coordinates of its endpointsendpointsendpoints.

The midpointmidpointmidpoint M of the line segment AB, where A = (xIndeks dolny A_A, yIndeks dolny A_A) and B = (xIndeks dolny B_B, yIndeks dolny B_B) has coordinates equal to

2. Choose the option ‘length of line segment”. Give ways of calculating the length of the line segment AB, where A = (xIndeks dolny A_A, yIndeks dolny A_A) and B = (xIndeks dolny B_B, yIndeks dolny B_B).

After having completed the exercise, students, guided by the teacher, write the formula for the length
of the line segment

The teacher asks a question – can this formula also be sued when the line segment is parallel to one
of the axes of the coordinate system, for example to the axis Ox and how to calculate this length?

The teacher sums‑up students’ propositions by writing down formulas

$\left|AB\right|=\sqrt{(x_B-x_A{)}^2}$ and $\left|AB\right|=|x_B-x_A|$.

As a note, the teacher formulates the equality $\sqrt{a^2}=\left|a\right|$ for any real number a.

Students work individually, doing exercises.

Task 1

The point M is the midpoint of the line segment AB. Calculate coordinates of the point M and length of the line segment AB.

a) A = (0,1), B = (3,-3)

b) A = (2,-7), B = (-4,-1)

c) A = ($\frac{2}{13}$, $-\frac{7}{13}$), B = ($-\frac{3}{13}$, $\frac{5}{13}$)

d) A = ($\sqrt{3}+3,1-\sqrt{3}$), B = ($2-\sqrt{3},\sqrt{3}+2$)

Task 2

Points A=(6,-3) and B=(9,6) are vertices of the parallelogram ABCD, and point S=(3,3) is the point of symmetry of this parallelogram.

a) calculate coordinatescoordinatescoordinates of vertices C and D,

b) calculate the perimeter of this parallelogram,

c) prove that this parallelogram is a square.

Task 3

Points A = (1,3), B = (5,1), C = (11,5) and D = (xIndeks dolny D_D,yIndeks dolny D_D) are vertices of the parallelogram ABCD. Calculate coordinates of the vertex D. Apply properties of parallelogram that says the tetragon is a parallelogram
if and only if its diagonals cross at a point that is the midpoint of both diagonals.m3bc5b95181566da8_1527752256679_0Points A = (1,3), B = (5,1), C = (11,5) and D = (xIndeks dolny D_D,yIndeks dolny D_D) are vertices of the parallelogram ABCD. Calculate coordinates of the vertex D. Apply properties of parallelogram that says the tetragon is a parallelogram
if and only if its diagonals cross at a point that is the midpoint of both diagonals.

An extra task:

The point A = (-3,3) is the vertex of the parallelogram ABCD, point M = (4,3) is the midpointmidpointmidpoint of the side BC, and point N = (5,6) – the midpoint of the side CD of this parallelogram. Calculate coordinates of vertices B, C and D.

Lesson summarym3bc5b95181566da8_1528450119332_0Lesson summary

Students do the revision exercises. Then together they sum‑up the classes, by formulating the conclusions to memorise:

- coordinates of the midpointmidpointmidpoint of a line segment are arithmetic means of proper endpointsendpointsendpoints of this line segment. If M – the midpoint AB, where A = (xIndeks dolny A_A,yIndeks dolny A_A) and B = (xIndeks dolny B_B,yIndeks dolny B_B) then

- length of the line segment AB can be calculated from the formula

$\left|AB\right|=\sqrt{(x_B-x_A{)}^2+(y_B-y_A{)}^2}$, where $A=(x_A,y_A)$ and $B=(x_B,y_B)$.

Selected words and expressions used in the lesson plan

coordinatescoordinatescoordinates

distance formuladistance formuladistance formula

endpointsendpointsendpoints

length of a line segmentlength of a line segmentlength of a line segment

mean of coordinatesmean of coordinatesmean of coordinates

midpointmidpointmidpoint