Topicmd2c36346c621e947_1528449000663_0Topic

Definition of the absolute valueabsolute valueabsolute value

Levelmd2c36346c621e947_1528449084556_0Level

Third

Core curriculummd2c36346c621e947_1528449076687_0Core curriculum

I. Real numbersreal numbersReal numbers. The student:

7) uses the geometrical and algebraic interpretation of the absolute valueabsolute valueabsolute value, solves the equations and inequalities of the type: $\left|x+4\right|=5$, $\left|x-2\right|<3$, $\left|x+3\right|\ge 4$.

Timingmd2c36346c621e947_1528449068082_0Timing

45 minutes

General objectivemd2c36346c621e947_1528449523725_0General objective

Interpreting and manipulating information presented in both mathematical and popular science texts, as well as in the form of graphs, diagrams, tables.

Specific objectivesmd2c36346c621e947_1528449552113_0Specific objectives

1. Communicating in English, developing mathematics, scientific, technical and IT competences, developing learning skills.

2. Understanding the definition and properties of an absolute valueabsolute valueabsolute value.

3. Understanding the interpretation of a geometric absolute value.

Learning outcomesmd2c36346c621e947_1528450430307_0Learning outcomes

The student:

- student learns the definition and properties of the absolute value,

- student learns the geometric interpretation of the absolute valueabsolute valueabsolute value.

Methodsmd2c36346c621e947_1528449534267_0Methods

1. Wandering posters.

2. Case study.

Forms of workmd2c36346c621e947_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmd2c36346c621e947_1528450127855_0Introduction

Students, working in small groups, create posters. The main topic is real numbersreal numbersreal numbers and their subsets.

Each group received a sheet of paper from the teacher with a question to answer. Students write their answer and pass the poster to the next group to let them answer their question. There are as many rounds as there are groups of students. After the task is finished, a representative of each group reads the results of their work. Posters are placed on the board. The teacher, together with the students, analyzes the information collected.

Proceduremd2c36346c621e947_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to learn the definition and properties of the absolute valueabsolute valueabsolute value of the real number.

Discussion - How are the opposite numbersopposite numbersopposite numbers relative to each other on the number linenumber linenumber line? Students make hypotheses. They check them and formulate the conclusions.

Conclusion:

- The opposite numbers on the number line are on opposite sides of zero at the same distance from it.md2c36346c621e947_1527752263647_0- The opposite numbers on the number line are on opposite sides of zero at the same distance from it.

Task
Students analyze the Interactive Illustration showing the definition and properties of the absolute valueabsolute valueabsolute value. They write the definition.

[Interactive illustration]

Definition

The absolute value of the real number a (designation |a|) is:
- the number a, if a is a non‑negative number,
- the number opposite to a, if a is a negative number.md2c36346c621e947_1527752256679_0The absolute value of the real number a (designation |a|) is:
- the number a, if a is a non‑negative number,
- the number opposite to a, if a is a negative number.

Note that the absolute value is derived from its absolute properties:
- the absolute value of the number is positive or equal to 0, i.e. | x | ≥ 0, for any real number x,
- if | x | = 0, then x = 0,
- the absolute values of the opposite numbers are equal, i.e. | x | = | -x | for any real number x,
- the distance between the numbers a and b on the number line is equal to the absolute value of their difference | a - b |.md2c36346c621e947_1527712094602_0Note that the absolute value is derived from its absolute properties:
- the absolute value of the number is positive or equal to 0, i.e. | x | ≥ 0, for any real number x,
- if | x | = 0, then x = 0,
- the absolute values of the opposite numbers are equal, i.e. | x | = | -x | for any real number x,
- the distance between the numbers a and b on the number line is equal to the absolute value of their difference | a - b |.

Using the definition of an absolute value, students independently solve the problems given.

Task
Calculate the value of the expression. Evaluate whether the result of the calculation is a rational number:

a) $\left|3-\sqrt{3}\right|-\left|\sqrt{3}+1,5\right|$,

b) $\left|3-\mathrm{\pi }\right|-\left|\mathrm{\pi }-3\right|$,

c) $\left|\sqrt{3}-\sqrt{5}\right|+\left|\sqrt{5}-3\right|$,

d) $3·\left|1-\sqrt{7}\right|-3\sqrt{7}$.

Discussion - What is the geometric interpretation of the absolute valueabsolute valueabsolute value? Students make hypotheses. They check them and formulate the conclusions.

Conclusion:

- The absolute value of the real number a is equal to the distance of the point in the coordinate a from the zero point on the number linenumber linenumber line.

Using this conclusion, students work in pairs and solve simple equations with the absolute valueabsolute valueabsolute value.

Task
Calculate the equations:

a) $\left|x\right|=6$,

b) $\left|x\right|=0$,

c) $\left|x\right|=-2$.

Discussion – How do we calculate the distance between numbers on the number linedistance between numbers on the number linedistance between numbers on the number line? Students make hypotheses, check them and formulate a conclusion.

Conclusion:

- The distance of numbers a and b on the number line is equal to the absolute value of their difference | a – b |.

Task
Write the distance on the number linenumber linenumber line between the numbers using the absolute valueabsolute valueabsolute value symbol. Calculate this distance:

a) $-3and-25$,

b) $8and2,4$,

c) $-3\sqrt{3}and5\sqrt{3}+2$,

d) $-2\sqrt{5}and3-\sqrt{5}$.

Task
Calculate the value of the expression, if $x=-3$:

a) $|\left|x\right|-x|$,

b) $3x·\left|x-2\right|$.

Task
Calculate the value of the following expressions:

a) $|x-4|-|x-5|,ifx\in (-\infty ,0)$,

b) $|x-5|-|x+6|,if,x\in (5,\infty )$,

c) $|6-x|+|3+x|,ifx\in (-3,6)$.

After solving all the tasks, the students present the results obtained. The teacher evaluates their work and explains all doubts.

An extra task:
Calculate:

a) $\sqrt{{\left(3-\sqrt{3}\right)}^2}-\sqrt{{\left(\sqrt{3}-3\right)}^2}+\sqrt{1}$,

b) $\sqrt{{\left(\sqrt{2}-\sqrt{3}\right)}^2}+\sqrt{{\left(\sqrt{2}+\sqrt{3}\right)}^2}$.

Lesson summarymd2c36346c621e947_1528450119332_0Lesson summary

Students do the revision exercises.

They formulate conclusions to remember.

Remember:

- The absolute valueabsolute valueabsolute value of the real number a is:

• the number a, if a is a non‑negative number,

• the number opposite to a, if a is a negative number.

- The absolute value of the real number a is equal to the distance of the point in the coordinate a from the zero point on the number line.
- The distance of numbers a and b on the number linenumber linenumber line is equal to the absolute valueabsolute valueabsolute value of their difference | a – b |.

Selected words and expressions used in the lesson plan

absolute valueabsolute valueabsolute value

distance between numbers on the number linedistance between numbers on the number linedistance between numbers on the number line

geometric interpretation of the absolute powergeometric interpretation of the absolute powergeometric interpretation of the absolute power

number linenumber linenumber line

opposite numbersopposite numbersopposite numbers

real numbersreal numbersreal numbers