Topicm15d60f4a8f8b66dc_1528449000663_0Topic

Square and cube roots

Levelm15d60f4a8f8b66dc_1528449084556_0Level

Second

Core curriculumm15d60f4a8f8b66dc_1528449076687_0Core curriculum

II. Roots. The student:

1) calculates values of square and cube roots of numbers that are respectively squares and cubes of rational numbers.

Timingm15d60f4a8f8b66dc_1528449068082_0Timing

45 minutes

General objectivem15d60f4a8f8b66dc_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesm15d60f4a8f8b66dc_1528449552113_0Specific objectives

1. Calculating values of square and cube roots.

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

Learning outcomesm15d60f4a8f8b66dc_1528450430307_0Learning outcomes

The student:

- calculates values of square and cube roots.

Methodsm15d60f4a8f8b66dc_1528449534267_0Methods

1. Discussion.

2. Map of associations.

Forms of workm15d60f4a8f8b66dc_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm15d60f4a8f8b66dc_1528450127855_0Introduction

The teacher informs students that during this class they will learn to calculate square and cube roots of given numbers.

Procedurem15d60f4a8f8b66dc_1528446435040_0Procedure

Task
In the second row, write such non‑negative numbers that their squares are values in the first row.

[Table 1]

Looking for non‑negative numbers based on their squares is called calculating the square root.

Task
In the second row, write such non‑negative numbers that their cubes are values in the first row.

[Table 2]

Looking for non‑negative numbers based on their cubes is called calculating the cube root.

Task
Students work individually, using computers. Their task is to get to know values of squares, cubes, square roots and cube roots of consecutive natural numbersnatural numbersnatural numbers from 1 to 100.

Students are divided into groups, which obtain a piece of paper on which they create the mental mapmental mapmental map describing the concept of the roots. Then they present their maps and together write down a definition.

[Geogebra applet]

Definition of the square root.

- The square root of the given non‑negative number $a$ is a non‑negative number $b$, whose square is equal to number $a$ . Such root is marked as the symbol $\sqrt{a}$.
$\sqrt{a}=b$, when $b^2=a$ for $a,b\ge 0$m15d60f4a8f8b66dc_1527752263647_0- The square root of the given non‑negative number $a$ is a non‑negative number $b$, whose square is equal to number $a$ . Such root is marked as the symbol $\sqrt{a}$.
$\sqrt{a}=b$, when $b^2=a$ for $a,b\ge 0$

For example:

$\sqrt{16}=4$, because $4^2=16$

$\sqrt{0,36}=0,6$, because $0,6^2=0,36$

$\sqrt{1}=1$, because $1^2=1$

$\sqrt{49}=7$, because $7^2=49$

$\sqrt{\frac{4}{9}}=\frac{2}{3}$, because ${\left(\frac{2}{3}\right)}^2=\frac{4}{9}$

Definition of the cube root.

- The cube root of the given number $a$ is a number $b$, whose cube is equal to number $a$. Such root is marked as the symbol $\sqrt[3]{a}$.
$\sqrt[3]{a}=b$, when $b^3=a$ for any numbers $a$ and $b$

For example:

$\sqrt[3]{27}=3$, because $3^2=27$

$\sqrt[3]{-\frac{1}{64}}=-\frac{1}{4}$, because ${(-\frac{1}{4})}^3=-\frac{1}{64}$

$\sqrt[3]{8}=2$, because $2^3=8$

$\sqrt[3]{1000}=10$, because ${10}^3=1000$

$\sqrt[3]{-125}=-5$, because ${(-5)}^3=-125$

Students use shaped abilities in exercises.

Task
Calculate.

a) $\sqrt{81},\sqrt{225},\sqrt{0,04},\sqrt{0,64},\sqrt{\frac{25}{144}},\sqrt{\frac{121}{169}}$

b) $\sqrt[3]{27},\sqrt[3]{-64},\sqrt[3]{-0,001},\sqrt[3]{0,008},\sqrt[3]{-\frac{27}{64}},\sqrt[3]{-\frac{125}{1000}}$

Task
Calculate the length of the side of a square whose area is equal to:m15d60f4a8f8b66dc_1527752256679_0Calculate the length of the side of a square whose area is equal to:

a) $64m^2$

b) $0,0036c{m}^2$

c) $1d{m}^2$

d) $\frac{9}{49}c{m}^2$

Task
Calculate the length of the side of a cube whose volume is equal to.

a) $0,027c{m}^3$

b) $\frac{64}{216}{m}^3$

c) $1d{m}^3$

An extra task:
Calculate $\sqrt{3\sqrt{3\sqrt{3\sqrt{9}}}}$.

Lesson summarym15d60f4a8f8b66dc_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they sum‑up the classes, by formulating the conclusions to memorise.

Definition of the square root.

- The square root of the given non‑negative number $a$ is a non‑negative number $b$, whose square is equal to number $a$. Such root is marked as the symbol $\sqrt{a}$.
$\sqrt{a}=b$, when $b^2=a$ for $a,b\ge 0$

Definition of the cube root.

- The cube root of the given number $a$ is a number $b$, whose cube is equal to number $a$. Such root is marked as the symbol $\sqrt[3]{a}$.
$\sqrt[3]{a}=b$, when $b^3=a$ for any numbers $a$ and $b$

Selected words and expressions used in the lesson plan

cube of a numbercube of a numbercube of a number

cube root of a numbercube root of a numbercube root of a number

natural numbersnatural numbersnatural numbers

mental mapmental mapmental map

square of a numbersquare of a numbersquare of a number

square root of a numbersquare root of a numbersquare root of a number