Topicmcdea1deddf16f0cd_1528449000663_0Topic

Calculating values of square and cube roots

Levelmcdea1deddf16f0cd_1528449084556_0Level

Second

Core curriculummcdea1deddf16f0cd_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.

Timingmcdea1deddf16f0cd_1528449068082_0Timing

45 minutes

General objectivemcdea1deddf16f0cd_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesmcdea1deddf16f0cd_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 outcomesmcdea1deddf16f0cd_1528450430307_0Learning outcomes

1. Calculates values of square and cube roots.

Methodsmcdea1deddf16f0cd_1528449534267_0Methods

1. Discussion.

2. Thematical contest.

Forms of workmcdea1deddf16f0cd_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmcdea1deddf16f0cd_1528450127855_0Introduction

Students revise the concept of rational numberrational numberrational number, definition of square and cube root and prime factorization.

Proceduremcdea1deddf16f0cd_1528446435040_0Procedure

Students analyse ways of calculating the values of square and cube roots. Then they do similar calculations on their own.

Example:

We calculate $\sqrt{576}.$

In order to this we perform the prime factorizationprime factorizationprime factorization of the number under the square.

[Illustration 1]

Therefore $\sqrt{576}=\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3}=\sqrt{2^2\cdot 2^2\cdot 2^2\cdot 3^2}=\sqrt{2^2}\cdot \sqrt{2^2}\cdot \sqrt{2^2}\cdot \sqrt{3^2}=2\cdot 2\cdot 2\cdot 3=24$.

Therefore $\sqrt{576}=24.$

Example:

We calculate $\sqrt[3]{3375}$.

In order to this we perform the prime factorizationprime factorizationprime factorization of the number under the square.

[Illustration 2]

Therefore $\sqrt[3]{3375}=\sqrt[3]{3\cdot 3\cdot 3\cdot 5\cdot 5\cdot 5}=\sqrt[3]{3^3\cdot 5^3}=\sqrt[3]{3^3}\cdot \sqrt[3]{5^3}=3\cdot 5=15$.

Therefore $\sqrt[3]{3375}=15.$

Task
Calculate.

a) $\sqrt{676}$

b) $\sqrt{1024}$

Task
Calculate.

a) $\sqrt[3]{1728}$

b) $\sqrt[3]{5832}$

Discussion – does $\sqrt[3]{-8}$ have the same value as $-\sqrt[3]{8}$?

Students check their assumptions by giving proper examples.

Conclusion:

- For any number a there is the equality $\sqrt[3]{-a}=-\sqrt[3]{a}.$mcdea1deddf16f0cd_1527752263647_0- For any number a there is the equality $\sqrt[3]{-a}=-\sqrt[3]{a}.$

Students do the exercises in pairs. They have the time limit. The pair that does all exercises first, gets two ‘pluses’.

Task
Insert proper sign in blank spaces <, >, =.

a) $\sqrt{64}...\sqrt[3]{64}$

b) $\sqrt{36}...\sqrt{49}$

c) $\sqrt{4}...\sqrt[3]{8}$

d) $-\sqrt{36}...-\sqrt[3]{-216}$

e) $\sqrt[3]{-1000}...-\sqrt[3]{-512}$

Task
Calculate the value of the expression.

a) $\sqrt{36+48}$

b) $\sqrt{169}-\sqrt{121}$

c) $\sqrt[3]{8}+\sqrt{49}$

d) $\sqrt[3]{-64}+\sqrt[3]{-1000}$

e) $\sqrt[3]{\frac{1}{216}}+\sqrt{\frac{1}{36}}-\sqrt[3]{-216}+\sqrt{36}$

f) $\sqrt{5\frac{4}{9}}·\frac{\sqrt{81}}{7}$

Task
Students work individually, using computers. Their task is to observe which values of square roots are expressed with rational numbers.

[Geogebra applet]

Task
Choose a number that is not a rational number.mcdea1deddf16f0cd_1527752256679_0Choose a number that is not a rational number.

$\sqrt{1\frac{7}{9}},\sqrt[3]{-1331},\sqrt{8},\sqrt{196}.$

An extra task:
Give an example of number $x$, for which the value of the expression $\sqrt{x^2+48}$ is a rational numberrational numberrational number.

Lesson summarymcdea1deddf16f0cd_1528450119332_0Lesson summary

Students do the revision exercises.

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

- For any number a there is the equality $\sqrt[3]{-a}=-\sqrt[3]{a}.$

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

prime factorizationprime factorizationprime factorization

rational numberrational numberrational number

square of a numbersquare of a numbersquare of a number

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