Topicmbbfaae6c5b86797f_1528449000663_0Topic

Properties of rootsproperties of rootsProperties of roots

Levelmbbfaae6c5b86797f_1528449084556_0Level

Second

Core curriculummbbfaae6c5b86797f_1528449076687_0Core curriculum

II. Roots. The student:

4) calculates the root of product and quotient of two numbers, removes and factors in perfect square factors;

5) divides roots of the same degree.

Timingmbbfaae6c5b86797f_1528449068082_0Timing

45 minutes

General objectivembbfaae6c5b86797f_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesmbbfaae6c5b86797f_1528449552113_0Specific objectives

1. Applying theorems about the product and the quotient of roots.

2. Calculating values of expressions containing roots.

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

Learning outcomesmbbfaae6c5b86797f_1528450430307_0Learning outcomes

The student:

- applies theorems about the product and the quotient of roots,

- calculates values of expressions containing roots.

Methodsmbbfaae6c5b86797f_1528449534267_0Methods

1. Discussion.

2. Thematical contest.

Forms of workmbbfaae6c5b86797f_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmbbfaae6c5b86797f_1528450127855_0Introduction

The teacher introduced the subject of the lesson – theorems about operations on roots.

Students will also calculate values of expressions containing square and cube roots.

Procedurembbfaae6c5b86797f_1528446435040_0Procedure

The teacher divides the class into 3 persons groups. Each group gets one example to solve: ${\left(\sqrt{25}\right)}^2,\sqrt{{25}^2},\sqrt{25}\cdot \sqrt{25}$.

Predicted results:

1 group - ${\left(\sqrt{25}\right)}^2=5^2=25$

2 group - $\left(\sqrt{{25}^2}\right)=625=25$

3 group - $\sqrt{25}\cdot \sqrt{25}=5\cdot 5=25$

Each group should obtain the same result.

Discussion – will we also obtain the same results while creating similar examples for cube rootcube rootcube root? Students solve the problem in group, by analysing their own examples.

Together they write down the conclusion:
- For any non‑negative number a the following equalities are true:mbbfaae6c5b86797f_1527752263647_0Together they write down the conclusion:
- For any non‑negative number a the following equalities are true:

- For any number a the following equalities are true:mbbfaae6c5b86797f_1527752263647_0- For any number a the following equalities are true:

Task
Students use the computers to get to know theorems about the product of square roots and cube roots.

[Slideshow]

They check if similar formulaformulaformula can be applied to the quotient of roots. They confirm their theory on specific examples.

Conclusion:

- For any non‑negative numbers a and b the following equalities are true:mbbfaae6c5b86797f_1527752256679_0- For any non‑negative numbers a and b the following equalities are true:

- For any numbers a and b the following equalities are true:mbbfaae6c5b86797f_1527752256679_0- For any numbers a and b the following equalities are true:

The teacher announced thematical contest. The class is divided into 4 persons groups. Each students of the group gets one question from each task. The group that answers correctly all questions, gets a mark.

Task
Calculate.

a) $\sqrt{3}·\sqrt{3}$

b) $\sqrt{2,4}·\sqrt{2,4}$

c) ${\left(\sqrt{81}\right)}^2$

d) ${\left(\sqrt{6,4}\right)}^2$

Task
Calculate.

a) $\sqrt[3]{5}·\sqrt[3]{5}·\sqrt[3]{5}$

b) $\sqrt[3]{25}·\sqrt[3]{25}·\sqrt[3]{25}$

c) ${\left(\sqrt[3]{27}\right)}^3$

d) $\sqrt[3]{125}·{\left(\sqrt[3]{125}\right)}^3$

Task
Do the operations.

a) $\sqrt{36·4}$

b) $\sqrt{0,16·81}$

c) $\sqrt[3]{8·64}$

d) $\sqrt[3]{27·0,001}$

Task
Do the operations.

a) $\sqrt{49:16}$

b) $\sqrt{1,44:0,09}$

c) $\sqrt{0,36:81}$

d) $\sqrt[3]{216:125}$

Task
Calculate.

a) $\sqrt{2}·\sqrt{32}$

b) $\sqrt{10}·\sqrt{14,4}$

c) $\sqrt[3]{4}·\sqrt[3]{16}$

d) $\sqrt[3]{9}·\sqrt[3]{81}$

Task
Calculate.

a) $\sqrt{27}:\sqrt{3}$

b) $\sqrt{72}:\sqrt{2}$

c) $\sqrt[3]{625}:\sqrt[3]{25}$

d) $\sqrt[3]{0,005}:\sqrt[3]{5}$

Task
Insert such number in the dotted space that the equality is true.

a) ${\left(\sqrt{5}\right)}^4=5^{...}$

b) ${\left(\sqrt{2}\right)}^6=2^{...}$

c) ${\left(\sqrt[3]{3}\right)}^6=3^{...}$

d) ${\left(\sqrt[3]{6}\right)}^9=6^{...}$

Task
Insert such numbernumbernumber in the dotted space that the equality is true.

a) $\sqrt{2^{...}}=2^4$

b) $\sqrt{7^{...}}=7^6$

c) $\sqrt[3]{{12}^{...}}={12}^6$

d) $\sqrt[3]{{25}^{...}}={25}^9$

An extra task:
Prove that the value of the expression $\frac{10}{\sqrt[3]{0,05}}-2\cdot \sqrt[3]{2500}$ is a natural numbernatural numbernatural number.

Lesson summarymbbfaae6c5b86797f_1528450119332_0Lesson summary

Students do the revision exercises.

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

- For any non‑negative numbernumbernumber a the following equalities are true.

- For any numbernumbernumber a the following equalities are true.

- For any non‑negative numbers a and b the following equalities are true.

- For any numbers a and b the following equalities are true.

Selected words and expressions used in the lesson plan

cube rootcube rootcube root

degree of the rootdegree of the rootdegree of the root

formulaformulaformula

natural numbernatural numbernatural number

numbernumbernumber

number under the rootnumber under the rootnumber under the root

properties of rootsproperties of rootsproperties of roots

quotient of rootsquotient of rootsquotient of roots

root of productsroot of productsroot of products

square rootsquare rootsquare root