Topicm4637699966cb5ad8_1528449000663_0Topic

The sequence of numbers

Levelm4637699966cb5ad8_1528449084556_0Level

Third

Core curriculumm4637699966cb5ad8_1528449076687_0Core curriculum

VI. The sequence of numbers. The student:

1) calculates the terms of a sequence expressed as a general formula;

2) calculates the first term of a sequence defined by recursive formula, as in the examples:

a) $\left\{\begin{array}{l}{a}_1=0,001\\ a_{n+1}=a_n+\frac{1}{2}{a}_n\left(1-a_n\right)\end{array}\right.$

b)$\left\{\begin{array}{l}{a}_1=1\\ a_2=1\\ a_{n+2}=a_{n+1}+a_n\end{array}\right.$

3) in simple cases examines if it is an increasing or decreasing sequence.

Timingm4637699966cb5ad8_1528449068082_0Timing

45 minutes

General objectivem4637699966cb5ad8_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesm4637699966cb5ad8_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Gives properties of the sequence of numbers.

3. Finds terms of a sequence.

Learning outcomesm4637699966cb5ad8_1528450430307_0Learning outcomes

The student:

- can give properties of the sequence of numbers,

- can finds terms of a sequence.

Methodsm4637699966cb5ad8_1528449534267_0Methods

1. Incomplete sentences.

2. Situation analysis.

Forms of workm4637699966cb5ad8_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm4637699966cb5ad8_1528450127855_0Introduction

The students use the incomplete sentences technique to recollect their knowledge about the functionfunctionfunction.

- The functionfunctionfunction is an association in which…

- The domain of a functionfunctionfunction is…

- The set of outputs of a functionfunctionfunction is…

- The functionfunctionfunction can be described in many ways ….

- The graph of a functionfunctionfunction is…

When the students finish their task, the teacher gets the students’ knowledge in order and explains any doubts.

Procedurem4637699966cb5ad8_1528446435040_0Procedure

The teacher informs the students that the aim of the class will be getting to know the properties of the functionfunctionfunction whose domain are positive natural numbersnatural numbersnatural numbers.

The definition of the sequencesequencesequence.

The sequence of numberssequence of numberssequence of numbers is a functionfunctionfunction defined in a set of positive natural numbersnatural numbersnatural numbers. The outputs of this functionfunctionfunction for consecutive natural numbersnatural numbersnatural numbers are called the terms of a sequencesequencesequence and denoted by (aIndeks dolny n_n).

Working in groups, the students think about the methods of describing sequencesmethods of describing sequencesmethods of describing sequences.

Task

Think how to describe a sequencesequencesequence. Remember that the sequence is a functionfunctionfunction. Formulate your conclusion.

The conclusion:

The sequencesequencesequence, just like any functionfunctionfunction, can be described by words, a table, a formula, a graph or ordered pairs.

The students work individually and analyse the graphs of sequences.

Task

[Geogebra applet]

Analyse the material presented in the applet. Sketch the fragments of the graphs of sequences in the given formulas:m4637699966cb5ad8_1527752263647_0Analyse the material presented in the applet. Sketch the fragments of the graphs of sequences in the given formulas:

a) aIndeks dolny n_n = 3 - n,

b) aIndeks dolny n_n = -4 + 2n,

c) aIndeks dolny n_n = (-1)Indeks górny nⁿ - n,

d) aIndeks dolny n_n = 2n - 2Indeks górny nⁿ,

e) aIndeks dolny n_n = nIndeks górny 2² - 2n + 3.

What have you observed? Formulate your conclusion.

The conclusion:

The graph of a sequencesequencesequence is drawn in the same way as the graph of an appropriate functionfunctionfunction. The graph is not a line, but a set of points.

The students use the information to do the tasks individually.

Task

Calculate the four initial terms of sequencesequencesequence ($a_{\mathrm{n}}$) and term $a_{11}$:

a) $a_n=\frac{n^2}{n+2}$,

b) $a_n=\frac{-2}{n(n+3)}$,

c) $a_n=n^4-n^3$.

Task

Draw the graph of sequencesequencesequence aIndeks dolny n_n = 2n - 1, knowing that n∈N and 1 ≤ n ≤ 10.

An extra task
Check, which terms of sequence $a_n=\frac{n+13}{n-2}$ are integer numbers.m4637699966cb5ad8_1527752256679_0An extra task
Check, which terms of sequence $a_n=\frac{n+13}{n-2}$ are integer numbers.

Lesson summarym4637699966cb5ad8_1528450119332_0Lesson summary

The students do the consolidation tasks. They formulate the conclusions to be remembered.

- The sequence of numberssequence of numberssequence of numbers is a functionfunctionfunction defined in a set of positive natural numbersnatural numbersnatural numbers.

- The outputs of this functionfunctionfunction for consecutive natural numbersnatural numbersnatural numbers are called the terms of a sequencesequencesequence.

Selected words and expressions used in the lesson plan

functionfunctionfunction

methods of describing sequencesmethods of describing sequencesmethods of describing sequences

natural numbersnatural numbersnatural numbers

sequencesequencesequence

sequence of numberssequence of numberssequence of numbers