Topicm692e007d06c00c91_1528449000663_0Topic

Number of elements of the finite set

Levelm692e007d06c00c91_1528449084556_0Level

Third

Core curriculumm692e007d06c00c91_1528449076687_0Core curriculum

XI. CombinatoricscombinatoricsCombinatorics. The basic level. The student:

1) calculates objects in simple combinatorics situationscombinatorics situationscombinatorics situations.

Timingm692e007d06c00c91_1528449068082_0Timing

45 minutes

General objectivem692e007d06c00c91_1528449523725_0General objective

Choosing and creating mathematical models to solve practical and theoretical problems.

Specific objectivesm692e007d06c00c91_1528449552113_0Specific objectives

1. Identifying number of objects in the finite set.

2. Calculating objects in simple combinatorics situations.

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

Learning outcomesm692e007d06c00c91_1528450430307_0Learning outcomes

The Student:

- identifies number of objects in the finite set,

- calculates objects in simple combinatorics situationscombinatorics situationscombinatorics situations.

Methodsm692e007d06c00c91_1528449534267_0Methods

1. Situational analysis.

2. JIGSAW.

Forms of workm692e007d06c00c91_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm692e007d06c00c91_1528450127855_0Introduction

Students revise concepts related with number sets.

The teacher introduces the subject of the lesson – determining number of elements of different number sets.

Procedurem692e007d06c00c91_1528446435040_0Procedure

Students work individually, using computers. Their task is to watch the interactive illustration, that explains the concept of the cardinality of a set.

[Interactive illustration]

After having completed the exercise, students present results of their observations:

- the cardinality of the set A is the number of all elements of the set A,
- the cardinality of the set A is denoted by: |A|.m692e007d06c00c91_1527752256679_0- the cardinality of the set A is the number of all elements of the set A,
- the cardinality of the set A is denoted by: |A|.

Students use obtained information in exercises, using the JIGSAW method.

The teacher divides students into 3 persons groups. Each member of the group gets different task from
the tasks below. After solving the tasks, students gather in groups that were doing the same task. They discuss the solutions and clarify any doubts.

Then, they return to the initial groups and present the solutions to other members.

Task 1

Identify how many numbers:

a) are two‑digit numbers and can be divided by 3 or by 25,
b) are three‑digit numbers greater than 250 and smaller than 600 and can be divided by 54.

Task 2

There are three senior year classes in one of the high schools.In these classes, 11 students go only for extra mathematics classes, 7 - only for extra Polish classes and 4 – only for extra English classes. Moreover, there are students that attend two different classes. An so: 9 students go for mathematics and English classes, 10 – English and Polish and 11 – Polish and mathematics. One student goes for aa three extra classes.
Give the cardinality of the sets that define groups of students at each extra classes and the cardinality of the set of students of all three senior classes.

Task 3

Read the equinumerosity principleequinumerosity principleequinumerosity principle in the frame and then check if defined below sets A, B are equinumerous:

A = {37, 38, 39 … 57},
B = {102, 103, 104 … 122}.

Definition

The equinumerosity principle:

Two sets A and B are equinumerous (have the same number of elements) if their elements can be clearly assigned to each other, meaning each element of the set A has exactly one element from the set B assigned and each element of the set B has exactly one element of the set A assigned.m692e007d06c00c91_1527752263647_0Two sets A and B are equinumerous (have the same number of elements) if their elements can be clearly assigned to each other, meaning each element of the set A has exactly one element from the set B assigned and each element of the set B has exactly one element of the set A assigned.

The teacher evaluates the students’ work and clarifies doubts.

An extra task

Give the cardinality of the setcardinality of the setcardinality of the set in which there are all three‑digit numbers, excluding those than can be divided by 2 or by 5.

Lesson summarym692e007d06c00c91_1528450119332_0Lesson summary

Students do the revision exercises.

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

- the cardinality of the set A is the number of all elements of the set A,
- the cardinality of the set A is denoted by: |A|.m692e007d06c00c91_1527752256679_0- the cardinality of the set A is the number of all elements of the set A,
- the cardinality of the set A is denoted by: |A|.

Selected words and expressions used in the lesson plan

cardinality of the setcardinality of the setcardinality of the set

combinatoricscombinatoricscombinatorics

combinatorics situationscombinatorics situationscombinatorics situations

equinumerosity principleequinumerosity principleequinumerosity principle

number of elements of a finite setnumber of elements of a finite setnumber of elements of a finite set

sets of numberssets of numberssets of numbers