Topicm539e58a9424c5e84_1528449000663_0Topic

Subsets of the finite set

Levelm539e58a9424c5e84_1528449084556_0Level

Third

Core curriculumm539e58a9424c5e84_1528449076687_0Core curriculum

XI.CombinatoricscombinatoricsCombinatorics.

The basic level. The student:

1. Calculates objects in simple combinatorics situationscombinatorics situationscombinatorics situations;

2. Objects applying rule of sum and rule of product (also together) for any number of activities, in situation no more difficult than:

a) calculating in how many four‑digit, odd, positive integers there is exactly one digit 1 and exactly one
digit 2,
b) calculating in how many four‑digit even, positive integers there is exactly one digit 0 and exactly one
digit 1.

Timingm539e58a9424c5e84_1528449068082_0Timing

45 minutes

General objectivem539e58a9424c5e84_1528449523725_0General objective

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

Specific objectivesm539e58a9424c5e84_1528449552113_0Specific objectives

1. Calculating objectscalculating objectsCalculating objects while applying the rule of sum and the rule of product.

2. Identifying the number of all two‑element subsetstwo‑element subsetstwo‑element subsets of the finite setfinite setfinite set.

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

Learning outcomesm539e58a9424c5e84_1528450430307_0Learning outcomes

The Student:

- calculates objects while applying the rule of sum and the rule of product,

- identifies the number of all two‑element subsets of the finite set.

Methodsm539e58a9424c5e84_1528449534267_0Methods

1. Situational analysis.

2. JIGSAW.

Forms of workm539e58a9424c5e84_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm539e58a9424c5e84_1528450127855_0Introduction

Students revise concepts of sets and subsets.

The teacher introduces the subject of the lesson - the number of all two‑element subsetstwo‑element subsetstwo‑element subsets of the finite setfinite setfinite set.

Procedurem539e58a9424c5e84_1528446435040_0Procedure

Students work in pairs. They do exercises and look for the formula for the number of all two‑elements subsets of the finite setfinite setfinite set.

Task 1

Write all two‑element subsetstwo‑element subsetstwo‑element subsets of the set:

a) A = {a,b,c,d}
b) A = {1,2,3,4,5}
c) A = {K,R,T,Z,W,Y}

Write how many subsets are there. Give the relations between the number of subsets and the number of elements in this set.

Students work individually, using computers. Their task is to watch the interactive illustration that presents this relation

[Interactive illustration]

After having finished the exercise, they present results of their observations.

To calculate the number of all two‑element subsetstwo‑element subsetstwo‑element subsets that can be chosen from the set $A=\{a_1,a_2,a_3,\dots ,a_n\}$we can use the formula:

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 2

How many ways are there to choose a 2‑people delegation from a 28‑people class?m539e58a9424c5e84_1527752256679_0How many ways are there to choose a 2‑people delegation from a 28‑people class?

Task 3

There are 12 girls and 8 boys in a class. A four‑people team is needed to take part in school competition. How many options of choice of such team are in this class?

Task 4

How many three‑digit natural numbers in which the ones digit is greater than the tens digit are there?

Task 5

Points were marked on parallel lines k and l, like on the drawing. How many triangles that meet the following conditions can be made?

- vertices of triangles are points marked on lines k and l,
- the base of the triangle is a line segment located on the line l.

[Illustration 1]

The teacher evaluates the students’ work and clarifies doubts.

An extra task:

Calculate how many natural, 11‑digit numbers whose product of digits is equal to 12 are there.

Lesson summarym539e58a9424c5e84_1528450119332_0Lesson summary

Students do the revision exercises. Then together they sum‑up the classes, by formulating the conclusions to memorise.

To calculate the number of all two‑element subsets that can be chosen from the set $A=\{a_1,a_2,a_3,\dots ,a_n\}$
we can use the formula:m539e58a9424c5e84_1527752263647_0To calculate the number of all two‑element subsets that can be chosen from the set $A=\{a_1,a_2,a_3,\dots ,a_n\}$
we can use the formula:

Selected words and expressions used in the lesson plan

calculating objectscalculating objectscalculating objects

combinatoricscombinatoricscombinatorics

combinatorics situationscombinatorics situationscombinatorics situations

finite setfinite setfinite set

number of all two‑element subsets from the set Anumber of all two‑element subsets from the set Anumber of all two‑element subsets from the set A

subsets of a finite setsubsets of a finite setsubsets of a finite set

two‑element subsetstwo‑element subsetstwo‑element subsets