Topicm3786dfccd7b84827_1528449000663_0Topic

Inverse proportionality

Levelm3786dfccd7b84827_1528449084556_0Level

The third

Core curriculumm3786dfccd7b84827_1528449076687_0Core curriculum

V. Functions. Student:

13. uses the function $f\left(x\right)=\frac{a}{x}$ and its graph to describe and interpret issues related to inversely proportional quantities, also in solving practical problems.

Timingm3786dfccd7b84827_1528449068082_0Timing

45 minutes

General objectivem3786dfccd7b84827_1528449523725_0General objective

Interpreting and manipulating information presented in both mathematical and popular science texts, as well as in the form of graphs, diagrams, tables.

Specific objectivesm3786dfccd7b84827_1528449552113_0Specific objectives

1. Communicating in English, developing mathematics skills, scientific, technical and IT competences; developing learning skills.

2. Determination of inversely proportional quantities.

3. Drawing a graph of inverse proportionality.

Learning outcomesm3786dfccd7b84827_1528450430307_0Learning outcomes

The student:

- determines inversely proportional quantities.

- draws a graph of inverse proportionality.

Methodsm3786dfccd7b84827_1528449534267_0Methods

1. Mind maps.

2. Situational analysis.

Forms of workm3786dfccd7b84827_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm3786dfccd7b84827_1528450127855_0Introduction

Students, working in groups, review the definition and properties of the direct proportionality.

They use the information to create a mind map.

After finishing the task, they present their work to their colleagues.

Procedurem3786dfccd7b84827_1528446435040_0Procedure

The teacher informs students that the aim of the lesson will be to learn about the inverse proportions.

Students working in groups solve introductory tasks.

Task

How many kilos of apples can we buy for PLN 40 if the price per 1 kilogram changes? Complete the table.

[Table]

What do you notice? Formulate the appropriate conclusion.

Conclusion
The higher the price of apples, the fewer kilograms of apples we can buy.m3786dfccd7b84827_1527752263647_0Conclusion
The higher the price of apples, the fewer kilograms of apples we can buy.

Students, working in groups, analyze the applet. They think about the relationship between the sides of a rectangle with a fixed surface area. They formulate the conclusion.

[Geogebra applet]

Conclusion
If we reduce the length of one side of the rectangle with a fixed surface area, the length of the second side increases proportionally.

The graph of dependencies between the sides of the rectangle has the shape of a curve whose asymptotes are the axes of the coordinate system.

The teacher informs students that two such quantities, whose product is constant, are called inversely proportional quantities.

Definition: Inversely proportional quantities.

We say that two positive quantities $x$ and $y$ are inversely proportional if, and only if, their product is constant.

Definition: Inverse Proportionalityinverse proportionalityInverse Proportionality

The function describing the relationship between positive inverse proportional values $x$ and $y$ is called inversely proportional, and the product $x$ ∙ $y$ = $a$ is called the inversely proportional coefficientproportional coefficientproportional coefficient.

The relationship between the inversely proportional quantities $x$ and $y$ can also be written in the form $y=\frac{a}{x}$

Using the information learned during the lesson, students independently solve the tasks.

Task
List all pairs of natural numbers that satisfy the dependence $x$ ∙ $y$ = 64m3786dfccd7b84827_1527752256679_0List all pairs of natural numbers that satisfy the dependence $x$ ∙ $y$ = 64

Task
A passenger car traveling at an average speedspeedspeed of $80 \frac{\mathrm{km}}{\mathrm{h}}$ covers a certain distancedistancedistance in 2 hours and 12 minutes. In what timetimetime will a motorcyclist cover this route with an average speedspeedspeed of $22 \frac{\mathrm{km}}{\mathrm{h}}$? At what

Task for volunteers
The food supplies collected in the 80‑people‑school canteen are enough for 6 days. How many days would this food last if the number of people at the school canteen increased by 40? (we assume that the portions will remain unchanged).

Lesson summarym3786dfccd7b84827_1528450119332_0Lesson summary

Students do the revision exercises.

Then they summarize the lesson and formulate the conclusion to remember:

Conclusion Two positive values $x$ and $y$ are inversely proportional if, and only if, their product is constant.

Selected words and expressions used in the lesson plan

distancedistancedistance

inverse proportionalityinverse proportionalityinverse proportionality

proportional coefficientproportional coefficientproportional coefficient

speedspeedspeed

timetimetime