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Inverse proportionality

Source: licencja: CC 0.

Proporcjonalność odwrotna

Learning objectives

You will determine inversely proportional quantities.

You will draw a graph of inverse proportionality.

Learning effect

  • You will learn to determine inversely proportional quantities.

  • You will learn to draw a graph of inverse proportionality.

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nagranie abstraktu

Think what you remember about proportionality and its properties. The aim of the lesson will be to learn about quantities which are inversely proportional.

Do the introductory tasks.

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

Price per kilo [PLN]

2

4

5

2,5

1,6

3,2

Number of kilos

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What do you notice? Formulate the appropriate conclusion.

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

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Aplet geogebra: Zależność między długościami boków prostokąta o stałym polu. Poniżej znajduje się galeria będąca wersją alternatywną dla aplikacji.
Source: GroMar, licencja: CC BY 3.0.
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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.

Search in the available resources what we call two such quantities, whose product is constant.

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

Inverse Proportionality
Definition: Inverse Proportionality
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The function describing the relationship between positive inverse proportional values x and y is called inversely proportional, and the product xy = 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=ax.

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Using the information learned during the lesson solve the tasks.

Task 2
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List all pairs of natural numbers that satisfy the dependence xy = 64.

Task 3
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A passenger car traveling at an average speedspeedspeed of 80kmh covers a certain distancedistancedistance in 2 hours and 12 minutes. In what timetimetime will a motorcyclist cover this route with an average speedspeedspeed of 22kmh? At what speedspeedspeed should the car go to cover this route in 4 hours?

Task 4
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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).

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Do the revision exercises.

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Remember:

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

Exercises

Exercise 1
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Wersja alternatywna ćwiczenia: Determine which sentences are true. Możliwe odpowiedzi: 1. The area of the circle and its radius length are inversely proportional quantities., 2. The sides of the rectangle with a area equal to 20 are inversely proportional., 3. The volume of the cube is inversely proportional to the length of its side., 4. The lengths of cathetus of the right-angled rectangular triangle with a surface area of 12 are inversely proportional.
zadanie
Source: GroMar, licencja: CC BY 3.0.
Exercise 2

The driver drove the 208km distance with an average speed of v [kmh] at some time t[h]. If he drove at a speed of 13kmh higher, then his route would be over 48 minutes shorter. Calculate v.

Exercise 3

Give three examples of inversely proportional quantities in English.

Exercise 4
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Wersja alternatywna ćwiczenia: Indicate which pairs of expressions or words are translated correctly. Możliwe odpowiedzi: 1. droga - distance, 2. prędkość - speed, 3. czas - time, 4. współczynnik proporcjonalności - inverse proportionality, 5. proporcjonalność odwrotna - proportional coefficient
Rzadanie
Source: GroMar, licencja: CC BY 3.0.
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Interaktywna gra, polegająca na łączeniu wyrazów w pary w ciągu jednej minuty. Czas zaczyna upływać wraz z rozpoczęciem gry. Jeden ruch to odkrywanie najpierw jednej potem drugiej karty z wyrazem. Każdy wyraz jest odczytywany. Kolejny ruch to odkrywanie trzeciej i czwartej karty. W ten sposób odsłuchasz wszystkie wyrazy. Nawigacja z poziomu klawiatury za pomocą strzałek, odsłuchiwanie wyrazów enterem lub spacją. Znajdź wszystkie pary wyrazów.
Source: Zespół autorski Politechniki Łódzkiej, licencja: CC BY 3.0.

Glossary

distance
distance

droga

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wymowa w języku angielskim: distance
inverse proportionality
inverse proportionality

proporcjonalność odwrotna

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wymowa w języku angielskim: inverse proportionality
proportional coefficient
proportional coefficient

współczynnik proporcjonalności

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wymowa w języku angielskim: proportional coefficient
speed
speed

prędkość

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wymowa w języku angielskim: speed
time
time

czas

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wymowa w języku angielskim: time

Keywords

inverse proportionalityinverse proportionalityinverse proportionality

speedspeedspeed

distancedistancedistance

timetimetime