2h1 Directly Proportional date:

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1 Opening activities: A. One week you go to the grocery store and buy 4 apples, 1 loaf of bread, and 1 dozen eggs. The following week you go to the grocery store and buy 6 apples, 2 loaves of bread, and 2 dozen eggs. Assuming that the prices of the items remain the same for both weeks, which week do you expect to spend more money on groceries? How do you know this is true? B. Some business people want to carry out an ambitious project: linking Montreal and New York by a daily high-speed train service. This would mean a person living in Montreal could work in New York, and vice versa, despite the 600 km separating both cities! Below are four categories of high-speed trains being considered by the committee in charge of the project. Which high-speed train should the committee choose? Comparative table of railway technologies: Feature Express Turbotrain Rapido Supertrain Means of Electric Diesel Magnetic Electric diesel propulsion Top speed 400 km/h 5 km/ min 500 km/h 100 m/s Number of seats 60 / wagon / 4 wagons 100 / wagon Cost of locomotive $15,000,000 $10,000,000 $25,000,000 $12,000,000 Lifespan of 20 years 20 years 25 years 20 years locomotive Cost of infrastructure $10,000 / m $2,500,000 / km $2,500,000 / km $2 Billion 1

2 Prerequisite Skills: Reviewing graphing: 2

3 Directly proportional relationships: A proportional relationship between two quantities occurs when the quantities vary directly with one another. This means that when the first variable is doubled, (or tripled, quadrupled halved, divided into thirds, fifths etc.) then the second variable is correspondingly doubled, (or tripled, quadrupled halved, divided into thirds, fifths etc.) Two variables are directly proportional when their ratio is a constant for all values of each variable. In other words, when one variable is divided by the other, the answer is always a constant. Two variables x and y are directly proportional if k = y x is always the same for each pair (x,y). k is called the scale factor, or the factor of proportionality. In German this is called the Proportionalitätsfaktor. EX.1. You are paid 5 for each hour that you babysit. a. Fill in the table. b. Calculate the scale factor, k for each pair (x,y). c. Is this a directly proportional situation? x (# of hours babysitting) y (money paid in Euros) 50 EX.2. Do the following tables represent directly proportional situations? Why or why not? a. b. Temperature ( o C) Evaporation Rate (mm/day)

4 2h1 Directly Proportional date: EX.3. The following tables represent directly proportional situations. i. Fill in the missing values. (watch the tutorial online about filling in tables of values for proportional situations). ii. Graph the relationship. a b EX.4. Which of the following is a proportional situation? a. Number of pages to the height of pile in a pile of printer paper b. Length of side of square Perimeter of the square c. x to (2x+10) d. z 2.5z e. Length of a side of a cube volume of the cube. f. Bananas are 2.40 a pound 4

5 2h1 Practice questions: Directly Proportional date: 1. Find the missing term in each of the following tables describing a proportional situation. a. x y b. c d Complete these tables of values associated with directly proportional situations. Graph each of them. a. x y b. x 2 4 y c. x d. x 2 y y e. x f. x y 44 y Auf sechs Tüten mit Weintrauben befinden sich die Angaben Zwei der Maßzahlen sind unleserlich geworden. a. Der Preis p der Tauben ist direkt proportional zur Masse m der Trauben. Begründe dies mithilfe der vier lesbaren Wertepaare. Gib den Proportionalitätsfaktor an und erkläre, was er bedeutet. b. Ergänze dann die beiden fehlenden Maßzahlen. 4. A 2.1 kg chicken costs a. What is the cost of a chicken weighing 1) 2.5 kg? 2) 1.4 kg? b. What is the mass of a chicken whose cost is: 1) 15.36? 2) 7.68? 5

6 5. Brigit changes 250 at a bank for $300 (US dollar). How many dollars would she have got for 350? Calculate your answer in 2 ways. 6. An ink jet printer has a performance of up to 6 pages a minute. a) What is the minimum number of minutes required to print out 20 pages? b) What is the maximum number of pages which can be printed out in 15 minutes? c) The printer has a definition of 600 dpi (dots per inch). That means that it prints out 600 dots in a horizontal line which is 1 inch (2.54cm) long. How many dots does it print in a 21cm line across an A4 sheet? 7. In a storm, 180cm 3 of rain fell on a measuring area of 900cm 2. How much rain fell on a town with an area of 5km 2? 8. Zeigen das die Größen x und y zueinander direkt proportional sind, und gib den Proportinalitätsfaktor an. x y a. Zeichne ein zur Tabelle passendes liniendiagramm. b. Lucas hat die Tabelle mit den Wertepaaren x1 = 7.5; y1 = 6 und x2 = -3.6; y2 = -2.4 ergänzt. Find zeichnerisch und rechnerisch heraus, ob sie passen. 9. Do the following tables represent directly proportional situations? If yes, what is the scale factor? a. Quantity of cookies Price ($) b. Lemons Pies In each case, decide if the description represents a proportional situation. Justify your answer using calculations. a. Pencils of a given brand sell for $0.25/ unit or $2.75/dozen. b. It takes Lea 45 min to hike Beaver Trail, which measures 2 km, and 2h 15min to hike Partridge Trail, which measures 6km. c. Last week Christiane s car burned 35 L of fuel. Last year, it burned 1820 L. 6

7 11. In each case, indicate if it is possible to answer the question using a proportion. Explain your answer. a. During the first moth of its life a baby grows 2 cm to reach a height of 27 cm. What will the baby s height be at 36 months? b. A student who studies 30 min/day scores an average of 72% on an exam. What will her average be if she studies 45 min/day? c. A battery is completely used up. Charging it for 90 min lets it regain one quarter of its maximum charge. How long does this battery need to charge in order to be completely recharged? 12. Each one of the following situations is a direct proportional situation a. A runner does 9 laps of the rack in 6 minutes. i. How long does it take him to do 12 laps? ii. How many laps does he do in 14 minutes? b. A car driver is traveling at an average speed of 80 km/h. i. What distance does he travel in 45 minutes? ii. How long does it take him to travel 220 km? c. A surface of 25 m 2 is painted with 6 liters of paint. i. What surface can be painted with 15 liters? ii. How many liters of paint is required to paint a surface of 70 m 2? d. You need 10 kilograms of flour to make 12 kg of bread. i. How many kilograms of flour are needed to make 45 kg of bread? ii. How many kilograms of bread can be made with 18 kg of flour? e. Three ice creams cost $5.25. i. How much doe eight ice creams cost? ii. How many ice creams can you buy with $15 7