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CONSTANT OF PROPORTIONALITY! WHAT IS IT? • A constant ratio in any proportional relationship • Really just another name for unit rate! Remember, to be constant means it never changes! WITH RATIO TABLES Find the constant of proportionality between the gallons and the miles. Miles 50 100 150 200 ÷ = ÷ = Gallons 2 4 ÷2 50 ? = 2 1 ÷2 25 miles per gallon 6 8 ÷ = This means our constant of proportionality is 25, so if we divide the miles by gallons we should always get 25. Let’s check! TRY THIS ONE! Find the constant of proportionality between the gallons and the miles. Number of Apples 9 27 36 Cost $3.00 $9.00 $12.00 ÷3 9 ? = 3 1 ÷3 Constant of proportionality = 3 Let’s Check! ÷ = ÷ = WITH GRAPHS Miles Find the constant of proportionality. 200 180 160 140 120 (2,90) 100 80 (1,45) 60 40 (0,0) 20 0 0 1 2 (4,180) Y 45 90 135 180 X 1 2 3 4 (3,135) To find our constant of proportionality we have to divide! 3 Hours 4 5 y x 45 1 90 135 180 3 2 4 45 = 1 135 = 3 45 45 90 = 2 180 = 4 Y 45 90 135 180 X 1 2 3 4 45 45 So, our constant of proportionality is 45. Do you see that our x value multiplied by our constant of proportionality is our y value? We could write this like: y=45(x) We will ALWAYS be able to write our constant of proportionality as an equation that looks like this: y=kx. And k will always be our constant of proportionality! Just like in our last example! y=45x. WHAT CAN WE DO WITH IT? • Population Density: Finding the constant of proportionality by dividing the population by the area. • Finding how many per 1 square mile POPULATION DENSITY Wildlife conservationists are concerned that the deer population might not be constant across the National Forest. The scientists found that there were 144 deer in a 16 square mile area of the forest. In another part of the forest, conservationists counted 117 in a 13 square mile area. Then a third conservationist counted 216 deer in a 24 square mile area of the forest. Do the conservationists need to be worried? Deer Square miles Do we have a constant of proportionality? 144 16 117 13 216 24 144 =9 16 117 =9 13 216 =9 24 POPULATION DENSITY So do the conservationists need to be worried? NOPE! The deer population is constant in the National Forest. Let’s write an equation! If we multiply our square miles by 9 we will have the number of deer. So our equation will look like: Deer = 9 x Miles OR y=9x Deer Square miles Constant of proportionality 144 16 9 117 13 9 216 24 9