Direct and Inverse Variation/Proportion

Which is it?? Table, Equation, or Graph Linear Nonlinear Not an inverse variation… just nonlinear Direct Variation Not a direct variation… just linear Inverse Variation 11/14/2018

Q3 Grade 7 Content Overview Vocabulary Variation and proportion Constant of proportionality Direct Proportion Indirect Proportion Equation Ratio Atlanta Public Schools Q3 Grade 7 Content Overview

What is Direct Variation/Proportion? Direct Proportion Direct Variation Directly Proportional A relationship between two variables (usually x and y) where one variable changes and the other variable changes in proportion to the first. Atlanta Public Schools Q3 Grade 7 Content Overview

Direct Variation/Proportion continued As the x values increase at a constant rate, the y values will also increase at a constant rate. As the x values decrease at a constant rate, the y values will decrease at a constant rate. X and y will have a proportional relationship. Atlanta Public Schools Q3 Grade 7 Content Overview

Constant of Proportionality When y is directly proportional to x (or is a direct variation), there will be a CONSTANT OF PROPORTIONALITY. In equations, this constant is represented by the letter k. y = kx The constant of proportionality is found by dividing the y values by the x values. y/x = k The constant of proportionality, or k, is the exact same thing as the slope (rise over run) that we previously learned about. Atlanta Public Schools Q3 Grade 7 Content Overview

Explain: Direct Proportion Consider the following direct proportion situation: It takes 2 hours for a company to manufacture each ladder that it makes Examine the table below: Ladders(x) 1 2 3 4 5 Hours(y) 2 4 6 8 10 ©Microsoft Word clipart Now look at the graph of the data Atlanta Public Schools Q3 Grade 7 Content Overview

Q3 Grade 7 Content Overview Graph Of Ladders Made against Time Taken Hours 2 4 6 8 10 12 Ladders Atlanta Public Schools Q3 Grade 7 Content Overview

Q3 Grade 7 Content Overview L T Key Facts about Direct Proportion All direct proportion graphs are straight line graphs passing through the origin Look at the table of values again: Ladders(x) 0 1 2 3 4 5 Hours(y) 2 4 6 8 10 If you divide the number of hours by the given number of ladders, what do you find each time? The answer is 2 every time. As x increases, y increases at a constant rate of 2x. This is a linear equation in the form y = kx where 2 is the constant of proportionality (k). Note also that the ratio of Ladders to Hours remains constant at x:2x. Atlanta Public Schools Q3 Grade 7 Content Overview

Inverse Variation/Proportion A relationship between two variables in which the product is a constant. When one variable increases the other decreases in proportion so that the product is unchanged. As x increases, y decreases. As x decreases, y increases. xy = k (Remember, k is the constant of proportionality) The graph of an inverse variation is not a straight line, but it is curved. The graph will never cross the x-axis. Atlanta Public Schools Q3 Grade 7 Content Overview

Explain: Indirect Proportion A farmer has enough cattle feed to feed 64 cows for 2 days. Examine the table below. As the number of cows increases the number of days of feed decreases. Cows 2 4 8 16 32 64 (x) Days 4 2 (y) 8 16 32 64 We are going to draw a graph of the table. The scale allows each axis to go at least up to 65. We estimate the position of the points (2,64) (4,32) etc. as accurately as possible. Atlanta Public Schools Q3 Grade 7 Content Overview

Q3 Grade 7 Content Overview 10 20 30 40 50 60 70 Graph of Number of Cows against Days of Feed (2,64) Days (4,32) (8,16) (16,8) (32,4) (64,2) 10 20 30 40 50 60 70 Cows Atlanta Public Schools Q3 Grade 7 Content Overview

Q3 Grade 7 Content Overview The graph is a typical inverse or indirect proportion graph Cows Days It shows us that as the number of cows increases, the number of days left of feed decreases. The reverse is also true. If we decrease the number of cows, we will increase the number of days of feed. Cows 2 4 8 16 32 64 (x) Days 4 2 (y) 8 16 32 64 This is a non-linear equation in the form y = k where k is still the constant of proportionality. However, y varies inversely to x. In this situation, what is k? x Atlanta Public Schools Q3 Grade 7 Content Overview