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Rates and Proportional Reasoning Today you will learn to: • determine unit rates and unit prices. M07.A-R.1.1.6 Use proportional relationships to solve multi-step ratio and percent problems. M07.A-R.1.1.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. Unit Rates and Proportional Relationships: Background In math, a coordinate system (or coordinate plane) is used to locate points. The coordinate plane is formed by the intersection of two number lines that meet at right angles at their zero points. • Origin – the point at which number lines intersect; (0,0) • x-axis – horizontal number line • y-axis – vertical number line An ordered pair of numbers is used to locate any point on a coordinate plane. For example, in (3,2), 3 is the x-coordinate, which corresponds to a number on the x-axis, and 2 is the y-coordinate, which corresponds to a number on the y-axis. 1 that y = mx. . Relationships Unit Rates and Proportional Proportional 2 ng for m, it can be seen that 11/ 6/ 13 4: Relationships: Graphs and Tables The equation y = mx is a linear equation. words, in a proportional relationship, the slope is equivalent to the ratio of Next Proportional Relationships A (directly) proportional relationship exists between two variables, ue is also referred to as the constant the unit Furthermore, whenofxproportionality equals zero,and/or y equals zero. x and y, if there is a nonzero constant, m, such (directly) proportional relationship exists that y =Amx. betweenthe twographical variables, representation of a proportional Therefore, In a proportional y = mx, m x and y, if relationship, there is a nonzero constant, m, such containing the point (0, 0). to as the constant of equation y =can mxbe is referred a linear equation. that y = mx. proportionality and/or the unit rate. Inxthe equation = mx,zero. m is the slope of the line. hermore, when equals zero, y y equals The equation = mxdescribes is a linear equation. A unity rate how many units of one type representation of quantity correspond to one relationship is a line efore, the graphical of a proportional Furthermore, x equals zero, equals Byofwhen solving for m, it ycan be zero. seen that . unit another type of quantity. aining the point (0, 0). Therefore, the graphical representation of a proportional relationship is a line Inmx, other words, in of a the proportional relationship, the slope e equation y = m is the slope line. ple containing 1: the point (0, 0). y to x. is Unit Rates and Proportional Relationships: Graphs Example 2: To find the unit rate, divide $120 by 6 hours. $120 ÷ 6 hours = $20 per hour Therefore, Steven earns $20 per hour. Which of the following graphs represents a proportional relationship? Solution: The graph of a proportional relationship is a line which contains the origin, (0, 0). Therefore, Graph B represents a proportional relationship. Unit Rates and Proportional Relationships: How to solve graphs Watch the following Khan Academy video showing how to solve a graphical rate problem. 4 Next A unit rate compares a quantity to one. Unit rates can be determined from proportional graphs, tables, equations, and verbal descriptions. Unit Rates and Proportional Relationships: Graphs Example 1: Myrtle drives the same number of miles to and from work each day, as shown on the graph below. Based on the graph, what is the unit rate of miles driven per day? Solution: Unit Rates and Proportional Relationships: How to solve tables Watch the following Khan Academy video showing how to solve a table rate problem. Unit Rates and Proportional Example 3: Relationships: Tables Therefore, Graph B represents a proportional relationship. 11/ 6/ 13 4:44 PM Determine if theSolution: following table represents a proportional relationship. Examine the relationship between the x and y values to see if there is a nonzero constant, m, such that y = mx. 1×[]=2 2×[]=4 3×[]=6 4×[]=8 5 × [ ] = 10 .com/ cfw/ content/ show- lesson/ b910b545?CFID= 42316365&CFTOKEN= 53009163&packID= a5188ec Each x-value, when multiplied by 2, will result in the corresponding y-value. Therefore, there is a nonzero constant, m, such that y = mx. In this relationship, the value of m is 2, and the corresponding equation is y = 2x. en the x and y values to the seetable if there a nonzero Thus, doesis represent a proportional relationship . Comment on Lesson Copyright © 2013 Edmentum - All rights reserved. Study Island Lesson Unit Rates and Proportional Relationships: Tables Example 2: 11/ 6/ 13 4:37 PM The table below shows the cost of grapes in the produce aisle at the grocery store. Pounds Cost 2 $4.30 4 $8.60 6 $12.90 8 $17.20 Based on the table, what is the price per pound of grapes? Solution: The price per pound of grapes can be modeled by the function y = kx, where x is the number of pounds of grapes, y is the total price, and k is the price per pound. Use point (2, 4.30) in the function y = kx to solve for k, which is the unit rate.