Transcript Unit Rates and Proportional Relationships

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
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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
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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.