Transcript Intro_to_Slope_Lesson_2

Chapter 14
Lesson 75
The Coordinate Plane
Lesson 75
RR.7
Understand that multiplication by rates and ratios
can be used to transform an input into an output.
Find the third when given two of the following: the
input, the rate or ratio, the output.
EE.18
Find solutions to linear equations with two
variables.
CP.1
Identify and plot ordered pairs on the coordinate
plane.
CP.2
Plot solutions to linear equations on the
coordinate plane.
Slide 1
Objectives
• Plot and interpret ordered pairs on the coordinate
plane.
• Plot values of quantities that represent a constant
ratio.
• Plot solutions to equations.
Lesson 75
Slide 2
Remember from Before
• How do you plot points to represent an
input and an output?
• How do you use a table to organize
solutions to an equation?
Lesson 75
Slide 3
Get Your Brain in Gear
1. Find 4 solutions to each equation.
Lesson 75
Slide 4
The system that uses points to represent input and
output values is called a coordinate plane.
We have used a horizontal number line.
This number line continues forever in both directions.
It never ends.
Lesson 75
Slide 5
We introduced a vertical number line where the
negative direction is down and the positive direction
is up:
This number line also continues on forever, but in the
up and down directions.
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Slide 6
We can place the vertical number line and the
horizontal number line on top of each other so they
share the same 0 point (the origin):
This forms a flat sheet that continues endlessly up,
down, to the left, and to the right. We call this a
coordinate plane.
Lesson 75
Slide 7
Check for Understanding
1. On the coordinate plane below are points marked A,
B, C, and D.
Which point has a positive
input and a positive output?
Which has a negative input
and a negative output?
Which point has a positive
input and a negative output?
What about a negative input
and a positive output?
Lesson 75
Slide 8
This shows an input of 1.5 and an
_
output of 2. Let’s represent this
as a point on a coordinate plane.
_
The first
We
Next
point
we find
find
where
the
the value
value
these1.5
two
2 on
on
lines
the
thecross
output
input represents
number
numberline.
line.
_
This is
both
anlocated
input ofhalfway
1.5 andbetween
an output
1 and
of 2:2.
Lesson 75
Slide 9
The blue lines are not necessary. However, when we
remove them, it’s harder to see which input and
output values the point represents:
We can label the point.
The standard way is to
use the following format:
(input, output)
Lesson 75
Slide 10
We write the input value first, and the output value
second.
Since the order matters, and there are two values (a
pair of values), we call this an ordered pair.
Each value in an ordered pair is call a coordinate.
The coordinates of an ordered pair tell us the
location of a point on the coordinate plane.
We can now represent any point on a plane using a
pair of numbers.
Also, any pair of numbers tells us a location on the
plane.
Lesson 75
Slide 11
Check for Understanding
2. Which of the following points most reasonably
_
represents ( 3.75, 2)?
_
What about (2.25, 2)?
Lesson 75
Slide 12
Check for Understanding
3. Draw a coordinate plane on graphing paper. Indicate
where the origin is located. Then plot each of the
following ordered pairs:
Lesson 75
Slide 13
Altogether, 5 unopened cans of soda weigh about 4
pounds:
How much does 1 can of soda weigh? What about 2 cans
of soda?
We use the rate:
Lesson 75
Slide 14
Let’s simplify the units:
Since the expression is in units of pounds, it verifies
that we set up the expression correctly.
Lesson 75
Slide 15
Let’s use the variable p to represent the number of
pounds that c cans weigh.
The following
Let’s
simplify the
is an
leftequivalent
expression:
equation:
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Slide 16
Let’s express
division:
4
5
in decimal notation using long
0.8 = p
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Slide 17
c = 1 and p = 0.8 form a solution to the equation. This
means that 1 can of soda weighs 0.8 pounds.
We can represent this solution on a coordinate plane.
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Slide 18
How many pounds do 2 cans of soda weigh?
To do this we solve the same equation but now with
c = 2:
We know that:
Lesson 75
Slide 19
c = 2 and p = 1.6 is another solution to our equation.
To represent this solution on the coordinate plane we
place a point at coordinates (2, 1.6):
Lesson 75
Slide 20
Next, let’s solve the equation when c is a negative
_
number such as 1:
p will be the opposite of 0.8:
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The solution is c = 1 and p = 0.8.
We represent this solution by plotting a point at location
_
_
( 1, 0.8):
Lesson 75
Slide 21
_
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What does it mean to have 1 can of soda weigh 0.8
pounds?
This solution means that if we remove 1 soda can from
something then we make that thing 0.8 pounds less
heavy.
Here are the solutions we have found so far:
Here are the solutions on the coordinate
plane:
Lesson 75
Slide 22
Check for Understanding
4. Find 5 different solutions to the equation below.
Organize the solutions in a table then plot the
solutions on a coordinate plane. Use the number of
seconds as the input and the number of meters as
the output.
Lesson 75
Slide 23
Check for Understanding
5. The equation below is related to the one we used,
but this one has a negative rate. Find 5 solutions to
this equation and plot them on a coordinate plane.
Again, use seconds as the input and meters as the
output. How do the points you plot here compare with
the points you plotted in the previous situation where
the rate was positive? Explain.
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Slide 24
Multiple Choice Practice
1. Estimate the coordinates for the point labeled p.
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Slide 25
Multiple Choice Practice
2. Estimate the coordinates for the point labeled k.
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Slide 26
Multiple Choice Practice
_
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3. Which point best represents ( 14, 12)?
Lesson 75
Slide 27
Find the Errors
Identify 4 things wrong with the way the student
drew the coordinate plane and plotted the point.
Demonstrate how to draw this correctly.
Lesson 75
Slide 28