Slide 1Lesson 75

The Coordinate Plane

EE.18 Find solutions to linear equations with two variables.

CP.1 Identify and plot ordered pairs on the coordinate plane.

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.

CP.2 Plot solutions to linear equations on the coordinate plane.

Chapter 14

Lesson 75

Slide 2Lesson 75

Objectives

• Plot and interpret ordered pairs on the coordinate plane.

• Plot values of quantities that represent a constant ratio.

• Plot solutions to equations.

Slide 3Lesson 75

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?

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Get Your Brain in Gear

1. Find 4 solutions to each equation.

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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.

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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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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.

Slide 8Lesson 75

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?

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This shows an input of 1.5 and an output of

_2. Let’s represent this

as a point on a coordinate plane.

We first find the value 1.5 on the input number line. This is located halfway between 1 and 2.Next we find the value

_2 on the output number line.The point where these two lines cross represents

both an input of 1.5 and an output of _2:

Slide 10Lesson 75

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)

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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.

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Check for Understanding

2. Which of the following points most reasonably represents (

_3.75, 2)?

What about (2.25, _2)?

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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:

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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:

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Let’s simplify the units:

Since the expression is in units of pounds, it verifies that we set up the expression correctly.

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Let’s use the variable p to represent the number of pounds that c cans weigh.

The following is an equivalent equation:Let’s simplify the left expression:

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Let’s express in decimal notation using long division:

0.8 = p

45

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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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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:

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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):

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Next, let’s solve the equation when c is a negative number such as

_1:

p will be the opposite of 0.8:

The solution is c = _1 and p =

_0.8.

We represent this solution by plotting a point at location (

_1,

_0.8):

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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:

Slide 23Lesson 75

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.

Slide 24Lesson 75

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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1. Estimate the coordinates for the point labeled p.

Multiple Choice Practice

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2. Estimate the coordinates for the point labeled k.

Multiple Choice Practice

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3. Which point best represents (_14,

_12)?

Multiple Choice Practice

Slide 28Lesson 75

Find the ErrorsIdentify 4 things wrong with the way the student drew the coordinate plane and plotted the point. Demonstrate how to draw this correctly.