Chapter 14 Lesson 74 Plotting Inputs and Outputs

Chapter 14 Lesson 74 Plotting Inputs and Outputs Lesson 74 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 • Use horizontal and vertical number lines to represent inputs and outputs. • Solve linear equations where an input is multiplied by a constant rate to produce an output. Organize the solutions in tables and plot the inputs and outputs on a coordinate plane. Lesson 74 Slide 2

Remember from Before • How do you multiply an input by a rate to produce an output? • How do you find the location of points on a number line? Lesson 74 Slide 3

Get Your Brain in Gear 1. Use mental math to find the solution to each equation. Lesson 74 Slide 4

A market sells pineapples for $2 each. At this rate, how many pineapples do we get for 6 dollars? We want to turn dollars into pineapples. Which rate do we use? What is the value of this expression? Now multiply 6 dollars by this rate: We can buy 3 pineapples for 6 dollars: Lesson 74 Slide 5

Back in lesson 17 we described this process as a machine. However, it’s more convenient to show the input and output on two separate number lines: input Lesson 74 Slide 6 output

Check for Understanding 1. Using the rate of 1 pineapple for 2 dollars, find the output for each of the following inputs. Each input gets multiplied by the rate to produce an output. a. 0 dollars b. 4 dollars c. 10 dollars 2. Find the missing rate to account for the following input and output values: Lesson 74 Slide 7

In the year 1637, a famous mathematician named René Descartes invented a powerful way to show the input and output of a machine like the one we discussed. The key idea was to show the output on a number line that goes up and down (vertical) instead of left and right (horizontal). Lesson 74 Slide 8

Let’s examine an input of 10 dollars which produces and output of 5 pineapples. René Descartes had the brilliant idea to put the output on a vertical number line where the positive direction is up: Lesson 74 Slide 9

Look what this lets us do. We can put both number lines on top of each other so that they share the same 0 point: Now we can represent an input of 10 dollars and an output of 5 pineapples using a single point. Lesson 74 Slide 10

Let’s represent our previous machine with an input of 6 dollars and an output of 3 pineapples with a single point. output value input value This point When The point we that place and has alla the point a input points likeofthis, above directly “ 6 dollars” it’sand called to the and below left plotting anand this output location right theofpoint. “ 3 correspond to pineapples” is an theoutput input pointofwhere of 6 3 dollars. pineapples. these two lines cross. Lesson 74 Slide 11

Check for Understanding 3. What input value and output value best represent the point below (both values are whole numbers)? 4. On the figure above, where would you plot the point that represents an input of 8 dollars and an output of 4 pineapples? Lesson 74 Slide 12

Let’s continue with our example where 1 pineapple costs $2. The following table shows how many pineapples we can get for different dollar amounts: dollars pineapples 0 0 2 1 4 2 6 3 8 4 10 5 Here are the points that correspond to an input of 0 dollars. Lesson 74 Slide 13

Let’s continue with our example where 1 pineapple costs $2. The following table shows how many pineapples we can get for different dollar amounts: dollars pineapples 0 0 2 1 4 2 6 3 8 4 10 5 We call Here The point arethis the that points represents thethat origin. correspond an input ofto 0 an andoutputofof 0 0 is pineapples. located where these two lines cross. Lesson 74 Slide 14

Let’s continue with our example where 1 pineapple costs $2. The following table shows how many pineapples we can get for different dollar amounts: dollars pineapples 0 0 2 1 4 2 6 3 8 4 10 5 Now plot an input of 2 dollars and an output of 1 pineapple. Where Here are thethe lines points intersect that represents anan input output input ofofof 2 1 dollars: 2 pineapple. dollars and an output of 1 pineapple. Lesson 74 Slide 15

Let’s continue with our example where 1 pineapple costs $2. The following table shows how many pineapples we can get for different dollar amounts: dollars pineapples 0 0 2 1 4 2 6 3 8 4 10 5 Now about Where How let’s graph willgraph inputinput 810 6 and and 4 output and output 43. 5? graph? 2. Lesson 74 Slide 16

The blue lines are not necessary. They just make it easier to see what input and output each point represents. Here are the same points without the blue lines: Lesson 74 Slide 17

Check for Understanding 5. Use graph paper to draw the input and output number lines. Label the input number line with units of hours and label the output number line with units of miles. Indicate the point that represents zero for both number lines as the origin. Plot the input and output values shown in the following table: Lesson 74 Slide 18

Multiple Choice Practice 1. Estimate the input and output value represented by the point shown below: Lesson 74 Slide 19

Multiple Choice Practice 2. The point that represents an input of zero and output of zero is called… Lesson 74 Slide 20

Find the Errors A student made 3 mistakes below. Identify and correct each mistake. Lesson 74 Slide 21