1. What’s slope got to do with it?

2. Today we are going to see how slope and y-intercept are actually used in many situations in the “real world”
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3. Our learning goals are to be able to:
A linear equation that describes a real world situation is called a Linear model
Our learning goals are to be able to:
Identify and explain the meaning of the slope as a rate of change in the context of a given model
Identify and explain the meaning of the y-intercept in the context of a given model

4. What do you remember about…. Slope Slope intercept form y-intercept
Instruct the students to write down each of the terms and then list or describe what they remember about each one. Tell them to leave a little extra space so that they can add other remarks during the class discussion.
They may do this individually or in pairs. Each student should write down their own responses.
After about 2 minutes, ask students to volunteer what they remember about slope. Answers will vary. The teacher should write all the student responses on the board or somewhere visible to all. Before going on to slope intercept form and y-intercept, go to the next slide to for the full explanation of slope. Go over all the bullets and formulas, being sure to point out how they are all different ways of stating the same basic concept of comparing the way the two variables are changing.

5. Remember what we already know about slope:
Slope is the ratio of vertical rise to horizontal run.
Slope is often described as “rise over run”.
Slope is also called the rate of change.
Given two points with coordinates (x1, y1) and (x2,y2) the formula for the slope of the line containing the points is
Students should write down all of the concepts about slope that they did not come up with on their own.
After they have finished, and the teacher has pointed out how all these interpretations are saying the same basic concept, then have students do the following examples to practice finding the slope accurately.
Find the slope of the line containing the points (3, -5) and (-1, -3).
Find the slope of the line containing the points (-4, 2) and (-7, -3)
Give the students one or two minutes to work out both examples. Then you can have them compare answers with their neighbor. Anticipate common errors such as subtracting x values in the numerator, being inconsistent with the order of the coordinates in the subtractions, and making subtraction errors.
1) Answer: -1/2
2) Answer: 5/3

6. What do you remember about…… y = mx + b
This is a youtube that some students may enjoy and may help review prior knowledge and add interest.
This may be used to remind students about slope intercept form. It shows a student rapping about slope intercept form.
Students should be prompted to volunteer to tell what they know about this equation. They should be able to tell that this is slope intercept form, it is the equation of a line, it has a slope of m and a y-intercept of b.
Some students may also know that any point in the form (x, y) that is on a line will also represent values that will make the equation true.

7. What do you remember about…… y = mx + b
This is the equation of a line. It tells us:
m is the slope of the line
It is the ratio between the change in y and the change in x.
It tells the “rate of change” of y and x
It shows how x and y are changing together.
It describes how steep the line is and how it is slanted.
b is the y-intercept
It tells us where the line crosses the y-axis
It shows us the value of y when x equals zero.

8. This equation can be used in a real world example about saving money.
Consider this linear equation……
y = 1,000x + 1,500
This equation can be used in a real world example about saving money.
x = the number of years since you opened a saving account
y = the total amount of money in the account ($).
How much money will you have after 2 years?
How much money will you have after 5 years?
How much did the amount in your account change over the 3 years?
How much was that per year?
Identify the slope and tell what it represents in this example.
What is the y – intercept and what does it represent?
At this point, the teacher should hand out the work sheet that accompanies the examples in the slides. This will enable students to more easily read the questions and write their responses in an organized format for future reference.

9. y = 1,000x + 1,500 How much money will you have after 2 years?
How much did the amount in your account change over those 3 years? How much was that per year?
Identify the slope and tell what it represents in this example.
What is the y – intercept and what does it represent?
Identify and explain what the slope represents
Slope = m = 1,000 and it represents that the money in the account increases $ 1,000 per year
( m= $1,000 / 1 year )
b) What does the y – intercept represent?
y- intercept =1,500 and it is the amount of money in dollars
that was in the account when it was opened. (year 0 )
After the students have read the introductory information, the teacher should ask the class how they can use the equation to answer the first question. Be sure to emphasize that the information we are given, 2 years, is a value for x, since x is defined as the number of years. Students should be able to conclude that by substituting 2 for x in the equation, they can answer the question.
Have the students write the answer to question 1 in a complete sentence: After 2 years, there will be $3,500 in the account.
Then have students follow the same process to answer question 2, and answer in a complete sentence: After 5 years, there will be $6,500 in the account.
Question 3 is to help students see how finding the “change in y” is simply subtracting the two y values that are the amounts at 2 and 5 years. To find the amount “PER YEAR” explain that “PER” means “EACH”, so they divide the total change in y ($3,000) by the number of years that have elapsed (5 – 2 = 3). This emphasizes the meaning of slope again.
Question 4: Students should be able to now make the connection that the slope that appears in the equation as the coefficient of x is the same value as the amount that the account changed each year. ($1,000). Students should write in a complete sentence that the slope is 1000, and it means that the amount of money in the account increased (changed by going up) by $1,000 each year.
Question 5: Students may be able to quickly identify the y-intercept as the constant term in the equation. TO answer the second part of the question, they need to write in a complete sentence that “the y-intercept is $1,500, which means that when x, the number of years, was ZERO, the amount in the account was $1,500. It must be emphasized that the y-intercept is the value of y when x equals zero.

10. In this “real world” example, let’s see what slope means:
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In this “real world” example, let’s see what slope means:
Given the linear equation y = -800 x + 8,000 where
x is the age of the car in years and
y is the value of the car in dollars.
What is the value of the car after 1 year?
What is the value of the car after 6 years?
How did the value of the car change between those 5 years?
Identify the value of the slope and explain what it represents in this example
What is the y – intercept and what does it represent in this example?
This example is for the students to try on their own with some guidance from the teacher. Instruct the students to read the equation and the definitions of the variables x and y.
Have the students complete the first three questions and then discuss their results.
Answers:
Substitute 1 in place of x: the value of the car after one year is $7,200.
Substitute 6 in place of x in the equation: the value of the car is $3,200 after 6 years.
The value of the car dropped (decreased, went down) by $4,000 in 5 years. Ask the students to explain how they got the $4,000 and how they got the tie of 5 years. In both cases, they are subtracting corresponding values of y and x, which goes back to the definition of slope.
and 5) Before going over the next 2 questions, prompt the students to think about this example and the definition of slope and y intercept and try to answer both question 4 and 5 in complete sentences.
The answers to questions 4 and 5 are on the next slide. Do not show them until the students respond on their own papers. IF they are having difficulty, you can refer back to the previous example, or ask for some student volunteers to explain their thinking.

11. The car was worth $8,000 when it was first bought.
Given y = x + 8,000
where x is the age of the car in years and
y is the cost of the car in dollars.
What is the value of the car after 1 year?
What is the value of the car after 6 years?
How did the value of the car change between those 5 years?
Identify the value of the slope; explain what it represents in this example.
Slope: m = -800
This is the “change in y” (value of the car) compared to the “change in x” (the age of the car).
Connect the numbers and the words:
The value of the car decreases by $800 every year
What is the y – intercept and what does it represent in this example?
The y-intercept is $ 8,000.
It represents the cost of the car when x (years) = 0.
The car was worth $8,000 when it was first bought.
( answers ) Discussion and clarification should occur so that everyone is understanding the main concepts and using the words to label their answers.

12. An electrician charges $30 for a home visit plus
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An electrician charges
$30 for a home visit plus
$60 for each hour of service.
Write the linear equation for this situation using C=the total cost and h=hours worked.
b) Identify the value of the slope; explain what slope represents in this example
c) What is the y-intercept and what does it
represent in this example?
IN this example the students will be asked to use the given information and write the linear equation that is the model for the situation. A good starting point is to have the class write the general equation for slope intercept form (y = mx + b) and then decide where 30, 60, C, and h fit in to the slope intercept form Ask: what variables are we using in place of x and y in this example? Which one is replacing x and which one is replacing y? Answer: since 60 is the amount charged for each hour worked, that corresponds to slope. Knowing that slope is the coefficient of x, then, then h, the number of hours worked must be used for x so that it is multiplied by 60.
C is the total cost of the visit, which will be not only the cost for all the hours worked but also the $30 for the home visit, so C is used in place of y.
The equation then is C = 60h + 30.
The answers to b and c are on the next slide. Be sure to have the students try to answer on their own papers first; they can confer with a neighbor and the teacher can circulate among the students to see how if they are needing assistance or additional explanation.
      

13. An electrician charges $30 for a home visit
plus $60 per hour of service.
a) Find the equation of the line for this situation
using C=the total cost, and h = hours worked
C = 60h + 30
b) Identify and explain what the slope represents:
Slope = 60 which represents the cost for each hour of service, or slope= $60/1 hour .
The slope is positive, that means that for each hour of service the cost will increase by $60.
c) What does the y-intercept represent?
The y intercept = 30. It means that every service call has an automatic cost of $30, before any time is worked (the number of hours = zero)
( answers )

14. 3) Find the equation of this line.  
   
The value of a used car decreases every year. (The older the car, the less it is worth.) A car that is 4 years old has a value of $5,200; when the same car is 7 years old, its value is $2,500.
Write two ordered pairs to represent the information in this example, using the form (x, v), with x = age of the car and y = the value of the car.
2) Find the slope of the line containing these two points.
3) Find the equation of this line.
4) What is the y-intercept? What does it represent?
5) What does the slope represent in this example?
In this example, the teacher will help the students.
Using the definitions of x and y, the ordered pairs are in the form (age in years, value of car) or (4, 5200) and (7, 2500)
Slope = -2700/3 or -900.
To find the equation of the line, students should recall the techniques learned prior to this lesson. By using either point and substituting into either slope intercept form or point slope form, they can find the equation of the line to be y = -900x (The equation should be transformed to slope intercept form.)
Y-intercept is It means the value of the car when 0 years have passed, or when the car was new.
Slope means change in value per year, or the value decreases $900 each year.

15. Let x= the age of the car, and v = the value of the car
From this, we see that both x (the car’s age) and v (the car’s value) are changing. And we can also tell that the value of the car depends on the age of the car.
Write two ordered pairs to represent the information in this
example, using the form (x, v). Answers: (4, 5200), (7, 2500)
2) Find the slope of the line containing these two points. m = -900
3) Find the equation of this line. v = -900 x
4) What is the y-intercept? What does it represent? The y-intercept is 8,800; the value of the car was $8,800 when the car was new.
5) What does the slope represent in this example?
Every year the car’s value drops by $900.

16. John just graduated from high school and started a new job.
He received some cash as gifts from his relatives.
Instead of spending this gift money, he decided to save it for the future.
He also decided to start saving a set amount
from his paycheck each week.
After 12 weeks, he had a total of $720.
After 30 weeks he had a total of $1,440.
This example is for the Independent practice. Read the example with the class, and then have the students read the information in the last two sentences. Make sure the students understand that John had a certain amount in gifts that he started with and then he started saving a certain amount each week from his paycheck. This should help the students see the difference between what will be the slope and what will be the y-intercept.

17. Choose and define two variables for this example.
Try this one on your own
Choose and define two variables for this example.
How are these variables changing?
Write two ordered pairs to represent the information in this example, using the form (w, d).
Find the slope of the line containing these two points.
Find the equation of this line.
What is the y-intercept? What does it represent?
What does the slope represent in this example?
The students should work independently on these questions. The teacher should circulate among the students, carefully observing their work and intervening at any time when a student is having difficulty or misunderstanding something.
The answers will be on the next slide, but should not be revealed until after the students complete their work and there is a whole class checking/sharing/discussing.

18. Determine and identify the two variables for this example.
The variables are number of weeks and amount of money saved. Let w = number of weeks, and let d = number of dollars saved.
2) How are these variables changing?
As the number of weeks increases, the amount of money saved by John also increases.
3) Write two ordered pairs to represent the information in this example, using the form (w, d).
(12, 720) and (30, 1,440).
4) Find the slope of the line containing these two points.
m = 40.

19. 5 )Find the equation of this line. y = 40x + 240.
6 )What is the y-intercept? What does it represent?
The y intercept is 240.
It means that when the number of weeks was 0, John had $240.
So we can say that John received $240 in graduation gifts.
7 )What does the slope represent in this example?
Use both the number value and the verbal meaning of the variables to the slope.
There was an increase of $40 per week.
This is the meaning of the slope: it telsl what 2 quantities
are changing and by how much (amount saved and weeks).
( answers )

20. A taxi cab ride costs a $10 flat fee plus $3 per mile traveled.
A linear equation
can represent the total cost of a taxi ride.
a) Write an equation that shows how the miles traveled and the total cost of the taxi ride are related.
b) Identify and explain what the slope means
c) What does the y-intercept mean?
For this example, students should do the entire problem independently. The teacher can provide prompts if needed, or suggest that the students refer to the previous examples that were scaffolded into smaller interim steps.
The answers are on the next slide, and should not be displayed until everyone has completed the example and perhaps had time to compare with a neighbor or share responses and methods used in a whole class discussion.

21. a) Write an equation: C = 3d + 10
Write an equation that shows how miles traveled and the total cost of the taxi ride are related.
a) Write an equation: C = 3d + 10
This uses d for distance in miles and
C for total cost of the taxi ride.
b) Slope = 3. The slope represents
the rate of change of the miles and the cost of the service.
Slope (m) = $ 3 / 1 mile. This means that for every 1 mile, the cost will increase by $3. As the distance increases, the cost of the service will increase. (Positive slope)
c) The y intercept is 10.
That means that even if the distance is ZERO miles, you must to pay a fee of $10 for the taxi cab ride.

22. EXIT TICKET 3 things you learned about 2 examples of how slope is used
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3 things you learned about
the meaning of slope
2 examples of how slope is used
in the real world
1 thing you learned about the meaning of y-intercept
Students should complete the exit ticket on their own notebook paper and hand it in.