Fuel Efficiency

Engage  10min

The teacher will present a picture on the board of a vending machine.  The teacher will share a story about a vending machine that with 1 quarter input you get 3 sodas.  This is the common vending machine input/output function model. q=3s. The teacher will propose other inputs q=s, q=4s, 4q=s, etc.  The teacher will fill out a table and encourage the kids to do the same.  The teacher might want to use x and y for the charts.

We can describe the coke machine with a mathematical function and use the number of quarters paired with the number of sodas bought in an ordered pair and plot them on a two dimensional grid.  Then we translate the sentence, ÒFor each quarter I buy three sodasÓ into the math equation q=3s.  Teacher shows overhead projection of a y=3x.  Teacher explains that the vertical number line is y which represent the quarters spent and explains the x value.  Teacher will present another function y=2x and have the students make up a story about it.

      1.When we get 3 sodas for one quarter, can you guess from this chart, how many sodas you might get for 6 quarters?... 13 quarters?

2. How did you come up with that guess?

  For  overhead graph transparency:

1. Can this graph tell us how many sodas we can buy for $1.
2. Can this graph tell us how much money we need to buy  24 sodas?...19 sodas?
3. can this graph y=2x tell us a story too?  What might this graph be saying in words?

1.18 sodas

    39 sodas

2. I added one more to the  figure we got for 5 quarters. 

I multiplied 13 by 3

1. Yes. Look at the vertical number line and find the 1. And then go across to where you drew you lineÉ(teacher checks y=quarters or dollars?)
2. Yes.  I followed the horizontal line across to 24 and went upÉ..I just dividedÉI added.

(teacher focuses on the graph as a representation of the math the students are doing.)

3.I bought two for one fries at McDonaldÕs.  y=2x

Explore: 15min

 introduce slope, intercept, coefficient and the y=mx+b formula.    T. will show four graphs (two of which should not intersect at the origin) with a positive slope.  Teacher will show the coefficient,  m, and the equations for each of the lines.  Teacher will show four more graphs (two with a positive slope). Show equations.  Discuss why the negative slope might be.  Make a table of the negative slope equation and the positive slope equation.  Discuss the direction of the values larger, smaller, positive, or negative?  Define the slope for the students as the change in y over the change in x.  (rise over run) Use the phrase rate of change. Show a series of eight graphs (include parallel lines and flat lines) and ask about the shape of the graphs.

Evaluate

Show four equations and graphs to the students and ask them to tell match them with each other.  Add a second set and include some with y-intercepts.

Slope:

1. What do you notice about these lines? 
2. What do you notice about these equations?
3. Is there a relation between the line and these coefficients?
4. How are these new graphs similar to the earlier ones?
5. Why are these leaning the other way?
6. What happens as these positive slopes increase?
7. What happens to the lines as these negative slopes decrease?
8. What do you notice about lines with a zero slope?
9. What do you notice about lines with the same slope?
10. These two lines are parallel, why?

Expected Student

Responses/Misconceptions

Slope:

1. They are all leaning to the rightÉThey are all going uphill.
2. The coefficient is positive.
3. Yes, maybe positive goes uphillÉ the y-side of the equation gets bigger as the x gets bigger.
4. They are leaning the same wayÉ they are leaning different ways.
5. They are getting smallerÉ
6. The graph gets steeper.
7. The line gets flatter.
8. Zero slopes are flat.
9. TheyÕre parallel.
10.  They have the same slope

Explain  20

Teacher will project a calculator on the screen.  T. will show a picture of a graph and ask student to guess its formula. T. will input the students formula to check   (teachers should keep it simple using only y=mx for these first exercises.)  T. will show the students how to work the graphing calculator including Òy=Ó, ÒzoomÓ, ÒgraphÓ and Ò2^ndÓ, ÒX,T,nÓ buttons.(teacher is already an expert with these machines and must do a hands-on  review in minute detail to make sure they get to teach slope instead of calculators  for novices 101!  REMEMBER: Students who  donÕt know how to use a tool properly will invariably be frustrated, bored and possibly break the tool)  Teacher will pass out the calculators and after a brief lesson will ask them to try to duplicate the line y=4x using a different equation.

Evaluate

Student will partner up and work together to find ten different ways to say y=6x+10 and three lines that are parallel.  Partners will turn in the worksheet for credit.

1.How do  you know this is going to give you the same line I have?

Students will have used some mathematical form of 4x +0 as their coefficient in the equation.

2.How are all of these equations similar?

Students will present their equation to the teacher and the teacher will check their accuracy using the overhead projector.

1. open answers
2. The other numbers all add to zero.  The [coefficient] equals four.

Elaborate:  15 min

The delta y/ delta x formula 

I think you understand the tilt of the line that has a positive or negative slope and I think you can tell me which line has a higher slope if I show you two lines, but IÕm a little confusedÉ some lines are flat.

 Usually the farther you fly a plane the more the airline charges.  Right?  Well this week Southwest airline is fly any where they go for $99 dollars.  No matter how far you fly.  The bus costs 50 cents no matter how far you ride.  A taxi costs $2.50 to get in and $6.00 every tenth of a mile.  A taxi: (cost) = $6.00(miles) + $2.50.  That slope is 6.  Find ing the slope is easy  first you must take two ordered pairs from your equation  (use the ÒtableÓ button on your calculator) and you subtract the first y value from the second y value and divide the result by the first x value subtracted from the second x value.  You are making a ratio of the amount  of rise over the amount of run.  How are you going to remember (y2-y1)/(x2-x1)?

Remember that the $99 plane ride has a slope of zero (99-99)/ (x2-x1)

A flat line has no tilt, its slope is zero.

Evaluate

Student will do the attached worksheet.

1. So what is the slope of the line that describes how far IÕm flying? 

2. What does that line look like?

3.what is the equation of that line?

4. Where the X? ie the number of miles IÕm flying?

5. y=99 Is that a flat line? (put it into your calculators)

6. How about y=0X+99, is that the same equation?

1. none
2. ItÕs flat
3. y=$99
4. donÕt know
5. yes.
6. yesÉI donÕt know