I have been teaching lessons from the CUNY HSE Math Curriculum Framework: Problem Solving in Functions & Algebra, starting with The Commission Problem (link to previous post). The Cab Fare Problem is the next lesson I taught.  I presented the question to the students as follows.

The students need to fill in a table with the information that was provided from the question. From there they needed to plot the points on the graph and to connect the four coordinates. They had to figure out what the base fee and fee for each mile traveled, and how much it would cost for a passenger who travels 10 miles.

How I Solved the Problem

I reviewed the problem and read all the information about how many miles and the total cost for each person.  I used this information to create the table that was provided in the problem.

I made sure to place the amount designated for each person correctly into the table, checking the coordinates were in the (x, y) format.   I created the rule by looking at the distance and cost for mile 1 and 3.  I noticed the distance was a matter of two miles and the cost difference was three dollars. Since the difference was three dollars, I split that difference in half to get $1.50. I added that amount to $4.50 and got a total of $6.00, and added $1.50 to that amount to get $7.50.  I then checked to see if using $1.50 per mile and adding the $3 as a flat fee would work as the rule, going from the input to the output, and it did! I was able to create a rule to calculate the cost of the taxi ride: 1.50x + 3.

Then I took the ordered pairs and plotted them onto the graph. The graph already had the axes labeled for cost and miles traveled.  I made sure to number the x and y axes the same way. Knowing that the base fee was three dollars and the cost per mile is one dollar and fifty cents, I was able to find out that the final cost would be $18 for a passenger who travels 10 miles.

Anticipating Student Approaches

I expected the students to fill in the table from the information that was provided in the problem.  From the distance amount and cost amount, they would come up with the coordinates. They would take that information and place it on the graph.

I also expect that they would guess and check to figure out the amount for the base fee and the cost per mile and to come up with a rule that can be used with the different miles traveled.

Student Work

Jerry

Jerry has been attending my HSE class for about 8 months.  When he first attended classes, his basic skills were low.  As time has passed, he has been working really hard to improve and grow as a student.

When I gave Jerry The Cab Fare Problem, he took the information and started filling in the table.  When it came to filling in coordinate part of the table, he got mixed up when making the ordered pairs.  He wrote the y first and then the x. After having a discussion with him, he started to remember the other problems we worked on with creating the ordered pairs and realized that he needed to put the x first and then the y to create the coordinate.  He took that information and was able to plot the coordinates on the graph. It helped also that the graph had the axes labeled so that he knew the miles traveled is the x value and the cost is the y value. He figured out that the base fee is three dollars and the price per mile is one dollar and fifty cents by looking at the graph, going from mile one to mile two.  Jerry solved question four before he answered number three.

 

Jessica

Jessica started by filling in the table with the information for distance, cost, and coordinate.  She did a few calculations to determine the difference for the distance that was not in the table.  She calculated what the difference in cost would be from going five miles to six miles, seeing that the difference would be $1.50. As the previous student, Jessica figured out the base and cost per mile before she found out what the cost for the ten miles traveled amount.  She then plotted that coordinates on to the graph.

 

Tuana

When presented the problem, Tuana was able to fill in the table with the information that was provided.  She filled in the coordinates knowing that the x value was miles traveled and y was the cost. The only thing that I realized afterwards was that she did not place the ordered pairs into parentheses. Tuana then placed the coordinates onto the graph.  She graphed the coordinates for two miles traveled, which was not even asked to be done. Just as the previous students, she worked out the base and the cost per mile before she figured out what the cost would be for ten miles traveled. Once she got that figured out, Tuana got $18 and placed the coordinate on the graph.

 

Will

After Will was given the problem and filled out the information in the table, I asked him to reread the amounts that needed to be filled in.  He noticed that he mixed up Mark and Solange’s amount into the wrong spots. Instead of rewriting it, he just crossed out the names and placed the correct names to coincide with the amounts.  Once he fixed that, he was able to plot the points on the graph. He figured out that the amount should start at three dollars and add one dollar and fifty cents to the miles to get the cost for one mile traveled.  Since he had figured that information out, he knew that the base was three dollars and cost per mile was a dollar fifty. He came up with the rule of: 1.50(m) + 3.00 to figure out what the cost was for 10 miles. Will used a calculator to find the output for ten, got 18 and plotted it on the graph.

 

Neha

Similar to the other students, Neha did not have any problems filling in the table. She labeled x above Distance and y above Cost to remember which one is the input and output, which helped her to write her coordinates.  With that information, she plotted her points on the graph. She even labeled the x axis and y axis. She worked out the problem to find the base, cost per mile, and wrote out the rule. She verified the rule by checking it by plugging in three into the rule to see if she would get seven dollars and fifty cents, which she did.  Neha then used a calculator to find out the cost of ten miles by plugging 10 into the rule to get 18.

Final Thoughts

It was my first time teaching The Cab Fare Problem and was curious if the students were going to have difficulty with the problem.  As I keep teaching the material from the CUNY HSE Math Curriculum Framework, I am getting more comfortable with the material. I  previously taught functions when doing a long-term leave replacement in public school, but that was four years ago. When you do not always teach these lessons, it sometimes takes just a little bit to have it come back to you.  As teachers, we should not be afraid to teach lessons that we have not taught or have been awhile teaching the material. There is always resources available to guide you along with teaching. You can ask colleagues or even look it up on the Internet to see how others have taught it on Youtube.

The next lesson that I would recommend teaching is Unit 3: Rate of Change/Starting Amount lesson from the CUNY HSE Curriculum Framework. I believe it was helpful to have the students work on The Commission Problem prior to this one because they saw the table they created and it mimicked the one in The Cab Fare Problem.  It seemed that they were not having much difficulty with working out this problem and did not need much guidance from me. I believe it helped to do the Commission Problem prior to this lesson so they understood what needed to be done to work out the problem.