Starting with the 2018/2019 school year, I decided to try my hand at the CUNY HSE Math Curriculum Framework. I started with Unit 1 so that students would have a better understanding of functions and working through problems to determine the rule needed to complete the table. I taught the taxi problem from Unit 2 as the next lesson. This will also written up as a separate Math Memo.
The first question dealt with The Commission Problem:
I started by asking students to discuss which one would have a better deal on making money based on the numbers. Then I asked students how many fish tanks would both of them need to sell so that they will bring in the same amount of money.
These types of questions make the students think about what is being asked, and then to think about how they would solve it. Prior to beginning of the units, we discussed different problem-solving strategies including: guess and check, make a table/chart, draw a picture, look for a pattern, think of a similar problem, work backwards, and write/solve an equation (video about the strategies). I shared these strategies with students, so they know they can use different ways to work out the problem. We revisited the different strategies and rewrote them all on the board so they had something to reference to if they got stuck trying to work out the problem.
How I Solved the Problem
I solved this problem by creating a table each for Eric and Nancy:
The tables illustrate how much each one makes when they sell different numbers of fish tanks, which answers the question of how many fish tanks each person would need to sell to make the same amount of money.
I then created a rule for each person, using the input (x) to find the output (y) value. I was able to create rules from the story, knowing that Eric makes a salary of $1400/month and would add $75 for each fish tank sold, and Nancy makes $250 for each fish tank sold. The rules I used are: 75x + 1400 (Eric) and 250x (Nancy). Using these rules allowed me to plug in a number of fish tanks sold to see how much they each made in salary.
Lastly, when looking at the graph, I knew that Eric’s salary is $1475 when selling 1 tank and Nancy would be at $250 when she sells 1 tank. This information helped to know which line belonged to each person.
Anticipating Student Approaches
One approach I expected students to use is Guess and Check. Having the information on how much each person makes along with commissions, the student can pick different numbers to calculate for Eric and Nancy’s salary.
Another problem-solving strategy that the students might use is creating a table by using the input values of fish tanks sold to get the output value. Each table would start with selling 0 fish tanks to know the value of their salary.
In choosing any of these strategies, the student will be able to calculate the salary to see how many fish tanks they would need to sell to make the same salary. No matter which way they choose they will get the amount as long as the calculations are correct.
Supporting Struggling Students
Allowing students time to work with the problem will give them time to think critically to determine their best approach to solve the problem. After allowing some time to pass, I would refer them to look at our list of problem-solving strategies to help those who do not know where to begin. If the student still does not recognize which strategy might useful, I would recommend that they create a table to visualize the scenario differently.
When creating their table, the students can write on top what each one designates, which will help them remember which is the input and output. This will also encourage the students to reread the problem to see exactly what it is being asked what needs to be done. During this process, I will also ask them questions that will support their critical thinking on solving the problem. For example:
- Which would represent the input and output value?
- Would Eric make his salary every day or does he get paid only one time at the end of the month?
- How many fish tanks would Nancy need to sell where she was making more than Eric?
Student Work
Debra’s Work
Debra has been in my HSE class for almost a year now. She has been making progress on working on her basic calculation and has shown progress throughout this time. After thinking about the information that was presented on the problem, Debra decided that Eric is correct on getting a better deal on their salaries.
It took Debra a little bit on a way to work out the problem to find out how many tanks that Eric and Nancy would need to sell to make the same salary in the month. I suggested that she look at our list of problem-solving strategies to help her in solving the problem. She then realized that she could create a table. She remembered we used this strategy when we solved “Maxine’s Rules for Love.” She created one for Nancy and one for Eric, labeling the input and output on top. It took her a few minutes to start her calculations but she started to understand quickly. Debra was able to create Nancy’s table without any problems. When she was constructing Eric’s table, at first she forgot that Eric made a monthly salary plus commission. She began with 0 for his output when he sold 0 fish tanks.
I suggested that she reread the original problem. She then went back and realized that she put in the wrong amount for not selling any fish tanks. Debra was able to fill in her table realizing that she needed to multiply the number of fish tanks with the input value and then add his monthly salary to get the output.
When she received the graph of Eric and Nancy’s salary, she was able to decipher which line belonged to who. She was able to label the x-axis with # of fish tanks sold and the y-axis with salary. By looking at the graph, she was able to see by drawing a line up to see how much they would make each if they sold 16 fish tanks. With figuring out the rule, she was able to plug in the information to check if her calculations were the same as the graph.
Debra did not change her mind, and still thought that Eric had the better deal with his salary. If I could talk to Debra now, I might ask her, “When would Nancy be making more money than Eric?” It can help for her to see how many fish tanks Nancy sold to make more money than Eric.
Will’s Work
Will has been attending my HSE class for about nine months. He comes in with strong basic skills but needs to be able to apply them more to certain tasks; at times he can get distracted easily. Will stated that Eric has the better deal with salary because unless Nancy sells tanks, she is not being paid.
He created a table for Eric and Nancy to designate if they were to sell 0 fish tanks during the week. It represented a scenario where Eric sold none and would make $1400 while Nancy would make $0 without any sales in the week. He also created another one showing the days of the week if Eric sells one tank and if Nancy sells one, how much they would be making. I asked Will to reread the question to see if it made sense that Eric would me making his base salary every day of the week. He realized after he completed his table it did not make sense and said it would be only once a month, not every day. He did not show his calculations but he was able to figure out on the graph that they both needed to sell 8 fish tanks to bring in the same amount of money. He also was able to see the amount of money they make from the graph if they sold a total of 16 fish tanks on the graph. With his future work it will benefit not only him but myself to see all his calculations to make sure that he fully understands the process to work out the problem.
Anjana’s Work
Anjana has been attending my HSE class for six months. She picks up on material that is presented to her very quickly. She has a good grasp on her basic skills that she is able to apply when solving problems. When given the problem about Eric and Nancy’s salaries, she stated that Eric makes a better deal because he will still be making a salary even if he does not sell any fish tanks.
Anjana was able to go straight to work and created a table for Eric and Nancy without having to refer back to the problem-solving strategies that were listed on the board. She figured out the rule and wrote it above each of the tables. To figure out the problem, she created two different tables and went up to 12 fish tanks sold for each to see their salaries in finding that they needed to sell 8 fish tanks each to make the same amount of money. She was able to label on the graph whose line belonged to who, but did not label the x-axis and y-axis. Anjana verbally told me how much Eric and Nancy would make if they sold 16 fish tanks.
Neha’s Work
Neha has been in class for only one month. She has basic skills and when she is presented with work she is able to work through the problems. With the “The Commission Problem”, she stated that Eric has the better deal. If he does not sell any tanks he will still earn money but if Nancy does not sell any tanks she will earn $0.
At first she did not know how to begin, but I referenced our list of problem-solving strategies and she chose to create a table. Neha realized she needed to make two tables: one for Eric and one for Nancy. After creating her table, she was able to see that when Eric and Nancy sold eight fish tanks they made the same salary. She did not show all her calculations on working out the problem because she used a calculator for her calculations. The tables did have the correct amounts for each of the inputs and outputs. After looking at the table, she was able to see that they both needed to sell 8 fish tanks to make the same salary.
When given the graph, Neha was able to state which line represented who on the graph. She was able to figure out how much money they would make when selling 12 and 16 fish tanks and the difference between the two. Neha, did see that Nancy would be making more money than Eric when they sold 16 fish tanks, but questioned, “Would they really sell that many per month?” I noticed that she did not circle the correct points on the graph for the amounts until after she handed it in.
She did not change her opinion that Eric has the better job, with stating that if they both do not sell any fish tanks, Eric will still make his base salary.
Jorge’s Work
Jorge has been attending class for 1 month. He comes in with some basic skills that will be a good foundation for future learning. After reading the scenario, he quickly determined that Eric was right and Nancy was wrong. He stated that if Eric does not sell any fish tanks, he will still make a salary where Nancy will not make anything.
As he started to work on the problem, he did a combination of guess and check and creating a table, which he was the only one who worked out the problem this way. When making his first table, he combined both of them. I gave him a little time to see how this will not work to find out the answer. After giving him that time, he realized that it did not make sense.
Jorge created two tables for Eric and Nancy. To find out that going from the input to the output he had the total and then added the commission to that total. Some of his calculations were scattered around, so it was hard to follow along, but he was able to get the correct amount for how many fish tanks they needed to sell to make the same salary. He figured out what the rule was for each table, and remembered that we discussed how to write the function rule and wrote it down.
Jorge asked a question: “Is it realistic?” for both of them selling fish tanks.
Victor’s Work
Victor has been attending my class for about a month now, and came in with good basic skills for math. Just as the same as the other students, he stated that Eric’s deal is better because he gets guaranteed money with his salary.
When given the problem, he starting creating a table for Eric, knowing to label it in (x) and out (y). He calculated only the commission part. Asking him to reread the problem, he realized that he forgot to include base salary into his calculations. Victor then created another table for Eric, he still did not include the base salary when Eric sold 0 fish tanks and he added 1475 each time to the output. I asked him to take a look to see if that makes sense and found that he should not have kept on adding that amount to the previous line. Again, I referred him to reread the problem, and this time he was able to create the tables for both Eric and Nancy.
When given the graph, he was able to say that both lines intercept at 2,000 to prove that his final calculations showed that they needed to sell 8 fish tanks to make the same amount of salary. I asked if he changed his mind on who makes the better salary, he said he did not and that Eric still has the better salary. 
Kevin’s Work
Kevin has been attending my class for less than a month. His basic skills are strong and is able to his work for the most part without any problems. When given the problem he states that Eric is definitely getting paid not matter what and Nancy only gets paid if she sells the fish tanks.
Kevin did write down all the problem-solving strategies down on his sheet so that he could choose which way he wanted to work out the problem. The strategy that he chose was guess and check. He multiplied the commission by eight for both Eric and Nancy, and then added the 1400 to Eric’s total to find out that they need to sell 8 fish tanks to make the same salary. I asked him how did he know to start with 8 and basically it was a guess.
He did change his mind on who made a better salary to Nancy because looking how her line goes up more when she keeps on selling more fish tanks. Just before class was over, Kevin discussed why he changed his mind on who had a better salary. The other students listened and you can see in their faces that they were processing the information and nodded their heads agreeing with him.
Kevin also posed some good questions, such as:
- Does she always sell this many tanks?
- Why do they both work doing the same thing?
- Do people really buy tanks this much?
His thought process is thorough and is able to understand Nancy’s line changes at a faster pace for every fish tank she sells, as where Eric is still making money but goes up at a slower pace.
Daisy’s Work
Daisy has been in my HSE class for almost a year. She has a good grasp of basic skills for working out problems. Daisy was the only student that stated Nancy has a better deal than Eric. She did state that if Nancy does not sell any fish tanks she would not be making any money, but then stated that she thought that she would not actually sell more than one tank a month.
Looking over the strategies, she created a table to work out the problem to figure out how many fish tanks needed to be sold so that they both will make the same salary. I noticed that when she first started the table for Eric, she wrote an output of 1475 for 0 tanks. She then continued the table by adding 75 to each of the outputs. I told her to reread the situation and she realized that if he sells 0 he would still make $1400, so she corrected each of the outputs to get the correct amount.
Daisy only needed a little guidance to get the correct outputs for her table. When creating the table for Nancy, she did not have any problems getting the correct outputs. Daisy was only the second student who remembered to label her x and y axises. She wrote rules for both Eric and Nancy to figure out their income. She then tested the functions by plugging in 12 into both rules.
Final Thoughts
It was my first time teaching “The Commission Problem” and did not know what to expect. I was actually a little fearful to begin teaching the CUNY HSE Math Curriculum Framework. The past year I feel I have been teaching to the TABE test but came to the conclusion that it was time to move forward and get past my fear. Even though I knew about functions, it was not something that I taught all the time. As I continue teaching the lessons in the CUNY HSE Math Curriculum Framework, it started to all come back and I realized there was nothing to fear about it. It helped me to work out the problems before I presented the lessons to the students and to anticipate what type of questions they may have about functions. I actually really enjoy teaching this topic and to watch the students grow in their knowledge of functions.
As we always hear, we teach to the test. Teaching to the test can be a good thing, but also a bad thing. Yes, it will help students get those gains when they take the TABE test (for the computation portion of the test), but will it eventually help them know the full array of topics that will be on the HSE test to help them pass that section? Last year, I felt it was a necessity to teach these topics, but after looking over the CUNY Math Curriculum, they will be able to utilize those concepts as they work through the problems using the different operations such as addition, subtraction, multiplication, division, fractions, and so forth.
When giving the students The Commission Problem, I did not know which strategy that they were going to use. After working out the problem, I realized that having more experience with algebraic expressions helped me with working out the problem by creating the rule. Even though we had just started working with functions, it was interesting to see how the students approached each problem. Seeing their frustration as they were unsure where to begin bothered me, but when they were redirected to the different problem-solving strategies they use, I could see the wheels turning and started the process of tackling the problem. I think the thing that bothered me the most is that they were struggling to begin the problem. After allowing a little time to pass, and the redirection on starting the problem aided in working it out. When the students were finished, we continued to discuss it so I made sure that they had a good grasp of the concept.
When I teach this again, I would make sure to check to make sure that the students labeled the x-axis and y-axis. Another thing that I would change would be to ask students who thought Eric had a better deal to look further at the graph when they both sold more than 8 fish tanks.
When finishing the problem, it was rewarding to see that the students were able to work out the problem even though at first some of them were struggling. It also helped that all of the students were working on the same problem together. Having a mixed ability classroom can be beneficial. Once the higher level students are finished and that their work is checked over, they feel it is rewarding to help their fellow peers to have a better understanding on how to work out the problem. They did not give them the answers, but asked questions that made them think about the problem more, so that they could find the answer on their own.
