I don’t have my own class these days, so I’m always grateful to teachers who let me visit as a guest teacher. Angelo Ditta and Will Croxton from LaGuardia Community College in Long Island City, Queens generously allowed me to teach a combined class at their campus on the last day of class so that I could try out this new problem. The students were at a high school equivalency level, but with different levels of math abilities, as is typical of most adult education classes. The students at La Guardia are a wonderful mix of native New Yorkers and people from around the world. It’s an amazing place.

## De-Evolution of a Math Problem

The original version of this task was inspired by questions on the TASC Readiness Math test where students are presented with functions modeling workplace situations.

I started by writing a TASC-style problem:

I like the problem for a few reasons. Since it’s in a work context and is about money, students will probably find it interesting and may bring up background knowledge and questions about calculating the amount of money that shows up on their paychecks. I also like that the problem includes a table of data which can relate to other possible views of a function (rule/equation, graph). If a class is learning functions, I would expect students to wonder what this data would look like as a rule or as a graph.

In the end, however, I decided not teach a class around this version of the problem. Practice questions like these may be good for assessment, but not for teaching. Since they don’t allow for different points of entry, they aren’t accessible to all students. They might be good for students who can use concepts they already know, but they have a limited value in helping students learn math. Students would either already understand the question or would be completely confused by it and the class would quickly split between those who know and those who don’t.

I’ve been playing around with removing information in order to make problems more interesting and accessible to a wider range of students (inspired by the fantastic work of Annie Fetter and others at the Math Forum). My first thought was just to remove the answer choices from this question, so that it would look like this:

This opens up the problem a bit, especially if the question didn’t ask for *which* function, but *what* function, since there might be more than one function possible. However, students who don’t understand the question wouldn’t be able to start. My guess is that many students would be thrown by *t* and *P(t)*, not to mention the fact that deriving a function from a table of data is pretty challenging for students in high school equivalency classes. The question actually becomes much more challenging than the original since even students who understand the formal notation wouldn’t be able to plug in numbers to find a function that works for this table of data.

So, what I removed the question as well?

This is getting better. This setup could certainly work with a notice/wonder activity, since there are numerous questions you could ask: What is his average salary? What is the most he has worked in a week? How much more did he make when he worked 2 more hours? etc. However, I decided that I wanted the problem to be a little more mysterious. If I remove the information about the hourly wage and the weekly deduction, Kareem’s net pay becomes a little strange. If you divide the pay by the hours, you don’t get the same numbers for different lines:

325/20 = $16.25

430/26 = $16.54

What? How is he being paid a different hourly wage when he works more hours? So, this seems interesting. Challenging, but interesting.

## Introducing the Problem

I decided that I still wanted students to try the TASC-style question at the end of the class, but I would get there with notice/wonder and an open-ended task based on the same situation. I wanted students to have an opportunity to discover rate of change (slope) and starting amount (y-intercept) from looking at a table, but I didn’t want to signal that it is necessary to use a formula or think about this explicitly as a function in order to make sense of the situation. My hope was that students would brainstorm a number of possible questions that could be asked about a data set of hours and wages, and that they would use different ways to answer those questions, and develop evidence as a group that could then be used to understand the formal representations of functions in the TASC practice problem at the end.

I decided on a question that I wanted students to explore (*How is Kareem’s net pay being calculated?*) and put together an opening notice/wonder sheet for students:

In brainstorming and discussing their noticings and wonderings, my hope was that students get situated, notice some inconsistencies and wonder how his net pay is being calculated. The notice/wonder launch allows for exploration and diverse responses. Since no specific question has been asked yet, students have time to build a shared understanding of the situation before rushing towards a solution. I wasn’t sure if they would ask my question exactly, but I thought they would be ready for it when I posed it.

I hope that students will notice interesting things about the pay history table such as:

- The hours are all even numbers
- The dollars all end in 5 or 0.
- Kareem gets paid every week. This is about a month and a half’s paychecks.
- The right column shows Net Pay.

And I hope they ask some of the following questions:

- What is Kareem’s hourly pay?
- What is his average pay?
- How much does he make a year?
- How much would he make if he worked 40 hours?

As well as questions like:

- How much does he pay in rent? Groceries?
- Why doesn’t he work full-time?
- Where is this, because he gets paid pretty well for a checkout clerk?

The core problem of the class was presented in a handout like this:

## How I Solved It

To answer the question of how Kareem’s net pay is calculated, I re-ordered the table by hours and noticed that the pay goes up $35 for every 2 extra hours worked and that the net pay for 28 hours is missing. I added $35 to $430 to get $465 for 28 hours of work.

From this, I determined that Kareem gets paid $17.50 for every 1 hour he works. However, $17.50 x 20 = $350, not $325. I tried multiplying $17.50 by 22 and got $385, not $360. Both amounts were $25 more than what Kareem got on the paycheck. I realized that if I multiplied the number of hours by $17.50 and then subtracted $25 from the answer, I ended up with the dollar amount in the Net Pay column.*

I then wrote the following equation and tested it against each row to see that it produced the correct net pay for different numbers of hours.

*Of course, I knew this already from writing the problem, but this is how I would have approached the table of data.

## Anticipating Student Approaches

The main challenge I expect is that students will not recognize a difference between net pay and gross pay, and as a result will calculate an hourly rate based on the net pay for each number of hours. My guess is they will see the variation in hourly rates and not know what to do next. I think it would be helpful to have them think in advance about different ways of being paid, so they might come up with the idea of money being added or subtracted to account for hourly wage not being consistent. I also want students to see how the net pay changes when 2 more hours are worked. They should see that the net pay goes up by $35 whenever Kareem works 2 more hours. Hopefully, they will take the next step to see what the actual hourly wage is.

I plan to start the class with a pair/share discussion about jobs students have had in the past: *How did you get paid? How often? Cash or check? Did you ever get extra money? Or money taken out of your check?* I’ll clarify net pay vs. gross pay if a student doesn’t do it during the notice/wonder conversation.

In first approaching the problem, I think students might guess what Kareem’s hourly wage is and try multiplying that amount by the hours to see if equals his pay. I expect them to start with Dec. 11th (20 hours, $325). They might guess that his hourly wage is $10/hr and adjust from there.

20 x $10/hr. = $200 (too low)

20 x $15/hr. = $300 (too low)

20 x $16/hr. = $320 (too low)

20 x $17/hr. = $340 (too high)

20 x $16.50/hr. = $330 (too high)

20 x $16.25/hr. = $325 (right amount)

Other students may just divide $325 by 20 hours to get $16.25/hr.

My guess is that students who get $16.25 will think they are done. My planned response: “Cool! Does that work for the other pay checks?”

Students would then try to figure out the hourly wage for Dec. 4th (26 hours, $430) or Nov. 27 (32 hours, $535):

26 x $16.25/hr. = $422.50 ($7.50 too low)

32 x $16.25/hr. = $520 ($15 too low)

I expect this will be perplexing for students. How could the hourly wage change for a different number of hours?

### Supporting Productive Struggle

I expect that students will find this problem challenging. Because of this, I want them to work in groups and lean on each other. My first priority is that students are interested in the problem and make sense of it from different angles before tackling the main question I have for them. It would be great if they came up with my exact question (*How is Kareem’s net pay being calculated?*) during the notice/wonder conversation, but it’s not essential. I expect their questions will similar.

Update: Here are the notes from the board with students’ noticings and wonderings. They didn’t ask my question, so I simply said, “These are great questions! I have one for you: *How is Kareem’s net pay being calculated?* Work for 5 minutes on your own before talking with a partner.”

My goal is to allow students to struggle with the problem. I’m going to use support questions to give students hints in case the struggle doesn’t seem productive. The push questions are for students to explore after they have answered the central question. It’s a strategy to help students work on their own and need me a little less.* My plan is to share a question with a group of students only if they need one to keep working productively.

There are some strategies I think will be helpful. I want students to see the growth in net pay as more hours are worked. I’d like them to analyze the table of Kareem’s pay history. I’m hoping they reorder the table and add some missing data. The first few support questions will move them in this direction if they don’t do this on their own.

### SUPPORT

*How much money would Kareem make if he worked 28 hours?*(So that students will see that the net pay grows by $35 for those 2 hours of extra work.)*What do you notice if you put Kareem’s pay in order of hours worked?*(So that students will see that the net pay grows by $35 every time the hours go up by 2 hours)*Gross pay – calculated wages before deductions such as taxes;*

*Net pay – what you actually get on your paycheck after deductions*(Reminder that something is being taken away from the pay we calculate with an hourly wage)*If Kareem worked 16 hours, how much would he get paid?*(So that students will subtract 35 for every 2 fewer hours worked)

### PUSH

*How much money would Kareem get in net pay if he worked 40 hours?*(So that students will extend the table and start to look for a rule)*How much is being deducted from Kareem’s salary each week?*(See if students can define the rule explicitly rather than just determine the net pay by adding or subtracting from the value above)*For the week of October 30th, Kareem was paid $587.50. How many hours did he work?*(See if students can move from output to input, in effect reversing the rule)*Imagine Kareem gets a $.50/hr raise. How much would his net pay be for 34 hours of work?*(See if students can explicitly define the rule in order to adjust it)*Choose a Wonder question we haven’t answered yet and try to answer it.*(Support*students’ curiosity and encourage independent exploration.)*

I would be thrilled if some students look at differences in the amount of net pay as the hours increase and recognize this as slope. I would be very impressed if they then realized that $25 is taken away from the gross pay every week in order to determine net pay. I doubt that any students will be able to put this all together into an equation to calculate net pay for any number of hours, but as a visiting teacher, I don’t know these students’ math skills well and someone might surprise me.

* The idea for support and push slips came from a great workshop by Rachael Eriksen Brown at the 2016 NCTM Regional in Philadelphia.

Note: I suggested students use calculators from the beginning. Some used calculators on their phones, though I recommended they use the TI-30XS since they would be able to use it on the actual exam.

## Student Work

### Rafia’s Work

I really like how much information Rafia was able to draw from the table. Her analysis brought a lot of useful information to the surface:

- She calculated the hourly rate for each date by dividing the net pay by the number of hours
- She ordered the hours in increasing size and saw the growth (+35) from 20 hours to 22 hours.
- She totaled the net pay for all 6 weeks ($2545)
- She also factored each of the hours in interesting ways. She factored 20, 26, 32, and 24 by 4, factored 22 by 3 and 30 by 5. I don’t understand where she was going with this, but I love that she was attacking the table of data in multiple ways.

Rafia worked with a group that was able to determine the net pay for 28 and 40 hours, but it looks like she didn’t get there herself. My guess is that she would be able to do it with a little help since she had already found the change in net pay for an extra 2 hours of work.

## Juan’s Work

After working on my first support question, *How much money would Kareem make if he worked 28 hours?*, Juan realized that Kareem’s wages were going up by $35 for every 2 extra hours of work and that his pay must be $17.50 per hour. Juan then used $17.50/hr to calculate Kareem’s gross pay for each week. He realized that this didn’t match the net pay column and thought a tax could be the explanation. Using 28 hours as his sample, he calculated the gross pay as $490 (28 x $17.50), then looked for the difference between this and the net pay of $465, which he determined using the +35 growth in the net pay column.

It’s hard to know where he went from this. It seems like he made the mistake of thinking $30 was the missing amount instead of $25, but calculated a correct estimate of 5% being taken out. I’m not sure if he got this by dividing 465 by 490, which produces .949 with approximately 5% missing or if he divided the correct difference of 25 by 490 (5.1%).

It seems to me that Juan was very close to cracking the code, but didn’t see the constant deduction from gross pay to get net pay for each quantity of hours. I wish I had seen his work during class. I wonder what he would have noticed if we had talked through his calculations of gross pay and looked at the corresponding net pay totals.

### Jeffrey’s Work

Jeffrey started by totaling the money for all 6 weeks. He then divided by 6 to get an average net pay per week of $424.167. It’s interesting that he rounded to 1/10 of a penny rather than a full cent. Later in the class, I saw that a number of students had questions about how and when to round the answers they got.

If you divide his net pay by the number of hours, Kareem seems to make about $16 for any number of hours worked. I think Jeffrey used $16 as Kareem’s approximate hourly wage and multiplied it by 28 hours to get $448. In his next move, Jeffrey seems to be responding to a question implied by the text above the wage table: *Is Kareem getting paid* *correctly*? Jeffrey compared $448 (approximately $450) to $424 to conclude that Kareem is not being paid enough, the logic being that he usually makes about $424 each week and if he works 28 hours he should make $448. There are a few misunderstandings here:

- Kareem would make $465, not $448. Jeffrey actually shows above that he had figured that out. It would have been interesting to ask him to talk about these two calculations.
- The average of $424/week is a result of different amounts of hours worked. This is how much he would make if he worked an average number of 25.7 hours. If he worked 28 hours, we would expect him to make more money, but it wouldn’t mean that he was being paid unfairly for fewer hours.

I didn’t get to see Jeffrey’s work during class and, to be honest, I’m not sure how I would have responded. I think the thread to pull on would be about the difference between his two calculations for 28 hours of work ($465 and $448). I might also ask him about the “35” he wrote next to the chart. I might be able to get him to build up from 26 (or down from 30) to confirm the net pay for 28 hours.

### Tsering’s Work

Tsering started by calculating the apparent hourly pay for each quantity of hours worked, which was common across many students’ work, and, similar to Jeffrey, decided on the approximate average of $16/hour. Though she made a mistake in writing this as hours/week, I think she was clear that this was dollars/hour. When I gave her group the question about net pay for 28 hours of work, she multiplied 28 by 16 to get $448, similar to Jeffrey. She then multiplied 40 by 16 to $640 as the net pay for 40 hours of work.

Interestingly, the table at the bottom of her worksheet shows an understanding of the differences in hours and dollar amounts, and includes the correct net pay of $465 for 28 hours of work. My guess is that she produced this table after the earlier calculations and didn’t have time to reconcile what she discovered. I wonder what she would have noticed had she continued looking at differences in the table to determine net pay for 40 hours of work, as compared to her previous calculations.

### Orlando’s Work

Along with calculating apparent hourly wages and realizing the actual hourly wage must be $17.50/hr, Orlando concluded that a 4.762% tax is being taken out of Kareem’s paycheck. It’s difficult to know how he determined this, but my guess is he made the following logical steps:

- 30 hrs. x $17.50 = $525
- Kareem was paid $500, not $525, so $25 was taken out
- $25/$525 = 0.047619048

Orlando clearly knows a lot. Besides determining the hourly wage in a few different ways and recognizing the growth in the table, he recognized the missing $25 as a part of the $525 whole, then calculated the percent this represented. If I had time, I might have asked him whether we should expect this tax withholding percentage to be the same for all any number of hours worked. If you follow Jeffrey’s logic in calculating a possible tax being taken out for 24 hours of work, for example, the tax percentage would not be 4.762%. I would want him to see that and decide what that means.

### Sefa’s Work

I was blown away by a couple aspects of Sefa’s work. During our group discussion of the problem towards the end of class, she shared the function at the bottom of the page in a slightly different form. Verbally, she told me to write this function…

*P(t)* = *n* X 17.5 + 325

and explained that it worked when *t* is greater than 20. I copied it to the board, a little reluctantly because I didn’t understand it and was pretty sure it didn’t work. Thinking she was confused, I said usually the two variables would match each other (*t* and *n*). She said okay, and I changed the *n* to *t*.

*P(t)* = *t* X 17.5 + 325

Not really sure what to do, I asked the class how we could know whether Sefa’s function would calculate Kareem’s net pay. They said that we should try different numbers of hours and see if we came up with the same net pay as the table. So, we started with 22 hours, since that was more than 20 (Sefa’s requirement). Seeing that the numbers were going to be way too big, I started writing on the board…

*P(22)* = *22* X 17.50 + …

“No,” Sefa cut in, “Only the part more than 20. It should be 2.”

Oh… It started to make sense to me.

*P(*2*)* = 2 X 17.50 + 325

*P(*2*)* = 35 + 325

*P(*2*)* = 360

We used 28 hours as our next example.

*P(*8*)* = 8 X 17.50 + 325

*P(*8*)* = 140 + 325

*P(*8*)* = 465

Sefa’s function worked! You figure out how many hours there are more than 20, then multiply by 17.5 and add 325. Looking at her work later, I realized that I may have misunderstood the first function she shared with the class. What if *t* stands for Kareem’s total hours and *n* stands for hours more than 20, and the situation is represented in two equations?

*P(t) = 17.5n + 325
n *=

*t*– 20

I should have just asked her to explain her function rather than making a suggestion before I understood what she was saying. That’s a good lesson.

If that weren’t enough, Sefa also came up with another way of estimating Kareem’s net pay for 28 hours.

Sefa took the apparent hourly wage for 32 hours ($16.71) and the apparent hourly wage for 26 hours ($16.54) and averaged them to get an estimate of the hourly wage for 28 hours ($16.63), which she then multiplied by 28 hours to get $465.50, which is only $.50 off the actual pay. I think she made a mistake in choosing 32 hours. If she had used the apparent hourly pay for 30 hours ($16.67) in her calculations, she would have been off only by 6 cents.

## Discussing the Problem

After students worked on the problem for about 20 minutes, we came together for a group discussion of the table in increasing order of hours worked. Since no one had come to a full solution on their own, I asked each group to share one thing they learned. They shared the following:

- The net pay goes up by 35 dollars
- He would make $465 for 28 hours of work
- 20 times 16.25 is 325
- His hourly wage is $17.50/hr because 35 divided by 2 is 17.5
- If he worked 21 hours, he would make $343.50
- When he works more than 20 hours, he makes $17.50/hr and $16.25/hr when he works less than 20 hours
- Sefa shared her function (described above)

## Test Practice Question

Time was running out in our class period, so I gave them the test practice question to try on their own. I was happy to see that the majority of students got the question right. I added a question at the bottom: *Why did you choose this answer?* It provides some interesting information and I will definitely be adding a similar question for test practice in the future.

### Wrong Answer

This is the only example I saw of a wrong answer. The student chose A) *P(t)* = 17.5*t*.

### Right Answer with Partial Explanation

These answers were correct, but the explanation seemed incomplete to me. Each of these students used 20 hours as an example to show that *P(t)* = 17.5*t* – 25 is the correct function, but *P(t)* = 16.25*t* would work also for the input of 20 hours. I would like these students to think about other inputs to show they understand that a function has to work for every input-output pair in a table of data.

This last example shows evidence of confusion about what *P(t)* means:

- Instead of understanding that
*P(t)*means “the function of*t*” or “the amount of payment received for different amounts of work in hours,” the student substitutes a value for both*t*(which is not necessarily incorrect) and then a value for*P*(which is incorrect). The value, 325, corresponds to*P(20)*, not*P*by itself. - The student then multiplies 325 by 20 to get 6500 because she has probably been taught that when numbers or variables are in parentheses next to each other, it means they are separate terms that are being multiplied.

Interestingly, the student gets the right answer, but clearly has some misconceptions that should be cleared up. If I were teaching the next class, I think it would be important to bring this example to the group for discussion. My Favorite No could be a good way to start conversation, since the student shows understanding of procedures (substitution, representing multiplication) that would be correct in other contexts.

### Right Answer with More Complete Explanation

These students gave a more complete explanation, either in their answer to my question at the bottom or in their annotation of the problem. A few students talked about how the weekly deduction means something is being subtracted in the function, which eliminates A) and B) as possible correct answers. Other students mention that they checked *P(t)* = 17.5*t* – 25 to see if it calculated the correct net pay for every quantity of hours in the table.

## Final Thoughts

The paycheck problem as I’m presenting it here is challenging for HSE students, but they were able to figure out a lot and some students really surprised me with their creativity. The problem provoked interesting conversation, challenged students and led towards a number of math topics and skills: algebra/functions, percent, average, rounding decimals, estimation, etc. I saw students teaching each other how to calculate the hourly wage from the pay and the number of hours. Other students showed each other how to calculate average. One group worked on determining percent when you know the part and the whole. A few students learned how to round up or down and thought about what decimal to round to when calculating dollar amounts. In short, there was a lot of math going on, and students seemed engaged by the problem.

I can’t imagine much of this would have happened if I had given it to students in its original form. Carving up (or whittling down?) standard practice test questions could be a promising approach to test preparation. This could allow for an open problem-solving approach even when we need to get to brass tacks — what will this look like on the test? — position by the end of the class or module.

The class was about an hour and 45 minutes (students gave up 10 minutes of their break!), but we ran out of time. I would have like a full 30 minutes for their independent problem-solving after noticing and wondering, supported by support and push questions. We had about 5 minutes of group discussion before they took the practice test question. I would have preferred 15-20 minutes. When I teach this again, I would like to have 3 hours, maybe across a couple class periods.

I wrote a couple support activities that would be interesting to try to between problem-solving and test practice:

Write Questions to Match Calculations: Since the TASC emphasizes understanding of mathematical formulas and calculations, rather than the ability to compute answers, why not give students the calculations and see if they can figure out what questions the calculations answer? This reversal was inspired by Fawn Nguyen’s blog post about a problem/strategy Don Steward shared. Here’s a sample of what it looks like:

Graphing Hours and Net Pay: I used Desmos.com to plot some of the hours/net pay combinations on a graph, leaving room for students to complete a table and plot other points. Ideally, students would be able to identify the rate of change and starting amount on the graph.

When I teach this problem again, I’d like to focus group discussion after problem solving on this string of questions:

- How much would Kareem make after 28 hours?
- How much does Kareem make in 2 extra hours of work?
- How much does Kareem make in an hour?
- 20 times 17.5 equals 350, not 325. Hmm?
- 22 times 17.5 equals 385, not 360. Hmm?
- (If students think Kareem makes $16.25/hr when he works 20 hours or less) How much does Kareem make if he works 18 hours? $292.50 or $290?
- How much does Kareem make if he works 16 hours? $260 or $255?

My guess is this would lead to other interesting conclusions and misconceptions to explore. I’ll report back when I know more.

P.S. I forgot to ask students why that $25 is being taken out of each check! That seems like an important question to reconnect the function to the real world context. When I wrote the problem, my intention was that this would be for a monthly subway MetroCard paid through TransitChek, though it’s actually not quite enough money.