The cookie problem does not require any advanced mathematics techniques, but it is quite challenging nonetheless. Thus, it is a “problem” for my students, but it has few barriers to entry and is approachable for everyone in my classes. Without giving any parameters for a solution, students can come up with a variety of ways to solve the problem. In the end, those various representations present a verdant opportunity for discussing algebra, algebraic thinking, diagrams, as well as mean and median and central tendency, and arithmetic series. So, altogether, it is a simple but rich problem.
How I Solved It
To solve this problem, I used a variable to stand for the number of cookies on day 1. Then, for each successive day, I added 6 more cookies than were eaten the day before. I did this for five days and let the entire equation = 100, since that is how many cookies Tim ate in 5 days.
Let x = the number of cookies on day 1. Then x + 6 would be the number of cookies he ate on day 2, x + 6 + 6 would be the number of cookies he ate on day 3–and so on. Therefore:
x + (x + 6) + (x + 6 + 6) + (x + 6 + 6 + 6) + (x + 6 + 6 + 6 + 6) = 100
5x + 60 = 100
5x = 40
x = 8 cookies.
Then, to make sure this is correct, I started with 8 cookies and added 6 more cookies each day successively, hoping that it would add up to 100. Doing that, I got:
8 + 14 + 20 + 26 + 32 = 100
100 = 100, so the solution checks out.
Anticipating Student Approaches
First, if we divide 100 by 5, we would be able to determine how many cookies Tim would have to eat each day on average. Knowing that, we can use the middle day–day 3–as a balancing point, and then add or subtract cookies on the other days. Let’s say that the five days Tim ate the cookies were Monday, Tuesday, Wednesday, Thursday, and Friday.
The average number of cookies eaten in the 5 days is 100 ÷ 5 = 20. This means that on Wednesday, Tim ate 20 cookies. Now I’ll add six cookies to find the number he ate on Thursday, and I’ll subtract 6 cookies to find the number he ate on Tuesday. I’ll then repeat the process to figure out how many cookies he ate on Monday and Friday.
Monday: 14 – 6 = 8 cookies
Tuesday: 20 – 6 = 14 cookies
Wednesday: 20 cookies (average)
Thursday: 20 + 6 = 26 cookies
Friday: 26 + 6 = 32 cookies
To check the answer again, I would calculate: (20 – 6 – 6) + (20 – 6) + (20) + (20 + 6) + (20 + 6 + 6) = 100.
Another method students might use is guess and check. I could take an educated guess, say 10, then added 6 cookies to each day thereafter to produce 10 + 16 + 22 + 28 + 34 = 110. Then I would know to adjust my guess by going down. Next I’d try starting with 8, and solve the problem. I anticipate some students having trouble organizing their guesses, and so I might recommend that they organize their information into a table.
My Goal for Student Learning
From working on this problem I want students to get that math, first and foremost, is about problem solving. Algebra, on the other hand, is simply a way to represent problems symbolically so that we may solve problems systematically, and we hope, more efficiently. But, problem solving can often be a creative endeavor. Using the various representations and solutions to this one problem I hope to establish a simple yet challenging example of how our various solutions can help us build a connection to symbolic representation and algebra in general. Simple two-step algebra problems do not convince some students of the value of solving problems algebraically. Those students often feel like they can solve the same problem without algebra more easily. By presenting them with this problem, I feel like we have a better case for using algebra. We can simplify this problem to a simple two-step problem, but it does not appear that way in the beginning. For this reason, I hope that this problem will present a greater opportunity for growth in algebraic thinking and symbolic representation.
Supporting Productive Struggle
I anticipate three major challenges for my students:
- explaining their reasoning;
- checking their solution;
- giving up when their solution doesn’t make sense.
To support the problem-solving efforts of those students and encourage perseverance, I will circulate around the room and ask each student questions that will help bring them to the next step. Most important, I believe I will need to encourage students to check to see if they answered the question. Students tend to come up with a solution, and/or perform algorithms only to end with their answer. But they still need to consider ways to figure out if there answer is correct and if it makes sense in the context of the problem. I plan to check in with students by asking a question like: “Does that answer fit with the problem of eating 6 more cookies than the day before and ending on 100 cookies after 5 days?”
I would ask students that finish to explain the impact that each of the following changes have on the mean number of cookies eaten by Tim each day and the number of cookies he eats on the first day if we change the original question to read:
- Tim ate 100 cookies in 5 days. Each day he ate 5 more than the day before. How many cookies did he eat on the first day?
- Tim ate 100 cookies in 5 days. Each day he ate more than the day before. How many cookies did he eat on the first day?
- Tim ate 120 cookies in 5 days. Each day he ate 6 more than the day before. How many cookies did he eat on the first day?
Student Work on the Cookie Problem
Jessica started out by drawing a diagram of 100 cookies. She started by guessing that Tim ate 10 cookies on the first day by circling 10 cookies. Then she subsequently circled 6 more cookies than the day before. So, she circled 10, 16, 22, 28, and then 24 cookies since that was all of the cookies left that she could circle as the rest of the cookies had already been “eaten”. However, next to the diagram, she found the sum of 10 + 16 + 22 + 28 + 34 = 110. With my assistance, she was able to determine that starting with 10 cookies on the first day, resulted in 10 too many cookies in the end. So, she decided to divide the remainder equally among the 5 days. Finding 10/5 = 2 she determined that that meant that she had Tim eat 2 too many cookies each day. As a result, she reduced each day by 2 cookies. She confirmed her answer by finding that 8 + 14 + 20 + 26 + 32 = 100 cookies!
I chose to talk about this sample of student work because she nicely illustrates her reasoning with a diagram that is helpful to other students and she clearly describes her thinking. Furthermore, she demonstrated that following her guess of 10 cookies she had a system for dealing with the difference between the result of her first guess and the desired outcome. That is 110 – 100 = 10, so we can split the difference with 10/5 = 2, then subtract 2 from each day. Not only did she guess and check, but she strategically moved from one result to her next move, subtract each by 2.
Jessica’s approach was similar to others’ in that she employed a guess and then a check. However, hers was atypical in that she started with a diagram, and she reasoned that the difference between her outcome of 110 and the desired outcome of 100 should be evenly distributed among the 5 days.
Ambrose started out by dividing the 100 total cookies evenly among the 5 days to give us 20 cookies/day. From there he seems to change strategies when he doesn’t seem to know what to do with the 20 cookies a day average. Instead, he then reasons, incorrectly, but with purpose, that Tim will eat a total of 30 “additional” cookies by finding 6 “additional” cookies a day for 5 days. It follows that since Tim ate 30 “additional” cookies, 100 – 30 = 70 cookies “he would have eaten if he never added the sixes.” To determine how many cookies he then ate per day “if he never added the sixes”, Ambrose determines that 70/5 = 14, therefore Tim “probably ate 14 on the first day.”
Following this answer, Ambrose does check to make sure 14 cookies on the first day will result in 100 cookies after 5 days eating 6 more than the day before. Determining that starting with 14 cookies on the first day will result in 130 cookies after 5 days, Ambrose reasons that his answer of 14 must be too high. As a result, he adjusts first to 7 cookies, finds that ends in 95 cookies, and then he adjusts his answer to 8 cookies on the first day.
I’m interested in Ambrose’s work because he demonstrated a strategic approach to determining the answer as opposed to a guess and check method. He employed algebraic thinking in attempting to work backwards from the total of 100 cookies. While his strategy did not initially produce the answer of 8, he demonstrated a thoughtful and meaning approach to the problem. From the result of that attempt, Ambrose correctly determined that he should reduce his answer, and then subsequently increase that answer to produce the desired outcome.
Ambrose’s work reflects the thinking of several other students, but it is atypical in that he fluidly switched strategies three times. He tried 100/5, then switched to 100 – 30, then he went to a guess-and-check method from his results. I love that each decision was made with a reason. Other students more commonly tried one strategy and kept pushing forward with that method until it worked or didn’t. From a class discussion standpoint Ambrose’s solution leaves a lot to talk about. Why did his initial attempt not result in 100? What could he change? What shouldn’t he change?
Monica immediately attempted to solve the problem using an algebraic equation. She correctly used five terms to represent five days with a sum of 100 cookies. However, all of the days, but 1, are represented with the same term. Therefore, this initial equation could not reflect the stated pattern of Tim eating 6 more cookies than the day before. Monica’s equation results in an answer of 4. She does not clearly state that this represents day 1, but she does check her work. Unfortunately, her check only reflects the fact that her computation is correct, not that her answer fits Tim’s stated cookie-eating habit.
I appreciated Monica’s work because she was the only student in the class to attempt to use variables and an algebraic solution. I find it interesting that she tried a check of sorts, but this check did not check to see if she answered the question, only that she produced 100 cookies. There is much for other students to learn from this example. Did this work or not? She got 100? What could Monica change? Why is 6n not 6 “more than the day before?” Why did the check confirm her work, but she didn’t get 8, the correct answer?
I learned that almost none of my students felt comfortable enough with writing algebraic equations to choose that as an initial solution strategy for this problem. Many of my students attempted calculations with little concern as to what those calculations were calculating. For example, a student would find 100/6 = 16.7, but they would have no idea what that number represented. Then, they would perform further operations on that number. Several students did not attempt to check if their answer solved the problem. Also, having students explain their work helped them expose to themselves whether or not performing certain operations makes sense or has a strategic purpose.
What I Might Change
After teaching the problem, I added some additional directions for the next time that I use it. I added: “Your explanation should be complete enough that someone reading it could follow your steps and inspect why you decided to take those steps.”
This problem was far more challenging for my students than I expected in general. I thought that most of my students would eventually come up with an answer through trial-and-error if they had enough time to experiment. However, several students felt that they had exhausted all ideas and possible solutions without much perseverance.
Not focusing on algebra initially was a great help for my students. In the past, I may have tried to steer the class to an algebraic solution earlier on. However, with this problem, I promoted guess-and-check to many students. This allowed them to make sense of the problem and persevere. In the end, they were in a better position to accept an algebraic equation that could model the situation.
For those students that performed 100/5 = 20, it was helpful to ask, “If every day Tim ate 20 cookies, he’d eat 100 cookies after 5 days, correct? But, according to the problem, he can’t eat the same amount every day. If we added cookies to every day, what would happen to the total? If we subtracted cookies from every day, what would happen to the total? Therefore, what should we do if we can’t add to all of the days and we can’t subtract from all of the days?”
It was also helpful to just get students to guess. One of the biggest barriers that some students have is writing down that first thought. Many of my students don’t want to guess for fear of it being wrong. So, they suffer from “paralysis by analysis.” Encouraging them to guess, and then check, is a big step for some of my students.
The greatest benefits for students came from:
- observing that other students use guess-and-check to initially make sense of the problem;
- observing the problem-solving methods of others;
- observing that an algebraic solution helps provide an efficient strategy for modeling a real-world situation;
- observing how the mean can serve as a balancing point, or center, for data.
I wanted students to walk away with a renewed sense of empowerment that I feel a lot of them have lost in mathematics over the years. In the elementary school level, students are accustomed to the process of trial and error, it is natural and helpful to understanding. However, as students progress through algebra and different rote procedures for solving problems I believe they become disconnected from making sense of problems and problem solving in general. Instead of a problem that needs to be worked through, they focus on a series of steps and how to precisely perform those procedures. In this problem, I wanted to talk about procedures or solution methods after they made sense of the problem, not before. I wanted students to do whatever they needed to do to make sense of this problem and come up with some solution.
In the end, every student had a firm understanding as to why the answer is 8, and they have at least one way to arrive at that solution. Most of the students can now solve the problems in more than one way, and the entire class was in a better position to accept the algebraic solution to the problem. I assessed this with subsequent problems and summative assessments and ongoing conversations.
Advice for Teachers
This question can take more time and discussion to completely cover than I thought, and it was actually much more difficult than I expected it to be. In one class, we spent well over an hour on this one question. Just make sure to allow ample time to talk explore your students’ thinking and their problem-solving choices, and great things can come from this question.