I like this problem because I think it’s a good one for introducing problem-solving strategies with a new influx of students that had just begun my class. Because the computational skills required to successfully solve the problem are low, I felt this would remove an anxiety barrier some of my newer students may present with. Additionally, I had just received a few new students who have very low TABE reading levels and much higher math scores. I felt this “less wordy” problem would allow us to focus more on developing the problem-solving skills, without overwhelming them at the same time with a longer story problem. For my students who have been with me longer, many still struggle with organizing their work, recognizing patterns, or persisting long enough to make their guesses count. This problem could provide room for growth for everyone.
How I Solved It
As few students in my class are comfortable writing expressions or working algebraically, I chose to work the problem out using guess-and-check, combined with a table to organize my data. These are strategies that students could use successfully to solve the problem.
I began at the outer limit 98 + 0 = 98 and 98 – 0 = 98, then tried 97 – 1 = 96, then 96 – 2 = 94, knowing these guesses are not close but thinking my students may need some examples of what was meant the problem is actually asking them to do. I also notice the pattern that for any change of the numbers by 1, the difference changes by 2. I then decided this was too slow and decided to try numbers that are much closer together 50 – 48 = 2 or 52 – 46 = 6. Obviously this difference is way too low. At this point I decides to try some numbers that were halfway between these sets of guesses: 60 – 38 = 22; 58 – 40 = 18.
Now I know I am close. Since I know that a change in the numbers of 1 will result in a change in the difference of 2 I decide to try 56 – 42 = 14.
Anticipating Student Approaches
The problem could be solved algebraically. I have one student in the class that I think will try this, so I’ll work it out just in case.
I think that most students, though, will use either guess and check, or a table like the one above, rather than algebra.
My Goal for Student Learning
I want my students to improve their ability to organize their work. They often come quite close to answering complex, multistep problems but cannot find or see any meaning in their work, because it is all over the place. My students also need to improve their pattern recognition skills, which would be helpful in making more strategic guesses. Additionally for some students the biggest challenge will be the blank page, as they are reluctant to try anything unless explicitly shown how to do something first.
Supporting Productive Struggle
Many students will not know where to start with this problem. I will try to mitigate this by discussing different problem-solving strategies and asking them to choose one that they would like to try. For the truly reluctant, I may need to ask them to think of some numbers that would add up to 98, then ask what the difference of those numbers would be (just to get them started so the page is not blank). I think that some of my students who are weaker readers may misinterpret the question. So we may need to spend some time deciphering what “sum” and “difference” means and possibly trying one or two “guesses” together so they understand what the objective is.
Students could write up what they tried on chart paper, and then I would ask them to present their work and explain why they made the guesses that they did. There are several other number sets that students could try if they are flush with success and eager for more. Ideally, at least some of these additional number sets should be saved for after discussion of the different methods everyone tried, so students revise their strategy or can try others. Students also could create their own number sets to challenge their peers.
Here are a few other’s I plan to give as extensions:
- The sum of two numbers is 33 and the difference of the same two numbers is 11. Can you figure out what the two numbers are?
- The sum of two numbers is 65 and the difference of the same two numbers is 19.
- Can you figure out what the two numbers are?
- The sum of two numbers is 243 and the difference of the same two numbers is 69. Can you figure out what the two numbers are?
Isaiah chose to utilize guess and check as his strategy, one most of his peers chose. While we took some time together as a class to discuss what the question meant, from his first few guesses it was clear Isaiah misunderstood the question. He did not realize both conditions must use the same two numbers. Isaiah is one of my newer students who has a significantly lower reading level than his math, which often makes understanding problems difficult for him. As Isaiah is not easily embarrassed, and since quite a few students were struggling to get started anyway, I elected to stop and ask his peers for help. We looked at the number sets he was trying and had his peers identify if what he was trying was what question meant. Despite verbal clarification from his peers, Isaiah still seemed unsure until we drew the following on the board:
____ + ____ = 98
____ – ____ = 14
After this, Isaiah’s guess and check method was more appropriate, though his guesses were random and he did not seem to be looking for patterns. He did eventually stumble upon the answer, though he did almost give up halfway through and required encouragement to persist. Isaiah’s approach was typical of many of his peers who chose to guess and check, so it was a good example of how this strategy could work, while allowing for discussion of missed opportunities and how to promote more strategic guessing.
Adam elected to use a table. Adam has been with me for a while and still suffers from “blank page anxiety.” He is very bright but doubts himself and is still uncomfortable if not given explicit directions. While his work above looks clear and organized, it does not show the 15 minutes he spent staring at the blank page and struggling with how to get started. Eventually Adam admitted he had no idea how to set up the table. He knew he wanted to make a table, stating this would help him see patterns, but he was anxious about setting it up wrong. It was not until I asked one of his peers to show Adam how he was setting up his own table, and I showed a different way I set up my table, that Adam was able to select a format and get to work. As you can see above, Adam was correct that the table would help him to organize and see patterns in his guesses. He was one of only two students who chose to use a table or chart, though he needed support in constructing one. After we shared Adam’s process, the majority of students chose to use one of the two tables shared out for their second attempt at this type of problem. Clearly many of my students need additional support and practice in constructing tables in many different contexts for this strategy to be useful to them.
SK was the only student who elected to use algebra to approach this question. He had significant schooling in his home country, but that was some time ago and much is misremembered. This is often complicated by translation issues or alternate non-US math notation, which often requires some muddling through together. SK also is reluctant to use simpler methods as he views them as “lesser than.” While SK wrote equations initially, he could not quite remember what to do with them and resorted to guess and check.
He was really focused on splitting 98 in half, but could not quite make that work for the difference. He really tried to get me to show him how to do it algebraically, then tried to wait me out. I strongly and repeatedly suggested he try a different method if his first choice was not working. After waiting me out did not work, he eventually elected to try draw a picture, which you can see below.
His picture took the form of a number line starting with the 49 and 49 (98 cut in half) that he was fixating on algebraically. He made the misstep of trying 35 and 63 (forgetting that changing both numbers by 14 would make a difference of 28), but after a challenge to prove those numbers worked he caught his error. I chose to talk about SK’s method, as it gave the class an opportunity to see two strategies no other students tried (algebra and drawing a picture). The drawing allowed students another way to visualize what was happening with the numbers.
After we explored all the students’ attempts, I allowed one of the HSE-level students from the classroom next door to demonstrate how to do the algebraic equation correctly, as SK was anxious to find out how the algebra could work. The student was impressed that Adam’s method was so much easier! This allowed students to see how some of the simple strategies they can utilize today could help solve what appears to be complex algebra problems that the higher-level math students are doing. It also provided a much-needed confidence boost to Adam, who was disappointed that he was not yet in the HSE class.
Advice for Teachers
Problems like this one are great for adding tools to students’ problem-solving toolkits. Students often have ideas about how algebra is “real math” and everything else is somehow lesser, like SK thought. But this problem shows that algebra isn’t always the most efficient pathway to a solution. Moreover, the students who were successful without using algebra felt empowered by their success with the problem, and they were able to successfully solve the extension questions. Problems like this have a low entry point so that all students can get involved and engaged, but they also open the door to having talking about topics as complicated as systems of equations. Every class is different, so just meet your students where they are and there’s a lot of great discussion that can come out.