The Multiples of Nine Problem

I recently discovered this problem, and I really like it for a number of reasons. First, it requires a little bit of vocabulary in order to get started. Students will have to know what a multiple is, they will have to know what digits are—and more specifically, how digits can differ from numbers—and they’ll have to understand the difference between even and odd numbers. I also like how nonintimidating it looks at first glance. “How hard could it be to find a multiple of 9 that has only even digits? I shouldn’t have to count up very far.” Because the problem doesn’t look lengthy or challenging, it comes as a surprise when the correct answer is actually the 32nd multiple of nine. I anticipate a lot of students writing out 9, 18, 27, 36, 45, 54, etc, and then getting frustrated or giving up when they don’t get to the answer fairly quickly.

Second, I like that the problem requires students to perform basic calculations and that it requires precision in order to get the answer right without making mistakes. Also, the repetition involved in either adding 9 over and over or multiplying by 9 over and over is helpful for students with lower math abilities, and it still provides good practice for students who are more comfortable working with numbers. Moreover, the 9 times tables are interesting because of the pattern that arises in the tens digit and the ones digit. My hope is that as students start listing out the multiples of 9, they will be able to see the pattern and work with it. I like exposing my students to several different ways of thinking about multiplication. My hope is that they find one that works for them.

And finally, I really like this problem because there are good extension questions. If a student finishes early, they can find the next smallest multiple, and then the next one. Once everyone has had plenty of time to work, the class can talk about divisibility tests, and they could work on finding all the three-digit multiples of 9 that have only even digits. And so on.

How I Solved It

I knew that none of the two-digit multiples of 9 contained only even numbers. I also knew that any multiples of 9 that were between 100 and 199 wouldn’t work, because they all would have a 1—an odd number—as the leading digit. I started working under the assumption that the correct number would be somewhere in the 200s, so I picked a nice, round number and started from there. I calculated 9 × 30 = 270. Because this had a 7 in it, I knew that it couldn’t be the right answer, but I noticed that if I were to add 18 to 270, I would get 288. Thus, 9 × 32 = 288 was my tentative answer.

I couldn’t commit to this answer because there might be a smaller multiple of 9 that was located in the 200s and also had only even digits. So I went back to 270 and began counting down by 18. I counted down by 18 instead of 9 because the correct number has to be an even number times 9 (so that I would have an even product). So the numbers I checked were 9 × 28 = 252, 9 × 26 = 234, and 9 × 24 = 216. None of these worked, so the correct answer must be 288.

Anticipating Student Approaches

I could just write out all the multiples of 9 and keep going until I found one that had only even digits, which is the method that I think most students will try. This method feels a little risky because, if I were just counting up by 9 to the next multiple rather than multiplying each time, it would be easy to make a mistake somewhere. Even if I were go through and multiply 9 by several numbers, it’s likely that I would miss a number at some point.

Another way to solve this involves knowing the divisibility test for 9. If the sum of the digits in a number add up to a multiple of 9, then the number itself is divisible by 9. The sum of the digits in this problem couldn’t be 9, though, because the sum of even numbers can never be odd. The smallest multiple of 9 with only even digits must be the smallest combination of three even numbers that add up to 18. It would have to be 288.

Also, when the multiples of 9 are organized into a table, an interesting pattern emerges. By looking at the digit sums and the changes to the ones and tens digit, we see some interesting things.

I don’t expect to see student use a table, at least initially, but if a student is having trouble organizing their work, I will definitely suggest it.

My Goal for Student Learning

This problem is intended for a class of new students with low math levels, many of whom struggle with multiplication, and it is going to take a while for most of them to finish. My goal is for them to stick with the problem and not get discouraged as the numbers start getting bigger and bigger. I am giving this problem during the first week of class, and my sense is that the students aren’t used to struggling with math problems for long periods of time. Another goal is for students to come up with an organized approach to tackling this problem. That is, I would like to see some students create tables or lists rather than simply start multiplying 9 by randomly chosen numbers. Also, because it’s so early in the cycle, I would like to see my students feel comfortable talking about their work and the work of their peers.

Supporting Productive Struggle

The first challenge I anticipate involves the vocabulary and phrasing of the problem. Because I will be working on this problem with a group of new students, they have only recently been introduced to multiples and factors. They will likely need a quick refresher. Similarly, I expect to see students struggle with the idea of “even digits,” and we may have to talk about it as a group to make sure that everyone is on the same page before we get started.

I also expect students to have a hard time organizing their work, and I expect to see some mistakes with basic computation as the multiples get higher and higher. This will require me to intervene somewhat to help students spot their mistakes—either with adding or multiplying, depending on their approach. I think that some students will want to give up after they’ve found the first fifteen or twenty multiples of 9. They might think that it’s a trick question and that there actually aren’t any multiples of 9 that have only even digits.

To support students who are struggling with this problem, I will help them identify mathematical mistakes so that they can correct them as they go along. I won’t tell them that they’ve made a mistake though; instead, I’ll ask them to talk about how they got from one number to the next so that they can see the mistake for themselves. I think that some students will notice the pattern in multiples of 9 (increasing tens digit, decreasing ones digit), and so I will help them to articulate it and apply it to the work that they are doing. For those students who work all the way through it and don’t see the pattern, I will ask them to look over their work and talk to me about the changes they see to the digits. I expect that some students will try to guess-and-check their way through the problem, which could potentially make it take a very long time. I will talk to these students about ways they might be able to organize their guesses so that they don’t lose track of the work they’ve already done.

Extension Questions

If some students finish early, I would ask them to find the next smallest multiple of 9 with only even digits, and then the next one, and the next one, and so on. It might seem a little tedious at first, but if I support it well, I can help students to understand how the divisibility test for 9 works. This is something that I don’t think many, if any, students will know.

All of the multiples of 9 that are less than 1000 and have only even digits are: 468, 486, 648, 666, 684, 828, and 882. Even if a student only got to 468 and 486, I could start having the conversation with them about how any number with a digit sum equal to a multiple of 9 must itself be a multiple of 9. Since no combination of even numbers can sum to 9, they must have to sum to 18. From there, students can work on finding the other possibilities.

Student Work

Fidel’s Approach

Fidel is one of the strongest students in this group. He attends every class session, asks good questions, and works hard on every problem that he encounters. Even this early in the cycle, Fidel’s classmates have come to recognize him as one of the leaders in the class, and they often rely on him to help them out when they are struggling. However, Fidel had a really hard time with this problem.

First off, he needed a reminder on the difference between odd and even numbers, and after we talked about it as a group, he wrote them down just to be sure. Then he started working. If you look closely at Fidel’s work, you’ll see that he started out by writing all of the multiples of 9, but then he erased them. When I asked why, he explained that when he got above 100, he noticed that all of the multiples would have a 1 in them and therefore couldn’t be correct. This is where he gave up on the list and decided to try guessing and checking. His guesses look a little disorganized, but there is a method to them. He was trying to locate multiples of 9 that were in the 200s. His first guesses were much too big, but he kept making adjustments. He erased most of these, but he left a few and, after a while he found 9 × 32 = 288.

What was interesting about Fidel’s work is how he noticed some important qualities about the numbers—namely, that the correct answer would have to start with a 2, 4, 6, or 8—but he didn’t come up with a good way of organizing the work that he was doing. Because he guessed and checked, several students finished the problem before him and began working on the extension questions. This was a case where the strongest student in the class struggled the most because the problem-solving strategy he chose may not have been the most appropriate one.

Jean Marie’s Approach

Of all the students in the class, Jean Marie probably has most difficulty with math. She performs all basic calculations on her fingers, and she has very little confidence in her ability to grow as a math student. This was the first extended problem that she had done on her own.

From the outset, Jean Marie was frustrated by this problem because she noticed that it had to do with times tables, and she reminded me several times that she doesn’t know her nines. You’ll even see at the top of the page that she was drawing circles for the first couple multiples of 9. While everyone else was working on their own, I spent a lot of time sitting with Jean Marie and talking her through the problem. She started with 9 × 1 = 9 but then couldn’t remember 9 × 2. So we talked about how she would figure it out. She seemed a little embarrassed when telling me that she would count on her fingers. But when I told her that her method was fine, she went back to work. She counted up to 18, and then counted up another 9 to 27, and so on. From here, she was able to work on her own, but she tried to give up about every five minutes. It took a lot of encouragement to get Jean Marie through this problem, and she made a lot of mistakes. I made the decision to help her identify her mistakes so that she wouldn’t get more frustrated as she got further and realized she had been working with incorrect numbers.

In the end, with a lot of support, Jean Marie did arrive at the correct answer. I liked how well-organized her method was, and I really appreciated her ability to stick with a problem that was so challenging and frustrating to her. She even finished before Fidel did! And it was a really important moment for her. She wrestled with a problem that she thought she could never do, and she was successful.

Feliciano’s Approach

This was the day when I learned that Feliciano is incredibly good with numbers and loves doing math. He did this problem on his second day in class, and since he was the first to finish, I got to talk to him about some of the extension questions that I was hoping to use.

Feliciano started out by listing the multiples that he knew off the top of his head, and then he worked additively from there. This approach was largely typical of what most students did. Each time he arrived at a new multiple of 9, he added 9, wrote the next one down, and repeated. By following this pattern, Feliciano got to 288 pretty quickly, so I asked him to find the next multiple of 9. He kept working additively for a while before figuring out that 9 × 52 was equal to 468. Here, Feliciano stopped and looked a little more closely at the relationship between 288 and 468. In the middle of the page, he adds 2 to the hundreds digit in 288 and subtracts two from the tens digit, which gives him 468. He repeats the process again to get 648.

At this point, Feliciano and I talked about why this worked. He couldn’t articulate the divisibility test for 9, but he was working with it intuitively when he found 468 and 648. After we talked about how the digits needed to add up to 18, he was able to find all of the other combinations, which are scattered around the page. I’m glad I got the chance to see how this problem worked with a student who already had such solid number sense. Feliciano was very engaged with the problem, and he enjoyed getting to learn and talk about the divisibility test.

Final Thoughts

I really liked the way this problem played out in class. For most of the students, this was only the second problem-solving activity that they had done. Because they were new to struggling with math problems, I hoped that working on this one would encourage persistence and help them to come up with strategies for organization. For the most part, we met those goals. We also took the time to talk about the patterns that appear in multiples of 9, as well as the divisibility test, which is show in the board work below. Through working on this problem and its extensions, I learned that with enough preparation, there are interesting questions that can be asked about any mathematical idea—even one as basic as multiples.

 

What I Might Change

I wouldn’t change much about how I did this problem. If I do it again early in the cycle, though, I might try reviewing the different problem-solving strategies that we had discussed before doing this problem. That way, students would have to make a more conscious choice between using a table/chart and trying to guess and check. Unfortunately, this time, a handful of students spun their wheels guessing and checking when they could have used a more effective method. Still, I think they benefited somewhat from doing it the “wrong way” before moving on to a better way.

I might also give out hundred charts to students who really struggle with their times tables. It could help them to get started, and it could also help them to identify a pattern that will help them remember their nines in the future. And lastly, if I do this problem early in a class cycle again, I might ask students to write a reflection of what it was like working on the problem.

Unexpected Challenges

I gave this problem again in another class—one with a wider range of math levels—and found that it was a little difficult to manage all of the students. Some students finished the problem quickly, while others needed me to sit with them and keep them working, give them feedback on their work, etc. This made it challenging to keep the higher-level students engaged while still supporting the students who needed individual attention.

Student Takeaways

My students liked this problem, and it fit in well with the work on factors and multiples that we were doing in class earlier in the week. They enjoyed trying out and discussing some of the problem-solving strategies that we had been working on as a class. They also got to hear about different solution methods from their peers, and they had the opportunity to share their frustrations with the problem, as well as the sequence of steps they took to break through that frustration. For one student in particular—Jean Marie—this problem was a major breakthrough. For the first time in class, she stuck with something, got angry at it, settled back down, tried again, failed, tried again, and finally succeeded. She hasn’t given up on a problem since. This is a great exercise to do with students who need to learn how to stick with something. It has a very low entry point, but the discussion can go a lot of different ways.

My students were also able to see the importance of pattern recognition in math. Recognizing the patterns for multiples of 9 helped several students write out all of the multiples quickly, rather than adding repeatedly. After we finished this activity, “Look for a pattern” was added to our list of problem-solving strategies, and it has since helped students succeed in other difficult problems.

Advice for Teachers

This is a good low-entry problem for students who are new to your class, but it could be used at any point in the cycle as a warmup exercise. Teachers should be prepared for students to get frustrated and give up, but they should also be prepared with extra questions for students who breeze through the exercise. The problem works best if you allow plenty of time for the class as a whole to debrief, especially because students need to see that there is a bigger takeaway from doing the problem than just crunching numbers. And there’s a lot of rich territory on which to have that discussion. Talk about organizing information, talk about patterns, talk about divisibility tests, and emphasize key vocabulary. Help your students understand that their struggle was worthwhile.

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