What Are Good Math Questions to Ask Students?

To begin Good Questions for Math Teaching, K–6,  Peter Sullivan and Pat Lilburn list three features of the good questions they’ve included in the book. These features sing to me.

  • Good questions require more than remembering a fact or reproducing a skill.
  • It’s possible for students to learn by answering the question.
  • There may be several acceptable answers.

Then they provide more than 300 questions organized into 16 topics (money, fractions, decimals, place value, counting and ordering, operations, weight, volume, area, time, length and perimeter, locations and position, two-dimensional shapes, three-dimensional shapes, chance, and data) with questions for each topic organized into Grades K–2, 3–4, and 5–6 (except for decimals where there are questions only for grades 3–4 and 5–6).

I looked at the questions they posed for operations, the cornerstone of the K–6 math curriculum. Here’s a sampling of the 23 questions they included for that topic, with rationales and tips for using each.

Grades K–2
1. A basketball player scored 9 points in two games. What could her scores in each of the games be?
This question helps children think about a problem for which there’s a range of possible answers. For young children, decomposing numbers within 10 is an important reasoning strategy. There are ten possible answers, for whether the players scored 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 points in Game 1. Changing the number of points is a way to tailor the question for individual needs.

2. I put some counters into groups with the same number in each group. I can’t remember what I did, but I do remember that I had 12 counters. What might the groups have been?
It’s important to supply children with counters to work on this question. There are four possible answers―two groups of 6, three groups of 4, four groups of 3, and six groups of 2. It’s fine if children don’t report all four possibilities, but for an extra challenge, you might ask a child to find all of the possible answers.

Grades 3–4
1. Five numbers added together make an odd number. What do you know about the numbers?
This question highlights some features of odd and even numbers. I love a question like this that both provides the opportunity to do a good deal of calculating, with numbers chosen by the child, and also asks children to make a generalization. In order for the sum to be odd, either one, three or five of the numbers must be odd. Would that also be true if the question were to explore six numbers that add to an odd sum?

2. What might the missing numbers be?

Again, this is a question that shows children that problems often have more than one answer. There are a variety of possibilities for choosing the missing numbers in the ones place for the addends, as long as they then determine a sum that has a 2 in the ones place. For a challenge, ask children to find all possibilities, or perhaps three or five possible solutions. If this problem is appealing, you might try a game version I described in a previous blog post: Four Strikes and You’re Out. I’ve found that’s always a favorite.

Grades 5–6
1. A school has 400 students. They all come to school by bus, and each bus carries the same number of students. How many students might there be on each bus?
This question doesn’t tell how many buses there are. That’s an open issue. When I watch children work on this, I’m interested in whether they use multiplication or division. My goal is for them to be able to use both and understand the inverse relationship between multiplication and division. For an easier version of this question: The answer to a division question is 5. What might the question be?

2. (a) How could you calculate 23 x 4 on your calculator if the 4 button is broken? (b) How could you calculate 23 x 21 if the 2 button on your calculator is broken?
The (a) question helps reinforce the relationship between multiplication and addition and most students press 23 + 23 + 23 + 23 = to get the answer of 92.  But that’s not the only way. I’ve found that children who give that solution don’t typically think there are other ways, for example pressing 23 x 2 + 23 x 2 =. After showing this alternative, they may be able to find other solutions. The (b) question is typically more challenging.

At the beginning of their book, the authors give tips for making up what they describe as good questions and also describe how to use good questions in the classroom. These will be useful when you’re back in school with your students.

And for middle school students, there’s a companion book written by Lainie Schuster and Nancy Canavan Anderson.



18 thoughts on “What Are Good Math Questions to Ask Students?

  1. I know this definitely opened my eyes open to questions that I can use in my classroom to allow students to have more inquiry learning. One important thing that you talked about were counters, which are the manipulatives, students need movement rather big or little.

  2. These problems are great!! I am going to find a way to incorporate them. Could you possibly create problems I can use with my gifted learners? They are 5 th graders, but their instructional level is now the beginning of grade 7. Thanks

    • Truth is that I didn’t create these, the authors of the Good Questions books did. The book for Grades 5-8 looks perfect for your need, organized into Number, Proportional Reasoning, Fractions/Decimals/Percents, Geometry, Albegraic Thinking, Data/Probability, Measurement. Pick a topic and I’ll send a sampler.

  3. Love these books and have used them in PD with teachers. Sometimes in coaching situations we take the questions from observations or teacher book to rewrite and make better questions following these guidelines! Such valuable resources from Math Solutions!

  4. Haven’t thought about these books in awhile. The questions always provoked amazing conversations when we included them in our work with kiddos and their teachers. So grateful for the reminder. Will be sending your post on to lots of math friends. Thank you.

    • Revisiting these was a reminder of how much is on my professional book shelf that I’d benefit from. Thanks for your comment.

    • My first teaching job was 9th graders (five classes of Algebra that left me sort of blithering by the end of the day). But I’ve been focusing on younger students. What about Desmos???? I love their Marble Slides, and more.

  5. Hello!

    I just ordered the middle school version. Could you send a sampler to me from the fraction section? Working on creating some online lessons for dividing fractions. I also left your book, my bible, at school and don’t have access 🙁

    Thank you Marilyn!

  6. It is very good blog with full information, we are looking forward to connect with you in future. Best home tutors are provided by TheTuitionTeacher.com in Delhi and Lucknow.

  7. Even though as educators we are aware of the type of questions that kindle both curiosity and the deeper understanding of the subject, we all need reiteration. I loved reading this blog and am sharing this with all my co teachers.
    Thank you Marilyn!
    Is it possible to get a sample of this book?

    • Ah, such a good question. It made me wonder about how you were thinking about “larger.” More surface area on the screen, for sure. But when I recently got a larger TV, my focus was on the maximum size I could fit into the cabinet space. Also, I wondered if the proportions of all TV screens are the same. And then my mind shifted to enlarging and reducing when I make copies. What does a 25% enlargement really do to the image? (Students could actually tinker with that on our printers.) So many more questions. Finally, I got back to work, but I’m still thinking. Thanks for the comment.

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