Rich Math Problems – Part 2
by Karen Rothschild
Strategies for Creating Rich Math Problems
The strategies below^{*} are ways to take routine tasks—like those often found in textbooks—and adapt them so that students have more opportunities to think deeply about the topics they are learning.
 Provide the answer. Ask for the question.

 Example 1: Instead of, “Round 5.94 to the nearest tenth,” try, “A number was rounded to 6. What could be the number?”
 Example 2: Instead of, “Find the sum of 3,6, and 8,” try, “The sum of three numbers is 17. What could be the numbers?”
 Example 3: Instead of, “What is the perimeter of a pool whose length is 50 yards and width is 25 yards?” try, “The distance around the rectangular swimming pool at the park is 150 yards. How long and how wide could the pool be?”
 Ask students to choose the numbers in a task.

 Example 4: Instead of a problem with all but one of the quantities given, try multiple quantities unknown, such as: “Taylor walked ☐ blocks to school. After school she walked ☐ blocks to the store, and then ☐ blocks to get home. She walked a total of ☐ blocks.”
 Example 5: Use any digits between 09 in the boxes to make a correct equation. You may only use a digit once in the equation ☐☐ + ☐ = ☐☐ – ☐ What is the smallest value that could be on each side of the equation? What is the largest value?
 Ask for similarities or differences.

 Example 6: Instead of, “Name 3 numbers that are multiples of 5, and three that are not,” try, “How are 5 and 100 alike? How are they different? Find as many ways as you can.”
 Example 7: Instead of, “Describe the characteristics of right triangles,” try, “How are these shapes alike? ◤▲How are they different?”
 Ask for contexts for numerical expressions.

 Example 8: Instead of “Find the number of plants in a garden with 6 rows of 4 plants each,” try, “Create a realworld question where you might have to find 6×4 to answer the question. Find the product and answer your question.”
 Ask for a mathematical sentence that includes certain numbers and words.

 Example 9: Create a sentence that includes the numbers 3 and 4 along with the words “more” and “and.” You’ll have to use some other words too.
 Provide a real world situation that requires mathematics. Provide areas of ambiguity so students can make choices.

 Example 10: Instead of, “Use the menu to find the cost of 2 hot dogs and a soda,” try, “You and 3 friends have $50 to spend on lunch. Use the menu to decide what to buy and how much it will cost.”
What Makes These Problems “Rich?”
All of the problem types above are rich problems because they provide:
 Opportunities to engage the problem solver in nonroutine ways of thinking about mathematics,
 An opportunity for productive struggle, and
 An opportunity for students to communicate their thinking about mathematical ideas.
Other characteristics of these problems include:
 Several correct answers.
 A “low floor and high ceiling,” meaning that all students can get started and all students can reach a point of struggle.
 An opportunity to practice routine skills in the service of engaging with a complex problem.
 An opportunity for formative assessment. Choices students make when working on these problems can give teachers information about students’ developmental levels, their neurodevelopmental strengths and challenges, and their learning preferences.
 A level of complexity that may require an extended amount of time to solve. Many of these examples can become longer investigations by asking students to find all possible solutions, or to find the greatest or least possible solution.
 An opportunity to look for patterns. Depending on how many solutions students find, several of the examples offer opportunities to look for patterns.
 An opportunity for students to choose from a range of tools and strategies to solve the problem based on their own neurodevelopmental strengths.
 An opportunity to discover a new (for the student) mathematical idea through working on the problem.
 An opportunity for students to engage their everyday knowledge of the real world
(Examples 3, 4, 8, and 10)
Teaching Math with Rich Problems
While an appropriate problem is crucial to immersing students in deep thinking about mathematics, student engagement in conceptual learning is also heavily influenced by pedagogical practices. In order to facilitate students’ grappling with ideas and mathematical reasoning, tasks should focus on learning goals and be constructed and presented in ways that, as much as possible, mitigate or eliminate challenges related to memory, language (including reading and writing), psychosocial factors, etc. when they are not essential to the goals of the tasks. When specific challenges are essential to mathematical goals, supports should be provided for those who need them. Problems should be presented in an accessible manner and students should be encouraged to represent their thinking in a variety of ways. A mathematical idea or problem solution can often be represented in a sophisticated and powerful way using visual representations and/or physical models.
Rich problems, in contrast to mere applications of skills, often require trying things out, and making mistakes. Students must feel comfortable taking risks and using mistakes to learn. Often, they must be explicitly told that mistakes are both “expected and respected.” Furthermore, learning from mistakes takes time, and teachers need to provide the time for attempts, failures, and reattempts. Perseverance comes from this iterative process. Teachers can facilitate deep thinking by suggesting avenues to explore, but not providing the kind of information that leads directly to an answer or use of only a single strategy. Teachers can also model making mistakes and learning from them.
Teachers can also promote communication skills by recognizing students as the authority on their own work. Students who learn to convince themselves and others of the validity of their work understand mathematics more deeply, develop strong communication skills, and become more confident in their capacity to learn. Some useful questions are: ”What do you see?” “What does that remind you of?” “What are you thinking about?” “Will that work?” “What can’t work?” “Why does that make sense?” “Can you convince me?” Teachers can also anticipate common misconceptions students make in particular situations and be prepared with topicspecific questions.
In sum, rich problems offer ALL students an opportunity to engage deeply in the mathematical ideas they are learning, to become fluent users of mathematics in a variety of settings, and to communicate their thinking. Success in using rich problems in the classroom depends on choosing appropriate problems, presenting them in ways that are accessible to students with a variety of neurodevelopmental strengths and challenges, and creating a classroom culture that encourages students to explore ideas and possibilities, make mistakes, and learn from each other through sharing their work.
References
The list of strategies and many of the examples are derived from:
Small, M. (2020). Good questions: Great ways to differentiate mathematics instruction. 4th edition. New York: Teachers College Press.
Open Middle (n.d.). www.openmiddle.com
Teach, Learn, Share (n.d.).
www.cristinamilos.education/2014/09/13/openendedtasksandquestionsinmathematics
Math for All is a professional development program that brings general and special education teachers together to enhance their skills in
planning and adapting mathematics lessons to ensure that all students achieve highquality learning outcomes in mathematics.