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**Thought for the day: **Some C*reative Math Prompts* are problems in disguise (without directions)! Many students predict the "hidden" question. However, the door is always open to new interpretations!

**Concepts**: patterns that connect multiplication and division; creating new multiplications and division strategies; connection between division and fractions; organizing data in order to find patterns and keep track of solutions

**Examples of noticing and wondering**

*I notice *that all four numbers in the picture are even.*I notice *that all three rows are the same.*I notice *64 is divisible by 8, so that might be good to try first.*I notice *that there are four circles between the 64 and the 2.*I notice *that multiplying by 4 and then dividing by 8 always has the same answer as dividing by 2, no matter what number I start with.

*I wonder *why there are three copies of the same thing.*I wonder *if it is possible to turn 64 into 2 using only x 4 and ÷ 8.*I wonder *how many ways I can turn 64 into 2 using only x 4 and ÷ 8.*I wonder *if it is okay to put fractions in the circles.*I wonder *if I can do it backwards (turn 2 into 64).

**Notes**

This Creative Math Prompt is part of an extended activity called "Hopping Home" on the Deep Math Projects page of this website.

Some teachers and students may notice that the prompt reminds them of a format commonly known as "frames and arrows." The details are different, but students' task is to turn the left number (64) into the right number (2) using only the two operations. The goal is to do it in five steps, inserting the temporary answer in the circle each step of the way.

Most students will divide by 8 first (64 ÷ 8) in order to work with a smaller number. In fact, although there are 10 solutions to the problem, most students will produce the same two or three answers. For some students, this may be enough. For additional challenges, they may make more copies of the problem and search for as many solutions as they can find. Some of the solutions involve three-digit numbers in the thinking process, and others involve fractions.

Don't worry if students have not been taught how to handle these kinds of calculations. Let them use their number sense to develop their own strategies as they are able. For example, 1 ÷ 8 is different than 8 ÷ 1. 1 ÷ 8 involves dividing 1 into 8 equal parts. How large is each part? (1/8) There is no need for you to teach processes for dividing fractions in order for them to discover this answer! They simply need to think carefully about what division means.

This prompt is easy to extend by asking new questions. Use more or fewer circles. Use numbers other than than 4 and 8. (There may often be no solution. Figuring out why is a great challenge!) Choose different starting and ending numbers. Think about how you can know when you have found all possible solutions!

Keep in mind that, while I meant for the prompt to suggest a particular problem, students may have other ideas. Follow up on their responses! When a student suggests a certain task, it is likely to be (1) something they will be interested in, and (2) something that they are prepared to learn from.

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