# Once Upon a Time

In the problem *Once Upon a Time*, students use measurement, number properties, and circular geometry to solve problems involving time and angles. The mathematical topics that underlie this problem are time measurement, including conversion between years, months, weeks, days, hours, minutes, and seconds; modular arithmetic that involves divisibility and remainders; pattern recognition; as well as circles and angular measurement. In each level, students must make sense of the problem and persevere in solving it (**MP.1**). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.

PRE-K

In this task, students will examine the parts of a clock. They are then presented with the task of determining how many minutes will pass before the large hand catches up to the small hand on a clock. Their task involves counting up and understanding how a clock measures time.

LEVEL A

In this level, students are presented with the task of determining how many minutes will pass before the large hand catches up to the small hand on a clock. Their task involves counting up and understanding how a clock measures time.

This level supports Common Core standard **3.MD.A.1** by having students measure time intervals in minutes and solve problems involving addition and subtraction of time intervals.

LEVEL B

In this level, the students are asked to convert their age from years into seasons, months, and weeks. Students are also asked to determine what day number the current date is in the year.

This level supports Common Core standard **4.OA.A.3** by having students solve problems using the four operations. Students will also need to be able to express units of time (years) in terms of a smaller unit (days, weeks, months).

LEVEL C

In this level, students are given a problem that requires an understanding of divisibility and may be determined by using knowledge of relatively prime factors.

This level extends the work of Common Core standards **6.NS.B.2 **and** 6.NS.B.4**. Students apply understanding of divisibility, common factors, and multiples to solve the problem.

LEVEL D

In this level, students are presented with a problem that involves three different-sized alarm clocks that ring at varied intervals. The task is to determine if or when the three clocks chime simultaneously.

This level supports Common Core standard **8.EE.C.8b** as students can write equations and solve systems of linear equations in two variables algebraically to determine if or when three different-sized alarm clocks chime simultaneously.

LEVEL E

In this level, students are asked to determine the times in a day that the minute and hour hands of a clock form an angle of 48 degrees.

This level supports Common Core standard **A-CED.A.1 **as students can write equations to describe the motion of the hands of a clock. They use these equations and their knowledge of the central angle of a circle between two radii to find when the hands are 48 degrees apart. Common Core standard **G-C.A.2** is supported as students make sense of the clock and the motion of its hands through angles in a circle. The problem is complicated by the fact that time is measured in intervals of 60. As they persist in analyzing the problem, they will notice patterns and regularity (**MP.8**) in the times for each hour of the day.

PROBLEM OF THE MONTH

Download the complete packet of *Once Upon a Time *Levels A-E here.

You can learn more about how to implement these problems in a school-wide Problem of the Month initiative in “Jumpstarting a Schoolwide Culture of Mathematical Thinking: Problems of the Month,” a practitioner’s guide. Download the guide as iBook with embedded videos or Download as PDF without embedded videos.

SOLUTIONS

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