# GPS: Hopping into Fractions

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• 1 Wayland Middle School in Wayland, Massachusetts
• 2 Southern Illinois University, Carbondale
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This is the final GPS document for February 2020

The PK–2 grade band task may be more easily presented by acting out the problem situation for the class. Teachers of younger students may wish to start with the bowl of carrots before Jai eats half of the them and emphasize the connection of the concept of 1/2 to the idea of equal sharing. Second-grade teachers may wish to open the scene when Mama inspects the bowl after Jai's snack. Changing the number of carrots left in the bowl and repeating the scenario several times may lead to student observations about doubling or multiplying by 2.

The 3–5 grade band task extends the problem so that two family members are each eating a fraction of the carrot sticks. One type of diagram students might use to solve this problem includes a rectangle representing the full bowl of carrots at the beginning of the scenario. Students then divide the rectangle to show the carrots eaten by each family member, similar to the diagram shown in figure 1.

Although repeating the problem with different numbers of remaining carrot sticks suggests numerically that the original number of sticks is 6 times the number left in the bowl, this type of diagram allows students to more directly visualize the 1/6 relationship.

The task for the 6–8 grade band lends itself to a folded paper model. Teachers can give each student a long strip of paper. Jai takes 1/10 of the carrots, so students can fold their strips into 10 equal sections. Kai takes 1/3 of the remaining carrots, and 9 parts are left, so she takes 3 of those pieces as shown in figure 2.

Because Mama takes 1/4 of what's left, folding the 6 remaining parts in half is advisable, making 12 smaller parts, 3 of which go to Mama, labeled with M's below. Pa takes 2/3 of the remaining 9 parts because 2/3 of 9 is 6. Therefore, the leftover 12 carrots together fill the last 3 boxes.

For the 9–12 grade band, the task has students experiment with different combinations of fractions that result in sensible new versions of this problem. Some of these older students might try to use traditional algorithms to add their chosen fractions without realizing that this is inappropriate because the fractions do not represent fractions of a same-size collection of carrot sticks. This presents an opportunity to address misconceptions about fractions. High school teachers may wish to begin with students solving the middle school task as an example before moving on to its generalization in this grade band.

The Hare family always keeps a bowl of carrot sticks on the table for family members to snack on between meals. Jai Hare was on the way to trumpet lessons and took 1/2 of the carrot sticks as he walked by. When Mama Hare went into the kitchen, she saw 5 carrots were left.

Act out this problem or draw a picture to figure out how many carrots Jai ate and how many carrots were in the bowl before Jai's snack.

The next day, after Mama adds more carrot sticks to the bowl, Jai is on the way to soccer practice.

He again eats 1/2 of the carrot sticks. This time, 7 carrot sticks remain. How many did Jai eat? How many carrot sticks were in the bowl when he walked into the room? How many had Mama added to the bowl?

The Hare family always keeps a bowl of carrot sticks on the table for family members to snack on between meals. Jai Hare was on the way to trumpet lessons and took 1/2 of the carrot sticks as he walked by. His sister, Kai Hare, grabbed 2/3 of what was left in the bowl on her way to basketball practice. When Mama Hare went into the kitchen, she counted 9 carrots left in the bowl.

How many carrots were eaten by each member of the Hare family? How many carrots were there to begin with?

Mama Hare looks at the remaining carrots again and realizes that she miscounted. If there were really 11 carrots left in the bowl, how many did each family member eat? How many carrots were there before today's snacks? The Hare family always keeps a bowl of carrot sticks on the table for family members to snack on between meals. Jai Hare was on the way to trumpet lessons and took 1/10 of the carrot sticks as he walked by. His sister, Kai Hare, grabbed 1/3 of the remaining carrots on her way to basketball practice. Mama Hare snacked on 1/4 of what Kai left in the bowl. When Pa Hare arrived, he ate 2/3 of what was left by Mama and then began making supper. When the Hare family sat down to eat supper, there were 12 carrot sticks left. How many carrot sticks did each family member eat?

Suppose that the next day, the same sequence of events transpired, except that Pa ate 7/9 of the remaining carrots when it was his turn. There were still 12 left over carrot sticks at dinner time. In this scenario, how many carrot sticks did each family member eat? What other fractions of the remaining carrots could Pa have eaten that lead to solutions to the problem in which every family member ate a whole number of carrot sticks?

The Hare family always keeps a bowl of carrot sticks on the table for family members to snack on between meals. Jai Hare was on the way to trumpet lessons and took 1/x of the carrot sticks as he walked by. His sister, Kai Hare, grabbed 1/y of the remaining carrots on her way to basketball practice. Mama Hare snacked on 1/z of what Kai left in the bowl. When Pa Hare arrived to begin making supper, 12 carrot sticks were left.

Let c be the number of carrot sticks in the bowl just before Jai entered the kitchen. Find several ordered quadruples (x, y, z, c) that represent solutions to this problem situation in which each family number ate a whole number of carrot sticks. Compare and discuss your solutions with your classmates. What patterns can be found among the ordered quadruples you found?

Kelly Hagan, kelly_hagan@wayland.k12.ma.us, teaches sixth grade at Wayland Middle School in Wayland, Massachusetts. She loves exploring a variety of ways to teach a new concept to students.

Cheng-Yao Lin, cylin@siu.edu, is a full professor of mathematics education at Southern Illinois University in Carbondale. His research interests include the teaching and learning of mathematics, preservice and in-service professional development for mathematics teachers, problem solving, and the impact of technology on mathematics content and pedagogy.

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## Footnotes

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. In this month's Growing Problem Solvers, we focused on students' understanding of fractions, specifically fractions of a group of objects. Fractions can be challenging for students of all ages, and these tasks, all contextualized problems involving a family of rabbits, lend themselves to students creating diagrams to model the problem.

Mathematics Teacher: Learning and Teaching PK-12

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