Everyday Mary and Laura walk to Mrs. Peterson's house to visit her. She always gives them each a Swedish cookie. One day when Mary and Laura walked home, they decided they would share their cookies with Baby Carrie who was at home and too little to join them. Laura thought she and Mary should each eat 1/2 of their cookies and give their other cookie halves to Baby Carrie. Mary argued that would not be fair. They need your advice. How should the girls share their cookies with Baby Carrie so that each sister gets her fair share?
This problem was built into a unit that focused on number sense, specifically fractions. The students had been introduced to fractions through the creation of fraction bars and then were engaged in numerous exploratory activities around fractions. This created the background knowledge that the students brought to this task. This task was challenging for most of my third and fourth grade students. It allowed students to use objects to determine how to divide a whole. Of course, the object being shared was relevant to their own lives - cookies! Specifically, the task gave me an opportunity to see who had a good level of understanding of fractions and how to draw fractions to show parts of a whole. The students who did this problem all tackled it a little bit differently. Some used fraction pies to manipulate the pieces. Others began by sketching and then working to divide their two cookies on paper. The students who were most successful did a combination of the two. They began with the manipulatives and then worked them into drawings in their own work. 60 minutes The task would obviously be timely if students were reading Little House on the Prairie, but could also be incorporated into a unit on Sweden or holidays around the world. For students with special needs, the numbers in the task could be easily adapted, making the task more into a division problem. For more gifted students, again, the numbers could be changed to make manipulating the denominators more challenging. Fraction pieces (preferably fraction pies) Two cookies split among three people equals 2/3 each because 2/3 + 2/3 + 2/3 = 6/3 = 2 whole cookies. This solution best maintains the mathematical integrity of the task.
Novices will understand enough to draw two cookies, but will be unable to find a solution that divides them equally. The student's communication will be lacking so that we will be unable to follow the student's thought process. Little or no mathematical language will be used to communicate and the student will avoid the use of fractional notation.
Apprentices will divide the cookies, addressing fairness, but not in a way that preserves the mathematical integrity of the task. The Apprentice may use fractional notation and/or other mathematical language and will have diagrams that communicate an approach.
Practitioners will arrive at a solution that preserves the mathematical integrity of the task, finding a solution that truly divides the two cookies evenly among three people. The Practitioner will use language of fractions and other math language to communicate and diagrams will assist in communicating the solution.
Experts will arrive at a correct solution and will go beyond the requirements of the task either by recreating the task and solving it again or by making mathematically relevant observations. The Expert will use precise math language and will have well labeled diagrams to communicate a solution.
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