Trina was playing with her new puppy last night. She began to think about what she had read in a book about dogs. It said that for every year a dog lives it actually is the same as 7 human years. She looked at her 4 1/2 month old puppy and wondered how many human years old her puppy was? Using as much math language and good reasoning as you can, figure out how many human years old Trina's puppy is?
Trina, one of my sixth grade students, called me one night all excited because she knew she had a math "dilemma" for the class. I was pleased that she recognized that a fair amount of mathematics was necessary to solve this problem and that it would be challenging for the class to solve. I actually was surprised at how difficult the problem was for my students. We were working on adding and subtracting fractions, so this problem dealing with fractions was quite timely. My students had enough mathematics to solve the problem, but used many different strategies because it was a problem that they had never encountered before.
At first, the task reads fairly simple. However, many students could not think of a way to get started, so this task makes students persevere. It also makes them try different approaches as they begin sorting out the problem. It makes many students begin converting fractions to decimals, so they can work with a calculator. Some students wanted to figure out the equivalent in dog age of one human day. They knew that months had different days, but they figured they would be very close. Some that got really frustrated rounded 4 1/2 months to six months and found 3 1/2 years old in dog age. This was reasonable, but I encouraged them to try to get a more accurate answer. Because we do a lot of problem solving with charts, some kids made a chart and found that very successful for this problem, although it might not have been as successful with other fractional parts of a year. 60 minutes It took 45 minutes to solve the problem and another 15 minutes to pull their answers together and report. This problem can lead very nicely to a discussion of different life spans and conjectures about why some animals live longer than others. I let my students work in pairs to solve this problem. You might ask your students to think about how graphing could lead them to a solution.
(4.5 months/12) x 7 years = 31.5/12 years = 2.625 years = 2 5/8 years
This student has set up the facts that s/he knows about the calendar and is trying different algorithms to solve the problem with little or no reasoning. There is no evidence of a strategy and no explanation of the reasons for the algorithms tried. There is no mathematical representation.
This student uses a strategy that would work, but uses faulty reasoning in changing a decimal fraction of a year to months. S/he correctly divides to find the dog age equivalent to one human month. However, s/he incorrectly assumes that .6 of a year is six months. The rest of the solution is based on that faulty reasoning.
This student's strategy shows s/he has an understanding of the problem and the major concepts necessary for a solution. Their chart shows equivalent dog years for fractions of human years. This strategy of taking half of each human year leads to a solution of this problem (it may not have gotten a solution to other age puppies). There is effective mathematical reasoning. The student sees that halving the human age would also halve the dog years. The explanation is clear and the chart is appropriate use of mathematical representation. The student also uses correct mathematical notation.
This student shows a deep understanding of the problem including the ability to identify the appropriate mathematical concepts. This student realizes that no matter how old the dog is, you need to multiply the age by seven. S/he realizes that the age of the dog is 4.5/12 of a year. Since s/he is unfamiliar with multiplying fractions, s/he used his/her knowledge of the fraction line as division and found the decimal equivalent of 4.5/12. This is a very efficient and sophisticated strategy that employs refined and complex reasoning. There is a clear and effective explanation and the student reached for a generalization that would solve any month old dog. The graph also actively communicates how to estimate the dog age of any living dog.
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