# Ages and Timelines: Subtraction on the Open Number Line (Contexts for Learning Mathematics)

## Catherine Twomey Fosnot

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Ages and Timelines: Subtraction on the Open Number Line is one of eight units in the Contexts for Learning Mathematics' Investigating Number Sense, Addition, and Subtraction (K - 3)    This unit begins with the story of Carlos, an eight-year-old boy who is fascinated by his great-grandfather's thick, beautiful silver hair. His great-grandfather lives in Puerto Rico and Carlos is preparing to meet him for the first time. Having only seen photos of him as a much younger man, Carlos wonders how old his great-grandfather is and how many years it will take before he might have hair like that, too. As Carlos begins to investigate these questions, his whole family becomes involved in exploring age differences and figuring out how old they each were when Carlos was born. When Carlos shares his investigation with his teacher, the whole school gets involved in the project.   This story context sets the stage for a series of investigations in this unit. Children interview their family members and compare age differences. Timelines are introduced as a context for using the open number line - a helpful model used as a tool to explore and represent strategies for addition and subtraction. This unit will focus on the open number line as a model for subtraction.   In contrast to a number line with counting numbers written below, an “open” number line is just an empty line used to record children's addition and subtraction strategies. Only the numbers that children use are recorded and the addition and subtraction are recorded as leaps or jumps. For example, if a child's strategy for adding 8 + 79 is 79 + 1 + 7, using a landmark number of 80, it would be recorded on the open number line   The recording would be similar if a child solves 87 - 8 by first removing 7 and then 1.Modeling children's thinking on the open number line helps them move beyond counting one by one for addition and subtraction, to strategies such as taking leaps of ten, decomposing, and/or using landmark numbers. Use of the open number line also encourages discussion of the relationship between addition and subtraction and of the relationship between various problems in which the operation of subtraction can be employed - such as removal, comparative difference, and finding a missing addend.   As the unit progresses, timelines are used to record years of birth, rather than ages. This change in context challenges learners to grapple with larger numbers and with the changing places of the part-whole relations of numbers on the number line. For example, first the number 79 may be marked on the number line as 8 less than 87; then it may be the difference between 2005 and 1926. Several minilessons for subtraction are also included in the unit. These are structured using strings of related problems as a way to guide learners more explicitly toward computational fluency with subtraction.   Note: It is expected that children will have had substantial experience with number lines prior to this unit. If this is not the case, you might want to take a look at the unit Measuring for the Art Show, to see how the number line model can be developed.   To learn more visit http://www.contextsforlearning.com

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Cathy Fosnot is a leading voice in mathematics education. She is Professor Emerita of Education at the City College of New York and the founder of Mathematics in the City, a national center for professional development located at the college. In 2007 she established New Perspectives on Learning to devote her time to speaking around the world, fostering school change through professional learning and support, and producing new resources and materials. Watch Cathy talk about the importance of context in math. Follow Cathy on Twitter @ctfosnot

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