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total number of students in the line to find the length of time it takes for each individual pass. Or a teacher can ask students to predict the length of time for each pass and have the students multiply this number by the total number of students in the school to find the total time. In this situation, the slope shows the constant rate of change, which us es multiplication. Some students understand the rate of change as repeated addition. Using skip counting as shown in Phase 1, helps these students. Since Ms. Hakkenberg has added these activities to her curriculum, more of her students understand the concept of slope, as indicated in the class assess ment. In Activity 1, when her students answer the question, “ How long does it take to fill the tank and then empty the tank? ” they experience the big ideas of both positive and negative rates of change, and learn to represent them with linear functions. Ac tivity 2 serves to review and solidify the students ’ Activity 1 learning and gives them practice match ing specific points on the coordinate plane to their own positions in terms of time vs. distance. Finally, the students see slope as both multiplication and repeated addition as they skip count and make pre dictions about ball passing time in Activity 3. Ms. Hakkenberg finds that her students are now com fortable comparing graphs and describing shifts up or down or describing a line as steeper or less steep than another. In addition, they are quick to grasp whether a slope is negative or positive, and give the slope ratio and units within a real - world context. With the help of the activities and tools described above, Ms. Hakkenberg and her classes now expe rience the joy of conceptual understanding and en joy their time together on the slope. Summary

https://drive.google.com/drive/folders/1fsI2p1 AlZBvQgFPmih3WJKXJIfgwY9oh?usp=shar ing Kalb, B., Peters, R., & Meyer, D. (2018, May 28). If your content is the Aspirin, how do you cre ate the headache? [Audio blog post]. Vrain Waves: Teaching Conversations with Minds Shaping Education. Retrieved from https:// itunes.apple.com/us/podcast/vrain - waves teaching - conversations - minds - shaping education/id1365316994?mt=2 Kaput, K. (2018). Evidence for Student - Centered Learning. Saint Paul, MN: Education Evolv ing. Leinwand, S. (2009). Accessible mathematics . Portsmouth, NH: Heinemann. Meyer, D. (2011, July 8). Dan Meyer ’ s Three - Act Math Tasks. Google Sheets. Retrieved from https://docs.google.com/spreadsheets/ d/1jXSt_CoDzyDFeJimZxnhgwOVsWkTQEsf qouLWNNC6Z4/pub?output=html. Teuscher, D., & Reys, R. E. (2010). Slope, rate of change, and steepness: Do students understand these concepts? Mathematics Teacher, 103 (7), 519 - 524. Virginia Department of Education. (2017). 2017 Mathematics institute: Professional develop ment resources. [Unpublished PowerPoint]. Retrieved from http://www.doe.virginia.gov/ instruction/mathematics/ professional_development/institutes/ index.shtml, 32 Wolbert, W. (2017). A different pitch to slope. Mathematics Teacher, 110 (9), 674 - 679.

Acknowledgements

Many thanks to Research Experience for Teachers (RET) NSF Grants #1301037 and #1609089 for support, inspiration, and time to work on these ac tivities.

References

Bazak, S. (March 3, 2016). Algebra I: Teaching linear regression (personal communication). Doubet, K. J., & Hockett, J. A. (2015). Differentiation in middle and high school . Alexandria, VA: Association for Su pervision and Curriculum Development. Fisher, D., & Hakkenberg, D. (2019, July 15). Fisher and Hakkenberg Slope. Retrieved from

Dawn D. Hakkenberg Mathematics,

Diana M. Fisher System Science, Portland State University fisherd@pdx.edu

Roanoke City Public Schools ddhakkenberg@gmail.com

Virginia Mathematics Teacher vol. 46, no. 2

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