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Moving From Natural Numbers to Signed Numbers

This presentation explores the key concepts students encounter when transitioning from natural numbers to signed numbers, focusing on graphing, directed differences, and understanding negative and positive values. By analyzing student responses and visualizing expressions on graphs, we address common misconceptions and teaching strategies. The use of two-color counters as visual aids helps in grasping the concepts of addition and subtraction with signed numbers. Engage with practical examples, dialogue around math problems, and innovative teaching approaches to enhance student understanding in mathematics.

germane-atkinson
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Moving From Natural Numbers to Signed Numbers

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  1. Moving From Natural Numbers to Signed Numbers Presented at V2CTM

  2. What math problem do you think this student is answering?

  3. Graph the line the student was working with.

  4. (–9,+7) (–3,+1) Did your graph look something like this?

  5. Where do the different parts of this expression show up on your graph?

  6. (–9,+7) (–3,+1) Where can students “see” the differences?

  7. (–9,+7) 1 – 7 (–3,+1)

  8. (–9,+7) 1 – 7 (–3,+1) – 3 – –9 What question are the differences, –6 and +6, answering?

  9. (–9,+7) 1 – 7 (–3,+1) – 3 – –9 What would happen if the student was starting with (–3,+1)?

  10. (–9,+7) (–3,+1) What would happen if the student was starting with (–3,+1)?

  11. – 9 – –3 (–9,+7) 1 – 7 7 – 1 – 3 – –9 (–3,+1) What would happen if the student was starting with (–3,+1)?

  12. How do we usually tell students to think about – 3 – –9?

  13. But – 3 + 9 doesn’t show up anywhere on the graph! • +9 doesn’t even show up! • Now the student has trouble making sense of the slope formula…

  14. Moving towards algebra, there are TWO new ways our students need to think: • We can work with negative numbers. • Subtraction can be a directed difference.

  15. Working with the counters as counting numbers. • 7 + 3 7 – 3 • What would addition mean? • What would the opposite of addition be? • What would directed difference mean?

  16. Think of two story problems for • 12 – 8 • One that uses take-away subtraction • One that uses a directed difference

  17. Two-Color Counters as Signed Numbers • What represents a positive number? • What represents a negative number? • What represents addition? • –5 + –10 +3 + +6 –11 + +2

  18. Two-Color Counters as Signed Numbers • What represents a positive number? • Yellow counters • What represents a negative number? • What represents addition? • –5 + –10 +3 + +6 –11 + +2

  19. Two-Color Counters as Signed Numbers • What represents a positive number? • Yellow counters • What represents a negative number? • Red counters • What represents addition? • –5 + –10 +3 + +6 –11 + +2

  20. Two-Color Counters as Signed Numbers • What represents a positive number? • Yellow counters • What represents a negative number? • Red counters • What represents addition? • Adding more counters • –5 + –10 +3 + +6 –11 + +2

  21. Two-Color Counters as Signed Numbers • What would the opposite of addition be? • –5 – –10 • +3 – +6 • +11 – +2 • –4 – +9

  22. Two-Color Counters as Signed Numbers • What would the opposite of addition be? • Taking away counters • –5 – –10 • +3 – +6 • +11 – +2 • –4 – +9

  23. Two-Color Counters as Signed Numbers • What would a directed difference be? • How to get from the second number to the first • –5 – –10 • +3 – +6 • +11 – +2 • –4 – +9

  24. Number Line Trips as Signed Numbers • What would a positive number represent? • What would a negative number represent? • What would addition represent? • –5 + –10 +3 + +6 –11 + +2

  25. Number Line Trips as Signed Numbers • What would a positive number represent? • A trip to the right • What would a negative number represent? • What would addition represent? • –5 + –10 +3 + +6 –11 + +2

  26. Number Line Trips as Signed Numbers • What would a positive number represent? • A trip to the right • What would a negative number represent? • A trip to the left • What would addition represent? • –5 + –10 +3 + +6 –11 + +2

  27. Number Line Trips as Signed Numbers • What would a positive number represent? • A trip to the right • What would a negative number represent? • A trip to the left • What would addition represent? • Continuing on to another trip (head-to-tail) • –5 + –10 +3 + +6 –11 + +2

  28. Number Line Trips as Signed Numbers • What would be the opposite of addition? • “Undoing” the next trip (head-to-head) • What would a directed difference be? • Getting from the end of second trip to the first • –5 – –10 +3 – +6 +11 – +2

  29. REMEMBER… • Moving towards algebra, there are TWO new ways our students need to think: • We can work with negative numbers. • Subtraction can be a directed difference.

  30. Thank you!! • Please contact me with any questions: • Katy Ulrich • culrich@vt.edu

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