1. Gaining A Deeper Understanding of Place Value Dianna J. Spence Gregg Velatini North Georgia College & State University

2. Clarification The exercises we introduce today are FOR YOU (not for your students!) We do not expect (or recommend) that you use these Base 6 activities with your elementary students. You may choose to do similar Base 10 activities with your students.

3. Why? “If not for our students, why do them?” ◦ To strengthen your own…  …understanding of number systems  …awareness of students’ perspectives & challenges “Why not just show us these activities in Base 10?” ◦ Because you know too much! ◦ Helpful (as a teacher) to experience lack of familiarity and limitations similar to those of your students

4. Learning a Number System Suppose our hands looked like this… Imagine learning to count on your fingers

5. Finger Counting ONETWO THREE FOUR FIVESIX 12 3 4 510 “TEN”

6. Help is on the way! Base Six Blocks unit rod

7. Counting with Blocks 1 = 23 4 5 10 11 12 13 14 15 20 21

8. Representing 2-Digit Numbers Build these numbers with your blocks *All numbers below are written in Base 6 24 51 40 35 55

9. What comes next? What is the next number after 45? What is the next number after 55? 50 100

10. Base Six Blocks Revisited unit rodflat cube 110100 1000 “giant rod” 10000

11. Representing 3-Digit Numbers Build these numbers with your blocks *All numbers below are written in Base 6 204 120 134 325 152

12. Identifying Numbers Write the number represented in each picture (answers should be in Base 6).

13. Modeling Simple Addition 23 +12 35

14. Modeling Addition with Trades 23 +15 42

15. More Addition with Trades 42 +34 120

16. Practice: Addition 32 +33 21 +15 30 +43 102 +45 214 +54 10540113 151312

17. Modeling Simple Subtraction 45 -24 21

18. Modeling Subtraction with Trades 41 -24 13

19. Practice: Subtraction 52 -33 21 -15 50 -43 103 -45 200 -54 1523 14102

20. Modeling Simple Multiplication 2 ×4 12

21. Modeling 2-Digit Multiplication 12 ×11 132 Notice Partial Products: 100 (flat) 20 (rods at top) 10 (rods at right) 2 (units)

22. 2-Digit Multiplication w/ Trades 14 ×21 334