1. The system of numeration we use today is derived from an ancient Hindu system. It was picked up and developed by Arab traders in the ninth and tenth centuries and quickly spread through Europe. The superiority of the Hindu-Arabic numeration system over others is largely due to the fact that it is based on the highly sophisticated concept of place value. Place Value

2. 2 Please first play the video clip found at http://www.math.harvard.edu/~knill/mathmovies/swf/inthenavy_28.html This clip is from the 1941 movie “In the Navy”, starring Bud Abbott and Lou Costello. My favourite quote from this movie clip is: Smokey Adams (Bud Abbott): Did you ever go to school, stupid? Seaman Pomeroy Watson (Lou Costello): Yeah, and I come out the same way.

3. Place Value3 How can Bud Abbott convince Lou Costello that 7 x 13 is not 28? Perhaps Lou could relate to the concept of poker chips. Place-value is often a difficult concept for some, but for some reason few have a problem with using poker chips (or coloured tiles for those squeamish about using something possibly related to gambling) whose values depend on their colours, when you simply tell them what the exchange values of white, blue, and red chips are. (e.g., a blue chip is worth, or the same as, ten white ones; a red one is worth ten blue ones.)

4. Place Value4 Using poker chips to model place value. 1 blue chip is worth 10 white chips

5. Place Value5 Divide 28 by 7 Since the 2 blues can’t be divided into 7 groups, swap them for 2 x 10 (i.e. 20) whites. The 28 white chips can easily be divided into 7 groups of 4.

6. Place Value6 13 x 7 In total, 7 blues and 21 whites.

7. Place Value7 13 x 7 In total, 7 blues and 21 whites. Swap 20 (2 x 10) whites for 2 blues.

8. Place Value8 13 x 7 = 91 (not 28!)

9. Place Value9 When teaching students After students are comfortable and good at manipulating concrete materials, the teacher can point out that when numbers are written numerically, the columns are like the different colour poker chips. The first column is like white poker chips, telling you how many "ones" you have, and the second column is like blue poker chips, telling you how many 10's (or chips worth ten) you have....etc.. This would be a good time to tell them that in fact the columns are even named like the poker chips -- the one's column, the ten's column, the hundred's column, etc..

10. Place Value10 10 Quick Questions:

11. Place Value11 1. Which is the number 1 more than 75 ? 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

12. Place Value12 1. 76 is the number 1 more than 75. 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

13. Place Value13 2. Which is the number 1 less than 40 ? 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

14. Place Value14 2. 39 is the number 1 less than 40. 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

15. Place Value15 3. Which is the number 10 more than 34 ? 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

16. Place Value16 3. 44 is the number 10 more than 34. 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

17. Place Value17 4. What is the number 10 less than 78 ? 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

18. Place Value18 4. 68 is the number 10 less than 78. 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

19. Place Value19 5. Find the number which is 1 ten and 1 more than 44. 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

20. Place Value20 5. 55 is the number which is 1 ten and 1 more than 44. 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

21. Place Value21 6. Find the number which is 1 ten and 1 less than 80. 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

22. Place Value22 6. 69 is the number which is 1 ten and 1 less than 80. 0123456789 10111213141516171819 20212223242526272829 30313233343536373839 40414243444546474849 50515253545556575859 60616263646566676869 70717273747576777879 80818283848586878889 90919293949596979899

23. Place Value23 7. What is 4 + 50 ? a) 45 b) 54 c) 450 d) 540 e) 504

24. Place Value24 7. 4 + 50 = a) - b) 54 c) - d) - e) -

25. Place Value25 8. Which is the larger number ? 3773

26. Place Value26 8. 73 is the larger number. 3773

27. Place Value27 9. What number is represented below ? a) 7 b) 34 c) 43 d) 304 e) 403

28. Place Value28 9. 43 is represented below. a) - b) - c) 43 d) - e) -

29. Place Value29 10. Which number has two more tens than 40 has, and the same amount in the ones place as 17 ? a) 57 b) 59 c) 67 d) 77 e) 79

30. Place Value30 10. 67 has two more tens than 40 has, and the same amount in the ones place as 17. a) - b) - c) 67 d) - e) -

31. Place Value31 A final word from Richard Garlikov (from his paper “The Concept and Teaching of Place Value”): “… I believe teaching involves more than just letting students (re-)invent things for themselves. A teacher must at least lead or guide in some form or other. How math(ematics), or anything, is taught is normally crucial to how well and how efficiently it is learned. It has taken civilization thousands of years, much ingenious creativity, and not a little fortuitous insight to develop many of the concepts and much of the knowledge it has; and children can not be expected to discover or invent for themselves many of those concepts or much of that knowledge without adults teaching them correctly, in person or in books or other media. Intellectual and scientific discovery is not transmitted genetically, and it is unrealistic to expect 25 years of an individual's biological development to recapitulate 25 centuries of collective intellectual accomplishment without significant help. Though many people can discover many things for themselves, it is virtually impossible for anyone to re-invent by himself enough of the significant ideas from the past to be competent in a given field, math(ematics) being no exception. Potential learning is generally severely impeded without teaching. And it is possibly impeded even more by bad teaching, since bad teaching tends to dampen curiosity and motivation, and since wrong information, just like bad habits, may be harder to build from than would be no information, and no habits at all.”

32. Place Value32 References Mathematics Explained for primary teachers (third edition, 2006) by Derek Haylock. http://www.math.harvard.edu/~knill/math movies/swf/inthenavy_28.html http://www.math.harvard.edu/~knill/math movies/swf/inthenavy_28.html http://www.garlikov.com/PlaceValue.html http://www.homeschoolmath.net/teaching/ pv/tens_ones_problems.php http://www.homeschoolmath.net/teaching/ pv/tens_ones_problems.php