1. Our Base 10 Number System Presented by Frank H. Osborne, Ph. D. © 2015 EMSE 3123 Math and Science in Education 1

2. Our Base 10 Number System Our base 10 number system is known as the Hindu-Arabic system. It uses ten digits to represent quantities. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 These numerals are symbols that can be used to write any quantity. Other symbols can be substituted. These are the ones that we commonly use--everyone is accustomed to using them. 2

3. Other Number Systems Other number systems are also used, especially when working with computers. Binary system (2 symbols) –0, 1 Octal system (8 symbols) –0, 1, 2, 3, 4, 5, 6, 7 Hexadecimal system (16 symbols) –0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F 3

4. How Number Systems Work Imagine we are cave people with no formal number system. We use marks to represent objects. Here is an example. I I I I I I I I I I I I I I I This is cumbersome and difficult to visualize when there are large numbers. 4

5. How Number Systems Work Having enough of this method, we decide to begin grouping them together. For some reason we decide that we should group them by six. What we get is: We can represent these using our numbers. There are two groups with three singles. In base six we would have: 23 (read as “two-three”, not twenty- three) 5

6. How Number Systems Work The number 23 indicates that there are: 2 groups of six items 3 single units We could use this thinking to make a base six number system. (none) = 0 I = 1 II = 2 III = 3 IIII = 4 IIIII = 5 6

7. How Number Systems Work Adding one more number means we need to group the objects in groups of six. The first number tells the number of groups and the second tells the number of ungrouped items left over. IIIIII = 10 IIIIII I = 11 IIIIII II = 12 IIIIII III = 13 IIIIII IIII = 14 IIIIII IIIII =15 7

8. How Number Systems Work Adding one more number means we need to group the objects in additional groups of six. IIIIII IIIIII = 20 IIIIII IIIIII I = 21 IIIIII IIIIII II = 22 IIIIII IIIIII III = 23 IIIIII IIIIII IIII = 24 IIIIII IIIIII IIIII = 25 8

9. How Number Systems Work Eventually we have five groups of six objects with five left over. We write this as: 55 (which is thirty-five in base 10). 9

10. How Number Systems Work Now we add one more object. How do we write the next number? We cannot use digits higher than 5 in base six. So we make one large group containing six groups with six items each. Result: We have the large group of six sixes, no groups of six and no units. 10

11. How Number Systems Work We write this number as: 100 Groups of six sixes Groups of six Units 1 0 0 We now turn our attention to base 4. 11

12. Base Four Number System The Base 4 number system has four symbols (0, 1, 2, and 3). That’s all. Number of Items Symbolic Representation (none) 0 I 1 II 2 III 3 12

13. Base Four Number System Number of Items Symbolic Representation (none) 0 I 1 II 2 III 3 Suppose you add one more to get IIII? That is a group of four so you put them together as a group and carry them to a space where you put groups of four. 13

14. Making Groups of Four Each time you have IIII, group them together. 14

15. Making Groups of Four Each time you have IIII, group them together. 15

16. Making Groups of Four When you add the next one, you will need to add new column in which to place groups of Four Fours. 16

17. Making Groups of Four When you add the next one, you will need to add new column in which to place groups of Four Fours. 17

18. Making Groups of Four This system can continue indefinitely. Eventually you could get the following: Groups of Groups Ones Four Fours of Four 4 4 3 What would happen when you added another one? 18

19. Making Groups of Four You would add a new column which would contain groups of sixteen fours. Groups of Groups of Groups Ones Sixteen Fours Four Fours of Four 1 0 0 0 This is the general way in which any base numbering system will work. The number of symbols represents the base. We group object when we have used up all of the symbols. 19

20. Base Ten Number System The Base 4 Activity illustrates the thinking processes that are necessary when learning the base 10 system. You may have noted special difficulties in base 4 with adding one unit to 3, 33, etc. Children face these difficulties when adding an additional unit to 9, 19, etc. in the base ten system. 20

21. Base Ten Number System The way to teach these concepts is by using manipulatives as your main teaching aid. Pencils can be used. Count them out in groups of 10 and wrap each group of 10 with a rubber band or place in a cup Once you have ten groups of 10, wrap it with a larger rubber band to make a group of one hundred or collect the 10 cups and place in the box. 21

22. Base Ten Number System Activity: Put two sheets of paper on the table. Label one "Tens" and the other "Ones" Place objects on the "Ones" sheet. Whenever 10 is reached, bundle the objects and place them on the "Tens" sheet. 22

23. Base Ten Number System Eventually add a third sheet labeled "Hundreds". Bundles of 10 tens go on it. 23

24. Base Ten Number System Example: Use pencils, toothpicks or interlocking cubes. Place any large number in a tray and have the children regroup them into groups of tens and ones. Later on, use more objects and add a tray for hundreds. 24

25. Base Ten Number System Children need to do these things over and over, with increasing complexity. –Counting and numbering to 10 –Counting and numbering to 20 and higher –Sums that are 10 –Groups of 10 There are various really good publications in bookstores organized by grade level. 25

26. Base Ten Number System Example: Children find groups of 10. 26

27. Base Ten Number System Dienes Blocks can be used to represent quantities in base 10. They also can be used to visualize the concept of volume and three-dimensionality. 27

28. Base Ten Number System Example: Construct 10 x 10 grids on paper or use Dienes blocks. Use to show numbers such as 233. 28

29. Base Ten Number System In 4 th or 5 th grade, larger numbers can be introduced. Dienes blocks can be used to represent larger numbers from 0 to 9,999. What number is shown below? 29

30. Base Ten Number System What would 5,234 look like? 30

31. Base Ten Number System What would 5,234 look like? 31

32. The End 32