1. Not Whole Numbers II: Decimals Presented by Frank H. Osborne, Ph. D. © 2015 EMSE 3123 Math and Science in Education 1

2. Rational Numbers as Decimals The decimal system should be a natural outgrowth of the place value ideas covered earlier a well as fractions. We have already prepared children for decimals when we introduced denominators of 10 or 100. 2

3. Rational Numbers as Decimals Preparation for understanding tenths. 3

4. Rational Numbers as Decimals Preparation for understanding hundredths. 4

5. Rational Numbers as Decimals Examples: we indicate decimal numbers.15 and.23 on the grid by shading. Students should be able to order any set of decimals from smallest to largest by shading them on the grid. 5

6. Rational Numbers as Decimals Each grid is a unit square. For each grid, a. What fraction is shaded? b. What decimal part is shaded? 6

7. Rational Numbers as Decimals Each grid is a unit square. For each grid, a. What fraction is shaded? b. What decimal part is shaded? 7

8. Rational Numbers as Decimals Decimals permit the place-value notation to be extended to rational numbers. Remember that we expressed whole numbers such as 138 with manipulatives. 8

9. Rational Numbers as Decimals Now we can add in some of the smaller parts (say 5 of the 1/10 to the 138 we already have). To indicate we are moving into rational numbers we use a dot (.) (decimal point) as an indicator. This is 138.5 9

10. Rational Numbers as Decimals We can also add hundredths. If we add 2 of the hundredths pieces to the 138.5, we get 138.52. It can continue on indefinitely, with the next pieces being 1/1000, then 1/10,000, etc. 10

11. Rational Numbers as Decimals An alternative is to use Dienes Blocks. When we started with these, the smallest equaled one unit but any block can be used to represent the whole. For example: 11

12. Rational Numbers as Decimals Let us use the large Dienes block cube as one. How would you represent 4.326? 12

13. Rational Numbers as Decimals Let us use the large Dienes block cube as one. How would you represent 4.326? 13

14. Decimal Addition and Subtraction We use manipulatives for addition and subtraction of decimals in the same way that was used for whole numbers. Start with expressing each term using manipulatives, then combine and regroup if adding, or take away with borrowing if necessary for subtraction. Students should be comfortable with this and easily use play money, trading chips, Dienes bocks, or Cuisenaire rods. 14

15. Decimal Addition and Subtraction With play money use only pennies (1/100 of a dollar), dimes (1/10 of a dollar), dollars, tens or hundreds. Money is ideal for decimal addition or subtraction. We could express $14.57 as 15

16. Decimal Addition and Subtraction With Dienes blocks, we could do the same presentation. 16

17. Decimal Addition and Subtraction Let us add $14.57 and $25.64. We combine and exchange for the next higher block when we get ten. In the cents column, the 7 hundredths plus the 4 hundredths give us one tenth and one hundredth left over. In the tenths column we have 5 tenths plus 6 tenths plus the 1 tenth carried over for a total of 12 tenths. We make 1 whole with 2 tenths left over. 17

18. Decimal Addition and Subtraction Let us add $14.57 and $25.64. We combine and exchange for the next higher block when we get ten. The 1 whole is carried and added to the 4 and 5 to make 10 giving us 1 ten to carry. Finally we have 1 ten plus 2 tens plus the 1 ten that was carried for a total of 4 tens. The result is $40.11. 18

19. Decimal Addition and Subtraction Result: Conclusion: Adding and subtracting decimals is not any different from adding and subtracting whole numbers. 19

20. Decimal Addition and Subtraction We can also use trading chips as an illustration. We will start by making the yellow chip to be one. We can add whole numbers such as 2475 + 3566 = 6011. + = We combine the chips and regroup. 20

21. Decimal Addition and Subtraction + =  21

22. Decimal Addition and Subtraction Result: Instead of calling the yellow chip one we could have made the green chip to be one. This would put a decimal point between green and blue. We would have added $24.75 and $35.66 to make the total to be $60.41. 22

23. Decimal Addition and Subtraction Subtraction of decimal numbers can be performed in a similar fashion. This example is 24.75 from 35.66. 35.66 – 24.75 = - = 23

24. Decimal Addition and Subtraction Here are the steps. Take away 5 yellow. Then, exchange 1 green for 10 blue—take away 7 blue. Take away 4 green. Finally, take away 2 reds.  24

25. Decimal Addition and Subtraction  So, our answer to the problem is 35.66 – 24.75 = 10.91 25

26. Multiplying Decimals Let us multiply 3.2 x 2. To assist us, we will let a flat = 1, a rod will = 0.1 and a unit = 0.01. A way to visualize multiplying decimals is to build an array with these items. 26

27. Multiplying Decimals An array is a rectangular arrangement of a quantity in rows and columns. Use the pattern of directions in multiplying. 27

28. Multiplying Decimals 28

29. Multiplying Decimals Each product is found on the array. 29

30. Multiplying Decimals Here is another example: 4.6 x 1.3 = 30

31. Multiplying Decimals Here is another example: 4.6 x 1.3 = 5.98 31

32. Dividing Decimals We have seen how we can apply whole number techniques of addition and subtraction to decimals. Similarly, we can apply the techniques of multiplying and dividing of whole numbers to decimals. We start by multiplying 3 x 7 = 21. 32

33. Dividing Decimals However, if the 3 x 7 is part of a whole unit grid, then the numbers inside are actually.3 and.7 as shown. 33

34. Dividing Decimals Each tiny square is 1/100 of the whole so we can move six parts and show that we have 21/100. Also, we can see that 21/100 is the same as 2/10 + 1/100. 34

35. Dividing Decimals By the time children get this far they should be proficient in whole number operations, including the use of algorithms. They should also have some experience in using manipulatives as applied to decimal operations. Ultimately, they will realize that there is really no difference between decimal operations and whole number operations. 35

36. Dividing Decimals As each fraction has a decimal equivalent, division of decimals proceeds in the same way as division of fractions and whole numbers. When children begin division of decimals, they should already be quite familiar with the meaning of division. How much is 0.2 ÷ 0.05? 36

37. Dividing Decimals How much is 0.2 ÷ 0.05? We know that 0.2 ÷ 0.05 is the same as 2/10 ÷ 5/100 or, “How many.05’s fit into.2?” We see that.05 fits into.2 four times, therefore 0.2 ÷ 0.05 = 4. 37

38. The End 38