1. Singapore Math Approach
Decimal Numbers

2. Please make a name tent

3. Some of the Tools used in Singapore Math
Place Value Strips
Place Value Disks
Place Value Charts
Number Bond Cards
Part-Whole Cards
Decimal Tiles
Decimal Strips

4. The big thing to remember about decimals…
It’s all about location, location, location.

5. Read the following number
34,
Write the decimal numbers using both the common fraction and the decimal fraction form
Elementary teachers – What would each number look like in expanded notation?
Secondary teachers – What would each number look like in scientific notation?

6. Developing the place value concept
All concepts begin with some concrete material such as base ten blocks.

7. Let’s Assume students have worked with base ten blocks.
Concrete

8. Then we move onto drawing pictures of the base ten blocks (Pictorial / Representation)
Concrete

9. Students then use words to describe the value.
Concrete
Word
Twenty Three AND
Two tens and Three

10. And Write the number in symbolic form (Abstract)
Concrete
The Number 23
Word
Twenty Three AND
Two tens and Three

11. Finally students should be able to go between any of the forms
Concrete
The Number 23
Word
Twenty Three AND
Two tens and Three

12. Developing the concept of a decimal What is 0.1?
Point out that multiple shapes are used.

13. What is 0.5?

14. The first thing student have to do to begin a study on decimals is to accept that decimals are another way to write fractions.

15. Common Misconception to address
The role of the decimal point is not to separate a number into two numbers. It is a warning that the single number is composed of a whole number part and a fractional part. The decimal point is a mark that lets a person know where the fractional part begins.

16. Scattered Grouped Exchange

17. How does that activity connect to the number line?

18. The number line helps to begin to prepare students to perform operations and compare decimals

19. Conceptual versus careless errors 63.5
We are trying to distinguish between the place, the name of the digit, and the value of the digit when in its particular place.
What errors might you expect students to make when asked to name the digit in the tenths place or to give the value of the digit 6?
The number in the tenths position is 5, not 0.5. The value of the digit 6 is 60, not 6 or 10.
Would these errors be conceptual or careless? What might be appropriate pedagogical responses to such errors?

20. Last idea before we move on: Decimal equivalence to fractions
The decimal form of 7/10 is easy, BUT give me an understanding how to get the decimal equivalence of ½ and 3/5.

21. Hundredths and Thousandths
Bring in a meter stick to show this on a number line

22. Checking for an understanding of place value with Base Ten Blocks

23. 100-Square

24. Number discs

25. Show these numbers using the number discs

26. Students need to understand that value also has a location.
The number line is less concrete and begins the shift towards representation and abstract use. This also reinforces the importance of place value. To plot a decimal value we need to first locate the tenths, then the hundredths, etc.

27. This will help feed the addition algorithm
Point out the use of the language one hundredths less than 52 hundredths

28. Here are six options for concrete materials for illustrating decimal fraction ideas.
Paper strips
Number discs
Base-ten blocks
Money
100-square grids
Number line
If you were to choose three for use in your classroom, which ones would you choose?
Are there others that you use that are not in this list?
On what basis did you make your choices?

29. Now What

30. Expanding on Conceptual Ideas from earlier studies
aka: Making Connections

31. Hindu-Arabic numeration
Additive property 75 =
Multiplicative property 75 = 7 x x 1

32. What are some different ways to describe these numbers?

33. Place Value Charts

34. How many ten’s are there in 120?
In the number 630
In the number 6.3
How many tens are there?
How many ones are there?
How many tenths are there?
How many ones are there?
How many tenths are there?
How many thousandths are there?

35. Counting patterns
These students have reverted to counting albeit in the tens place but counting by ones without thought to regrouping by tens. This carries over to decimals.

36. What should the student have written?
What objective is being tested in each sequence?
What error pattern do we observe?
Did the student continue any of the sequences correctly?
What does the student appear to know?
What would you suggest as a follow-up?
Do we need to return to the use of concrete materials?

37. Equivalence as a tool
Before we can discuss ordering and comparing decimals and decimal fractions we need to establish the notion of decimal equivalences.

38. How do we get a decimal equivalence from a fraction?

39. The annexation of zero(s)
If I annex a zero to the number 4 it becomes ten times bigger than the original. If I annex a zero to the number 0.4 it maintains its value even though it looks different What’s the deal?
What other concept has the student used in this situation?
To provide a valid answer to the question, what should the student have said?
How would you help the student re-evaluate the response?

40. Consider the numbers: ½ 0.5 2/4 0.50
Are these numbers equivalent? What other equivalent numbers could we add to the collection? Mary and Juan added the numbers 3/6 and 10/20 to the set. Are they correct?
Here is a number but the denominator has been smudged. 50/. What is the denominator? Ali added to the set of numbers. Is he correct? How would that number be expressed as a fraction?

41. **** Building an Algorithm ****
Two basic fractional equivalency to begin to build the algorithm.
½ = 5/ and ¼ = 25/100
How do we know what 1/8 is in decimal form from previous work?
What about connecting the first two to develop a pattern?

42. Ordering and comparing decimals
Which is bigger 0.3 or 0.299? How did you approach this comparison?
What reasoning would students use to correctly answer the question, but the reasoning be incorrect?
How would reasoning that 300 is bigger than 299 be incorrect reasoning?
How would you help a student struggling with comparing decimal values?

43. Compare and order the numbers 5.237 and 5.291

44. This method help over come the misguiding notion that bigger numbers are longer.

45. Staircases

46. Staircases

47. Activity: Number Card Line
Which is largest / smallest? How do you know? Which card is 0.01 less than 0.88? Is one card exactly 0.01 more than another? How could you use the idea of number cards in your class?

48. Clothesline

49. Rounding off decimals to the nearest whole number
Application of what it means to be nearer. Conversation about place value and its influence on rounding based on the number line argument.

50. Using the number line to teach Rounding to other decimal places.

51. Round the number 6. 48 to the nearest whole number
Round the number 6.48 to the nearest whole number. How would you help susie?

52. Key Benchmark - Closeness

53. Key Benchmark – Between-ness
Name a fraction between 2/7 and 4/7.
Name a fraction between 1/3 and ½.
Name two different (non-equivalent) fractions between 1/3 and ½.
How many fractions are between 1/3 and ½?
Name a decimal fraction between 0.3 and 0.5.
Name a decimal fraction between 0.3 and 0.4.
Name two decimal fractions between 0.3 and 0.4.
How many decimal fractions are between 0.3 and 0.4?

54. What time is it?

55. Addition of decimals
Students who have a good understanding of addition by place value have no problem learning to add decimals.

56. Add to 42.6

57. Misconceptions
The importance of rounding and estimating

58. Add 4.1 and 5.68
The importance of the annexation of zero

59. Subtraction

60. Design a task Plan a lesson for the example “subtract 0.4 from 5”.
What materials will you use?
How will you address the missing decimal point?
When would you expect students to annex the zero?
What is the sequencing in your steps to provide an exemplary sequencing of the steps?

61. Multiplication
Multiplication always begins as repeated addition.

62. Bar model and area tools
What is the product of 0.5 and 0.25?

63. Using the 100-grid square
The square can be used to show the product of 0.2 x 0.5.
I can also use it to show 2 x .32.

64. The sequencing of multiplying decimals
Decimal Number Discs
Number Line
Area Model
What tool could be use for each?

65. Area Models to Arrays Concrete model: Base-ten blocks Pictorial:
Area Array

66. Multi-digit multiplication Step 1: Multiplying by 10, 100, 1000

67. Multi-digit multiplication Step 2: Multiplying by Multiples of 10, 100, 1000

68. Multi-digit multiplication Step 3: Multiplying by any two-digit number

69. Where’s the Point?

70. Word Problems

72. What types of problems are represented by division?
Equal Sharing – How many in a group?
Measurement – How many groups?
Missing factor

73. Division of decimals as sharing

74. Divide 0.35 by 7

75. 3 methods for dividing 2.4 by 40

76. Thinking about misconceptions

77. Division as repeated Subtraction

78. A visual model of this strategy. 0.5 ÷ 6
The question to be asked is, “In 6, how many 0.5s are there?” or “How many times can we subtract 0.5 from 6?”

79. Types of real world problems that this could help students think about.

80. What about dividing by a decimal?
Divide 3.2 by 6.5
Divide 6.12 by 0.4

81. Does multiplication always make a number bigger
Does multiplication always make a number bigger? Does division always make a number smaller?

82. Evaluations
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83. This Concludes Our Broadcast Day