Decimals with a comprehensive K-5 curriculum from © E. Paul Goldenberg 2008, revised July 2011 This presentation may be shown for professional development

Decimals with

a comprehensive K-5 curriculum from

© E. Paul Goldenberg 2008, revised July 2011This presentation may be shown for professional development purposes only, and may not be sold, distributed, or altered.

200 300

251 252

251 252

What do students need to know?

Zooming in on the number lineZooming in on the number line Arranging numbers in orderArranging numbers in order Adding and subtractingAdding and subtracting Multiplying and dividing by 10Multiplying and dividing by 10 Multiplying decimalsMultiplying decimals

Zooming in with a weak glass

30 1 2 4 5 6 7

2 2½ 3

Zooming in with a weak glass

300 10 20 40 50 60 70

20 25 30

300 10 20 40 50 60 70

Zooming In

2320 21 22 24 25 26 27 28 29 30

Ten “spaces” (nine new numbers, labeled 1 to 9)

Grade 4 Student Letter

Numbers between numbers

Multiply a small number by itself. Predict.Multiply a small number by itself. Predict. Square a small number (≤15). Tell me theSquare a small number (≤15). Tell me the

result. I’ll “guess” your starting number.result. I’ll “guess” your starting number. Let’s reverse it. Let’s reverse it. I got 36I got 36 I got 49I got 49 I got 43I got 43 Hmmm… Between 6 and 7?Hmmm… Between 6 and 7?

30 1 2 4 5 6 7

Zooming In More

6.36.1 6.2 6.4 6.5 6.6 6.7 6.8 6.96 7

Ten “spaces”Ten “spaces” (nine new numbers, labeled 1 to 9)

Zooming In…

7 49

6 36

n

6.9

6.8

6.7

6.6

6.5

6.4

6.3

6.2

6.1

n × n

Zooming In…

7 49

47.61

46.24

44.89

43.56

42.25

40.96

39.69

38.44

37.21

6 36

n

6.9

6.8

6.7

6.6

6.5

6.4

6.3

6.2

6.1

n × n

6.56.2 6.3 6.4 6.6 6.7 6.8 6.9

6.5.26.5.1

Zooming In Still More

6.51 6.52 6.53 6.54 6.55 6.56 6.57 6.58 6.596.5 6.6

Ten “spaces”Ten “spaces” (nine new numbers, labeled 1 to 9)

Zooming In…

7 49

47.61

46.24

44.89

43.56

42.25

40.96

39.69

38.44

37.21

6 36

n

6.9

6.8

6.7

6.6

6.5

6.4

6.3

6.2

6.1

n × n6.6 43.56

6.5 42.25

6.59

6.58

6.57

6.56

6.55

6.54

6.53

6.52

6.516.5.1

6.5.2

n n × n

Zooming In…

7 49

47.61

46.24

44.89

43.56

42.25

40.96

39.69

38.44

37.21

6 36

n

6.9

6.8

6.7

6.6

6.5

6.4

6.3

6.2

6.1

n × n6.6 43.56

6.5 42.25

6.59

6.58

6.57

6.56

6.55

6.54

6.53

6.52

6.51 42.3801

42.5104

42.6409

42.7716

42.9025

43.0336

43.1649

43.2964

43.4281

n n × n

Zooming In Even More

6.56 43.0336

6.55 42.9025

n

6.559

6.558

6.557

6.556

6.555

6.554

6.553

6.552

6.551

n × n



• Which number is biggest?Which number is biggest?

• Where is the units digit? Hmm… Can’t tell yet…Where is the units digit? Hmm… Can’t tell yet…

• What if the others have more hidden stuff?What if the others have more hidden stuff?

• OK. Which is the OK. Which is the smallestsmallest??

• Is that it? Is that it? What if there are hidden digits?What if there are hidden digits?

• Sort the rest in order…Sort the rest in order…What do we What do we already already know?know?

• What do we know now?What do we know now?

• And now?And now?

• And now?And now?

3.053.05

3.0423.042

3.04200013.0420001

3.042003.04200

3.04183.0418

“Alphabetical order” to compare decimals Teach and Practice

Reading Decimals: 98.6° F

Five reasons why we should Five reasons why we should not not insist on insist on pronouncing 6.53 as “six and 53 hundredths”pronouncing 6.53 as “six and 53 hundredths”

n How do you write “6 How do you write “6 5353//100100” as a decimal?” as a decimal?

n How many units? Tenths? Hundredths?How many units? Tenths? Hundredths?n 5353//100100 sounds bigger than sounds bigger than 66//1010 but easy order is but easy order is

one of the main one of the main pointspoints of decimals of decimalsn It’s a nuisance to pronounce 3.14159It’s a nuisance to pronounce 3.14159n Nobody, except in U.S. schools, does it.Nobody, except in U.S. schools, does it.



Grade 4 of Think Math!



So 3.6 and 3.17 are just kinds of 3

What’s 3 + 4? Picture it on number line!What’s 3 + 4? Picture it on number line! What’s 3.6 + 4?What’s 3.6 + 4? What’s 3 + 4.1?What’s 3 + 4.1? What’s 3.5 + 0.1?What’s 3.5 + 0.1? What’s 3.5 + 3.5?What’s 3.5 + 3.5? What’s 3.5 + 3.6? What’s 3.5 + 3.6? 3.5 + 3.5 + 0.1 3.5 + 3.5 + 0.1

4 + 3 =4 + 3 =

4.4 + 3 =4.4 + 3 =

4 + 3.7 =4 + 3.7 =

0.40.4 ++ 0.70.7 ==

4.44.4 ++ 3.73.7 ==

44 + 37 =44 + 37 =

77.47.7

1.18.181

43

What “tens” is it between?

40 50

40

43

How far from the nearest tens?

40 50

43 50

3 7

50 70

43 71

7143 50 70

How far is 43 from 71?28

71– 43

7 20 1

4.3

What “ones” is it between?

4 5

4

4.3

How far from the nearest “ones”?

4 5

4.3 5

0.3 0.7

5 7

4.3 7.1

How far is 4.3 from 7.1?2.8

7.1– 4.3

.7 2 .1

7.14.3 5 7







Decimals & Place Value

Grade 5

Why multiply and divide by 10?

Practice reading numbersPractice reading numbers Reviewing place valueReviewing place value Seeing the pattern on the right side of theSeeing the pattern on the right side of the Seeing that × and ÷ by 10 undo each otherSeeing that × and ÷ by 10 undo each other And…And…



Why multiply and divide by 10?

Practice reading numbersPractice reading numbers Reviewing place valueReviewing place value Seeing the pattern on the right side of theSeeing the pattern on the right side of the Seeing that × and ÷ by 10 undo each otherSeeing that × and ÷ by 10 undo each other Seeing Seeing why why 28 × 32 helps with 2.8 × 3.228 × 32 helps with 2.8 × 3.2 2.8 × 10 × 3.2 × 10, then divide by 1002.8 × 10 × 3.2 × 10, then divide by 100

TM! does decimals the way scientists, engineers, builders, mathematicians, doctors, statisticians … do.

In pronunciationIn pronunciation As an extension of place valueAs an extension of place value As an approximationAs an approximation As an exact numberAs an exact number




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Decimals with a comprehensive K-5 curriculum from © E. Paul Goldenberg 2008, revised July 2011 This presentation may be shown for professional development