1. Decimals with a comprehensive K-5 curriculum from © E. Paul Goldenberg 2008, revised July 2011 This presentation may be shown for professional development purposes only, and may not be sold, distributed, or altered. 200300 251252 251252

2. What do students need to know? Zooming in on the number line Zooming in on the number line Arranging numbers in order Arranging numbers in order Adding and subtracting Adding and subtracting Multiplying and dividing by 10 Multiplying and dividing by 10 Multiplying decimals Multiplying decimals

3. Zooming in with a weak glass 30124567 2 2½ 3

4. Zooming in with a weak glass 300102040506070 20 25 30

5. 0102040506070 Zooming In 2320212224252627282930 Ten “spaces” (nine new numbers, labeled 1 to 9)

6. Grade 4 Student Letter

7. Numbers between numbers Multiply a small number by itself. Predict. Multiply a small number by itself. Predict. Square a small number (≤15). Tell me the result. I’ll “guess” your starting number. Square a small number (≤15). Tell me the result. I’ll “guess” your starting number. Let’s reverse it. Let’s reverse it. I got 36 I got 36 I got 49 I got 49 I got 43 I got 43 Hmmm… Between 6 and 7? Hmmm… Between 6 and 7?

8. 30124567 Zooming In More 6.36.16.26.46.56.66.76.86.9 67 Ten “spaces”Ten “spaces” (nine new numbers, labeled 1 to 9)

9. Zooming In… 749 636 n 6.9 6.8 6.7 6.6 6.5 6.4 6.3 6.2 6.1 n × n

10. Zooming In… 749 47.61 46.24 44.89 43.56 42.25 40.96 39.69 38.44 37.21 636 n 6.9 6.8 6.7 6.6 6.5 6.4 6.3 6.2 6.1 n × n

11. 6.56.26.36.46.66.76.86.9 6.5.26.5.1 Zooming In Still More 6.516.526.536.546.556.566.576.586.59 6.56.6 Ten “spaces”Ten “spaces” (nine new numbers, labeled 1 to 9)

12. Zooming In… 749 47.61 46.24 44.89 43.56 42.25 40.96 39.69 38.44 37.21 636 n 6.9 6.8 6.7 6.6 6.5 6.4 6.3 6.2 6.1 n × n 6.643.56 6.542.25 6.59 6.58 6.57 6.56 6.55 6.54 6.53 6.52 6.516.5.1 6.5.2 nn × n

13. Zooming In… 749 47.61 46.24 44.89 43.56 42.25 40.96 39.69 38.44 37.21 636 n 6.9 6.8 6.7 6.6 6.5 6.4 6.3 6.2 6.1 n × n 6.643.56 6.542.25 6.59 6.58 6.57 6.56 6.55 6.54 6.53 6.52 6.5142.3801 42.5104 42.6409 42.7716 42.9025 43.0336 43.1649 43.2964 43.4281 nn × n

14. Zooming In Even More 6.5643.0336 6.5542.9025 n 6.559 6.558 6.557 6.556 6.555 6.554 6.553 6.552 6.551 n × n

15. What do students need to know? Zooming in on the number line Zooming in on the number line Arranging numbers in order Arranging numbers in order Adding and subtracting Adding and subtracting Multiplying and dividing by 10 Multiplying and dividing by 10 Multiplying decimals Multiplying decimals

16. 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 smallest? OK. Which is the smallest? Is that it? What if there are hidden digits? Is that it? What if there are hidden digits? Sort the rest in order… What do we already know? Sort the rest in order… What do we already know? What do we know now? What do we know now? And now? And now? 3.053.0423.04200013.042003.0418 “Alphabetical order” to compare decimals Teach and Practice

17. Reading Decimals: 98.6° F Five reasons why we should not insist on pronouncing 6.53 as “six and 53 hundredths” 1. How do you write “6 53 / 100 ” as a decimal? 2. How many units? Tenths? Hundredths? 3. 53 / 100 sounds bigger than 6 / 10 but easy order is one of the main points of decimals 4. It’s a nuisance to pronounce 3.14159 5. Nobody, except in U.S. schools, does it.

18. What do students need to know? Zooming in on the number line Zooming in on the number line Arranging numbers in order Arranging numbers in order Adding and subtracting Adding and subtracting Multiplying and dividing by 10 Multiplying and dividing by 10 Multiplying decimals Multiplying decimals

19. Grade 4 of Think Math!

22. 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?  3.5 + 3.5 + 0.1 What’s 3.5 + 3.6?  3.5 + 3.5 + 0.1

23. 4 + 3 = 4.4 + 3 = 4 + 3.7 = 0.4 + 0.7 = 4.4 + 3.7 = 44 + 37 = 7 7. 4 7.7 1.1 8.1 81

24. 43 What “tens” is it between? 4050

25. 40 43 How far from the nearest tens? 4050 4350 37

26. 70 4371 43 50 70 How far is 43 from 71? 28 71 – 43 7201

27. 4.3 What “ones” is it between? 45

28. 4 4.3 How far from the nearest “ones”? 45 4.35 0.30.7

29. 5 7 4.37.1 How far is 4.3 from 7.1? 2.8 7.1 – 4.3.72.1 7.14.3 5 7

30. Grade 4 of Think Math!

34. What do students need to know? Zooming in on the number line Zooming in on the number line Arranging numbers in order Arranging numbers in order Adding and subtracting Adding and subtracting Multiplying and dividing by 10 Multiplying and dividing by 10 Multiplying decimals Multiplying decimals

35. Decimals & Place Value Grade 5

36. Why multiply and divide by 10? Practice reading numbers Practice reading numbers Reviewing place value Reviewing place value Seeing the pattern on the right side of the Seeing the pattern on the right side of the Seeing that × and ÷ by 10 undo each other Seeing that × and ÷ by 10 undo each other And… And…

37. What do students need to know? Zooming in on the number line Zooming in on the number line Arranging numbers in order Arranging numbers in order Adding and subtracting Adding and subtracting Multiplying and dividing by 10 Multiplying and dividing by 10 Multiplying decimals Multiplying decimals

38. Why multiply and divide by 10? Practice reading numbers Practice reading numbers Reviewing place value Reviewing place value Seeing the pattern on the right side of the Seeing the pattern on the right side of the Seeing that × and ÷ by 10 undo each other Seeing that × and ÷ by 10 undo each other Seeing why 28 × 32 helps with 2.8 × 3.2 Seeing why 28 × 32 helps with 2.8 × 3.2 2.8 × 10 × 3.2 × 10, then divide by 100 2.8 × 10 × 3.2 × 10, then divide by 100

39. TM! does decimals the way scientists, engineers, builders, mathematicians, doctors, statisticians … do. In pronunciation In pronunciation As an extension of place value As an extension of place value As an approximation As an approximation As an exact number As an exact number

40. Grade 4 of Think Math!

43. Think Math! Information Exchange http://thinkmath.edc.org/ More to be found on