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Zooming In On Number Lines: connecting whole numbers, decimals, fractions

Zooming In On Number Lines: connecting whole numbers, decimals, fractions. Cindy Carter The Rashi School Newton, Massachusetts NCTM, San Diego, April 2010. http://thinkmath.edc.org.

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Zooming In On Number Lines: connecting whole numbers, decimals, fractions

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  1. Zooming In On Number Lines:connecting whole numbers, decimals, fractions Cindy Carter The Rashi School Newton, Massachusetts NCTM, San Diego, April 2010 http://thinkmath.edc.org Created in PowerPoint 2008Some parts of this presentation may not play perfectly on earlier versions of PowerPoint. Fonts may align differently on different machines.

  2. Children(all children, all programs, all times, in school and out)are theory builders • Provide environment, smart tools, opportunity • Start with what they know • Support multiple ways of thinking (Differentiating instruction without differentiating students!)

  3. Mathematics(in school and out, all times, all topics)is the study of one discipline • All topics related by underlying mathematics • Smart tools make that relationship clear Don’t reinvent the wheel

  4. Better at basic skills Understand underlying mathematics better, too

  5. Today’s talk – Number Linea smart tool Whole numbers – early grades building a strong and useful foundation Decimals & Fractions – look a lot the same using the number line to take advantage of knowledge about whole numbers

  6. The Number Line is special • Unlike Cuisenaire rods, base 10 blocks, …the number line is not just a school tool: mathematicians use it, too. • All the numbers that students encounter through elementary and middle school live on the number line: counting numbers, zero, fractions, decimals, negative numbers.

  7. The Number Line Just another way to count?

  8. Counting and measuring are different! We use numbers for counting and measuring Counting starts at 1.It uses only “counting numbers”(no fractions, decimals, negatives) Measuring starts at 0.It uses fractions, decimals, and negative numbers. Counters and base-10 blocks support counting, but not measuring. The number line supports measuring also.

  9. Number lines represent measuring The number line is like a ruler Numbers are addresses They also name their distance from zero Distance is an important meaning of numbers Addition or subtraction? moving Sign (+ or –) indicates direction

  10. Kindergarten Video link to come

  11. Kindergarten Video link to come

  12. 0 1 2 3 4 5 6 7 8 9 10 11 One image of + and – • A child stands on the number line at 3 • Teacher (or another student) writes on the board + 3 – 2 • Where did the child land? X X X Grade K

  13. First grade • Building standard notation

  14. 2nd grade • Building mental math

  15. Shannon’s Grade 2, pairs to 30 Video link to come

  16. Mental Number Linelearning to think mathematically –1,000,000 –229 +1,000,030 +259 30 30 • Mathematics takes place in the mind • The right tools let young children develop that skill!

  17. Mental Number Linelearning to think mathematically • 2nd grade not ready for using the standard algorithm with such large numbers • Are ready for mathematical competence and confidence • Doing math is exciting! • Building foundations for the standard algorithm

  18. Folded Number Lines for bigger numbers

  19. Number Line Hotelfolding is practical and … • Start at 47 and go down 10 • What is 10 higher than 23?

  20. Number Line Hotel • How are the numbers in one column alike? • How are the numbers in one row alike?

  21. Number Line Hotel • Where are the 40s?

  22. Conventional hundreds chartbigger numbers, but…

  23. Number Line • Supports different models of how to do math • Already talked about subtraction as moving backwards or down on the number line • Subtraction as distance • Example from 4th grade

  24. Grade 4, revisiting subtracting Video link to come

  25. 3rd Grade 76 – 47 29 Subtraction of larger numbers using the distance model Thinking about 76 and 47 as addresses Subtraction – finding the distance between

  26. How far is 47 from 76? 29 47 76 70 50 76 – 47 3 20 6 47 70 76 50 Grade 3

  27. What “tens” is it between? 47 40 50 Buidling those 2 skills With 2nd graders

  28. How far from the nearest tens? 47 40 50 Pairs to 10 7 3 40 50 47 2nd graders get comfortable with tens neighbors and numbers between

  29. Grade 2, finding 70 1/2 Video link to come

  30. Another example of subtraction as distance I have saved $37, but I need $62 to buy the … • How can we do 62 – 37? • It means “how far from 37 to 62?” • Find the nearest multiples of 10 between them • So 62 – 37 = 25 37 62 37 40 60 62 3 2 20

  31. All numbers live on a number line! Numbers serve as location and as distance Addition is adding distances Subtraction measures the distance between Decimals and fractions live there, too.

  32. 0 0 10 1 20 2 30 3 4 40 50 5 60 6 70 7 Zooming In Grade 4

  33. 0 1 2 3 4 5 6 7 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 Zooming In More Decimals are an extension of whole numbers Grade 4

  34. 0 .1 .2 .3 .4 .5 .6 .7 .1 .11 .12 .13 .14 .15 .16 .17 .18 .19 .2 Zooming In Still More Grade 4

  35. 0 1 2 3 4 5 6 7 Fractions, too! 2½ 2 3 4th graders building the notion of zooming in on the number line numbers are preprinted

  36. Grade 4, zooming in on decimals Video link to come

  37. Grade 4, zooming in on decimals Video link to come

  38. And we zoomed in more… Video link to come

  39. Grade 4 of Think Math!

  40. How far is 4.7 from 7.6? 2.9 4.7 7.6 7 5 7.6 – 4.7 .3 2 .6 4.7 7 7.6 5 Grades 4 & 5

  41. What “ones” is it between? 4.7 4 5

  42. How far from the nearest “ones”? 4.7 4 5 Pairs to 1 0.7 0.3 4 5 4.7

  43. How far is 2 from 8 ? 5 2 8 1 2 1 2 4 3 1 2 1 1 2 5 5 5 5 5 5 5 5 5 5 5 8 3 8 – 2 5 8 3 2 8

  44. Same story Find the nearest whole numbers Find the distance to each Skill practice – Pairs to 1

  45. How far is 19 from 58? 39 19 58 Addresses on the number line 50 20 58 Tens between – 19 1 30 8 19 50 58 20 Zoom in Distances Add it up

  46. How far is 1.9 from 5.8? 3.9 1.9 5.8 Addresses on the number line 5 2 5.8 Ones between – 1.9 .1 3 .8 1.9 5 5.8 2 Zoom in Distances Add it up

  47. How far is 3 from 9 ? 5 3 9 4 5 Addresses on the number line 4 5 4 4 5 4 5 6 2 7 7 7 7 7 7 7 7 7 7 7 9 4 9 – 3 Ones between 5 9 4 3 9 Zoom in Distances Add it up

  48. Adding/Subtracting: moving What’s 16 + 32? (start: find the 10s row) Address: 16 Move: up 3 rows, right 2 Number Line Hotel

  49. Adding/Subtracting: moving What’s 1.6 + 3.2? (start: find the 1s row) Address: 1.6 Move: up 3 rows, right .2 Decimal Number Line Hotel

  50. Adding/Subtracting: moving 30 2 16 46 48 3 .2 1.6 4.6 4.8 What’s 16 + 32? What’s 1.6 + 3.2?

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