1. Making Sense of Math: Early Number Concepts Amy Lewis Math Specialist IU1 Center for STEM Education

2. Goals for the course Use a variety of tools to deepen understanding of place value and explore number relationships to connect number concept meanings and representations. Participate collaboratively in solving problems in other base systems to strengthen reasoning skills. Connect new understandings of numbers to classroom practice.

3. Pre-Test Please take a few minutes to complete the pre-test. Although you should do the best that you can, please do not feel pressure to get all of the questions perfect. This is only a measure of growth from the start of the course until the end.

4. Day 1: Use physical models to develop number sense in our Base-10 system through number construction and deconstruction. Examine benchmark numbers and use them to deepen understanding of place value. Reflect upon how to use these strategies in the classroom.

5. What does it mean for students to have “number sense”? Number Meaning Relationships Magnitude Operation Sense Real Life Number Sense - Applications Howden, 1989

6. Sense of Number… …in its most fundamental form, entails an ability to immediately identify the numerical value associated with small quantities; …should extend to numbers written in fraction, decimal, and exponential forms. …when lacking, interferes with learning algorithms and number facts and prevents use of strategies to verify if solutions to problems are reasonable. NMAP, page 27, March 2008

7. Big Idea How can we make the concept of our place-value system visible, concrete, and relevant to give students a better sense of number?

8. How Many Stars?

9. What does this response tell us?

10. How Many Stars? What does this response tell us? What do we do about it?

11. Groups and Leftovers For each row of the chart –Grab a handful of beans (less than 100 beans) –Group the beans into groups of whatever is in the first column of the table. For example, if the number in the first column is 7, put your beans into groups of 7 and then fill in the rest of the information. –Grab a new handful of beans for the next row. What patterns do you notice?

12. Two-Handed Math A tool that we use in the early years is to have students show values with their fingers. –Show me 4 fingers –7 fingers Let’s extend this… –Show me 12 fingers.

13. Two-Handed Math Preferred method for this activity: How would we show 1, 3, and 7 using this method?

14. Two-Handed Math What about… –14 fingers? –16 fingers? –18 fingers? How many more from 18 are needed for 20? What about… –24 fingers? –34 fingers? The Wave! 

15. Two-Handed Math What are possible extensions? –If you wanted to make 73, how many people would you need? –Musical numbers –34 + 20 –43 + 25 –48 + 37 –31 – 12 Other possibilities?

16. Ten-Frame Math “Since 10 plays such a large role in our numerations system and because two fives make up 10, it is very useful to develop relationships for the numbers 1 to 10 to the important anchors of 5 and 10.” John Van de Walle

17. Five-Frame Tell-About Show 3 on your Five-Frame. What can you tell us about 3 from looking at your mat? List as many things as you can. Share with your partner.

18. Crazy Mixed-Up Numbers Use your Ten Frame to show 6. Change your number to a 4. –How did you accomplish this? –How might a student accomplish this? Change your new number to an 8. –How did you accomplish this? –How might a student accomplish this? What is being assessed in this task? How do we address what we find?

19. How many dots do you see? How do you know? There are 4 and 3 more. 3 are empty. I see 6 and 1 more. I counted fast – 2, 4, 6, 7. I counted faster – 3 and 3 is 6, and 1 more is 7.

20. Ten-Frame Flash Cards Practice the flashcards with your partner. How could we modify this task? –“War” –Say the Ten Fact What characteristics of numbers are students learning in this activity? –Relationships of numbers to 5 and 10 –Concept of 0 –Odds and Evens

21. Make 10 on the Ten Frame Randomly select 2 flashcards. Model each number on your Double Ten Frame mat. Decide on the easiest way to show (without counting) what the total is.

22. Ten-Frame Activities Five-Frame Tell-About Crazy Mixed Up Numbers How many dots do you see? Ten-Frame Flash Cards Make 10 on the Ten-Frame Say the 10 Fact

23. Ten-Frame Article http://makingsenseofmath.iu1.wikispaces.net /Early+Number+Conceptshttp://makingsenseofmath.iu1.wikispaces.net /Early+Number+Concepts

24. Three Other Ways 463 Show this using your base-10 blocks. Find and record at least three other ways to show this number. How many 1s are in this number? How many 10s are in this number? How many 100s are in this number?

25. 7536 How many 1000s are in this number? How many 100s are in this number? How many 10s are in this number? How many 1s are in this number? How is this understanding richer than, “What is the digit in the 10s place?”

26. Thinking Bigger Where do we encounter bigger numbers in daily life?

27. How Much Is It? On October 12, 2009, at 11:15:26 p.m EST, the US National Debt Clock read: $12,516,385,125,567.99 How do you read that number? How big is this number?

28. How Much Is It? The National Debt has increased an average of $3.95 billion per day since September 28, 2007. Write this number out. Is it: –$3,000,000,000.95? – $30,000,000.95? –$3,950,000,000.00? –$3,000,000,095.00?

29. How Much Is It? 100 years ago—July 1, 1910 —the National Debt was: $2,652 million How do you write that number? How does it compare to our current debt of almost $12 trillion? –A lot less? –Less, but only a little less? –About the same? –More, but only a little more? –A lot more?

30. Is This Possible? Ms. Hope E. Ternal entered her third grade class on Monday morning counting: 999,997... 999,998... 999,999... 1,000,000! “ Whew, I made it, ” she said. “ After school on Friday I started counting, and did not stop all weekend. I wanted to see if I could get to one million — and I made it! ” Is this possible? Could Ms. Ternal have started counting on Friday and gotten to 1 million by Monday? Assume that she counts at a rate of one number per second.

31. How Different Is It? Jim was thinking about counting from 1 to different numbers: A thousand A million A billion “They’re all big numbers. They come in order— thousands, millions, billions — so I think counting to each one will take a little bit longer than the one before, but not a lot.” Is Jim correct? Why or why not?

32. How Different Is It? Assume you count at a rate of one number per second. How long will it take to count from 1 to one thousand? How long will it take to count from 1 to one million? How long will it take to count from 1 to one billion?

33. Counting Bigger If you start counting at a rate of one number per second: It will take about 17 minutes to get to a thousand; It will take you 12 days to count to a million; It will take you 31 years to count to a billion.

34. Understanding Positional Systems

35. Smart by Shel Silverstein My dad gave me one dollar bill 'Cause I'm his smartest son, And I swapped it for two shiny quarters 'Cause two is more than one! And then I took the quarters And traded them to Lou For three dimes -- I guess he don't know That three is more than two!

36. Just then, along came old blind Bates And just 'cause he can't see He gave me four nickels for my three dimes, And four is more than three! And I took the nickels to Hiram Coombs Down at the seed-feed store, And the fool gave me five pennies for them, And five is more than four! Smart by Shel Silverstein

37. And then I went and showed my dad, And he got red in the cheeks And closed his eyes and shook his head-- Too proud of me to speak! Smart by Shel Silverstein

38. Big Ideas of the Base-Ten System The position of the digits in numbers determines what they represent—their value. Each place value to the left of another is ten times greater than the one to the right. (e.g., 10 x 10 = 100) There are standard “trade rules”: –Right to left: 10 for 1 –Left to right: 1 for 10

39. Thinking Even Bigger Representing Large Numbers

40. Thinking Even Bigger In this activity you will try to extend what you know about representing place values for smaller numbers—ones, tens, hundreds—to larger numbers—up to millions, billions, and even trillions.

41. Representing Relatively Small Numbers To start, think about the relationships among your base-ten blocks: How many units cubes are need to make a long? What are the dimensions of a long? How many longs are needed to make one flat? How many unit cubes are in one flat? What are the dimensions of a flat? How many flats are needed to make a super cube? How many unit cubes in one super cube? What are the dimensions of a super cube?

42. Thinking Even Bigger Ms. Take thinks that having only four types of blocks in a set is too limiting for her gifted students. “I can’t imagine why no one has ever expanded the set to include larger blocks.”

43. Thinking Even Bigger With a partner or group, design an extended set of base- ten blocks. What would the next blocks in the set look like? –What does each new block look like? Build or draw a diagram. –What does each represent? –What should you name each one? –What are the dimensions of each? Extend the set as far as you can. What patterns do you notice? Record your results on poster paper.

44. Poster Presentations

45. Place Value Chart What patterns do you notice?

46. Thinking Bigger A million pennies Wall 5 ft x 4 ft x 1 ft thick with a 9 in cube stepstool Height stacked: 0.99 miles

47. Thinking Bigger Ten million pennies 6 ft x 6 ft x 6 ft Height stacked: 9.88 miles

48. 10-Billion Pennies 90 ft x 11 ft x 250 ft If pennies stacked, height: 9,864 miles

49. Thinking Bigger Hundred billion pennies 126.72 ft x 126.72 ft x 127.72 ft Height Stacked 98,660 miles 8,969 acres laid flat

50. Looking at Models Each group will examine examples of physical models that can be used to represent the base-10 number system. For each one, –How does this model work? –How can the model be used to deepen students’ understanding of number? –What are the benefits and limitations of this model? Place your answers on a poster paper.

51. Looking at Models Two Hands Counters and Cups Bundles of Sticks Interlocking cubes Pre-grouped Base-Ten Models Base-Ten Blocks Bean Sticks

52. Looking at Models How do we know when to use the different models with our students? How can we make these models available to students?

53. Finding Meaning Answer the following problem: Derrick has 23 play cars. For his birthday, he gets 34 more cars. How many play cars does he have now? What does the 7 digit mean in the answer? What does the 5 digit mean in the answer? How would your students answer that question?

54. Finding Meaning Let’s visit NCTM Illuminations to see more. http://illuminations.nctm.org/Reflections_preK-2.html

55. Looking Back What mathematics have we explored today? How have these activities shaped your understanding of place value? How would you describe the cognitive demand of the tasks we explored today?

56. Homework Complete the following problems. Look at how the students solved these problems. What place value concepts do they seem to struggle with? How would you address these struggles?

57. Making Sense of Math Wiki http://makingsenseofmath.iu1.wikispaces.net Join the wiki to ensure that you can post comments on the discussion board.

58. Final Project To receive 1 CPE Credit for this course, participants must complete a Final Project. Each participant can choose a Final Project from the following three choices.

59. Final Project Student Work –Collect 5 pieces of student work that demonstrate varying levels of place value misunderstandings on 1 of the provided tasks. –Identify the mathematical misconceptions in the work. –For each artifact, write and implement an action plan that describes how you are going to use the strategies used in this course to address the misunderstanding. –Reflect on the successes and challenges faced when implementing each action plan. Rubric is posted on the wiki.

60. Final Project Unit of Instruction –How might you teach early number concepts different based on your learning from this course? –Write a unit of instruction to incorporate the number sense strategies into your mathematics instruction. –Describe an action plan for implementation. Rubric is posted on the wiki.

61. Final Project Self-Study –Do you have an idea/topic for a project you’d like to explore which is not listed above? –Please discuss your idea with the instructor in order to receive permission to pursue your own line of study.

62. Questions? Amy Lewis alewis@washjeff.edu (724) 250-3330