1. Page 1 www.tie.net Building Understanding of the Number System Through Hands-On Experiences Marcia Torgrude K-12 Math Specialist mtorgrude@tie.net

2. Page 2 Develop understanding and ideas to promote deeper understanding of the number system within the Common Core Develop hands-on strategies to build understanding of place value. Develop hands-on strategies to help promote understanding of fractions. Use tools to help students work fluently with rational numbers. Experience online tools for the number system Outcomes for Today

3. Page 3 I Have….. Who Has Let’s play! What does this have to do with learning? Where does it fit the common core standards? What about the Standards of Mathematical Practice? Search I Have…Who Has online.

4. Page 4 Standards of Mathematical Practice

5. Page 5 What does it mean to “do mathematics?” What does it mean to “do mathematics?” The Standards of Mathematical Practice are descriptions of the fundamental skills needed to “do” mathematics.

6. Page 6 What does it mean to “do mathematics?” What does it mean to “do mathematics?” Standards of Practice describe what it means for students to demonstrate proficiency in mathematics. They are our new “basic skills.” Content Standards are the “what” of mathematics

7. Page 7 We must get past the idea of mathematics as a collection of algorithms, steps, or procedures. Just getting answers, although important, is not “doing mathematics.” “Doing mathematics”

8. Page 8 Working with Whole Numbers Adding Subtracting Multiplying Dividing With Base 10 Blocks

9. Page 9 “Doing mathematics” Using Modeling to Make Sense of Mathematical Procedures Modeling addition with Base 10 blocks Place It!

10. Page 10 “Doing mathematics” Using Modeling to Make Sense of Mathematical Procedures Modeling subtraction with Base 10 blocks ONLY BUILD the beginning number 302 − 178 412 - 189

11. Page 11 Multiplication and Division Identify strategies that individuals can use to solve multi-digit multiplication and division problems in sense-making ways Connect concepts to “standard algorithms” Discuss teaching strategies that enhance a child’s understanding

12. Page 12 Practice Concrete Multiplication What does multiplication look like using base ten blocks?

13. Page 13 Let’s Try this without the blocks 143x23

14. Page 14 Making Connections through diagram – 23 x 143 200080060 3001209 100 + 40 + 3 20 + 3

15. Page 15 Making Connections – Getting to the algorithm 300+20+6 X 10+9 3000 200 60 2700 180 54 6194 326 x 19 54 180 2700 60 200 3000 6194 326 x 19 2934 326 6194

16. Page 16 Understanding the abstract Do you think that using base-10 blocks helps to give meaning to the multiplication algorithm? How? One common concern when using models is that students will not make connections between the concrete models, their representations, and the mathematical concept. Did we make the connections? How?

17. Page 17 Practice Concrete Division What does division look like using base ten blocks?

18. Page 18 American Idol is back! If they travel to 11 different cities and can only take a total of 132 people to Hollywood, how many people can be selected from each city? How can we use the base ten block and the array model to help us with division?

19. Page 19 Understanding the abstract Do you think that using base-10 blocks helps to give meaning to the division algorithm? How?

20. Page 20 Another Strategy for Division Use of friendly or “benchmark” numbers Partial quotient division: Multiplication for division – use what we know

21. Page 21 Partial Quotient Division Our family took a trip and my dad told me we drove a total of 2,112 miles in 6 days. How many miles per day did we travel on the average? What friendly numbers did you use?

22. Page 22 Virtual Whole Number Tools http://illuminations.nctm.org/ –Activities –Calculation Nation Tens Frame - http://illuminations.nctm.org/ActivityDetail.aspx?ID=75http://illuminations.nctm.org/ActivityDetail.aspx?ID=75 Grouping and Grazing - http://illuminations.nctm.org/ActivityDetail.aspx?ID=218http://illuminations.nctm.org/ActivityDetail.aspx?ID=218 Adding with base 10 Blocks - http://nlvm.usu.edu/en/nav/frames_asid_154_g_1_t_1.html?from=category_g_1_t_1. html http://nlvm.usu.edu/en/nav/frames_asid_154_g_1_t_1.html?from=category_g_1_t_1. html Subtracting with Base 10 Blocks - http://nlvm.usu.edu/en/nav/frames_asid_155_g_1_t_1.html?from=category_g_1_t_1. html http://nlvm.usu.edu/en/nav/frames_asid_155_g_1_t_1.html?from=category_g_1_t_1. html Primary Krypto - http://illuminations.nctm.org/ActivityDetail.aspx?ID=173http://illuminations.nctm.org/ActivityDetail.aspx?ID=173 Product Game - http://illuminations.nctm.org/ActivityDetail.aspx?ID=29http://illuminations.nctm.org/ActivityDetail.aspx?ID=29 Times Table -http:// illuminations.nctm.org/ActivityDetail.aspx?ID=155http:// illuminations.nctm.org/ActivityDetail.aspx?ID=155

23. Page 23 What were the goals of the activities? What common core standards have we been working on? What Standards of Mathematical Practice were present during the activities? Small Group Discussion

24. Page 24 Working with Fractions Equivalence Addition Subtraction Multiplication Division

25. Page 25 Fraction Equivalence, Adding, and Subtracting Using Pattern Blocks

26. Page 26 Fraction Equivalence, Adding, and Subtracting Using Pattern Blocks

27. Page 27 Modeling equivalence, adding, subtracting, multiplying, and dividing Using Cuisenaire Rods http://www.learner.org/vod/vod_window.html? pid=1853http://www.learner.org/vod/vod_window.html? pid=1853 http://www.teachersdomain.org/assets/wgbh/r ttt12/rttt12_int_cuisenaire/index.htmlhttp://www.teachersdomain.org/assets/wgbh/r ttt12/rttt12_int_cuisenaire/index.html

28. Page 28 Fraction Using Cuisenaire Rods http://www.learner.org/courses/learningmath/number /session8/part_b/modeling.htmlhttp://www.learner.org/courses/learningmath/number /session8/part_b/modeling.html If we are trying to work with fourths and thirds what will our new whole need to be? Using Cuisenaire rods model: 1/3 + 1/4 1/3 - 1/4 1/3 x 1/4 1/3 ÷ 1/4

29. Page 29 Why do we invert and multiply to divide? How does this work? ¾ ÷ 5/6= B How many 5/6 are in ¾?  x 5/6 = ¾ B x 5/6 = ¾ I need to multiply 5/6 by its reciprocal to solve for B or box. B x 5/6 x 6/5 = ¾ x 6/5 B = ¾ x 6/5

30. Page 30 Fractions, Decimals, and Percents Make sense through grids

31. Page 31 Virtual Fractions Equivalent Fractions - http://illuminations.nctm.org/ActivityDetail.aspx?ID=80http://illuminations.nctm.org/ActivityDetail.aspx?ID=80 Fraction Models - http://illuminations.nctm.org/ActivityDetail.aspx?ID=11http://illuminations.nctm.org/ActivityDetail.aspx?ID=11 Fraction Game - http://illuminations.nctm.org/ActivityDetail.aspx?ID=18http://illuminations.nctm.org/ActivityDetail.aspx?ID=18 Fraction Pieces - http://nlvm.usu.edu/en/nav/frames_asid_274_g_3_t_1.html?open=activities&from= category_g_3_t_1.html http://nlvm.usu.edu/en/nav/frames_asid_274_g_3_t_1.html?open=activities&from= category_g_3_t_1.html Fraction Adding - http://nlvm.usu.edu/en/nav/frames_asid_106_g_3_t_1.html?from=category_g_3_t_ 1.html http://nlvm.usu.edu/en/nav/frames_asid_106_g_3_t_1.html?from=category_g_3_t_ 1.html Fraction Comparing - http://nlvm.usu.edu/en/nav/frames_asid_159_g_3_t_1.html?from=category_g_3_t_ 1.html http://nlvm.usu.edu/en/nav/frames_asid_159_g_3_t_1.html?from=category_g_3_t_ 1.html Fraction Equivalence - http://nlvm.usu.edu/en/nav/frames_asid_105_g_3_t_1.html?from=category_g_3_t_ 1.html http://nlvm.usu.edu/en/nav/frames_asid_105_g_3_t_1.html?from=category_g_3_t_ 1.html Fraction Rectangle Multiplication - http://nlvm.usu.edu/en/nav/frames_asid_194_g_3_t_1.html?from=category_g_3_t_ 1.html http://nlvm.usu.edu/en/nav/frames_asid_194_g_3_t_1.html?from=category_g_3_t_ 1.html

32. Page 32 What were the goals of the activities? What common core standards have we been working on? What Standards of Mathematical Practice were present during the activities? Small Group Discussion

33. Page 33 Rational Numbers Integers –Charge Model –Linear Model

34. Page 34 · Charge Model Use your positive/negative counters to represent the following numbers using at least the number of tiles listed. You can challenge yourself by using more than the minimum number of tiles. Be prepared to share and prove your solution. Ways to build understanding of Integers

35. Page 35 · Linear Model Matt earns merits and demerits at his school. One day he earned 3 merits for his math game, 2 demerits for being late to class, 1 merit for being courteous, 5 demerits for arguing with his teacher, and 2 merits for helping another student. If he began the day with 4 merits, how many did he have at the end of the day? Ways to build understanding of Integers

36. Page 36 Model the following problems with your counters and sketch your work using a plus sign for positive and a negative sign for negative counters: 3 + 5 +3 + (-5) -3 + 5 -3 + (-5) What do you notice? Make some generalizations about the rules for adding integers. Now consider: -3 - (-5) What generalization can you make? Ways to build understanding of Integers

37. Page 37 Charge and Linear Model Solve this problem using both methods: He ather started the month with $12. She spent $5 on a game, but realized that she forgot to pay her annual club dues so she wrote a check for $15 because her dad said he would loan her enough money to cover the check. How much does Heather have to borrow from her dad? Ways to build understanding of Integers

38. Page 38 How is this different from the way students built their understanding of positive/negative integers in the past? What common core standard have we been working on? What Standards of Mathematical Practice were present during the activity? Ways to build understanding of Integers

39. Page 39 Concrete Algebra Explore Build Add Subtract Multiply Divide

40. Page 40 Connecting Number System to Algebra

41. Page 41 Making Connections through diagram – 23 x 143 200080060 3001209 100 + 40 + 3 20 + 3

42. Page 42 Virtual Algebra Illuminations Algebra Tiles - http://illuminations.nctm.org/ActivityDetail.aspx?ID=216 http://illuminations.nctm.org/ActivityDetail.aspx?ID=216 NLVM algebra Tiles - http://nlvm.usu.edu/en/nav/frames_asid_189_g_3_t_2.html?op en=activities&from=category_g_3_t_2.html http://nlvm.usu.edu/en/nav/frames_asid_189_g_3_t_2.html?op en=activities&from=category_g_3_t_2.html NLVM Scales -Positives http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?op en=instructions&from=category_g_3_t_2.html http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?op en=instructions&from=category_g_3_t_2.html NLVM Scales – Negatives http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?op en=instructions&from=category_g_3_t_2.htmlhttp://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?op en=instructions&from=category_g_3_t_2.html Pan Balance - Numbers http://illuminations.nctm.org/ActivityDetail.aspx?id=26 Pan Balance - Expressions http://illuminations.nctm.org/ActivityDetail.aspx?ID=10

43. Page 43 Thank You - Exit Card I am reaffirmed because I already… The big idea I will work on is… I still need help with….