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

2. www.tie.net Develop understanding and ideas to promote deeper understanding of the number system. 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

3. www.tie.net 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.

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5. www.tie.net Person facing screen – gives clues Person facing away – guesses the word or phrase

6. www.tie.net Reasoning Tools Abstract Precision Structure

7. www.tie.net Standards of Mathematical Practice Persevere Questioning Concrete Modeling

8. www.tie.net The Practice Standards are descriptions of the fundamental skills needed to “do” mathematics.

9. www.tie.net Practice Standards describe what it means for students to demonstrate proficiency in mathematics. They are our new “basic skills.” Content Standards are the “what” of mathematics

10. www.tie.net 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.”

11. www.tie.net Adding Subtracting Multiplying Dividing With Base 10 Blocks

12. www.tie.net Using Modeling to Make Sense of Mathematical Procedures Modeling addition with Base 10 blocks Race to 100

13. www.tie.net Using Modeling to Make Sense of Mathematical Procedures Modeling subtraction with Base 10 blocks 302 − 178 412 - 189

14. www.tie.net 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

15. www.tie.net What does multiplication look like using base ten blocks?

16. www.tie.net 143x23

17. www.tie.net 200080060 3001209 100 + 40 + 3 20 + 3

18. www.tie.net 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

19. www.tie.net 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. Base-10 blocks as an area model emphasize the distributive property and provide a visual representation to the partial products of the multiplication algorithm.

20. www.tie.net What does division look like using base ten blocks?

21. www.tie.net How can we use the base ten block array model to help us with division?

22. www.tie.net Do you think that using base-10 blocks helps to give meaning to the division algorithm? How?

23. www.tie.net Use of friendly or “benchmark” numbers Partial quotient division Multiplication for division – use what we know

24. www.tie.net 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=218 http://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

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26. www.tie.net Equivalence Addition Subtraction Multiplication Division

27. www.tie.net Using Pattern Blocks Using Cuisenaire Rods

28. www.tie.net Using Arrays

29. www.tie.net Using Cuisenaire Rods Using Sets

30. www.tie.net http://www.learner.org/courses/learningmath/number/session8/part_b/modelin g.html http:// www.learner.org/courses/learningmath/number/session8/part_b/try.html Another Strategy for Division http://www.learner.org/courses/learningmath/number/session8/video.html

31. www.tie.net 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

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33. www.tie.net Integers – Charge Model – Linear Model

34. www.tie.net · 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. www.tie.net · 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. www.tie.net 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. www.tie.net 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. www.tie.net 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?

39. www.tie.net Explore Build Add Subtract Multiply Divide

40. www.tie.net Connecting Number System to Algebra

41. www.tie.net 200080060 3001209 100 + 40 + 3 20 + 3

42. www.tie.net 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.htm l?open=activities&from=category_g_3_t_2.html http://nlvm.usu.edu/en/nav/frames_asid_189_g_3_t_2.htm l?open=activities&from=category_g_3_t_2.html NLVM Scales -Positives http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.htm l?open=instructions&from=category_g_3_t_2.html http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.htm l?open=instructions&from=category_g_3_t_2.html NLVM Scales – Negatives http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.htm l?open=instructions&from=category_g_3_t_2.html http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.htm l?open=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. www.tie.net 3 Ideas you will use with your learners 2 Aha’s you had today 1 Question you still have – Please provide your email for a response