1. Page 1 Concrete to Abstract Building Connections to and Conceptual Understanding of the Number System Marcia Torgrude mtorgrude@tie.net mtorgrude@tie.net

2. Page 2 Practice Concrete Multiplication What does multiplication look like using base ten blocks?

3. Page 3 Let’s Try this without the blocks 143x23

4. Page 4 Making Connections through diagram – 23 x 143 200080060 3001209 100 + 40 + 3 20 + 3

5. Page 5 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 3260 6194 326x19

6. Page 6 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?

7. Page 7 Concrete Algebra Explore Build Add Subtract Multiply Divide

8. Page 8 Connecting Number System to Algebra

9. Page 9 Making Connections through diagram – 23 x 143 200080060 3001209 100 + 40 + 3 20 + 3

10. Page 10 Rational Numbers Integers –Charge Model –Linear Model

11. Page 11 · 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

12. Page 12 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

13. Page 13 · 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

14. Page 14 Charge and Linear Model Solve this problem using both methods: Heather 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

15. Page 15 How is this different from the way students built their understanding of algebraic conceps in the past? What common core standards have we been working with? What Standards of Mathematical Practice were present during the activity? Ways to build understanding of Algebraic Concepts

16. Page 16 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