1. Manipulatives — Concrete and Virtual — Across the Grades

2. Why? multiple representations analogies Hofstadter social nature of knowledge Lave, Wenger, Brown, Zygotsky, conversations/justifications experiences plausible reasoning Polya

3. More Why conceptual knowledge procedural knowledge

4. base 10 blocks 10-sided dice grid paper animations and slide shows paper-and-pencil sketches index cards, egg cartons, etc. What?

5. Concrete Manipulatives race games www.soesd.k12.or.us/files/race_games.pdf www.soesd.k12.or.us/files/place_value_mat.pdf www.soesd.k12.or.us/files/race_games.pdf www.soesd.k12.or.us/files/place_value_mat.pdf diffies www.soesd.k12.or.us/files/diffy_blank.pdf www.soesd.k12.or.us/files/diffy_blank.pdf length vs. area skip counting  arrays –multiplication facts –partial products

6. Virtual Manipulatives base 10 blocks at NLVM http://nlvm.usu.edu/en/nav/vlibrary.html http://nlvm.usu.edu/en/nav/vlibrary.html diffies http://nlvm.usu.edu/en/nav/frames_asid_326_g_2_t_1.html http://nlvm.usu.edu/en/nav/frames_asid_326_g_2_t_1.html length vs. area skip counting  arrays multiplication facts partial products

7. Assembling a base 10 kit

8. Subtraction—create your own http://nlvm.usu.edu/en/nav/frames_asid_155_g_3_t_1.html

9. Diffies—online http://nlvm.usu.edu/en/nav/frames_asid_326_g_2_t_1.html

10. Diffies — on paper http://www.soesd.k12.or.us/files/diffy_blank.pdf get random numbers with virtual 10-sided dice

11. Multiplication: number line  arrays counting  skip counting (length) skip counting  grid grid pieces stack up to  area 7 + 7 + 7 becomes

12. Grid patterns www.amblesideprimary.com/ambleweb/mentalmaths/scribblesquare.html

13. Grid patterns online www.amblesideprimary.com/ambleweb/mentalmaths/scribblesquare.html www.amblesideprimary.com/ambleweb/mentalmaths/scribbletable.html Draw your own!

14. Arrays

15. counting by fives facts with 5 x 5 = 25 6 x 7 (just a blank)

16. with base 10 blocks 12 23 12 x 23 6 30 40 200 276 200 6 40 30

17. Arrays and partial products http://nlvm.usu.edu/en/nav/frames_asid_189_g_3_t_2.html?open=activities specify the dimensions fill it in!

18. partial products x + 2 2x + 3 2x 2 6 4x 3x x 2 2x +3 6 3x 4x 2x 2 2x 2 +7x +6

19. What if x = 10 ? 12 23 200 6 40 30 12 x 23 6 30 40 200 276

20. 3535 1212 Here’s x 3535 1212 x= 3 10

21. Animations mimic procedural knowledge Do-it-yourself PowerPoints www.soesd.k12.or.us/support/training www.soesd.k12.or.us/support/training Flash, Java, and PowerPoints on the Web www.soesd.k12.or.us/files/manipulinks.pdf www.soesd.k12.or.us/files/manipulinks.pdf

22. 0 + – 3 - 5 3 - 5=- 2

23. Area of a triangle (base x height) 2 = 12345 67845 112345 5 3

24. Sliding side c along the base to the center of the square c also makes a parallelogram

25. 1¾ ÷ ½ = 3 ½

26. Graph: 33 +7 m =  3  7  or 3   7  +3 77

27. More Why + Goodbye pedagogical efficiencies We can do things with math in earlier grades that make math easier for students in the long run. analogy and automaticity Algebra is like arithmetic. Mathematicians know this. Our students need to know it too. Larry Francis Computer Information Services, Southern Oregon ESD larry_francis@soesd.k12.or.us and 541.858.6748 www.soesd.k12.or.us/support/trainingwww.soesd.k12.or.us/support/training and www.soesd.k12.or.us/mathwww.soesd.k12.or.us/math