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Manipulatives — Concrete and Virtual — Across the Grades

Manipulatives — Concrete and Virtual — Across the Grades

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Manipulatives — Concrete and Virtual — Across the Grades

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  1. Manipulatives—Concrete and Virtual—Across the Grades

  2. Why? • multiple representations • analogiesHofstadter • social nature of knowledgeLave, Wenger, Brown, Zygotsky, • conversations/justifications • experiences • plausible reasoningPolya

  3. More Why conceptualknowledge proceduralknowledge

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

  5. Concrete Manipulatives • race gameswww.soesd.k12.or.us/files/race_games.pdfwww.soesd.k12.or.us/files/place_value_mat.pdf • diffieswww.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 NLVMhttp://nlvm.usu.edu/en/nav/vlibrary.html • diffieshttp://nlvm.usu.edu/en/nav/frames_asid_326_g_2_t_1.html • length vs. areaskip 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. 7 + 7 + 7 becomes Multiplication:number line  arrays • counting  skip counting (length) • skip counting  grid • grid pieces stack up to  area

  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 fivesfacts with 5 x 5 = 25 (just a blank) 6 x 7

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

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

  18. 4x 6 2x2 3x partial products x 2 2x + 3 6 3x 4x 2x2 2x2 + 7x + 6 x + 2 2x + 3

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

  20. 3 5 1 2 3 5 1 2 3 10 Here’s x x =

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

  22. 3 - 5 0 – + = - 2 - 5 -2 3

  23. 5 1 2 3 4 5 6 7 8 4 5 3 11 2 3 4 5 Area of a triangle (base x height) = 2

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

  25. 1¾ ÷ ½ = 3 ½

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

  27. More Why + Goodbye • pedagogical efficienciesWe can do things with math in earlier grades that make math easier for students in the long run. • analogy and automaticityAlgebra 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/training and www.soesd.k12.or.us/math