Manipulatives —Concrete and Virtual—Across the Grades
Manipulatives Concrete and Virtual Across the Grades Why? multiple representations analogies Hofstadter social nature of knowledge Lave, Wenger, Brown, Zygotsky,
conversations/justifications experiences plausible reasoning Polya More Why conceptual knowledge
procedural knowledge What? base 10 blocks 10-sided dice grid paper animations and slide shows paper-and-pencil sketches index cards, egg cartons, etc.
Concrete Manipulatives race games 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 length vs. area skip counting arrays
multiplication facts partial products Virtual Manipulatives base 10 blocks at NLVM http://nlvm.usu.edu/en/nav/vlibrary.html diffies http://nlvm.usu.edu/en/nav/frames_asid_326_g_2_t_1.html length vs. area
skip counting arrays multiplication facts partial products Assembling a base 10 kit Subtractioncreate your own http://nlvm.usu.edu/en/nav/frames_asid_155_g_3_t_1.html Diffiesonline
http://nlvm.usu.edu/en/nav/frames_asid_326_g_2_t_1.html Diffies on paper http://www.soesd.k12.or.us/files/diffy_blank.pdf get random numbers with virtual 10-sided dice Multiplication: number line arrays counting skip counting (length)
skip counting grid grid pieces stack up to area 7 + 7 + 7 becomes Grid patterns www.amblesideprimary.com/ambleweb/mentalmaths/scribblesquare.html Grid patterns online www.amblesideprimary.com/ambleweb/mentalmaths/scribblesquare.html www.amblesideprimary.com/ambleweb/mentalmaths/scribbletable.html
Draw your own! Arrays counting by fives facts with 5 x 5 = 25 (just a blank) 6x7
with base 10 blocks 40 12 200 23 6
30 12 x 23 6 30 40 200 276 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 partial products x+2 x
2 2x + 3 6 3x 4x 2x + 3 4x 6
2x2 3x 2x2 2x2 + 7x + 6 What if x = 10 ? 12 23
40 200 6 30 12 x 23 6 30
40 200 276 Here s 1 x 2 1 x
2 3 5 3 5 = 3
10 Animations mimic procedural knowledge Do-it-yourself PowerPoints www.soesd.k12.or.us/support/training Flash, Java, and PowerPoints on the Web www.soesd.k12.or.us/files/manipulinks.pdf 3-5 = -2 -5
-2 3 0 + Area of a triangle =
(base x height) 2 1 2 36 11
5 3 4 5 7 8
4 5 2 3 4 5
Sliding side c along the base to the center of the square c also makes a parallelogram 1 = 3 Graph:
Y 7 +3 3 +7 X m = 3
7 or 3 7 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 [email protected] and 541.858.6748 www.soesd.k12.or.us/support/training and www.soesd.k12.or.us/math
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