Manipulatives — Concrete and Virtual — Across the Grades.

Manipulatives— Concrete and Virtual

—Across the Grades

Why?

• multiple representations• analogies

Hofstadter

• social nature of knowledgeLave, Wenger, Brown, Zygotsky,

• conversations/justifications

• experiences• plausible reasoning

Polya

More Why

conceptualknowledge

proceduralknowledge

• base 10 blocks

• 10-sided dice

• grid paper

• animations and slide shows

• paper-and-pencil sketches

• index cards, egg cartons, etc.

What?

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

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

Assembling a base 10 kit

Subtraction—create your own

http://nlvm.usu.edu/en/nav/frames_asid_155_g_3_t_1.html

Diffies—online

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 onlinewww.amblesideprimary.com/ambleweb/mentalmaths/scribblesquare.html www.amblesideprimary.com/ambleweb/mentalmaths/scribbletable.html

Draw your own!

Arrays

counting by fivesfacts with 5 x 5 = 25

6 x 7 (just a blank)

with base 10 blocks

12

23

12 x 23

63040

200276

200

640

30

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!

partial productsx +

2

2x + 3

2x2

64x

3x

x 22x + 3

63x 4x

2x2

2x2 + 7x + 6

What if x = 10 ?

12

23

200

640

30

12 x 23

63040

200276

35

12

Here’s

x

35

12

x = 310

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

0

+–

3

- 5

3 - 5 = - 2

-2

Area of a triangle

(base x height)

2=

1 2 3 4 5

6 7 8 4 5

11 2 3 4 5

5

3

Sliding side c along the baseto the center of the square

c

also makes a parallelogram

1¾ ÷ ½ = 3 ½

X

Y

Graph:

3

+7m = 3 7

or 3 7

+3

7

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 FrancisComputer Information Services, Southern Oregon ESD

[email protected] and 541.858.6748www.soesd.k12.or.us/support/training and www.soesd.k12.or.us/math