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Manipulatives — Concrete and Virtual — Across the Grades.

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Presentation on theme: "Manipulatives — Concrete and Virtual — Across the Grades."— Presentation transcript:

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 diffies length vs. area skip counting  arrays –multiplication facts –partial products

6 Virtual Manipulatives base 10 blocks at NLVM diffies length vs. area skip counting  arrays multiplication facts partial products

7 Assembling a base 10 kit

8 Subtraction—create your own

9 Diffies—online

10 Diffies — on paper 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 becomes

12 Grid patterns

13 Grid patterns online 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 x

17 Arrays and partial products specify the dimensions fill it in!

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

19 What if x = 10 ? x

20 Here’s x x= 3 10

21 Animations mimic procedural knowledge Do-it-yourself PowerPoints Flash, Java, and PowerPoints on the Web

22 0 + – =- 2

23 Area of a triangle (base x height) 2 =

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 and and


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