Multiplication and division with a comprehensive K-5 curriculum from © E. Paul Goldenberg 2008 This presentation may be shown for professional development purposes only, and may not be sold, distributed, or altered.

Foundations for multiplication The logic of addition: any-order any-grouping The logic of addition: any-order any-grouping The image of multiplication: intersections, arrays, and area The image of multiplication: intersections, arrays, and area The logic of multiplication: any-order any-grouping The logic of multiplication: any-order any-grouping

Sorting in Kindergarten Picture a young child with a small pile of buttons. Natural to sort. We help children refine and extend what is already natural.

Back to the very beginnings Children can also summarize. “Data” from the buttons. bluegray large small

large small bluegray If we substitute numbers for the original objects… Abstraction GR 1 (GR K – exposure)

A Cross Number Puzzle 5 Don’t always start with the question! GR 1

Building the addition algorithm Only multiples of 10 in yellow. Only less than 10 in blue = GR 2

Relating addition and subtraction Ultimately, building the addition and subtraction algorithms GR 2

Foundations for multiplication The logic of addition: any-order any-grouping The logic of addition: any-order any-grouping The image of multiplication: intersections, arrays, and area The image of multiplication: intersections, arrays, and area The logic of multiplication: any-order any-grouping The logic of multiplication: any-order any-grouping

Naming intersections, first grade Put a red house at the intersection of A street and N avenue. Where is the green house? How do we go from the green house to the school?

Combinatorics, 1 st week 2 nd grade How many two-letter words can you make, starting with a red letter and ending with a purple letter? How many two-letter words can you make, starting with a red letter and ending with a purple letter? aisnt

Multiplication, coordinates, phonics? aisnt asas inin atat GR 2

Multiplication, coordinates, phonics? wsillitink bp stickacking brtr GR 2

How will they learn the facts? Doubling and halving (also for fractions!) Doubling and halving (also for fractions!) Multiplying and dividing by 10 (decimals!) Multiplying and dividing by 10 (decimals!) Arrays, arrays, arrays Arrays, arrays, arrays Begins in Kindergaten Begins in GR 1 Throughout TM!

Intersections, arrays, area 3 4 = 4 3 GR 2

Similar questions, similar image Four skirts and three shirts: how many outfits? Five flavors of ice cream and four toppings: how many sundaes? (one scoop, one topping) How many 2-block towers can you make from four differently-colored Lego blocks? GR 2

Foundations for multiplication The logic of addition: any-order any-grouping The logic of addition: any-order any-grouping The image of multiplication: intersections, arrays, and area The image of multiplication: intersections, arrays, and area The logic of multiplication: any-order any-grouping The logic of multiplication: any-order any-grouping

Representing 22 × GR 3

Representing the algorithm GR 4

Representing the algorithm x Recording the process using all four partial products GR 4

Building the common algorithm GR 4 Summarizing across Summarizing down Product

Building the common algorithm × GR

Building the common algorithm × 1 GR 4

Getting ready for algebra GR x We record the process using all four partial products We don’t summarize!

In algebra (d + 2) (r + 7) = d r 2 7 dr 7d7d 2r2r 14 Middle School dr + 7d + 2r + 14 We record the process using all four partial products We don’t summarize!

Multiplication, zillions of dots × 17 = 374 GR 3

Multiplication, area × 17 = 374 GR 3

Representing division (not yet the algorithm) “Oh! Division is just unmultipli- cation!” “Oh! Division is just unmultipli- cation!” ÷ 17 = GR 3

The division algorithm If the multiplication algorithm is really clear, division comes almost for free. If the multiplication algorithm is really clear, division comes almost for free.

? × 17 = 374 Representing the algorithm GR 4

Building the standard algorithm Easy once you understand the underlying math Easy once you understand the underlying math Practice old skills to build a table of multiples Practice old skills to build a table of multiples GR 5 (GR 4 exposure)

Total = Left tensones Summary: 999 ÷ 37 = GR 5