Multiplication and division

with

a comprehensive K-5 curriculum from

© E. Paul Goldenberg 2008This presentation may be shown for professional development purposes only, and may not be sold, distributed, or altered.

Foundations for multiplication

The The logic logic of addition:of addition:any-order any-groupingany-order any-grouping

The The image image of multiplication: of multiplication: intersections, arrays, and areaintersections, arrays, and area

The The logiclogic of multiplication: of multiplication:any-order any-groupingany-order any-grouping

Sorting in Kindergarten

Picture a young child with Picture a young child with a small pile of buttons.a small pile of buttons.

Natural to sort.Natural to sort.

We help children refine We help children refine and extend what is already and extend what is already natural.natural.

6

4

7 3 10

Back to the very beginnings

Children can also summarize.Children can also summarize.

““Data” from the buttons.Data” from the buttons.

blue gray

large

small

large

small

blue gray

If we substitute numbers for the original objects…If we substitute numbers for the original objects…

Abstraction

6

4

7 3 10

6

4

7 3 10

4 2

3 1

GR 1 (GR K – exposure)

A Cross Number Puzzle

5

Don’t always start with the question!Don’t always start with the question!

21

8

13

912

7 6

3

GR 1

Building the addition algorithmOnly multiples of 10 in yellow. Only less than 10 in blue.Only multiples of 10 in yellow. Only less than 10 in blue.

63

38

25

1350

20 5

830

50 + 13 = 63

25 + 38

63

GR 2

Relating addition and subtraction

6

4

7 3 10

4 2

3 16

4

7 3 10

4 2

3 1

Ultimately, building the addition and subtraction algorithms

GR 2

Foundations for multiplication

The The logic logic of addition:of addition:any-order any-groupingany-order any-grouping

The The image image of multiplication: of multiplication: intersections, arrays, and areaintersections, arrays, and area

The The logiclogic of multiplication: of multiplication:any-order any-groupingany-order any-grouping

Naming intersections, first gradePut a red house at the intersection of A street and N avenue.

Where is the green house?

How do we go fromthe green house tothe school?

Combinatorics, 1st week 2nd grade

How many two-letter words can you make, How many two-letter words can you make, starting with a red letterstarting with a red letter and and ending with a purple letterending with a purple letter??

a i s n t

Multiplication, coordinates, phonics?

a i s n t

asin

at

GR 2

Multiplication, coordinates, phonics?

w s ill it inkb p

st ick ack ingbr tr

GR 2

How will they learn the facts?

Doubling and halving Doubling and halving (also for fractions!)(also for fractions!)

Multiplying and dividing by 10 Multiplying and dividing by 10 (decimals!)(decimals!)

Arrays, arrays, arraysArrays, 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 The logic logic of addition:of addition:any-order any-groupingany-order any-grouping

The The image image of multiplication: of multiplication: intersections, arrays, and areaintersections, arrays, and area

The The logiclogic of multiplication: of multiplication:any-order any-groupingany-order any-grouping

Representing 22 × 17

22

17

GR 3

Representing the algorithm

20

10

2

7

GR 4

Representing the algorithm

20

10

2

7

200

140

20

14

2217

200140

20

x

14374

Recording the process using all

four partial products

GR 4

Building the common algorithm

20

10

2

7

200

140

20

14

220

154

37434340

GR 4S

umm

ariz

ing

acro

ssSummarizing down

Product

Building the common algorithm

10

7

200

140

20

14

220

154

2217

154220374

×

1

20 2

GR 4

37434340

37434340

Building the common algorithm

20

10

2

7

200

140

20

14

220

154

172234

340374

×

1

GR 4

Getting ready for algebra

20

10

2

7

200

140

20

14

GR 4

2217

200140

20

x

14374

We record the process using all

four partial products

We don’t summarize!

In algebra (d + 2) (r + 7) =

d

r

2

7

dr

7d

2r

14

Middle School

dr + 7d + 2r + 14

We record the process using all

four partial products

We don’t summarize!

Multiplication, zillions of dots22

17 374

22 × 17 = 374

GR 3

Multiplication, area22

17 374

22 × 17 = 374

GR 3

Representing division (not yet the algorithm)

“ “Oh! Oh! Division is Division is just just unmultipli-unmultipli-cation!”cation!”

22

17 374

374 ÷ 17 = 222217 374

GR 3

The division algorithm

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

17 170 34 374

10 2

170

10 22

34

204-170

-170

-340

374

374374

? × 17 = 374

Representing the algorithm

GR 4

Building the standard algorithm

Easy once you Easy once you understand the understand the underlying mathunderlying math

Practice old skills Practice old skills to build a table of to build a table of multiplesmultiples

GR 5 (GR 4 exposure)

1 2 3 4 5 6 7 8 937 37 74 111 148 185 222 259 296 333

37

Total = 999

37 999

Left

Left

tens ones

Summary: 999 ÷ 37 =

20 20740

740259

259

77

2 5 9027

GR 5