4 + 5 = 9

3 + 1 = 4

7

Addition and Subtraction:From buttons to algebra

Challenge: can Challenge: can you find some you find some

that don’t work?that don’t work?

5x + 3y = 23

2x + 3y = 11

Is there anything interesting about Is there anything interesting about addition and subtraction sentences?addition and subtraction sentences?

Start with 2nd gradeStart with 2nd grade

Math could be spark curiosity!Math could be spark curiosity!

13+ =6 51 4

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

Back to the very beginnings

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

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

Detour: How’d they learn the facts?

One hand, plusOne hand, plus All about 10All about 10

Making itMaking it Adding or subtracting itAdding or subtracting itRememberingRemembering it! it!

Using itUsing it Adjusting with itAdjusting with it Four-hand additionFour-hand addition

And living on a number line…

Numbers as location and as distanceNumbers as location and as distance Subtraction as distance betweenSubtraction as distance between

And living on a number line…

Numbers as location and as distanceNumbers as location and as distance Subtraction as distance betweenSubtraction as distance between Fractions and decimals live there, too.Fractions and decimals live there, too.

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

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

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

25 + 38 = 63

The subtraction 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

25

38

63

-530

60 3

830

25 + 38 = 63 63 – 38 = 25

Uh oh!

The subtraction 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

25

38

63

520

60 3

830

25 + 38 = 63 63 – 38 = 25

50 13

The subtraction 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

25

38

63

223

53 10

830

25 + 38 = 63 63 – 38 = 25

The algebra connection: adding

4 2

3 1

10

4

6

37

4 + 2 = 6

3 + 1 = 4

10+ =7 3

The algebra connection: subtracting

7 3

3 1

6

4

10

24

7 + 3 = 10

3 + 1 = 4

6+ =4 2

The eighth-grade look

5x 3y

2x 3y 11

23 5x + 3y = 23

2x + 3y = 11

12+ =3x 0x = 4

3x 0 12

All from buttons!