Transcript From Buttons to Algebra: Learning the Ideas and Language

Addition and Subtraction:
From buttons to algebra
Challenge: can
you find some
that don’t work?
4+5=9
5x + 3y = 23
3+1=4
2x + 3y = 11
5
17 + 46 = 13
Math could be spark curiosity!
 Is there anything interesting about
addition and subtraction sentences?
 Start with 2nd grade
© E. Paul Goldenberg 2008
This 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
a small pile of buttons.
Natural to sort.
We help children refine
and extend what is already
natural.
Back to the very beginnings
blue
gray
6
small
Children can also summarize.
4
large
7
3
10
“Data” from the buttons.
Abstraction
If we substitute numbers for the original objects…
blue
gray
small
6
4
2
6
large
4
3
1
4
10
7
3
10
7
3
A Cross Number Puzzle
Don’t always start with the question!
7
6
13
5
3
8
12
9
21
Detour: How’d they learn the facts?
One hand, plus
 All about 10
 Making it
 Adding or subtracting it
 Remembering it!
 Using it
 Adjusting with it
 Four-hand addition

And living on a number line…
Numbers as location and as distance
 Subtraction as distance between

And living on a number line…
Numbers as location and as distance
 Subtraction as distance between
 Fractions and decimals live there, too.

Building the addition algorithm
Only multiples of 10 in yellow. Only less than 10 in blue.
20
5
25
30
8
38
50
13 63
Relating addition and subtraction
4
10
2
3
6
7
3
4
1
1
4
3
7
6
3
2
10
4
Ultimately, building the addition and subtraction algorithms
The addition algorithm
Only multiples of 10 in yellow. Only less than 10 in blue.
20
5
25
30
8
38
50
13 63
25 + 38 = 63
The subtraction algorithm
Only multiples of 10 in yellow. Only less than 10 in blue.
20
5
25
60
3
63
30
8
38
30
8
38
50
13 63
30
-5
25
25 + 38 = 63
63 – 38 = 25
The subtraction algorithm
Only multiples of 10 in yellow. Only less than 10 in blue.
20
5
25
50
60 13
3 63
30
8
38
30
8
38
50
13 63
20
5
25
25 + 38 = 63
63 – 38 = 25
The subtraction algorithm
Only multiples of 10 in yellow. Only less than 10 in blue.
20
5
25
53 10 63
30
8
38
30
8
38
50
13 63
23
2
25
25 + 38 = 63
63 – 38 = 25
The algebra connection: adding
4
2
6
4+2=6
3
1
4
3+1=4
7
3
10
7 + 3 = 10
The algebra connection: subtracting
7
3
10
7 + 3 = 10
3
1
4
3+1= 4
4
2
6
4 +2 = 6
The eighth-grade look
5x
3y 23
5x + 3y = 23
2x
3y
11
2x + 3y = 11
3x
0
12
3x + 0 = 12
x=4
All from buttons!