4 + 5 = 9 3 + 1 = 4 7 addition and subtraction: from buttons to algebra challenge: can you find some...
TRANSCRIPT
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!