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© Joan A. Cotter, Ph.D., 2012
How Visualization Enhances Montessori Mathematics PART 2
by Joan A. Cotter, [email protected]
8 16 24 32 40
PowerPoint Presentation & HandoutRightStartMath.com >Resources
Montessori FoundationConference
Friday, Nov 2, 2012Sarasota, Florida
35
25
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
6 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
6 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
6 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
6 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
6 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
9 3 =
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
9 3 =
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
9 3 =
30
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
9 3 =
30 – 3 = 27
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
4 8 =
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
4 8 =
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
4 8 =
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
4 8 =
20 + 12 = 32
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
7 7 =
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
7 7 =
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
7 7 =
25
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
7 7 =
25 + 10 + 10
© Joan A. Cotter, Ph.D., 2012
Multiplication StrategiesBasic facts
7 7 =
25 + 10 + 10 + 4 = 49
© Joan A. Cotter, Ph.D., 2012
Multiplication on the AL AbacusCommutative property
5 6 = 6 5
© Joan A. Cotter, Ph.D., 2012
Multiplication on the AL AbacusCommutative property
5 6 = 6 5
© Joan A. Cotter, Ph.D., 2012
Multiplication on the AL AbacusCommutative property
5 6 = 6 5
© Joan A. Cotter, Ph.D., 2012
Multiplication on the AL AbacusCommutative property
5 6 = 6 5
© Joan A. Cotter, Ph.D., 2012
7 8 =
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
© Joan A. Cotter, Ph.D., 2012
7 8 =
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
For facts > 5 5Multiplication on the AL Abacus
© Joan A. Cotter, Ph.D., 2012
7 8 =
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens:
© Joan A. Cotter, Ph.D., 2012
7 8 =
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens:
© Joan A. Cotter, Ph.D., 2012
7 8 =
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: 20+ 30
© Joan A. Cotter, Ph.D., 2012
7 8 =50 +
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: 20+ 30
50
© Joan A. Cotter, Ph.D., 2012
7 8 =50 +
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones:20+ 30
50
© Joan A. Cotter, Ph.D., 2012
7 8 =50 +
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones:20+ 30
50
© Joan A. Cotter, Ph.D., 2012
7 8 =50 +
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones: 3 2
20+ 30
50
© Joan A. Cotter, Ph.D., 2012
7 8 =50 + 6
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones: 3 26
20+ 30
50
© Joan A. Cotter, Ph.D., 2012
7 8 =50 + 6 = 56
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones: 3 26
20+ 30
50
© Joan A. Cotter, Ph.D., 2012
9 7 =
Multiplication on the AL AbacusFor facts > 5 5
© Joan A. Cotter, Ph.D., 2012
9 7 =
Multiplication on the AL AbacusFor facts > 5 5
© Joan A. Cotter, Ph.D., 2012
9 7 =
Multiplication on the AL AbacusFor facts > 5 5
Tens:
© Joan A. Cotter, Ph.D., 2012
9 7 =
Multiplication on the AL AbacusFor facts > 5 5
Tens:
© Joan A. Cotter, Ph.D., 2012
9 7 =
Multiplication on the AL AbacusFor facts > 5 5
Tens: 40+ 20
© Joan A. Cotter, Ph.D., 2012
9 7 =60 +
Multiplication on the AL AbacusFor facts > 5 5
Tens: 40+ 20
60
© Joan A. Cotter, Ph.D., 2012
9 7 =60 +
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones:40+ 20
60
© Joan A. Cotter, Ph.D., 2012
9 7 =60 +
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones:40+ 20
60
© Joan A. Cotter, Ph.D., 2012
9 7 =60 +
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones: 1 3
40+ 20
60
© Joan A. Cotter, Ph.D., 2012
9 7 =60 + 3
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones:40+ 20
60
1 3
3
© Joan A. Cotter, Ph.D., 2012
9 7 =60 + 3 = 63
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones: 1 3
3
40+ 20
60
© Joan A. Cotter, Ph.D., 2012
The Multiplication Board1 2 3 4 5 6 7 8 9 10
6
6 4
© Joan A. Cotter, Ph.D., 2012
1 2 3 4 5 6 7 8 9 10
6
The Multiplication Board
6 4
© Joan A. Cotter, Ph.D., 2012
The Multiplication Board1 2 3 4 5 6 7 8 9 10
7
7 7
© Joan A. Cotter, Ph.D., 2012
1 2 3 4 5 6 7 8 9 10
7
The Multiplication Board
7 7
© Joan A. Cotter, Ph.D., 2012
The Multiplication Board
7 7
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsTwos
2 4 6 8 10
12 14 16 18 20
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsTwos
2 4 6 8 10
12 14 16 18 20
The ones repeat in the second row.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsFours
4 8 12 16 20
24 28 32 36 40
The ones repeat in the second row.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
6 4
6 4 is the fourth number (multiple).
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80 8 7
8 7 is the seventh number (multiple).
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsNines
9 18 27 36 45
90 81 72 63 54
The second row is written in reverse order.Also the digits in each number add to 9.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: The tens are the same in each row.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the digits in the columns.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the digits in the columns.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the digits in the columns.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.
© Joan A. Cotter, Ph.D., 2012
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
Objective: To help the players learn the multiples patterns.
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
Object of the game: To be the first player to collect all ten cards of a multiple in order.
Objective: To help the players learn the multiples patterns.
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
The 7s envelope contains 10 cards, each with one of the numbers listed.
7 14 2128 35 4249 56 63
70
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
The 8s envelope contains 10 cards, each with one of the numbers listed.
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
14
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
40
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
8856
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7
14
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14
24
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
7 14 2128 35 4249 56 6370
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2012
7 14 2128 35 4249 56 6370
Multiples Memory
8 16 24 32 4048 56 64 72 80
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
Objective: To help the players master themultiplication facts.
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
Objective: To help the players master themultiplication facts.
Object of the game: To collect the most cards by matching the multiplier with the product.
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
Materials Needed:
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
Materials Needed:• Ten basic cards, numbered 1 to 10
3 4 5
8 9 10
2
7
1
6
© Joan A. Cotter, Ph.D., 2012
3
Multiplication Memory
Materials Needed:• Ten basic cards, numbered 1 to 10• A set of product cards (3s used here)
3 4 5
8 9 10
2
7
1
6
6 61827
91221
3
30
1524
© Joan A. Cotter, Ph.D., 2012
3
Multiplication Memory
Materials Needed:• Ten basic cards, numbered 1 to 10• A set of product cards (3s used here) • A stickie note with “3 x” written on it
3 4 5
8 9 10
2
7
1
6
3 x 6 61827
91221
3
30
1524
© Joan A. Cotter, Ph.D., 2012
3
Multiplication Memory
Materials Needed:• Ten basic cards, numbered 1 to 10• A set of product cards (3s used here) • A stickie with “3 x” written on it• A stickie with “=” written on it
3 4 5
8 9 10
2
7
1
6
3 x 6 61827
91221
3
30
1524
© Joan A. Cotter, Ph.D., 2012
3
Multiplication Memory
Materials Needed:• Ten basic cards, numbered 1 to 10• A set of product cards (3s used here) • A stickie with “3 x” written on it• A stickie with “=” written on it• A manipulative with groups of five
3 4 5
8 9 10
2
7
1
6
3 x 6 61827
91221
3
30
1524
© Joan A. Cotter, Ph.D., 2012
3 x
Multiplication Memory
6 61827
91221
3
30
1524
© Joan A. Cotter, Ph.D., 2012
3 x
Multiplication Memory
6 61827
91221
3
30
1524
© Joan A. Cotter, Ph.D., 2012
3 x
Multiplication Memory
6 61827
91221
3
30
1524
5
© Joan A. Cotter, Ph.D., 2012
3 x
Multiplication Memory
6 61827
91221
3
30
1524
5
© Joan A. Cotter, Ph.D., 2012
3 x
Multiplication Memory
6 61827
91221
3
30
1524
5
© Joan A. Cotter, Ph.D., 2012
3 x
Multiplication Memory
6 61827
91221
3
30
1524
5
3 taken 5 timesequals 15.
© Joan A. Cotter, Ph.D., 2012
3 x
Multiplication Memory
21
6 61827
91221
3
30
1524
5
3 taken 5 timesequals 15.
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
3 x
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
3 x
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
3 x7
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
3 x7
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
3 x7
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
7
3 taken 7 timesequals 21.
3 x
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
21
6 61827
91221
3
30
1524
73 x
3 taken 7 timesequals 21.
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
3 x
7 21
3 taken 7 timesequals 21.
© Joan A. Cotter, Ph.D., 2012
2
Multiplication Memory
6 61827
91221
3
30
1524
3 x
7 21
© Joan A. Cotter, Ph.D., 2012
2
Multiplication Memory
6 61827
91221
3
30
1524
33 x
7 21
© Joan A. Cotter, Ph.D., 2012
2
Multiplication Memory
6 61827
91221
3
30
1524
33 x
7 21
© Joan A. Cotter, Ph.D., 2012
2
Multiplication Memory
6 61827
91221
3
30
1524
33 x
7 21
3 taken 3 timesequals 9.
© Joan A. Cotter, Ph.D., 2012
2
Multiplication Memory
12
6 61827
91221
3
30
1524
33 x
7 21
3 taken 3 timesequals 9.
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
3 x
7 21
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
3 x
7 21
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
5
3 x
7 21
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
5
3 x
7 21
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
5
3 x
7 21
3 taken 5 timesequals 15.
© Joan A. Cotter, Ph.D., 2012
Multiplication Memory
6 61827
91221
3
30
1524
5
153 x
7 21
3 taken 5 timesequals 15.
© Joan A. Cotter, Ph.D., 2012
7
Multiplication Memory
6 61827
91221
3
30
1524
3 x
215 15
3 taken 5 timesequals 15.
© Joan A. Cotter, Ph.D., 2012
7
Multiplication Memory
6 61827
91221
3
30
1524
3 x
215 15
© Joan A. Cotter, Ph.D., 2012
1
Multiplication Memory
6 61827
91221
3
30
1524
3 x
38 24
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
A rectangle 3 6
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
A rectangle 3 6
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
A rectangle 3 6
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
A rectangle 3 6
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
A rectangle 3 6
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
4 7
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
4 7
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
9 3
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
9 3
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
6 6
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
6 6
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
4 7
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
4 7
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
4 7
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
7 9
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
7 9
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
7 9
9 7
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
7 9
9 7
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
squares
© Joan A. Cotter, Ph.D., 2012
Multiplication Tables
squares
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
© Joan A. Cotter, Ph.D., 2012
Fraction Chart1
12
12
13
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18
19
110
13
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19
14
15
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16171819
15
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17
17
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18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
12
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
12
13
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
12
13
14
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
12
13
14
15
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
12
13
14
15
16
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
12
13
14
15
1716
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
12
13
14
15
17
18
16
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
12
13
14
15
17
18
16
19
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
12
13
14
15
17
18
110
16
19
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
12
13
14
15
17
18
110
16
19
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
1
12
13
14
15
17
18
110
16
19
A hyperbola.
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
Without numbers.
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
One divided into 2 equal parts.
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
One divided into 2 equal parts.
One divided into 3 equal parts.
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
One divided into 2 equal parts.
One divided into 3 equal parts.
One divided into 4 equal parts.
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
One divided into 3 equal parts.
1 One whole
1
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
One divided into 3 equal parts.
1 One whole
divided into
1
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
One divided into 3 equal parts.
1 One whole
divided into3 three equal parts
1
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
One divided into 3 equal parts.
1 One
divided by3 three
1
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
One divided into 3 equal parts.
1 One
divided by3 three
In English, except for half, we use ordinal numbers to name fractions.
1
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
One divided into 3 equal parts.
1Read as “one-third.”
3
1
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
One divided into 3 equal parts.
1Read as “one-third.”
3
113
13
13
© Joan A. Cotter, Ph.D., 2012
Unit Fraction War
Objective: To help the children realize a unit fraction decreases as the denominator increases.
© Joan A. Cotter, Ph.D., 2012
Unit Fraction War
Object of the game: To collect all, or most, of the cards with the greater unit fraction.
Objective: To help the children realize a unit fraction decreases as the denominator increases.
© Joan A. Cotter, Ph.D., 2012
Unit Fraction War
© Joan A. Cotter, Ph.D., 2012
15
14
Unit Fraction War
© Joan A. Cotter, Ph.D., 2012
15
14
Unit Fraction War
© Joan A. Cotter, Ph.D., 2012
Unit Fraction War
© Joan A. Cotter, Ph.D., 2012
Unit Fraction War
118
© Joan A. Cotter, Ph.D., 2012
Unit Fraction War
118
© Joan A. Cotter, Ph.D., 2012
Unit Fraction War
© Joan A. Cotter, Ph.D., 2012
Unit Fraction War
16
16
© Joan A. Cotter, Ph.D., 2012
Unit Fraction War
16
16
© Joan A. Cotter, Ph.D., 2012
Unit Fraction War
16
16
13
14
© Joan A. Cotter, Ph.D., 2012
Unit Fraction War
© Joan A. Cotter, Ph.D., 2012
1
Fraction Naming1
13
13
13
© Joan A. Cotter, Ph.D., 2012
Fraction Naming11
13
13
13
© Joan A. Cotter, Ph.D., 2012
1113
13
13
Fraction Naming
© Joan A. Cotter, Ph.D., 2012
1113
13
13
Fraction Naming
1Two
3s
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
1Two is
323
s
1113
13
13
© Joan A. Cotter, Ph.D., 2012
Fraction Naming
Read as “two-thirds.”
1Two is
323
s
1113
13
13
© Joan A. Cotter, Ph.D., 2012
Fraction Chart1
12
12
13
14
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16
17
18
19
110
13
13
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19
14
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16171819
15
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18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many fourths in a whole?
© Joan A. Cotter, Ph.D., 2012
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
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19
14
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16
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16171819
15
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18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many fourths in a whole?
© Joan A. Cotter, Ph.D., 2012
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
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17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many fourths in a whole?
© Joan A. Cotter, Ph.D., 2012
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
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18
14
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16171819
15
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17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many fourths in a whole?
© Joan A. Cotter, Ph.D., 2012
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
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16171819
15
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18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many fourths in a whole?
© Joan A. Cotter, Ph.D., 2012
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
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18
14
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16171819
15
16
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17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many eighths in a whole?
© Joan A. Cotter, Ph.D., 2012
Concentrating on One Game
Objective: To help the children realize that 5 fifths, 8 eighths, and so forth, make a whole.
© Joan A. Cotter, Ph.D., 2012
Concentrating on One Game
Object of the game: To find the pairs that make a whole.
Objective: To help the children realize that 5 fifths, 8 eighths, and so forth, make a whole.
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
53
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
53
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
53
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
53
25
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
38
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
38
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
38
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
38
78
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
58
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
58
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
58
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
38
58
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
© Joan A. Cotter, Ph.D., 2012
Concentrating on One
© Joan A. Cotter, Ph.D., 2012
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Which is more, 3/4 or 4/5?
© Joan A. Cotter, Ph.D., 2012
Fraction Chart1
12
12
13
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15
16
17
18
19
110
13
13
14
15
16
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18
19
14
15
16
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18
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15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Which is more, 3/4 or 4/5?
© Joan A. Cotter, Ph.D., 2012
Fraction Chart
Which is more, 7/8 or 8/9?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
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18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
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18
19
14
15
16
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16171819
15
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18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
Which is more, 7/8 or 8/9?
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
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18
14
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16171819
15
16
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18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
An interesting pattern.
© Joan A. Cotter, Ph.D., 2012
Partial Chart
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
Partial Chart
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
Partial Chart
© Joan A. Cotter, Ph.D., 2012
Partial Chart
1 2 3 4 5 6
© Joan A. Cotter, Ph.D., 2012
Fraction War
Objective: To practice comparing ones, halves, fourths, and eighths in preparation for reading a ruler.
© Joan A. Cotter, Ph.D., 2012
Fraction War
Object of the game: To capture all the cards.
Objective: To practice comparing ones, halves, fourths, and eighths in preparation for reading a ruler.
© Joan A. Cotter, Ph.D., 2012
Fraction War1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
Fraction War
14
18
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
14
18
© Joan A. Cotter, Ph.D., 2012
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
14
18
© Joan A. Cotter, Ph.D., 2012
Fraction War1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
Fraction War
34
58
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
34
58
© Joan A. Cotter, Ph.D., 2012
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
34
58
© Joan A. Cotter, Ph.D., 2012
Fraction War1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
Fraction War
34
34
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
Fraction War
34
34
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
Fraction War
34
34
38
14
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
Fraction War1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
Fraction War1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
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17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half?
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half?
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
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17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half? Eighths?
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half? Eighths?
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
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18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half? Eighths? Sevenths?
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
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17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half? Eighths? Sevenths?
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
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16171819
15
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17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
What is 1/4 plus 1/8?
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
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17
18
14
15
16171819
15
16
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17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
What is 1/4 plus 1/8?
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
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17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
What is 1/4 plus 1/8?
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
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18
14
15
16171819
15
16
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17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
What is 1/4 plus 1/8? [3/8]
© Joan A. Cotter, Ph.D., 2012
112
12
13
14
15
16
17
18
19
110
13
13
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15
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19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
If the chance of rain is 3/4, the chance it won’t rain is 1/4.
© Joan A. Cotter, Ph.D., 2012
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Fraction Chart
If the chance of rain is 3/4, the chance it won’t rain is 1/4.
© Joan A. Cotter, Ph.D., 2012
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Fraction Chart
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Improper fraction, 9/8, is seen as 1 and 1/8.
© Joan A. Cotter, Ph.D., 2012
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Fraction Chart
What is 1/2 of 1/2?
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© Joan A. Cotter, Ph.D., 2012
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Fraction Chart
What is 1/2 of 1/2?
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© Joan A. Cotter, Ph.D., 2012
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Fraction Chart
What is 1/3 of 1/2?
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© Joan A. Cotter, Ph.D., 2012
Fraction Chart
What is 1/3 of 1/2?
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© Joan A. Cotter, Ph.D., 2012
Circle Model
© Joan A. Cotter, Ph.D., 2012
Circle Model
Are we comparing angles, arcs, or area?
© Joan A. Cotter, Ph.D., 2012
Circle Model
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Try to compare 4/5 and 5/6 with this model.
© Joan A. Cotter, Ph.D., 2012
Circle Model
Experts in visual literacy say that comparing quantities in pie charts is difficult because most people think linearly. It is easier to compare along a straight line than compare pie slices. askoxford.com
© Joan A. Cotter, Ph.D., 2012
Circle Model
Experts in visual literacy say that comparing quantities in pie charts is difficult because most people think linearly. It is easier to compare along a straight line than compare pie slices. askoxford.com
Specialists also suggest refraining from using more than one pie chart for comparison.
www.statcan.ca
© Joan A. Cotter, Ph.D., 2012
Circle ModelDifficulties
© Joan A. Cotter, Ph.D., 2012
• Perpetuates cultural myth fractions are < 1.
Circle ModelDifficulties
© Joan A. Cotter, Ph.D., 2012
• Perpetuates cultural myth fractions are < 1.
• Does not give the child the “big picture.”
Circle ModelDifficulties
© Joan A. Cotter, Ph.D., 2012
• Perpetuates cultural myth fractions are < 1.
• Does not give the child the “big picture.”
• Limits understanding of fractions: they are more than “a part of a whole or part of a set.”
Circle ModelDifficulties
© Joan A. Cotter, Ph.D., 2012
• Perpetuates cultural myth fractions are < 1.
• Does not give the child the “big picture.”
• Limits understanding of fractions: they are more than “a part of a whole or part of a set.”
• Makes it difficult for the child to see how fractions relate to each other.
Circle ModelDifficulties
© Joan A. Cotter, Ph.D., 2012
Simplifying Fractions
© Joan A. Cotter, Ph.D., 2012
1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 9 12 15 18 21 24 27
4 8 12 16 20 24 28 32 36
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6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
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Simplifying Fractions
© Joan A. Cotter, Ph.D., 2012
1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 9 12 15 18 21 24 27
4 8 12 16 20 24 28 32 36
10
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6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
© Joan A. Cotter, Ph.D., 2012
1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 9 12 15 18 21 24 27
4 8 12 16 20 24 28 32 36
10
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6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
© Joan A. Cotter, Ph.D., 2012
1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 9 12 15 18 21 24 27
4 8 12 16 20 24 28 32 36
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6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
© Joan A. Cotter, Ph.D., 2012
1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 9 12 15 18 21 24 27
4 8 12 16 20 24 28 32 36
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6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
21212828
© Joan A. Cotter, Ph.D., 2012
1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 9 12 15 18 21 24 27
4 8 12 16 20 24 28 32 36
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6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
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Simplifying Fractions
© Joan A. Cotter, Ph.D., 2012
1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 9 12 15 18 21 24 27
4 8 12 16 20 24 28 32 36
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6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
21212828
© Joan A. Cotter, Ph.D., 2012
1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 9 12 15 18 21 24 27
4 8 12 16 20 24 28 32 36
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6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
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Simplifying Fractions
21212828
© Joan A. Cotter, Ph.D., 2012
1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
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4 8 12 16 20 24 28 32 36
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7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
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Simplifying Fractions
21212828
45457272
© Joan A. Cotter, Ph.D., 2012
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7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
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Simplifying Fractions
21212828
45457272
© Joan A. Cotter, Ph.D., 2012
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7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
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Simplifying Fractions
21212828
45457272
© Joan A. Cotter, Ph.D., 2012
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4 8 12 16 20 24 28 32 36
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7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
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Simplifying Fractions
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© Joan A. Cotter, Ph.D., 2012
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Simplifying Fractions
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© Joan A. Cotter, Ph.D., 2012
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Simplifying Fractions
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© Joan A. Cotter, Ph.D., 2012
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8 16 24 32 40 48 56 64 72 80
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Simplifying Fractions
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© Joan A. Cotter, Ph.D., 2012
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Simplifying Fractions
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© Joan A. Cotter, Ph.D., 2012
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Simplifying Fractions
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© Joan A. Cotter, Ph.D., 2012
How Visualization Enhances Montessori Mathematics PART 2
by Joan A. Cotter, [email protected]
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PowerPoint Presentation & HandoutRightStartMath.com >Resources
Montessori FoundationConference
Friday, Nov 2, 2012Sarasota, Florida
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