IMF: Visualizing and Montessori Math PART 2

How Visualization Enhances Montessori Mathematics PART 2

by Joan A. Cotter, Ph.D.JoanCotter@RightStartMath.com

8 16 24 32 40

PowerPoint Presentation & HandoutRightStartMath.com >Resources

Montessori FoundationConference

Friday, Nov 2, 2012Sarasota, Florida

Multiplication StrategiesBasic facts

6 4 =(6 taken 4 times)

30 – 3 = 27

20 + 12 = 32

25 + 10 + 10

25 + 10 + 10 + 4 = 49

Multiplication on the AL AbacusCommutative property

5 6 = 6 5

This method was used in the Middle Ages, rather than memorize the facts > 5 5.

Multiplication on the AL AbacusFor facts > 5 5

For facts > 5 5Multiplication on the AL Abacus

Tens: 20+ 30

7 8 =50 +

Tens: 20+ 30

7 8 =50 +

Tens: Ones:20+ 30

7 8 =50 +

Tens: Ones:20+ 30

7 8 =50 +

Tens: Ones: 3 2

20+ 30

7 8 =50 + 6

Tens: Ones: 3 26

20+ 30

7 8 =50 + 6 = 56

Tens: Ones: 3 26

20+ 30

Tens: 40+ 20

9 7 =60 +

Tens: 40+ 20

9 7 =60 +

Tens: Ones:40+ 20

9 7 =60 +

Tens: Ones:40+ 20

9 7 =60 +

Tens: Ones: 1 3

40+ 20

9 7 =60 + 3

Tens: Ones:40+ 20

9 7 =60 + 3 = 63

Tens: Ones: 1 3

40+ 20

The Multiplication Board1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

The Multiplication Board

The Multiplication Board1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

Multiples PatternsTwos

2 4 6 8 10

12 14 16 18 20

Multiples PatternsTwos

2 4 6 8 10

12 14 16 18 20

The ones repeat in the second row.

Multiples PatternsFours

4 8 12 16 20

24 28 32 36 40

The ones repeat in the second row.

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 12 18 24 30

36 42 48 54 60

8 16 24 32 40

48 56 64 72 80

6 12 18 24 30

36 42 48 54 60

8 16 24 32 40

48 56 64 72 80

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.

6 12 18 24 30

36 42 48 54 60

8 16 24 32 40

48 56 64 72 80

6 12 18 24 30

36 42 48 54 60

8 16 24 32 40

48 56 64 72 80

6 12 18 24 30

36 42 48 54 60

8 16 24 32 40

48 56 64 72 80

6 12 18 24 30

36 42 48 54 60

8 16 24 32 40

48 56 64 72 80

6 12 18 24 30

36 42 48 54 60

8 16 24 32 40

48 56 64 72 80

6 4 is the fourth number (multiple).

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).

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.

Multiples PatternsThrees

12 15 18

21 24 27

The 3s have several patterns: Observe the ones.

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

The 3s have several patterns: The tens are the same in each row.

12 15 18

21 24 27

The 3s have several patterns: Add the digits in the columns.

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

The 3s have several patterns: Add the “opposites.”

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

Multiples PatternsSevens

7 14 21

28 35 42

49 56 63

The 7s have the 1, 2, 3… pattern.

7 14 21

28 35 42

49 56 63

7 14 21

28 35 42

49 56 63

7 14 21

28 35 42

49 56 63

7 14 21

28 35 42

49 56 63

Look at the tens.

7 14 21

28 35 42

49 56 63

Look at the tens.

7 14 21

28 35 42

49 56 63

Look at the tens.

Multiples Memory

Objective: To help the players learn the multiples patterns.

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.

Multiples Memory

The 7s envelope contains 10 cards, each with one of the numbers listed.

7 14 2128 35 4249 56 63

Multiples Memory

The 8s envelope contains 10 cards, each with one of the numbers listed.

8 16 24 32 4048 56 64 72 80

Multiples Memory

7 14 2128 35 4249 56 6370

8 16 24 32 4048 56 64 72 80

Multiples Memory

7 14 2128 35 4249 56 6370

8 16 24 32 4048 56 64 72 80

Multiples Memory

7 14 2128 35 4249 56 6370

8 16 24 32 4048 56 64 72 80

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

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

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

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

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

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

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

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

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

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

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

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

7 14 2128 35 4249 56 6370

Multiples Memory

8 16 24 32 4048 56 64 72 80

7 14 2128 35 4249 56 6370

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.

Materials Needed:

Materials Needed:• Ten basic cards, numbered 1 to 10

8 9 10

Materials Needed:• Ten basic cards, numbered 1 to 10• A set of product cards (3s used here)

8 9 10

6 61827

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

8 9 10

3 x 6 61827

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

8 9 10

3 x 6 61827

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

8 9 10

3 x 6 61827

6 61827

3 taken 5 timesequals 15.

6 61827

215 15

6 61827

215 15

6 61827

Multiplication Tables

A rectangle 3 6

squares

Fraction Chart

Fraction Chart1

16171819

Fraction Chart

Stairs (Unit fractions)

Fraction Chart

A hyperbola.

Fraction Chart

Without numbers.

Fraction Naming

One divided into 2 equal parts.

Fraction Naming

1 One whole

Fraction Naming

1 One whole

divided into

Fraction Naming

1 One whole

divided into3 three equal parts

Fraction Naming

divided by3 three

Fraction Naming

divided by3 three

In English, except for half, we use ordinal numbers to name fractions.

Fraction Naming

1Read as “one-third.”

Fraction Naming

1Read as “one-third.”

Unit Fraction War

Objective: To help the children realize a unit fraction decreases as the denominator increases.

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.

Unit Fraction War

Fraction Naming1

Fraction Naming11

Fraction Naming

1Two is

Fraction Naming

Read as “two-thirds.”

1Two is

Fraction Chart1

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How many fourths in a whole?

Fraction Chart1

16171819

Fraction Chart1

16171819

Fraction Chart1

16171819

Fraction Chart1

16171819

Fraction Chart1

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How many eighths in a whole?

Concentrating on One Game

Objective: To help the children realize that 5 fifths, 8 eighths, and so forth, make a whole.

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.

Concentrating on One

Fraction Chart1

16171819

Which is more, 3/4 or 4/5?

Fraction Chart1

16171819

Fraction Chart

16171819

Fraction Chart

16171819

Fraction Chart

An interesting pattern.

Partial Chart

1 2 3 4 5 6

Fraction War

Objective: To practice comparing ones, halves, fourths, and eighths in preparation for reading a ruler.

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.

Fraction War1

Fraction War

Fraction War1

Fraction War

Fraction War1

Fraction War

Fraction War1

16171819

Fraction Chart

How many fourths equal a half?

16171819

Fraction Chart

How many fourths equal a half?

16171819

Fraction Chart

How many fourths equal a half? Eighths?

16171819

Fraction Chart

How many fourths equal a half? Eighths?

16171819

Fraction Chart

How many fourths equal a half? Eighths? Sevenths?

16171819

Fraction Chart

How many fourths equal a half? Eighths? Sevenths?

16171819

Fraction Chart

What is 1/4 plus 1/8?

16171819

Fraction Chart

16171819

Fraction Chart

16171819

Fraction Chart

What is 1/4 plus 1/8? [3/8]

16171819

Fraction Chart

If the chance of rain is 3/4, the chance it won’t rain is 1/4.

16171819

Fraction Chart

If the chance of rain is 3/4, the chance it won’t rain is 1/4.

16171819

Fraction Chart

Improper fraction, 9/8, is seen as 1 and 1/8.

16171819

Fraction Chart

What is 1/2 of 1/2?

16171819

Fraction Chart

What is 1/2 of 1/2?

16171819

Fraction Chart

What is 1/3 of 1/2?

Fraction Chart

What is 1/3 of 1/2?

16171819

Circle Model

Are we comparing angles, arcs, or area?

Circle Model

Try to compare 4/5 and 5/6 with this model.

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

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

Circle ModelDifficulties

• 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.

Simplifying Fractions

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

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

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

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

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

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

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

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

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

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

21212828

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

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

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

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

21212828

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

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

21212828

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

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

21212828

45457272

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

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

21212828

45457272

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

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

21212828

45457272

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

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

12 12 1616

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

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

12 12 1616

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

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

12 12 1616

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

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

12 12 1616

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

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

12 12 1616

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

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

12 12 1616

How Visualization Enhances Montessori Mathematics PART 2

by Joan A. Cotter, Ph.D.JoanCotter@RightStartMath.com

8 16 24 32 40

PowerPoint Presentation & HandoutRightStartMath.com >Resources

Montessori FoundationConference

Friday, Nov 2, 2012Sarasota, Florida

IMF: Visualizing and Montessori Math PART 2

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Bibliography Montessori Adolescent Education · Maria Montessori) (London: The Montessori Society, 1939 ) 1941 Montessori, Maria, Door het Kind naar een Nieuwe Wereld, lectures translated