MCTM Strategies & Games

Teaching the Arithmetic Facts Using
Strategies and Games
by Joan A. Cotter, Ph.D.
JoanCotter@RightStartMath.com

MCTM
May 4, 2012
Duluth, Minnesota

7 3 8 16 24 32 40

PowerPoint Presentation & Handout
RightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012

Learning the Facts

© Joan A. Cotter, Ph.D., 2012

Learning the Facts
Limited success when:
• Based on counting.
Whether dots, fingers, number lines, or
counting words.


Learning the Facts
counting words.

• Based on rote memory.
Whether by flash cards or timed tests.


Learning the Facts
counting words.

• Based on rote memory.
Whether by flash cards or timed tests.

• Based on skip counting for multiplication facts.


Counting Model
From a child's perspective


Counting Model

Because we’re so familiar with 1, 2, 3, we’ll use letters.

A=1
B=2
C=3
D=4
E = 5, and so forth


Counting Model

F
+E


Counting Model

F
+E

A


Counting Model

F
+E

A B


Counting Model

F
+E

A B C


Counting Model

F
+E

A B C D E F


Counting Model

F
+E

A B C D E F A


Counting Model

F
+E

A B C D E F A B


Counting Model

F
+E

A B C D E F A B C D E


Counting Model

F
+E

A B C D E F A B C D E

What is the sum?
(It must be a letter.)

Counting Model

F
+E
K

A B C D E F G H I J K


Counting Model

E
+G

Add with your fingers.


Counting Model

H
+D

Add without your fingers.


Counting Model

Now memorize the facts!!

G
+D


Counting Model


H
+
G

F
+D


Counting Model


H
+
G

F
+D
D
+C

Counting Model


H
+
G

F
+D
D C
+C +G

Counting Model


H
E

+
G
I

F
+

+D
D C
+C +G

Counting Model

H
–E

Subtract with your fingers.


Counting Model

J
–F

Subtract without using your fingers.


Counting Model

Try skip counting by B’s to T:
B, D, . . . T.


Counting Model

Try skip counting by B’s to T:
B, D, . . . T.

What is D x E?


Memorizing Math


Memorizing Math
Some research:

Percentage Recall
Immediately After 1 day After 4 wks
Rote 32 23 8
Concept 69 69 58


Memorizing Math 9
+7
Flash cards:


Memorizing Math 9
+7
Flash cards:
• Are often used to teach rote.


Memorizing Math 9
+7
Flash cards:
• Are liked by those who don’t need them.


Memorizing Math 9
+7
Flash cards:
• Don’t work for those with learning disabilities.


Memorizing Math 9
+7
Flash cards:
• Give the false impression that math isn’t about
thinking.


Memorizing Math 9
+7
Flash cards:
thinking.
• Often produce stress – children under stress stop
learning.


Memorizing Math 9
+7
Flash cards:
thinking.
• Often produce stress – children under stress stop
learning.
• Are not concrete – they use abstract symbols.


AN ALTERNATIVE:

SUBITIZING
and
GAMES


Subitizing Quantities
Identifying without counting



• Five-month-old infants can subitize to 3.




• Three-year-olds can subitize to 5.




• Three-year-olds can subitize to 5.

• Five-year-olds can subitize 6 to 10 by
using five as a subbase.


Adding
Name the quantity (practice subitizing).


Adding
4+3=


Adding
4+3=7


Characteristics of a Good Game


• Produces learning through playing.


• Incorporates manipulatives.


• Teaches strategies.


• Encourages mental work.


• Detects errors; provides continuous assessment.


• Detects errors; provides continuous assessment.
• Is enjoyable.


Go to the Dump Game
Objective:
To learn the facts that total 10:
1+9
2+8
3+7
4+6
5+5


Go to the Dump Game
Objective:
To learn the facts that total 10:
1+9
2+8
3+7
4+6
5+5
Object of the game:
To collect the most pairs that equal ten.


Go to the Dump Game
6+ = 10


Go to the Dump Game
6 + 4 = 10


Go to the Dump Game

Starting


Go to the Dump Game

72 7 9 5

72 1 3 8 4 6 34 9

Starting


Go to the Dump Game

72 7 9 5

72 1 3 8 4 6 34 9

Finding pairs


Go to the Dump Game

72 7 9 5

4 6

72 1 3 8 34 9

Finding pairs


Go to the Dump Game

72 7 9 5

7 3 4 6

2 1 8 34 9

Finding pairs


Go to the Dump Game

72 7 9 5

2 8 4 6

1 34 9

Finding pairs


Go to the Dump Game

72 7 9 5

2 8 4 6

1 34 9

Playing


Go to the Dump Game
BlueCap, do you
have an3?
have a 3?
72 7 9 5

2 8 4 6

1 34 9

Playing


Go to the Dump Game
BlueCap, do you
have an3?
have a 3?
72 7 9 5 3

2 8 4 6

1 4 9

Playing


Go to the Dump Game
7 3 BlueCap, do you
have an3?
have a 3?
2 7 9 5

2 8 4 6

1 4 9

Playing


Go to the Dump Game
7 3 BlueCap, do you
have an3?
have a 8?
2 7 9 5

2 8 4 6

1 4 9

Playing


Go to the Dump Game
7 3 BlueCap, do you
have an3?
have a 8?
2 7 9 5

2 8 4 6

1 4 9

Go to the dump.
Playing


Go to the Dump Game
7 3 BlueCap, do you
have an3?
have a 8?
2 2 7 9 5

2 8 4 6

1 4 9

Go to the dump.
Playing


Go to the Dump Game
7 3

2 2 7 9 5

2 8 4 6

1 4 9

Playing


Go to the Dump Game
7 3

2 2 7 9 5

2 8 4 6

1 4 9

PinkCap, do you
Playing have a 6?

Go to the Dump Game
7 3

2 2 7 9 5

2 8 4 6

1 4 9

PinkCap, do you
Go to the dump.
Playing have a 6?

Go to the Dump Game
7 3

2 2 7 9 5

2 8 4 6

1 5 4 9

Playing


Go to the Dump Game
7 3

2 2 7 9 5

2 8 4 6

1 5 4 9

YellowCap, do
you have a 9? Playing


Go to the Dump Game
7 3

2 2 7 5

2 8 4 6

1 5 4 9

YellowCap, do


Go to the Dump Game
7 3

2 2 7 5

2 8 4 6

19 5 4 9

YellowCap, do


Go to the Dump Game
7 3

2 2 7 5

2
1 8
9 4 6

5 4 9

Playing


Go to the Dump Game
7 3

2 2 7 5

2
1 8
9 4 6

2 9 1 7 7 5 4 9

Playing


Go to the Dump Game
9 1

4 6 5 5

Winner?


Go to the Dump Game
9
1

4
6 5

Winner?


Go to the Dump Game

9
1

4
6 5

Winner?


Go to the Dump Game

Play it again.


Fact Strategies


Fact Strategies

• A strategy is a way to learn a new fact or
recall a forgotten fact.


Fact Strategies

• A strategy is a way to learn a new fact or
recall a forgotten fact.

• A visualizable representation is part of a
powerful strategy.


Fact Strategies
Complete the Ten

9+5=


Fact Strategies
Complete the Ten

9+5=

Take 1 from
the 5 and give
it to the 9.


Fact Strategies
Complete the Ten

9 + 5 = 14

Take 1 from
the 5 and give
it to the 9.


Fact Strategies
Two Fives

8+6=


Fact Strategies
Two Fives

8+6=
10 + 4 = 14


Fact Strategies
Two Fives

7+5=


Fact Strategies
Two Fives

7 + 5 = 12


Fact Strategies
Going Down

15 – 9 =


Fact Strategies
Going Down

15 – 9 =

Subtract 5;
then 4.


Fact Strategies
Going Down

15 – 9 = 6

Subtract 5;
then 4.


Fact Strategies
Subtract from 10

15 – 9 =


Fact Strategies
Subtract from 10

15 – 9 =

Subtract 9
from 10.


Fact Strategies
Subtract from 10

15 – 9 = 6

Subtract 9
from 10.


Fact Strategies
Going Up

15 – 9 =


Fact Strategies
Going Up

15 – 9 =

Start with 9;
go up to 15.


Fact Strategies
Going Up

15 – 9 =
1+5=6

Start with 9;
go up to 15.


Rows and Columns Game
Objective:
To find a total of 15 by adding 2, 3, or 4
cards in row or column.


Objective:
To find a total of 15 by adding 2, 3, or 4
cards in row or column.

Object of the game:
To collect the most cards.


8 7 1 9

6 4 3 3

2 2 5 6

6 3 8 8


1 9

6 4 3 3

6 3 8 8


7 6 1 9

6 4 3 3

2 1 5 1

6 3 8 8


1

6 4 3 3

1 5 1

3 8 8


Multiplication Strategies
Basic facts


Basic facts

6× 4=
(6 taken 4 times)


Basic facts

9× 3=


Basic facts

9× 3=
30


Basic facts

9× 3=
30 – 3 = 27


Basic facts

4× 8=


Basic facts

4× 8=
20 + 12 = 32


Basic facts

7× 7=


Basic facts

7× 7=
25


Basic facts

7× 7=
25 + 10 + 10


Basic facts

7× 7=
25 + 10 + 10
+ 4 = 49


Multiples Patterns
Twos

2 4 6 8 10
12 14 16 18 20


Multiples Patterns
Twos

2 4 6 8 10
12 14 16 18 20

The ones repeat in the second row.


Multiples Patterns
Fours

4 8 12 16 20
24 28 32 36 40

The ones repeat in the second row.


Multiples Patterns
Sixes and Eights

6 12 18 24 30
36 42 48 54 60

8 16 24 32 40
48 56 64 72 80


Multiples Patterns
Sixes 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.

Multiples Patterns
Sixes and Eights

6 12 18 24 30 6× 4

36 42 48 54 60

8 16 24 32 40
48 56 64 72 80

6 × 4 is the fourth number (multiple).

Multiples Patterns
Sixes 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).

Multiples Patterns
Nines

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 Patterns
Threes

3 6 9
2 15 18
21 24 27
30

The 3s have several patterns:
Observe the ones.

Multiples Patterns
Threes

3 6 9
12 15 18
21 24 27
30

Observe the ones.

Multiples Patterns
Threes

3 6 9
12 15 18
21 24 27
30

The tens are the same in each row.

Multiples Patterns
Threes

3 6 9
12 15 18
21 24 27
30

Add the digits in the columns.

Multiples Patterns
Threes

3 6 9
12 15 18
21 24 27
30

Add the “opposites.”

Multiples Patterns
Sevens

7 14 21
28 35 42
49 56 63
70

The 7s have the 1, 2, 3… pattern.


Multiples Patterns
Sevens

7 14 21
28 35 42
49 56 63
70

Look at the tens.


Multiples Memory


Multiples Memory
Objective:
To help the players learn the
multiples patterns.


Multiples Memory
Objective:
To help the players learn the multiples patterns.

Object of the game:
To be the first player to collect all ten
cards of a multiple in order.


Multiples Memory

7 14 21
28 35 42
49 56 63
70

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


Multiples Memory

8 16 24 32 40
48 56 64 72 80

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


Multiples Memory
7 14 21
28 35 42 8 16 24 32 40
49 56 63 48 56 64 72 80
70

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70


Multiples Memory
7 14 21
28 35 42
49 56 63
70

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70


Multiples Memory
7 14 21
28 35 42
49 56 63
70

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63 14
70


Multiples Memory
8 16 24 32 40
48 56 64 72 80

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70


Multiples Memory
8 16 24 32 40
48 56 64 72 80

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70
40


Multiples Memory
7 14 21
28 35 42
49 56 63
70

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70

8


Multiples Memory
8 16 24 32 40
48 56 64 72 80

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70

8


Multiples Memory
8 16 24 32 40
48 56 64 72 80

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70 8


Multiples Memory
8 16 24 32 40
48 56 64 72 80

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63 56
70 8


Multiples Memory
7 14 21
28 35 42
49 56 63
70

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70 8


Multiples Memory
7 14 21
28 35 42
49 56 63
70

7
7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70 8


Multiples Memory
7 14 21
28 35 42
49 56 63
70

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70 8

7


Multiples Memory
7 14 21
28 35 42
49 56 63
70

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63 14
70 8

7


Multiples Memory
7 14 21
28 35 42
49 56 63
70

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70 8

7 14


Multiples Memory
7 14 21
28 35 42
49 56 63
70

24
7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70 8

7 14


Multiples Memory
7 14 21
28 35 42 8 16 24 32 40
49 56 63 48 56 64 72 80
70

7 14 21 8 16 24 32 40
28 35 42 48 56 64 72 80
49 56 63
70 8

7 14


Multiplication Memory
Objective:
To help the players master the
multiplication facts.


Objective:
To help the players master the
multiplication facts.

Object of the game:
To collect the most cards by matching
the multiplier with the product.



Materials Needed:


1 2 3 4 5

6 7 8 9 10

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


3
1 2 3 4 5
3 6 9
12 15 18
6 7 8 9 10 21 24 27
30

Materials Needed:
• A set of product cards (3s used here)


3
1 2 3 4 5 3x
3 6 9
12 15 18
6 7 8 9 10 21 24 27
30

Materials Needed:
• A stickie note with “3 x” written on it


3
1 2 3 4 5 3x
3 6 9
12 15 18
6 7 8 9 10 21 24 27
30 =

Materials Needed:
• A stickie with “3 x” written on it
• A stickie with “=” written on it


3
1 2 3 4 5 3x
3 6 9
12 15 18
6 7 8 9 10 21 24 27
30 =

Materials Needed:
• A stickie with “3 x” written on it
• A stickie with “=” written on it
• A manipulative with groups of five



3x =

3 6 9
12 15 18
21 24 27
30


5
3x =

3 6 9
12 15 18
21 24 27
30


5
3x =

3 taken 5 times
equals 15.

3 6 9
12 15 18
21 24 27
30


5 21
3x =

3 taken 5 times
equals 15.

3 6 9
12 15 18
21 24 27
30



3x =
7

3 6 9
12 15 18
21 24 27
30



3x =
7

3 taken 7 times
equals 21.

3 6 9
12 15 18
21 24 27
30


21
3x =
7

3 taken 7 times
equals 21.

3 6 9
12 15 18
21 24 27
30



3x =

3 taken 7 times
equals 21.

3 6 9
12 15 18 7 21
21 24 27
30



3x =
2

3 6 9
12 15 18 7 21
21 24 27
30



3x =
2
3

3 6 9
12 15 18 7 21
21 24 27
30



3x =
2
3

3 taken 3 times
equals 9.

3 6 9
12 15 18 7 21
21 24 27
30



3x =
2
3 12

3 taken 3 times
equals 9.

3 6 9
12 15 18 7 21
21 24 27
30



3x =

3 6 9
12 15 18 7 21
21 24 27
30


5
3x =

3 6 9
12 15 18 7 21
21 24 27
30


5
3x =

3 taken 5 times
equals 15.

3 6 9
12 15 18 7 21
21 24 27
30


5
3x =
15

3 taken 5 times
equals 15.

3 6 9
12 15 18 7 21
21 24 27
30



3x =

3 taken 5 times
equals 15.

5 15
3 6 9
12 15 18 7 21
21 24 27
30



3x =

5 15
3 6 9
12 15 18 7 21
21 24 27
30



3x =

8 24
3 6 9
12 15 18 1 3
21 24 27
30


Framing the Future of Mathematics in
Minnesota

Math in Minnesota starts with the youngest.

Let’s build on their natural ability to subitize.

Keep joy in math; use games, not flash cards.

Help them to use their minds to visualize.


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