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© Joan A. Cotter, Ph.D., 2012
Teaching the Arithmetic Facts Using Strategies and Games
MCTMMay 4, 2012
Duluth, Minnesota
by Joan A. Cotter, Ph.D.JoanCotter@RightStartMath.com
7 3 8 16 24 32 40
PowerPoint Presentation & HandoutRightStartMath.com >Resources
© Joan A. Cotter, Ph.D., 2012
Learning the Facts
© Joan A. Cotter, Ph.D., 2012
Learning the Facts
• Based on counting.
Limited success when:
Whether dots, fingers, number lines, or counting words.
© Joan A. Cotter, Ph.D., 2012
Learning the Facts
• Based on counting.
Limited success when:
• Based on rote memory.
Whether dots, fingers, number lines, or counting words.
Whether by flash cards or timed tests.
© Joan A. Cotter, Ph.D., 2012
Learning the Facts
• Based on counting.
• Based on skip counting for multiplication facts.
Limited success when:
• Based on rote memory.
Whether dots, fingers, number lines, or counting words.
Whether by flash cards or timed tests.
© Joan A. Cotter, Ph.D., 2012
Counting ModelFrom a child's perspective
© Joan A. Cotter, Ph.D., 2012
Counting ModelFrom a child's perspective
Because we’re so familiar with 1, 2, 3, we’ll use letters.
A = 1B = 2C = 3D = 4E = 5, and so forth
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
F + E
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
A
F + E
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
A B
F + E
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
A CB
F + E
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
A FC D EB
F + E
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
AA FC D EB
F + E
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
A BA FC D EB
F + E
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
A C D EBA FC D EB
F + E
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
A C D EBA FC D EB
F + E
What is the sum?(It must be a letter.)
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
K
G I J KHA FC D EB
F + E
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
E + G
Add with your fingers.
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
H+ D
Add without your fingers.
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
Now memorize the facts!!
G + D
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
Now memorize the facts!!
G + D
H + F
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
Now memorize the facts!!
G + D
H + F
D + C
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
Now memorize the facts!!
G + D
H + F
C + G
D + C
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
E
+ I
Now memorize the facts!!
G + D
H + F
C + G
D + C
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
H – E
Subtract with your fingers.
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
J – F
Subtract without using your fingers.
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
Try skip counting by B’s to T: B, D, . . . T.
© Joan A. Cotter, Ph.D., 2012
Counting Model From a child's perspective
Try skip counting by B’s to T: B, D, . . . T.
What is D x E?
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote 32 23 8
Concept 69 69 58
Some research:
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote 32 23 8
Concept 69 69 58
Some research:
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote 32 23 8
Concept 69 69 58
Some research:
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote 32 23 8
Concept 69 69 58
Some research:
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote 32 23 8
Concept 69 69 58
Some research:
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote 32 23 8
Concept 69 69 58
Some research:
© Joan A. Cotter, Ph.D., 2012
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote 32 23 8
Concept 69 69 58
Some research:
© Joan A. Cotter, Ph.D., 2012
Memorizing Math 9 + 7Flash cards:
© Joan A. Cotter, Ph.D., 2012
• Are often used to teach rote.
Memorizing Math 9 + 7Flash cards:
© Joan A. Cotter, Ph.D., 2012
• Are often used to teach rote.
• Are liked by those who don’t need them.
Memorizing Math 9 + 7Flash cards:
© Joan A. Cotter, Ph.D., 2012
• Are often used to teach rote.
• Are liked by those who don’t need them.
• Don’t work for those with learning disabilities.
Memorizing Math 9 + 7Flash cards:
© Joan A. Cotter, Ph.D., 2012
• Are often used to teach rote.
• Are liked by those who don’t need them.
• Don’t work for those with learning disabilities.
• Give the false impression that math isn’t about thinking.
Memorizing Math 9 + 7Flash cards:
© Joan A. Cotter, Ph.D., 2012
• Are often used to teach rote.
• Are liked by those who don’t need them.
• Don’t work for those with learning disabilities.
• Give the false impression that math isn’t about thinking.
• Often produce stress – children under stress stop learning.
Memorizing Math 9 + 7Flash cards:
© Joan A. Cotter, Ph.D., 2012
• Are often used to teach rote.
• Are liked by those who don’t need them.
• Don’t work for those with learning disabilities.
• Give the false impression that math isn’t about thinking.
• Often produce stress – children under stress stop learning.
• Are not concrete – they use abstract symbols.
Memorizing MathFlash cards:
9 + 7
© Joan A. Cotter, Ph.D., 2012
AN ALTERNATIVE:
SUBITIZINGand
GAMES
© Joan A. Cotter, Ph.D., 2012
Subitizing QuantitiesIdentifying without counting
© Joan A. Cotter, Ph.D., 2012
Subitizing QuantitiesIdentifying without counting
• Five-month-old infants can subitize to 3.
© Joan A. Cotter, Ph.D., 2012
Subitizing QuantitiesIdentifying without counting
• Three-year-olds can subitize to 5.
• Five-month-old infants can subitize to 3.
© Joan A. Cotter, Ph.D., 2012
Subitizing QuantitiesIdentifying without counting
• Three-year-olds can subitize to 5.
• Five-year-olds can subitize 6 to 10 by using five as a subbase.
• Five-month-old infants can subitize to 3.
© Joan A. Cotter, Ph.D., 2012
AddingName the quantity (practice subitizing).
© Joan A. Cotter, Ph.D., 2012
AddingName the quantity (practice subitizing).
© Joan A. Cotter, Ph.D., 2012
AddingName the quantity (practice subitizing).
© Joan A. Cotter, Ph.D., 2012
Adding
4 + 3 =
© Joan A. Cotter, Ph.D., 2012
Adding
4 + 3 =
© Joan A. Cotter, Ph.D., 2012
Adding
4 + 3 =
© Joan A. Cotter, Ph.D., 2012
Adding
4 + 3 = 7
© Joan A. Cotter, Ph.D., 2012
Adding
4 + 3 =
© Joan A. Cotter, Ph.D., 2012
Characteristics of a Good Game
© Joan A. Cotter, Ph.D., 2012
Characteristics of a Good Game
• Produces learning through playing.
© Joan A. Cotter, Ph.D., 2012
Characteristics of a Good Game
• Produces learning through playing.
• Incorporates manipulatives.
© Joan A. Cotter, Ph.D., 2012
Characteristics of a Good Game
• Produces learning through playing.
• Incorporates manipulatives.
• Teaches strategies.
© Joan A. Cotter, Ph.D., 2012
Characteristics of a Good Game
• Produces learning through playing.
• Incorporates manipulatives.
• Teaches strategies.
• Encourages mental work.
© Joan A. Cotter, Ph.D., 2012
Characteristics of a Good Game
• Produces learning through playing.
• Incorporates manipulatives.
• Teaches strategies.
• Encourages mental work.
• Detects errors; provides continuous assessment.
© Joan A. Cotter, Ph.D., 2012
Characteristics of a Good Game
• Produces learning through playing.
• Incorporates manipulatives.
• Teaches strategies.
• Encourages mental work.
• Detects errors; provides continuous assessment.
• Is enjoyable.
© Joan A. Cotter, Ph.D., 2012
Go to the Dump GameObjective: To learn the facts that total 10:
1 + 92 + 83 + 74 + 65 + 5
© Joan A. Cotter, Ph.D., 2012
Go to the Dump GameObjective: To learn the facts that total 10:
1 + 92 + 83 + 74 + 65 + 5
Object of the game: To collect the most pairs that equal ten.
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
6 + = 10
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
6 + = 10
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
6 + 4 = 10
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Starting
© Joan A. Cotter, Ph.D., 2012
7 2 7 9 5
7 42 61 3 8 3 4 9
Go to the Dump Game
Starting
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 42 61 3 8 3 4 9
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 42 61 3 8 3 4 9
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 42 61 3 8 3 4 9
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 2 1 3 8 3 4 9
4 6
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 2 1 3 8 3 4 9
4 6
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 2 1 3 8 3 4 9
4 6
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Finding pairs
7 2 7 9 5
2 1 8 3 4 9
4 67 3
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Finding pairs
7 2 7 9 5
1 3 4 9
4 62 82 8
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Playing
7 2 7 9 5
1 3 4 9
4 62 82 8
© Joan A. Cotter, Ph.D., 2012
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 3?
Playing
7 2 7 9 5
1 3 4 9
4 62 82 8
© Joan A. Cotter, Ph.D., 2012
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 3?
Playing
7 2 7 9 5
1
3
4 9
4 62 82 8
© Joan A. Cotter, Ph.D., 2012
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 3?
Playing
2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2012
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 8?
Playing
2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2012
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 8?
Go to the dump.Playing
2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2012
2
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 8?
Go to the dump.Playing
2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Playing
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
PinkCap, do youhave a 6?Playing
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
PinkCap, do youhave a 6?PlayingGo to the dump.
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2012
5
Go to the Dump Game
Playing
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Playing
5
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
YellowCap, doyou have a 9? Playing
5
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
YellowCap, doyou have a 9? Playing
5
2 2 7 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
YellowCap, doyou have a 9? Playing
5
2 2 7 5
1 4 9
4 62 82 8
7 3
9
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Playing
5
2 2 7 5
4 9
4 62 81 9
7 3
© Joan A. Cotter, Ph.D., 2012
2 9 1 7 7
Go to the Dump Game
Playing
5
2 2 7 5
4 9
4 62 81 9
7 3
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Winner?
5 54 6
9 1
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Winner?
5546
91
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Winner?
5546
91
© Joan A. Cotter, Ph.D., 2012
Go to the Dump Game
Play it again.
© Joan A. Cotter, Ph.D., 2012
Fact Strategies
© Joan A. Cotter, Ph.D., 2012
Fact Strategies
• A strategy is a way to learn a new fact or recall a forgotten fact.
© Joan A. Cotter, Ph.D., 2012
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.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 = 14
Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
8 + 6 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
8 + 6 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
8 + 6 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
8 + 6 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
8 + 6 =
10 + 4 = 14
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
7 + 5 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
7 + 5 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
7 + 5 = 12
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 =
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 =
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 =
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 = 6
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesSubtract from 10
15 – 9 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesSubtract from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesSubtract from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesSubtract from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesSubtract from 10
15 – 9 = 6
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
1 + 5 = 6
Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2012
Rows and Columns Game
Objective: To find a total of 15 by adding 2, 3, or 4 cards in row or column.
© Joan A. Cotter, Ph.D., 2012
Rows and Columns Game
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.
© Joan A. Cotter, Ph.D., 2012
Rows and Columns Game8 7 1 9
6 4 3 3
2 2 5 6
6 3 8 8
© Joan A. Cotter, Ph.D., 2012
Rows and Columns Game8 7 1 9
6 4 3 3
2 2 5 6
6 3 8 8
© Joan A. Cotter, Ph.D., 2012
Rows and Columns Game8 7 1 9
6 4 3 3
2 2 5 6
6 3 8 8
© Joan A. Cotter, Ph.D., 2012
Rows and Columns Game1 9
6 4 3 3
6 3 8 8
© Joan A. Cotter, Ph.D., 2012
Rows and Columns Game1 9
6 4 3 3
6 3 8 8
7 6
2 1 5 1
© Joan A. Cotter, Ph.D., 2012
Rows and Columns Game1 9
6 4 3 3
6 3 8 8
7 6
2 1 5 1
© Joan A. Cotter, Ph.D., 2012
Rows and Columns Game1 9
6 4 3 3
6 3 8 8
7 6
2 1 5 1
© Joan A. Cotter, Ph.D., 2012
Rows and Columns Game1
6 4 3 3
3 8 8
1 5 1
© Joan A. Cotter, Ph.D., 2012
Rows and Columns Game
© 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
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
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.
© Joan A. Cotter, Ph.D., 2012
Teaching the Arithmetic Facts Using Strategies and Games
MCTMMay 4, 2012
Duluth, Minnesota
by Joan A. Cotter, Ph.D.JoanCotter@RightStartMath.com
7 3 8 16 24 32 40
PowerPoint Presentation & HandoutRightStartMath.com >Resources
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