Game of 21

1
Starting Notes
Please read Not All Manipulatives and Models
are Created Equally. Special Journal entry today.

• Game of 21
• Start with 21 small cubes.
• You make take 1, 2 or 3 blocks away on your turn.
• The goal is to take the last cube.

Where's The Math?
2
Throw then a ball they can hit!
3
12345..96979899
1 2 3 4 5 6 7 8 9
A Big String of Consecutive Numbers
4
Operations Actions
The Search for Efficient Tools
5
Using the methods that you were taught
in elementary school, compute the following on
paper.
267 133 42
216 x 24
3.
1.
846 35
212 - 45
4.
2.
6
The methods that you just used are called
Algorithms mathematical recipes

• As Teachers
• Work from our personal knowledge.
• Search for alternative processes.
• Recognize the individual differences
• observed in our students.

7
Active Learning
1 x 5 5
Fact Families in Arrays. (2-dimensional models)
2 x 5 10
3 x 5 15
Product is 12. What are the possible arrays?
2 NSN 13
8
Operations With Numbers
Assembly (Compose)
Disassembly (Decompose)
Automaticity
9
Addition (Assembly)
Subtraction (Disassembly)
Inverse
3 NSN 11
Repeated
Repeated
Multiplication (Assembly)
Division (Disassembly)
Inverse
Sequence of Operation Development
10
Rafi had 12 cookies on a plate. If he ate them 3
at a time, how many times could he go back for
cookies?
Subtraction 12 - 3 - 3 - 3 - 3 0 Solution 4
times
Addition 3 3 3 3 12 Solution 4 times
Multiplication 4 (trips) x 3 12 Solution 4
times
Division 12 ? 3 4 Solution 4 times
11
Kamii A standard algorithm not be taught
encourages students to give up own thinking
unteach place value

• Working With DifferentAlgorithms
• Help children develop a strong
• understanding of the operations.
• 2. Develop efficient strategies for fact
• retrieval.
• 3. Provide practice in the selection
• and use of these strategies.

12
Compose / Decompose Numbers
Use expanded notation to advantage. It takes the
students knowledge of place value and builds new
structure.

• 300 70 2
• 48 40 8

5 NSN 2
13
The Terms of Language

• Carry
• Trade
• Regroup
• squish

• Borrow
• Trade
• Regroup
• nab

Skill Represent numbers in many ways and
maintain equality.
14
Room to Play with Numbers
Sally is 12 years old and has a brother Pete who
is 7 years old.
12 - 7 5
2008 ...
12 5 - 7 5 5
2013 ...
15
Assembly Addition
Think Pair Share
267 133 42 442
Multi-Pathing
Model this on a place value mat. Regroup /
squish as needed.
Ways to Add
NSN 2-18
16
Strategies for Addition

• Think Turnarounds 14 21 21 14
• Commutative Property
• Think Compensation 16 29 15 30
• Think Compatible 3 32 17 32 20
• Associative Property
• Think Simplification 45 78 110 13

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
3NSN12
17
Disassembly Subtraction
Many nuances take away, compare, missing addend
212 - 45
Caution Build the model for the first
number. Now . Remove the second value.
Model this on a place value mat. Regroup /
nab as needed.
18
Disassembly Subtraction
Think Pair Share
212 - 45
Multi-Pathing
Model this on a place value mat. Regroup / nab
as needed.
19
More Playing with Numbers
23 20 20 3 -18 -18 - -
2 5
UM - WNI Ways to Subt
20
Strategies for Subtraction

• Think Related Facts 17 9 8 / 9 8 17
• Think Decomposition 34 23 34 20 3
• Think Commutative Problem 12 9 ? 9 12
• Think Equal Offset 68 14 70 16 (Careful!)

21
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22
Assembly Multiplication
24 x 216
Use a calculator to model repeated addition. 216
216 Then press the sign 22 more times
to add 216 a total of 24 times.
Reference Assembly/Disassembly of Numbers
Think Pair Share
NSN 3-12
Multi-Pathing
WNI- T3 2 digit multiplier-Ex3
23
Patterns in Arrays . Number Fact Families
24
3 x 4 3 groups of 4
5 x 4 5 groups of 4
2 x 4 2 groups of 4
25
6 x 5 30 6 groups of 5
3 x 2 x 5 30 3 groups of 2 groups of 5
26
Strategies for Multiplication
Think Pattern - Multiplication Tables Think
Turnarounds 5 x 7 7 x 5 Commutative
Property Think Distributive Property Follow
equality 9 x 14 9 x (10 4) 9 x 14 9 x 10
9 x 4
27
UNum Patterns Factor Arrays
28
Multiplication Curiosities Finger Facts
29
Multiplication Curiosities Napier's Bones
30
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31
Disassembly Division
Nothing more than FAIR sharing
The Division Mat A Model for Division
32
Gizinta
?
4 ) 1 4 8
Reference Assembly/Disassembly of Numbers
Think Pair Share
Multi-Pathing
33
48 ? 8 24?8 24?8
34
Strategies for Division

• Think Expanded notation 96 ? 4 (8016) ? 4
• Think multiplication 24 ? 6 4 // 4 x 6 24
• Think leftovers
• 32 ( 7 x ______ ) _______
• 55 ( 9 x ______ ) _______
• 22 ( 5 x ______ ) _______
• Division Mats / fair sharing

NSN 4-15
35
Maintenance
By late Formative Years, most operational skills
will require periodic maintenance to ensure
accuracy as well as improved processing.
63
9
x
7
63
7
9
36
1
6
2
3
9
4
5
8
7
Making 10 (again)
37
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38
"Imma Whiz"
39
Krypto

• How
• Make a set of cards numbered 1-25.
• Shuffle and deal 5 cards face up.
• Select a Target Card.
• Goal use operations (any order)
• to have the 5 cards equal the target value.

21
5
17
8
3
12

40
In one sentence, please respond to The CAU TION
found in the article Not All Manipulatives and
Models are Created Equally.
Special Journal entry today.
41
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