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Week 5 - Tuesday Algorithms
Addition:
1. American Standard
576
+ 279
855
2. Partial Sums
576
+279
1 5
14
+ 7
855
3. Partial Sums with Place Value
576
+ 279
1 5
1 4 0
+ 7 0 0
8 5 5
4. Left-to-Right
576
+ 279
700
140
+ 15
855
5. Expanded Notation
576 = 500 + 70 + 6
+ 279 = 200 + 70 + 9
855 = 800 + 50 + 5
6. Lattice Method
576
+ 279
Subtraction Algorithms
1.American Standard
476
- 289
287
2.Reverse Indian
576
- 289
3.Left-to-Right
576
- 289
300 (500-200)
(200)
90 (170 - 80) borrowing from 300
(80) (16 - 9) borrowing from 90
(7)
4.Expanded Notation:
576 = 500 + 70 + 6
-289= 200 + 80 + 9
287 = 200 + 80 + 7
5.Integer Subtraction Algorithm
576
-289
-3 290 - 3 = 287
-10
+300
Multiplication Algorithms
1.American Standard
1
2 3
x 1 4
1 9 2
+2 3 0
3 2 2
2.Place Value
2 3
x 1 4
322
4 x 3=12
4 x 20 = 80
10 x 3 = 30
10 x 20 = 200
322
3.Expanded Notation
(Take 23 14 times)
10
23 = 20 + 3
x 14 = x 10 + 4
100 90 + 2 <—— by taking my 23 4 times
200 + 30 + 0 <—— no ones (it takes that place since we are multiplying by 10s now)
300 + 20 + 2
4.Lattice Method
Lattice Method for Addition:
https://youtu.be/tYmF2GW0wwQ?si=3N3K6khH90-T4ko8
Week 4 Division
Division is about sharing or splitting
John has 15 cookies. He puts 3 cookies in each bag. How many bags can he fill?
John has 15 bags. He puts the cookies into 5 bags with the same number of cookies in each bag. How many cookies in each bag?
*They are the same but completely different to our first graders.
Long Division: Usually taught Standard American Algorithm
Alternative Algorithm:
How many boxes?
___12____
16| 197
-160 -----> 10 boxes
37 +
-32 -----> 2 boxes
5 12
Week 4: Tuesday
Addition Meaning and Properties:
Identity: a + 0 = a *When I add 0 to any number, the identity of the number does not change
Commutative (order): a + b = b + a *The order that you add does not matter
Associative (grouping): (a + b) + c = a + (b + c) *The way you group your numbers does not matter
Subtraction:
*As adults we subtract, a first grader would use addition
*Answer: This would be confusing to a first grader because the equation doesn't tell them to take away. It looks like an addition problem.
Multiplication:
3 x 4: 3 groups of 4 3 x 4 = 12
factors product
xxxx xxxx xxxx
Repeated Addition Cartesian Product: Combining Groups
Properties:
Identity: a x 1 = a *When I multiply by 1 the identity of the number does not change
Commutative (order): a x b = b x a *The order in which I multiply do not matter
Associative (grouping): (a x b) x c = a x (b x c) *The way we group our problem does not matter
Zero: a x 0 = 0 *Any number multiplied by zero equals zero
3 x 7 = 7 + 7 + 7
5 + 2
x x x x x x x
x x x x x x x
x x x x x x x
(3 x 5) (3 x 2)
a x (b + c) = (a x b) + (a x c)
3 x 7 = 3 x (5 + 2) = (3 x 5) + (3 x 2)
partial products
Distributive Property: When I multiply a number by the sum of two other numbers. It is the same as multiplying the number by each addend.
3 x 9 = 3 x (5 + 4) = (3 x 5) + (3 x 4) = 15 + 12 = 27
3 x 9 = 3 x (7 + 2) = (3 x 7) + (3 x 2) = 21 + 6 + 27
3 x 9 = 3 x (6 + 3) = (3 x 6) + (3 x 3) = 18 + 9 = 27
Addition Properties:
https://youtu.be/a0deCn5QNFI?si=WKsH9-tYEBDXmPHH
Week 3 Thursday:
Base 2 Digits Used: 0-1
ones 2^0
twos 2^1
fours 2^2
8s 2^3
16s 2^4
1111 base 2: (1 x 2^3) + (1 x 2^2) + (1 x 2^1) + (1 x 2^0)
1111 base 2: (1 x 8) + (1 x 4) + (1 x 2) + (1 x 1)
1111 base 2: 8 + 4 + 2 + 1 = 15
Which number is bigger?
43 base 5 __>___ 25 base 6 -----> 23 __>__ 17
43 base 5: (4 x 5^1) + (3 x 5^0) | 25 base 6: (2 x 6^1) + (5 x 6^0)
43 base 5: 20 + 3 | 25 base 6: 12 + 5
43 base 5: 23 | 25 base 6: 17
Compare:
23 base 5 __<___ 23 base 6 *Same digits but second base is larger
Week 3: Tuesday
Base 3: Digits Used: 0, 1, 2 Example:
ones 3^0 x x x = 10 base 3
3s 3^1 x x x|x x = 12 base 3
9s 3^2
27s 3^3
Examples:
1222 base 3: (1 x 3^3) + (2 x 3^2) + (2 x 3^1) + (2 x 3^0)
1222 base 3: (1 x 27) + (2 x 9) + (2 x 3) + (2 x 1)
1222 base 3: 27 + 18 + 6 + 2 = 53
1222 base 4: (1 x 4^3) + (2 x 4^2) + (2 x 4^1) + (2 x 4^0)
1222 base 4: (1 x 64) + (2 x 16) + (2 x 4) + (2 x 1)
1222 base 4: 64 + 32 + 8 + 2 = 106
Base 10 Base 5 Base 3
122.21 122.21 122.21
|| || ||
1/10| 1/5| 1/3|
1/100 1/25 1/9
Base 3 Number System Video:
https://youtu.be/X7HLvBzB9-I?si=0RDnQT48a4QpNoeV
Week 2 Thursday Numeration Systems
Base 10 System
3 3 3
| | |
| | ones
| tens
hundreds
Decimal system because it is a 1-10 relationship.
hundreds
| tens
| | ones
| | | tenths
3 7 5 . 2 5 ----- hundredths (1/10 x 1/10 = 1/100)
| |
1/10 |
1/100
Related to Money:
Digits used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
10 11
x x
x x
x x
x x
x x
x x
x x
x x
x x
x x x
Expanded Notation:
375 = 300 + 70 + 5
375 = (3 x 100) + (7 x10) + (5 x 1)
375 = (3 x 10^2) + (7 x 10^1) + (5 x 10^0)
Example:
1078 = 1000 + 0 + 70 + 8
1078 = (1 x 1000) + (0 x 100) + (7 x 10) + (8 x 1)
1078= (1 x 10^3) + (0 x 10^2) + (7 x 10^1) + (8 x 10^0)
Different Bases:
Base 5 Digits Used: 0, 1, 2, 3, 4 Expanded:
ones 5^0 1 1 1 base 5
fives 5^1 | | ones
25s 5^2 | fives
125s 5^3 25s
111 base 5: (1x 5^2) + (1 x 5^1) + (1 x 5^0)
111 base 5: (1 x 25) + (1 x 5) + (1 x 1)
111 base 5: 25 + 5 + 1 = 31
1023 base 5: (1 x 5^3) + (0 x 5^2) + (2 x 5^1) + (3 x 5^0)
1023 base 5: (1 x 125) + (0 x 25) + (2 x 5) + (3 x 1)
1023 base 5: 125 + 0 + 10 + 3 = 138
Khan Academy Video:
https://youtu.be/ku4KOFQ-bB4?si=PNILbWj735gYMFI3
Week 2: Tuesday
"How to Solve It" - 1945 George Polya
Problem:
How many handshakes? (Without repeating handshake)
x x x x x x Solve: 6 + 5 + 4 + 3 + 2 + 1 = 21 handshakes
Strategies: Whatever makes the most sense to the student
Bigger Number: Simplify
Problem: 4 3 cent stamps
3 7 cent stamps
How many different postage amounts can we put together?
Pattern:
Only 3 cents: Only 7 cents:
x x
x x x x
x x x x x x
x x x x
4 x 3 = 12
Simplify:
5 11 cents
x
6 9 cents
30
5 + 6 + 30 = 41
Cartesian Product:
x x 4 x 2 = 8
y y y y
Polyas 4 Steps & Describe: Test
Problem Solving Week 1:
Steps to Problem Solving:
Tag of War: Acrobats, Grandmas, and Ivan
R1: 4A = 5G
R2: I = 2G + A
R3: I + 3G ____ 4A <----- (2G + A) + 3G _____ 4A
Answer: I + 3G > 4A <------ 5G + A > 4A
Give: Time and Manipulatives (Hands on Tools)
Problem Solving Steps Video:
https://youtu.be/kn8frIzQupA?si=oMIOF-95AUMSO2b7
