1150 day 6
TRANSCRIPT
Integers and Divisibility
Counting (natural) numbers 1, 2, 3, …
Whole numbers 0, 1, 2, 3, …
Integers … -3, -2, -1, 0, 1, 2, 3, …
Absolute value - the distance between a number and zero
|4| =
| 2| =
4
2
Models for integer addition
Number Line Model
2 + –3 = 1
1 + 2 = 3
Charged field (chip) model for Addition
2 + 3 =
1 + 2 =
5
1
3 + 3 = 0
--
---
-+-
---
+++
Two numbers are additive inverses if their sum is zero.
Charged field (chip) model for Subtraction
2 ( 1) =
1 ( 2) =
1
3
--
+
“Take away”Start with 2 negativesTake away 1 negative1 negative left
Start with 1 positiveTake away 2 negatives3 positives left-
-
3
--
2 1 =
Start with 2 negativesTake away 1 positive3 negatives left
Number Line model for subtraction
2 1 = 3
-2-1
Negative numbers “face left” Positive numbers “face right”
Subtraction “walk backward”Addition “walk forward”
2 + ( 1) = 3
direction to face
how to walk
“Walk the line”
Relating subtraction to addition:a – b = a + (– b)
Number Line model for subtraction
2 ( 3) =
Negative numbers Positive numbers
Subtraction (backward)Addition (forward)
-2-(-3)
1
2 + (3) = 1
direction to face how to walk
Integer Multiplication
Charged Field (Chip) Model
3( 2) =
3(2) =
Add 3groups
of twonegatives
Take away3 groups
of twopositives
- -- -- -
6
+ +- - + +
- -
+ +- -
6
Number Line model for multiplication
3( 1) =
3( 1) =
3-1
Negative numbers Positive numbers
-(-1)
3
-1-1
3 groups
arrows
Of -1 arrows
Reverse3 groups
Of -1 arrows
-1 -(-1)-(-1)
Integer Multiplication – An investigation of patterns
3 · 3 =3 · 2 =3 · 1 =3 · 0 =
3 · 1 =3 · 2 =
3 · 3 =
9630369
3 · 3 =2 · 3 =1 · 3 =0 · 3 =1 · 3 =2 · 3 =
3 · 3 =
963
0369
Positive · Positive = PositivePositive · Negative = Negative
Positive · Negative = NegativeNegative · Negative = Positive
3
3
3
3
3
3
+3
+3
+3
+3
+3
+3
Same Signs – Positive answerDifferent Signs – Negative answer
Multiplication and Division
a · b = c means c b = a
Example: 3 · 4 = 12 means 12 4 = 3
Integer Division
pos · pos = pos so pos pos = pospos · neg = neg so neg neg = posneg · pos = neg so neg pos = negneg · neg = pos so pos neg = neg
Sign rules for division are identical to multiplication
Using the Difference of Squares formula to multiply
(a + b)(a – b) = a2 – b2
Multiply 42 · 38=(40 + 2)(40 – 2)= 402 – 22
= 1600 – 4= 1596
Multiply 107 · 93=(100 + 7)(100 – 7)= 1002 – 72
= 10000 – 49= 9951
Divisibility
If a and b are integers, then b divides a if there is an integer c such that a = b · c
Does 3 | 12
Does 6 | 12
Does 24 | 12
Yes
Yes
No
Because 12 = 3 · 4
Because 12 = 6 · 2
Because 12 = 24 · integer
Why?
Divisibility tests
234567891011
Even number (ends in 0, 2, 4, 6, 8)Sum of digits is divisible by 3Last two digits divisible by 4Ends in 0 or 5Divisible by both 2 and 3Cross out, double, subtractLast three digits divisible by 8Sum of digits divisible by 9Ends in 0Difference of alternate digits (ocean waves)
Test 5182 for divisibility by
2
3
4
5
6
Yes
No
No
No
No
Why?
5182 is even
5 + 1 + 8 + 2 = 16, and 3 | 18
4 | 82
5182 does not end in 0 or 5
Not divisible by both 2 and 3
Test 5182 for divisibility by
7
8
9
10
11
No
No
No
No
No
Why?
7 | 43
8 | 182
5 +1 + 8 + 2 = 16 and 9 | 16
5182 does not end in 0
11 | 10
5182- 4
514-843
5 1 8 2
13
3
13 – 3 = 10
Test 3,885,840 for divisibility by
2
3
4
5
6
Yes
Yes
Yes
Yes
Yes
Why?
3,885,840 is even
3+8+8+5+8+4+0=36, and 3 | 36
4 | 40
3,885,840 ends in 0
Divisible by both 2 and 3
Test 3,885,840 for divisibility by
7
8
9
10
11
Yes
Yes
Yes
Yes
No
Why?
7 | 21
8 | 840
9 | 36
Ends in 0
11 | 2
3,885,840- 0
388,584-8
38,850-0
3,885-10378
-1621
3 8 8 5 8 4 0
19
17 19 – 17 = 2