copyright 2014 scott storla. the integers copyright 2014 scott storla

Copyright 2014 Scott Storla

'Cool Cash' card confusion

A LOTTERY scratchcard has been withdrawn from sale by Camelot - because players couldn't understand it. The Cool Cash game - launched on Monday - was taken out of shops yesterday after some players failed to grasp whether or not they had won. To qualify for a prize, users had to scratch away a window to reveal a temperature lower than the figure displayed on each card. As the game had a winter theme, the temperature was usually below freezing. But the concept of comparing negative numbers proved too difficult for some. Camelot received dozens of complaints on the first day from players who could not understand how, for example, -5 is higher than -6. Tina Farrell, from Levenshulme, called Camelot after failing to win with several cards. The 23-year-old, who said she had left school without a maths GCSE, said: "On one of my cards it said I had to find temperatures lower than -8. The numbers I uncovered were -6 and -7 so I thought I had won, and so did the woman in the shop. But when she scanned the card the machine said I hadn't. "I phoned Camelot and they fobbed me off with some story that -6 is higher - not lower - than -8 but I'm not having it.


The Integers


Vocabulary

Integers

Absolute value


Positive

Money we have

Yards gained

Degrees above 0

Floors above ground

Above sea level

Negative

Money we owe

Yards lost

Degrees below 0

Floors below ground

Below sea level


Natural numbers

Definition – Integers

English : The natural numbers greater than

0, their negatives and 0.

... 3, 2, 1,0,1, 2, 3...


Procedure – Order of Operations

Beginning with the innermost grouping idea and working out;

Explicit Grouping ( ), [ ], { }

Implicit Grouping

Operations in the numerators and denominators of fractions

Operations in Radicands

Operations in Exponents

1. Start to the left and work right simplifying each operation beyond the basic four as you come to them.

2. Start again to the left and work right simplifying each multiplication or division as you come to them.

3. Start again to the left and work right simplifying each addition or subtraction as you come to them.


Operations and Operators

Operation Operator(s)

Addition +

Subtraction

Multiplication

Division

Power 2

Root

Absolute value

Logarithm log ln

Exponential 10 e


Absolute Value


Integers have two attributes,

a size (how much, how many) and

a sign (positive or negative).

For instance $2 has a size of 2 and it’s

positive, we have $2.

– $2 also has a size of 2 and it’s negative, we

owe $2.


3 3

3 3 3

Absolute value bars, return

the size of a number.

Procedure – Order of Operations

Starting with the innermost type of grouping and working out

1. Perform all operations beyond the basic four.

2. Multiply and divide, left to right.

3. Add and subtract, left to right.

Absolute value is both an operator and an implicit grouping symbol.


4 3 17

Simplify

12 17

5

5


13 1 10

14 10

14 10

4

Simplify


18 16 2 3

18 16 6

18 10

18

28

Simplify

10


Multiplying and Dividing Integers


Solver/Writer Pairs

1. The solver doesn’t hold a writing instrument. The writer does.

2. The solver tells the writer how to transform the problem. a) Count the number of operators. b) Discuss the order for the operations. c) Carry out the order describing one transformation per

line.

3. The writer only includes justified work that is explained well.

4. One solver will be called on to finish the problem using the recorded process.


Procedure – Multiplying or Dividing Two Integers

1. Multiply or divide the absolute values of the two integers.

2. If originally both integers had the same sign, then the result is positive. If they originally had different signs, then the result is negative.


4 3 5

12 5

60

Procedure – Multiplying or Dividing Two Integers

1. Multiply or divide the absolute values of the two integers.

2. If originally both integers had the same sign, then the result is positive. If they originally had different signs, then the result is negative.

Simplify


7 4 3 4

28 12

16

Simplify


8 6

2 3

2

8

4

Simplify


8 5 2 15

5 2

40 30

5 2

10

5 2

10

10

1

Simplify


Adding Integers With Similar Signs


You can only add or subtract like terms.

Terms are like if they count the same unit.


Procedure – Adding Two Integers with Similar Sign

1. When adding two integers with a common sign add their absolute values and use the common sign.

2 9 4

11 4

15

Simplify


2 3 2 3

5

11

Simplify

6


4 2 6 6 1

8 6 6

14 6

20

8 6 6 1

Simplify


Adding Integers With Different Signs


Procedure – Adding Two Integers with Different Signs

1. Subtract the smaller absolute value from the larger.

2. Attach the original sign of the number that had the larger absolute value.


10 5 2 4 12

5 2 4 12

7 4 12

11 12

1

Simplify


4 15 20 8 2

4 5 8 2

20 16

4

20 8 2

Simplify


112 2 6 2

12 2 6 2

12 2 4

12 8

4

Simplify


Subtracting Integers


Careful! Meanings for the symbol

Depending on the situation you may find it easier to think of the dash symbol, , as

subtraction, as the negative of, as an opposite or as a factor of 1 . Sometimes it’s

helpful to change your point of view within the same problem. With practice you will

develop ways of understanding which meaning is the most appropriate.


Procedure – Subtracting Two Integers

1. Change the subtraction to addition.

2. Change the number that follows the subtraction to its opposite.

3. Follow the procedure for adding two integers.


Procedure – Subtracting Two Integers

1. Change the subtraction to addition.

2. Change the number that follows the subtraction to its opposite.

3. Follow the procedure for adding two integers.

4 7

4 7

11

Simplify


12 2 6

12 2 6

10 6

16

Simplify

12 2 6


7 2 4 3

14 12

14 12

26

Simplify


6 4 7

6 4 71

1

6 1 3

6 3

9

Simplify


8 16 6 2

8 16 6 21

8 16 1 4

8 16 4

8 4

4

8 16 1 6 2

Simplify


2 2 2 2 2 4

Simplify

2 2 2 2 2 4

2 2 2 2 2

2 2 2 4

2 2 6

2 12

10


Integers


What’s the sign of the result?

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.

( 12)( 15) 17 40

45 1825 72

12 8

5 12 7

44

1432

3 10511 72

4 1

5 8

17 40 9 18



1. ( 12)( 15) 6. 17 40

2. 45 1825 72

7. 12 8

3. 5 12 7

8. 44

1432

4. 3 10511 72

9. 4 1

5 8

5. 17 40 10. 9 18



1. 6.

2. 7.

3. 8.

4. 9.

5. 10.

12 5 1 4

7 1813

25 72 4 1

5 8

5 12 3

1 42 3

8 12 4 15 8

8 12 1

94



1. 12 5 6. 1 4

2. 7 18

1325 72

7. 4 15 8

3. 5 12 3 8.

1 42 3

4. 8 12 9. 4 15 8

5. 8 12 10. 1

94



1. 6.

2. 7.

3. 8.

4. 9.

5. 10.

75 2

9 8 10

44 12 63 4 24 7 3

1 13

2 4 4 8 2

3 105 311 72 8

4 66 2

4 40 12 41 17 5



1. 7

5 29

6. 8 10

2. 44 12 6 7. 3 4 24 7 3

3. 1 1

32 4

8. 4 8 2

4. 3 105 311 72 8

9. 4 66 2

5. 4 40 12 10. 41 17 5



1. 12 15 6. 1 4

2. 458 887 7. 54 97

3. 31 22 8. 32 9

4. 82 12 9. 37 12

5. 8 12 10. 9 1



1. 5 17 2 6. 8 10

2. 44 12 6 7. 4 7 23

3. 12 4 30 8. 4 8 2

4. 11 3 55 9. 4 66 2

5. 4 40 12 10. 4 1 35

copyright 2014 scott storla. the integers copyright 2014 scott storla

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