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A LOTTERY scratchcard has been withdrawn from sale by Camelot - because players couldn't understand it.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.
Copyright 2014 Scott Storla
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
Copyright 2014 Scott Storla
Natural numbers
Definition – Integers
English : The natural numbers greater than
0, their negatives and 0.
... 3, 2, 1,0,1, 2, 3...
Copyright 2014 Scott Storla
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.
Copyright 2014 Scott Storla
Operations and Operators
Operation Operator(s)
Addition +
Subtraction
Multiplication
Division
Power 2
Root
Absolute value
Logarithm log ln
Exponential 10 e
Copyright 2014 Scott Storla
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.
Copyright 2014 Scott Storla
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.
Copyright 2014 Scott Storla
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.
Copyright 2014 Scott Storla
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.
Copyright 2014 Scott Storla
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
Copyright 2014 Scott Storla
You can only add or subtract like terms.
Terms are like if they count the same unit.
Copyright 2014 Scott Storla
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
Copyright 2014 Scott Storla
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.
Copyright 2014 Scott Storla
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.
Copyright 2014 Scott Storla
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.
Copyright 2014 Scott Storla
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
Copyright 2014 Scott Storla
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
Copyright 2014 Scott Storla
What’s the sign of the result?
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
Copyright 2014 Scott Storla
What’s the sign of the result?
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
Copyright 2014 Scott Storla
What’s the sign of the result?
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
Copyright 2014 Scott Storla
What’s the sign of the result?
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
Copyright 2014 Scott Storla
What’s the sign of the result?
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
Copyright 2014 Scott Storla
What’s the sign of the result?
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