What is the probability of rolling a negative number?

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Numbers are the mathematical values or figures used for the purpose of measuring or calculating quantities. It is also represented by numerals as 3, 4, 8, etc. Some other examples of numbers are whole numbers, integers, natural numbers, rational and irrational numbers, etc.

The system includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. is termed a number system. These types of numbers can be expressed in the form of figures as well as words accordingly.

For example, the numbers like 55 and 75 expressed in the form of figures can also be written as fifty-five and seventy-five.

A numeral system or Number system is defined as an elementary system to express numbers and figures. It is the unique way to represent numbers in arithmetic and algebraic structure.

It is used in various arithmetic values applicable to carry out various arithmetic operations like addition, subtraction, multiplication, etc which are applicable in daily lives for the purpose of calculation. The number value is determined by the digit, its place value in the number, and the base of the number system.

Generally, Numbers are also known as numerals which are the mathematical values used for counting, measurements, labelling, and measuring fundamental quantities.

Numbers have been classified into two types:

Positive numbers, Negative numbers: and further classified into different types such as natural number, rational number, whole numbers, decimal numbers, integers etc

Positive numbers: Positive numbers are the numbers which are greater than zero that lies on the right side of the number line. Example: 0, 1, 2, 3, 4…
Negative numbers: Negative numbers are the numbers which are less than zero that lies on the left side of the number line. Example: … -5, -4, -3, -2, -1, 0

Answer:

Negative positive is sometimes negative sometimes positive because when we add two digits with the different operand then the resulting sign will be of the higher digit . whether its negative or positive.
Example: If we add -8 & 7 then
– 8 + 7 = -1
If we reverse -7 + 8 = 1
in this way negative positive leads to different result depends on higher operand.
There are certain rules of negative positive
if we add two positive numbers then will get positive number
   (+) + (+) = +
if we add one positive and one negative number then higher operand will be the answer
   (+) + (-) = ± whichever is on higher side or
if we add two negative numbers then the result will be in negative
(-) + (-) = – or -5 + (-9) = -5 -9 = -14
if we multiply one positive and one negative number then the result will always be negative
  5 × (-5) = -25
if we divide one positive and one negative number then the result will always be negative
4/(-2) = -2 or (-4)/2 = -2

Sample Questions

Question 1: Add 5 and – 7?

Solution:

If we add one positive and one negative number then higher operand will be the answer .
add 5 & -7
= 5 + (-7)
first open the bracket
= 5 – 7
= -2

Question 2: Add 7 and – 5?

Solution:

If we add one positive and one negative number then higher operand will be the answer.
add 7 & – 5
= 7 + (-5)
first open the bracket
= 7 – 5
= 2

Question 3: Add – 8 & – 9.

Solution:

Add -8 & -9

= -8 + -(9)
First open the brackets
= -8 -9
= -17

Question 4: Multiply – 8 × 5?

Solution:

If we multiply one positive and one negative number then the result will always be negative
therefore = -8 × 5
= -40

Question 5: Divide -8 by 2?

Solution:

If we divide one positive and one negative number then the result will always be negative .
therefore
= -8/2
= -4

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. Sample Space showing all possible outcomes of rolling and adding two dice.

+	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

When you count the number of ways each outcome can be made the table becomes:

Poss. Totals	2	3	4	5	6	7	8	9	10	11	12
No. of Ways	1	2	3	4	5	6	5	4	3	2	1

No. of ways is also called Frequency.

Possibly the best way to explain this task is to tell the story of how it came to us. When Matthew Reames was teaching at St. Edmund's Junior School, England, he journeyed through many investigative challenges with his class. Examples are recorded in Task 45, Eric The Sheep, and Task 211, Soft Drink Crates. Then one day he wrote:

I have been looking at Sicherman Dice recently (these dice are evidently the only other set of dice using positive, non-zero integers to give the same possible outcomes as a 'regular' pair of dice). Rather than being the 'normal' 1 to 6, these are two different dice, one numbered 1, 2, 2, 3, 3, 4 and the other numbered 1, 3, 4, 5, 6, 8.
I used these with Year 6 and 7 classes (aged about 10 and 11 this time of year). In future lessons I would probably include this in the regular probability things, but this was one of those end-of-term lessons. We talked about the possible outcomes of flipping one coin, of rolling one number cube, then flipping two coins, and finally rolling two 'normal' dice. We made an outcome grid for addition (similar to the one on the Dice Differences task cameo but with addition rather than subtraction)
.

Then, I asked the class to find another set of dice that would give the same possible outcomes (not necessarily in the same order) on the chart - we talked about why. When allowed to use zero or negative numbers, there are a huge (possibly infinite) number of possibilities. There was a lot of good discussion about this!
Then, I said that they could not use zero or negative numbers (or fractions or decimals, etc - only integers). Though this required a bit more guidance from me (and I eventually had to tell them the final combination), there was good discussion about what they knew some of the numbers had to be (for example, each cube had to have a 1 otherwise they could not make a 2).

Taking Matthew's point here, we decided in preparing a task that was intended to be used by a pair of students relatively independently that asking them to rediscover Sicherman's dice was a big call for most. Hence the scaffold presentation on the card that gives the dice values and requires the students to discover that these two special dice result in the same frequency table.

+	1	3	4	5	6	8
1	2	4	5	6	7	9
2	3	5	6	7	8	10
2	3	5	6	7	8	10
3	4	6	7	8	9	11
3	4	6	7	8	9	11
4	5	7	8	9	10	12

Again, when you count the number of ways each outcome can be made the frequency table remains the same.

Possible Totals	2	3	4	5	6	7	8	9	10	11	12
Frequency	1	2	3	4	5	6	5	4	3	2	1

All of this was leading up to some further activities on the TI Nspire handhelds we were borrowing - talking about probability distribution and why, for a small number of rolls, the frequency graphs they made were so different but then got more and more identical as they increased the number of rolls.
Anyway, I wasn't sure if you already had a task similar to this or not, but I thought it was worth passing along just in case. My classes certainly had some fun trying to figure it out and there was a lot of excellent maths discussion along the way.

Matthew has also supplied this spreadsheet which has two examples of whole number dice with zero as one face. (Note: The spreadsheet has been zipped.)

Once students understand how these tables are made, you might want them to search for more pairs of dice with the same sample space (all possible outcomes) as the 'normal dice'. In doing so you might want them to make the tables by hand because of the arithmetic practice, or you might, as Matthew did, want to introduce them to the idea of using a spreadsheet as a tool.

Can you see how these two dice were created
from 'normal dice'?

Matthew's spreadsheet is an electronic Investigation Guide. You enter the dice numbers in the top row and left column and the sheet contains formulae which, for any cell, automatically sum the left column dice number for that cell with the top row dice number for that cell. It makes it very quick to try out hypotheses about the dice numbers.

Also, given that we are aiming for the sums 2 and 12 to occur only once in any table, if the appropriate numbers are put first and last in the column and row, then only four other numbers need to be found for each dice.

Matthew's sheet contains examples his class developed using whole numbers, decimals, fractions and negative numbers. Wow! Now there's millions of solutions.

Extensions

Roy Grice learnt about the Sicherman Dice challenge through the February 2013 Mathematics Centre eNews when we introduced the companion Maths300 lesson to this task. All Roy knew from the brief news item was that the challenge was to 'design two new cube dice which will produce the same frequency table' as the one for 1- 6 cube dice.

At the April annual Easter conference of ATM he passed on his thanks for the problem - he had thoroughly enjoyed the search - and offered these extra challenges:

What happens if we use four sided dice?
Are there any other dice with the Sicherman property?

When these challenges were offered later in April at a meeting of SMaL-Syd, Sweden, a couple of days later, one teacher offered one solution to the first of them. Unfortunately, neither the teacher's name, nor the solution is recorded in our correspondence.

Can you rediscover that solution?
How many solutions are there for the four sided dice?

Roy is sure that there are no solutions to the case of a five sided dice. In Roy's own words from an email in May 2013:

I remain convinced that there is no solution for a 1-5 dice but do let me know if one is found. Occasionally I still exercise my obsession and attempt to find a solution for a 1-8 dice but it is proving most difficult to track.